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Home » Library » Knowledge Base » Legal Institutions » Judiciary » What are the jurisdictions of the Court for Children? What are the restrictions on media reporting regarding a child involved in the court proceedings?
What are the jurisdictions of the Court for Children? What are the restrictions on media reporting regarding a child involved in the court proceedings?
Powers or Orders
A Court for Children has the jurisdiction to try all offences except offences punishable with death.
A Court for Children has the power to:
admonish and discharge the child;
discharge the child upon his executing a bond to be of good behaviour and to comply with such conditions as may be imposed by the Court;
order the child to be placed in the care of a relative or other fit and proper person -
for such period to be specified by the Court;
with such conditions as may be imposed by the Court;
order the child to pay a fine, compensation or costs;
make a probation order under Section 98 of the Child Act 2001 [Act 611];
order the child to be sent to an approved school or a Henry Gurney School;
order the child, if a male, to be whipped with not more than ten strokes of a light cane -
within the Court premises; and
in the presence, if he desires to be present, of the parent or guardian of the child;
impose on the child, if he is aged fourteen (14) years and above and the offence is punishable with imprisonment, any term of imprisonment which could be awarded by a Sessions Court.
If a child aged fourteen (14) years and above is found guilty of an offence, he shall not be ordered to be imprisoned if he can be suitably dealt with in any other way whether by probation, or fine, or being sent to an approved school.
However, if the child is ordered to be imprisoned, he shall not be allowed to associate with adult prisoners.
A Court for Children may order:
the detention of a child in a place of detention, probation hostel, approved school or Henry Gurney School, approved institution or centre;
the supervision of a child by a Social Welfare Officer or probation officer, as the case may be; or
any probation period,
to be extended beyond the date on which the child attains the age of eighteen (18) years.
For offences punishable by death penalty, in lieu of a death sentence, the child shall be detained in prison at the pleasure of the Ruler. The Board of Visiting Justices for that prison will review the case at least once a year and may recommend to the Ruler on the early release or further detention of the child.
Media Reporting
Any mass media reporting regarding:
any step taken against a child or in any criminal act or omission, be it at the pre-trial, trial or post-trial stage;
any child in respect of whom custody is taken under Part V of the Child Act 2001 [Act 611] which provides for children in need of care and protection;
any child in respect of whom any of the offences specified in the First Schedule has been or is suspected to have been committed; or
any proceedings under Part VI of the Child Act 2001 [Act 611] which provides for children in need of protection and rehabilitation,
shall not reveal the name, address or educational institution, or include any particulars that may lead to the identification of the child concerned as being the person against or in respect of whom action is taken or as being a witness to the action.
A picture of the child concerned in any of the matters mentioned above shall not be published in any newspaper or magazine or transmitted through any electronic medium, which includes broadcast by radio or television.
Any person who contravenes the above commits an offence and shall on conviction be liable to a fine not exceeding RM10,000 or to imprisonment for a term not exceeding five (5) years or to both.
Knowledge Base ID : 1511
Last Reviewed : February 28, 2014
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text:
\section{Introduction} \label{sec1}

It is widely accepted that Lie superalgebras and their representations \cite{Kac1977,Kac1978} play an essential
role in the analysis, utilization and ultimate understanding of supersymmetry in physical systems.
Such structures first came about in the setting of elementary particle physics (see
\cite{FayFer1977} and references therein) but have since been applied in a variety
of fields, including high energy physics \cite{CHR2012,dLMR2012} and condensed matter physics
\cite{BGM2012}. To provide some archetypes, Lie superalgebras appear as symmetry algebras of quantum
many-body boson-fermion systems, particularly those relating to the shell model of nuclear physics
\cite{BalBarIac1981}, they occur in the context of integrability in string theory
\cite{HoIwTs2009,Beis2012}, and also supersymmetric Yang-Mills theory \cite{BeiStau2003,Min2012}. In
most cases, it is not only the algebraic structure which is important, but also the representations.

Various aspects of the representation theory of basic classical Lie superalgebras (in
the sense of Kac) have been developed over the years, but the most relevant to the current work are
those that present matrix element formulae for various types of representations,
particularly in the articles
\cite{TolIstSmi1986,Palev1987,Palev1989,StoiVan2010,Molev2011,GIW1,GIW2}, all of which are related
to $gl(m|n)$.
Of these articles, \cite{GIW1,GIW2} exhibit work undertaken by the current authors in order to develop techniques that make use of
characteristic identities. Specifically, the first paper \cite{GIW1} constructed invariants and reduced matrix
elements associated with representations of $gl(m|n)$, and the second article made use of these
results to produce matrix element formulae for the type 1 unitary representations of $gl(m|n)$. In
\cite{GIW2}, we highlight the significance of unitary representations in the context of canonical subalgebra embeddings. In
particular, due to the inner product (positive definite sesquilinear form) inherited by the
subalgebra, complete reducibility of a unitary representation follows immediately down the
subalgebra chain.

By contrast, the representations of $osp(m|2k)$ are in general not unitary, particularly due to the fact that
there is not a positive definite sesquilinear form defined on the
irreducible modules. The one exceptional case is $osp(2|2k)$ (i.e. when $m=2$), which is precisely the Lie
superalgebra $C(k+1)$ in the notation of Kac's classification \cite{Kac1977}. The Lie superalgebra
$osp(2|2k)$ is in fact a type I basic classical Lie superalgebra \cite{Kac1978}, the set of which also
includes $gl(m|n)$. The type I basic classical Lie superalgebras are well known
to be the only Lie superalgebras that give rise to unitary representations \cite{SNR1977}, and such
unitary representations have been studied and classified by Gould and Zhang in a series of papers
\cite{ZhaGou19902,ZhaGou1990,GouZha1990}.

In this paper, we consider the canonical subalgebra chain
\begin{equation}
osp(m+1|2k)\supset osp(m|2k) \supset \cdots \supset osp(2|2k)\supset osp (1|2k) \cdots
\label{subchain}
\end{equation}
Since the representations are not unitary in general, the question of complete reducibility is not
as straightforward. The matrix element formulae that will arise from these calculations will still be
valid, however, so what becomes challenging is to be able to {\em interpret} such
formulae in a particular circumstance. This problem will be addressed in a forthcoming
article. The current article, however, develops the important reduced matrix element formulae that
enable such full matrix element formulae to be expressed. In a sense, the work done in the current
article is analogous to our previous article \cite{GIW1}, which dealt with $gl(m|n)$.

There has been a substantial amount of interest in the representation theory of $osp(m|2k)$. Apart from
Jarvis and Green's \cite{JarGre1979,GreJar1983} and Gould's \cite{Gould1987} earlier work on characteristic identities for Lie
superalgebras, there are important works that deal with representations of orthosymplectic
superalgebras via their connection with parastatistics such as
\cite{Stoi2013,Palev1982,Van1984,LievStoiVan2008} and a wide variety of perspectives on developing
and understanding the representation theory
\cite{FarmJar1983,FarmJar1984,Nish1990,Quesne1990,BenLSRam1998,Coul2013,LS1999,LS2000,LS2002}.
Notably, the recent paper of \cite{Stoi2013} has exhibited a very constructive approach to the
representation theory of orthosymplectic Lie superalgebras, which is ultimately where our interest
lies. The current paper is therefore dedicated to laying the foundations for further work in
developing matrix element formulae similar in nature to the Gelfand-Tsetlin type formulae
\cite{GT1950,GT1950b}.

It is worth remarking that we have chosen to work with the canonical subalgebra chain
(\ref{subchain}) above, since it most naturally generalises the subalgebra chain used in the
representation theory of the orthogonal Lie algebras. The alternative of proceeding down the
subalgebra chain
\begin{equation}
osp(m|2k+2)\supset osp(m|2k) \supset\cdots
\label{subchain2}
\end{equation}
gives rise to a state labelling problem
which will not be considered here.
As one can tell, however, continuing down the chain beyond what is written in
(\ref{subchain}) will take us to the symplectic Lie algebra $sp(2k)$. The problem for the
symplectic Lie algebra has been solved in a sense by Molev \cite{Molev1999} and also looked at by Gould
and Kalnins \cite{GouKal1985} and Gould \cite{Gould1989}. Note, however, that the subalgebra chain
(\ref{subchain2}) is particularly convenient when dealing with the case $osp(m|n)$ for $n\leq 4$, where no
state labelling problem arises.

The paper is organised as follows. After establishing notation, conventions, and some algebraic
structural aspects -- including root systems -- in Sections \ref{sec2} to \ref{sec8}, we look at
the representation theory of $osp(m|2k)$. In particular, in Section \ref{sec9} we introduce the
characteristic identities and in Section \ref{branchingcondition} give {\em branching conditions} as
a set of necessary conditions satisfied by the representations of the canonical subalgebra.
After discussing vector operators in Section \ref{sec11},
we derive formulae for the reduced matrix elements in Sections \ref{sec12} and \ref{sec13}.

\section{$osp(m|n=2k)$ as a subalgebra of $gl(m|n)$}
\label{sec2}

Using the same fundamental notation as in \cite{GIW1,GIW2}, we have even indices $\{ i\ |\ 1\leq
i\leq m\}$, odd indices $\{\mu\ |\ 1\leq \mu\leq n=2k\}$,
and the grading defined by
$$
(i)=0,\ \ (\mu)=1.
$$
Recall that $gl(m|n)$ is spanned by the elementary generators (in ungraded index notation)
$$
e_{pq},\ \ 1\leq p,q\leq m+n.
$$
We also recall, given an $(m+n)\times(m+n)$ homogeneous matrix $X$ so that
$$
X =
\begin{blockarray}{cccc}
&&&\\
&&&\\
\begin{block}{(cc|cc)}
 &A & 0&\\
\cline{2-3}
&0 & D&\\
\end{block}
&\underbrace{}_m & \underbrace{}_n&
\end{blockarray}
,\ \ (X)=0,\ \ \mbox{ or }\ \
X =
\begin{blockarray}{cccc}
&&&\\
&&&\\
\begin{block}{(cc|cc)}
 &0 & B&\\
\cline{2-3}
&C & 0&\\
\end{block}
&\underbrace{}_m & \underbrace{}_n&
\end{blockarray}
,\ \ (X)=1,
$$
the definition of {\em supertranspose}:
$$
\left( X^T\right)_{pq} = (-1)^{(X)(q)}X_{qp}.
$$
We note the following property:
$$
(XY)^T = (-1)^{(X)(Y)} Y^T X^T.
$$
To introduce the subalgebra $osp(m|n)$ we first need an even metric
\begin{equation}
g = \left( \begin{array}{c|c} g_0 & 0\\ \hline  0 & g_1 \end{array} \right)
	\label{gdef}
\end{equation}
with $g_0$ symmetric and $g_1$ anti-symmetric. Thus we have the symmetry
\begin{align}
	g_{pq} = (-1)^{(p)}g_{qp}
\label{gsym}
\end{align}
and note that $g_{pq}=0$ unless $(p)=(q)$. Letting $t$ denote the usual matrix transpose,
it is also worth noting that
\begin{align*}
g^T &= g^t \\
&= \left( \begin{array}{c|c} g_0 & 0\\ \hline  0 & -g_1 \end{array} \right) = \gamma g = g
\gamma
\end{align*}
where
\begin{equation}
\gamma = \left( \begin{array}{c|c} I & 0\\ \hline  0 & -I \end{array}
\right)
\label{gradaut}
\end{equation}
is the grading automorphism.

\noindent
{\em Remark}: In general if $X$ is an even matrix, $X^T = X^t$, i.e. graded transpose
reduces to normal transpose for even operators.
$\blacksquare$ \medskip

Then it is easily seen that the set of matrices leaving the form $g$ invariant, i.e.
$$
L = \{ x\ |\ x^Tg + gx=0\ \}
$$
where $T$ is supertransposition, forms a Lie superalgebra, called the orthosymplectic Lie
superalgebra, denoted $osp(m|n)$.

To obtain a convenient basis for this Lie superalgebra we note that
\begin{align*}
x\in L \ \Leftrightarrow \ 0&= x^tg+gx\\
&= x^Tg^T\gamma + gx\\
&= (gx)^T\gamma + gx
\end{align*}
Then $x\in L$ $\Leftrightarrow$ $(gx)^T\gamma = -gx$ or, using $\gamma^2=I$,
$$
x\in L \ \Leftrightarrow \ (gx)^T = -gx\gamma.
$$
We call a homogeneous matrix $X$ anti-supersymmetric if
$$
X^T = -X\gamma
$$
with $\gamma$ the grading automorphism. Then we must have
$$
X_{pq} = -X_{qp}
$$
unless {\em both} indices are odd in which case $X_{pq}=X_{qp}$. Therefore $X$ is
anti-supersymmetric if and only if
$$
X_{pq} = -(-1)^{(p)(q)}X_{qp}.
$$

The anti-supersymmetric matrices have a basis consisting of the matrices
$$
e_{pq}-(-1)^{(p)(q)}e_{qp},\ \ 1\leq p,q\leq m+n,
$$
i.e.
\begin{align*}
e_{ij}-e_{ji},&\ \ 1\leq i<j\leq m,\\
e_{i\mu} - e_{\mu i},&\ \ 1\leq i\leq m, \ 1\leq \mu\leq n,\\
e_{\mu\nu} + e_{\nu\mu},&\ \ 1\leq \mu,\nu\leq n.
\end{align*}
This gives rise to the $osp(m|n)$ generators:
\begin{align*}
  x\in L=osp(m|n) &\Leftrightarrow \ gx \mbox{ is anti-supersymmetric}\\
  &\Leftrightarrow \ x = g^{-1}\left( e_{pq}-(-1)^{(p)(q)}e_{qp} \right).
\end{align*}
Under the assumption $g$ is orthogonal, so $g^t=g^{-1}$, which we will do, we obtain the
$osp(m|n)$ generators
\begin{align}
	\sigma_{pq} &= g^t\left( e_{pq}-(-1)^{(p)(q)}e_{qp} \right) \nonumber\\
				  &= g^t_{rp}e_{rq} - (-1)^{(p)(q)}g^t_{rq}e_{rp} \nonumber \\
      &= g_{pr}e_{rq} - (-1)^{(p)(q)}g_{qr}e_{rp}, \label{ospgen}
\end{align}
where we have adopted the summation convention over repeated indices.
Here $e_{pq}$ are the $gl(m|n)$ generators satisfying the graded commutation relations
$$
[e_{pq},e_{rs}] = \delta_{rq}e_{ps} - (-1)^{((p)+(q))((r)+(s))}\delta_{ps}e_{rq}.
$$
Using this it can be shown that the generators (\ref{ospgen}) satisfy the symmetry
property
$$
\sigma_{pq} = -(-1)^{(p)(q)}\sigma_{qp}
$$
as well as the graded commutation relations
\begin{align}
	[\sigma_{pq},\sigma_{rs}] &= g_{rq}\sigma_{ps}-(-1)^{((p)+(q))((r)+(s))}g_{ps}\sigma_{rq}
\nonumber \\
& \ \ -(-1)^{(p)(q)}\left\{ g_{rp}\sigma_{qs}-(-1)^{((p)+(q))((r)+(s))}g_{qs}\sigma_{rp}
\right\}.
\label{osprels}
\end{align}

We shall also work with the generators
$$
\sigma^p_{\ q} = g^{pr}\sigma_{rq}
$$
where $g^{pr}\equiv \left(g^{-1}\right)_{pr}$. This gives the generators
\begin{align}
\sigma^p_{\ q} &= g^{pr}\sigma_{rq} \nonumber \\
&= g^{pr}\left\{ g_{rs}e_{sq} - (-1)^{(r)(q)}g_{qs}e_{sr} \right\} \nonumber\\
&= e_{pq} - (-1)^{(p)(q)}g^{pr}g_{qs}e_{sr} \label{altospgen}
\end{align}
which will depend on the explicit choice of metric $g$.

\section{Adjoint tensors} \label{sec3}

Operators $X_{pq}$ are said to form an $osp(m|n)$ adjoint tensor operator if they satisfy
the same relations as the $\sigma_{pq}$ under graded commutation:
\begin{align}
	[\sigma_{pq},X_{rs}] &= g_{rq}X_{ps}-(-1)^{((p)+(q))((r)+(s))}g_{ps}X_{rq}
\\
& \ \ -(-1)^{(p)(q)}\left\{ g_{rp}X_{qs}-(-1)^{((p)+(q))((r)+(s))}g_{qs}X_{rp}
\right\}
\label{adjtensor}
\end{align}
where the left hand side is the usual graded commutator. We immediately have the
following.

\begin{lemma}
Let $X_{pq}$ be an adjoint tensor. Then
\begin{itemize}
\item[(i)] The trace
$$
x \equiv X^p_{\ p} =\mbox{tr}(X) = g^{pq}X_{qp}
$$
is always an invariant, i.e. $[\sigma_{pq},x]=0$.
\item[(ii)] If $W_{pq}$ is another adjoint tensor, then so too is
$$
V_{pq}\equiv X_{pr}(-1)^{(r)}g^{rs}W_{sq}.
$$
\end{itemize}
\end{lemma}
\proof{
The result of (i) is easily obtained by substituting $x = g^{pq}X_{qp}$ into the
graded commutator $[\sigma_{pq},x]$ and making use of the symmetry property (\ref{gsym})
of the metric, along with $g^{sr}g_{rq} = \delta^s_{\ r}$. The result of (ii) also follows
by straightforward application of these properties, in addition to the graded derivation rule
$$
[A,BC] = [A,B]C + (-1)^{(A)(B)}B[A,C].
$$
}

The above suggests we define matrix powers
\begin{align*}
	\hat{\sigma}^p_{\ q} &= (-1)^{(p)}g^{pr}\sigma_{rq} = (-1)^{(p)}\sigma^p_{\ q},\\
\left( \hat{\sigma}^{N+1} \right)^p_{\ q} &= \hat{\sigma}^p_{\ r}\left( \hat{\sigma}^N \right)^r_{\ q}
= \left( \hat{\sigma}^N \right)^p_{\ r} \hat{\sigma}^r_{\ q}
\end{align*}
with $\left( \hat{\sigma}^0 \right)^p_{\ q} = \delta^p_{\ q}.$ Note that as a matrix
$\hat{\sigma}=\gamma\sigma,$ with $\gamma$ the grading automorphism. Then it is easily
seen that
$$
X_{pq} = (-1)^{(p)}g_{pr}\left( \hat{\sigma}^N \right)^r_{\ q}
$$
transforms as an adjoint tensor, so we arrive at the Casimir invariants
\begin{align*}
I_N &=  g^{pq}X_{qp}\\
    &=  g^{pq}(-1)^{(q)} g_{qr}\left( \hat{\sigma}^N \right)^r_{\ p}\\
    &=  (-1)^{(p)} \left( \hat{\sigma}^N \right)^p_{\ p}\\
&= \mbox{str}\left( \hat{\sigma}^N \right).
\end{align*}
In particular we have the second order invariant
$$
I_2 = \mbox{str}\left( \hat{\sigma}^2 \right)
$$
which we will see is actually twice the universal Casimir element, while the first order
invariant vanishes identically. Indeed
\begin{align*}
I_1 &= \mbox{str}\left( \hat{\sigma} \right)\\
 &= g^{pq}\sigma_{qp} \\
 &= g^{pq}\left( g_{qr}e_{rp} - (-1)^{(p)(q)}g_{pr}e_{rq} \right)\\
&= \left( e^p_{\ p} - g^{qp}g_{pr}e_{rq} \right)\\
&= e^p_{\ p} - e^q_{\ q} = 0.
\end{align*}

\section{Racah generators} \label{sec4}

To obtain the relations satisfied by the $osp(m|n)$ generators $\sigma^p_{\ q}$ we need to
make a choice for our metric $g$. We choose $g$ to give the Racah $o(m)$ generators which
respect the subalgebra embedding
$$
osp(m|n)\supset osp(m-1|n).
$$
Thus we take $g$ as in (\ref{gdef}) with
$$
g_0 = I \mbox{ (i.e. $m\times m$ identity matrix),}
$$
and
$$
g_1 =
\begin{blockarray}{cc}
\begin{block}{(c)c}
  \begin{array}{c|c}
    0 &
    \begin{array}{cccc} &&&1\\ && 1 & \\ & \iddots & & \\ 1 & && \end{array}
    \\
    \cline{1-2}
    \begin{array}{cccc} &&&-1\\ && -1 & \\ & \iddots & & \\ -1 & && \end{array}
    & 0
  \end{array}
&
  \begin{array}{c}
    \left.\vphantom{\begin{array}{cccc} &&&1\\ && 1 & \\ & \iddots & & \\ 1 & &&
    \end{array}}\right\} k
    \\
    \vphantom{\begin{array}{cccc} &&&-1\\ && -1 & \\ & \iddots & & \\ -1 & && \end{array}}
  \end{array}
\\
\end{block}
\underbrace{
\hphantom{
\begin{array}{cc|l}
0 &
\left. \begin{array}{cccc} &&&1\\ && 1 & \\ & \iddots & & \\ 1 & && \end{array}
\right\} & k \\
\cline{1-2}
\begin{array}{rrrr} &&&-1\\ && -1 & \\ & \iddots & & \\ -1 & && \end{array}
& 0 & \end{array}
}
}_n &
\end{blockarray}.
$$
Hence with this choice we have
\begin{align*}
g_{ij} &= \delta_{ij},\ \ 1\leq i,j\leq m,\\
\delta_{\mu\nu} &= \theta_\mu \delta_{\mu\overline{\nu}},\ \ \overline{\nu}-n+1-\nu,\
1\leq \mu,\nu\leq n,\\
g_{i\mu} &= g_{\mu i} = 0,\ \ 1\leq i\leq m,\ 1\leq \mu\leq n,
\end{align*}
where $\theta_{\mu}$ is the step function defined by
$$
\theta_\mu =
\left\{
\begin{array}{rl} 1, & \mu \leq k=n/2,\\
-1, & \mu > k.
\end{array}
\right.
$$
Then $g$ is indeed orthogonal since
$$
g^{-1} = \left( \begin{array}{c|c} I & 0\\ \hline 0 & g_1^t \end{array} \right) = g^t
$$
and satisfies
\begin{align*}
g^2 &= \left( \begin{array}{c|c} I & 0\\ \hline 0 & -I \end{array} \right) = \gamma \\
\Rightarrow g^{-1} &= \gamma g = g\gamma.
\end{align*}
Thus our generators $\sigma^p_{\ q}$ are given, from (\ref{altospgen}), by
$$
\sigma^p_{\ q} = e_{pq} - (-1)^{(p)(q)}g^{pr}g_{qs}e_{sr}.
$$
Now observe that
\begin{align*}
g^{pq} &= \left( g^{-1} \right)_{pq} = \left( g^t \right)_{pq} \\
&= g_{qp} = \theta_q\delta_{p\tilde{q}}
\end{align*}
where we now define for even indices $p=i$, $\theta_i=1$ and the opposite index
$\tilde{p}$ is given by
$$
\tilde{p} = \left\{
\begin{array}{rl}
i, &p=i,\\
\overline{\mu}, & p = \mu,
\end{array}
\right.
$$
i.e. $\tilde{i} = i,$ $\tilde{\mu} = \overline{\mu}$. In this notation we have
\begin{align*}
	\sigma^p_{\ q} &= e_{pq} - (-1)^{(p)(q)}g^{pr}g_{qs} e_{sr} \\
	&= e_{pq} - (-1)^{(p)(q)}\theta_r\delta_{p\tilde{r}}\theta_q\delta_{q\tilde{s}} e_{sr}\\
 &= e_{pq} - (-1)^{(p)(q)}\theta_{\tilde{p}}\theta_q e_{\tilde{q}\tilde{p}}.
\end{align*}
Finally observe that
$$
\theta_p\theta_{\tilde{p}} = (-1)^{(p)},\ \ \theta_p^2=1,
$$
so we arrive at
\begin{align}
	\sigma^p_{\ q} = e_{pq} - (-1)^{(p)((p)+(q))}\theta_p\theta_q e_{\tilde{q}\tilde{p}}
\label{Racahgen}
\end{align}
which are the Racah generators. They satisfy the symmetry relation
\begin{align}
	\sigma^p_{\ q} = -(-1)^{(p)((p)+(q))} \theta_p\theta_q \sigma^{\tilde{q}}_{\ \tilde{p}}.
\label{Racahsymm}
\end{align}
{\em Remark}:  Explicitly we have
\begin{align*}
\sigma^i_{\ j} &= e_{ij} - e_{ji}, \mbox{ (Racah generators of $o(m)$) } \\
\sigma^{\mu}_{\ \nu} &= e_{\mu\nu} -
\theta_{\mu}\theta_{\nu}e_{\overline{\nu}\overline{\mu}} \\
\sigma^i_{\mu} &= e_{i\mu} - \theta_\mu e_{\overline{\mu} i}\\
\sigma^{\mu}_i &= e_{\mu i} + \theta_\mu e_{i\overline{\mu}}.
\end{align*}
$\blacksquare$ \medskip

The above Racah generators satisfy the following graded commutation relations
\begin{align}
[\sigma^p_{\ q},\sigma^r_{\ s}] &= \delta^r_{\ q}\sigma^p_{\ s} -
	(-1)^{((p)+(q))((r)+(s))} \delta^p_{\ s}\sigma^r_{\ q}\nonumber\\
	&\ \ - (-1)^{(p)((p)+(q))}\theta_p\theta_q\left\{ \delta^r_{\
\tilde{p}}\sigma^{\tilde{q}}_{\ s} - (-1)^{((p)+(q))((r)+(s))}\delta^{\tilde{q}}_{\
s}\sigma^r_{\ \tilde{p}} \right\}
\label{Racahrels}
\end{align}
as we would expect in view of the symmetry relation (\ref{Racahsymm}).

\section{Cartan-Weyl generators} \label{sec5}

The trouble with the above Racah generators is that they are not in Cartan-Weyl form.
Thus it is necessary to find a Cartan subalgebra for $osp(m|n)$ in the above basis and
furthermore to find the Cartan-Weyl generators in terms of the Racah generators. This
problem was first considered for the Lie algebra $o(m)$ by Wong \cite{Wong1967} and Pang and Hecht
\cite{PangHecht1967}.

Here we consider a simplified approach to this problem based on previous work of Gould
\cite{Gould1978}. Following the latter we introduce the numerical $o(m)$ matrix which, for
$m=2h$ even, is given by
$$
M = \frac{1}{\sqrt{2}}
\begin{blockarray}{rrrrrcrrrr}
&&&&&&&&&\\
\begin{block}{(rrrrrcrrrr)}
1       &  0     & 0        & \cdots &  0     & 0      & \cdots & 0       & 0      & 1       \\
-i      &  0     & 0        &        & \vdots & \vdots &        & 0       & 0      & i       \\
 0      &  1     & 0        &        &        &        &        & 0       & 1      & 0       \\
 0      & -i     & 0        & \cdots &        &        & \cdots & 0       & i      & 0       \\
 \vdots &        & \ddots   &        &        &        &        & \iddots &        &  \vdots \\
        &        &          &   1    & 0      & 0      &   1    &         &        &         \\
        &        &          &  -i    & 0      & 0      &   i    &         &        &         \\
 \vdots &        &          &   0    & 1      & 1      &   0    &         &        &  \vdots \\
   0    & \cdots &          &   0    & -i     & i      &   0    &         & \cdots &     0   \\
\end{block}
       &&&&        \stackrel{\uparrow}{{\tiny \mbox{$h$} }} & \stackrel{\uparrow}{{\tiny
\mbox{$h+1$} }}    &&&&
\end{blockarray}
$$
i.e. for $1\leq j\leq h$,
\begin{align*}
M^{2j-1}_{\phantom{2j-1}\ j} &= 1/\sqrt{2} = M^{2j-1}_{\ \ m+1-j},\\
M^{2j}_{\phantom{2j}\ j} &= -i/\sqrt{2} = -M^{2j}_{\ \ m+1-j},
\end{align*}
with all other entries being zero. When $m=2h+1$ is odd we add an extra row and column to
give
$$
M = \frac{1}{\sqrt{2}}
\begin{blockarray}{rrrrrcrrrrr}
&&&&&&&&&&\\
\begin{block}{(rrrrrcrrrrr)}
1       &  0     & 0        & \cdots & 0  & 0      & 0 & \cdots & 0       & 0      & 1       \\
-i      &  0     & 0        &        &    & \vdots   &   &        & 0       & 0      & i       \\
 0      &  1     & 0        &        &    &        &   &        & 0       & 1      & 0       \\
 0      & -i     & 0        & \cdots &    &        &   & \cdots & 0       & i      & 0       \\
 \vdots &        & \ddots   &        &    &        &   &        & \iddots &        &  \vdots \\
        &        &          &   1    & 0  &        & 0 &   1    &         &        &         \\
        &        &          &  -i    & 0  &        & 0 &   i    &         &        &         \\
 \vdots &        &          &   0    & 1  & \vdots & 1 &   0    &         &        &  \vdots \\
   0    & \cdots &          &   0    & -i &    0   & i &   0    &         & \cdots &     0   \\
   0    & \cdots &          &       & 0 &    \sqrt{2}     & 0 &        &         & \cdots &     0   \\
\end{block}
       &&&&        & \stackrel{\uparrow}{{\tiny \mbox{$h+1$} }}  &  &&&&
\end{blockarray}
$$
so we have an additional non-zero entry $\displaystyle{M^m_{\ h+1}=1.}$

We note that the columns of the matrix $M$ form an orthonormal basis so that $M$ is a
unitary matrix, i.e.
$$
M^\dagger =  M^{-1}
$$
as may be easily shown. We note also the following additional symmetry:
\begin{align}
\overline{M}_{ij} = M_{i\overline{j}},\ \ 1\leq i,j\leq m
\label{Msymm}
\end{align}
where now $\overline{i}$ (the opposite index) is given by $\overline{i} = m+1-i.$
Combining these we have:

\begin{lemma} \label{lemmap11}
$$
M_{k\overline{i}} M_{kj} = \delta_{ij}
$$
i.e. the columns of $M$ are orthogonal in this sense. Similarly
$$
\left( M^{-1} \right)_{\overline{i}k}\left( M^{-1} \right)_{jk} = \delta_{ij}.
$$
\end{lemma}
\proof{
Since $M^{-1}=M^\dagger$ we have
\begin{align*}
\delta_{ij} &= \left(M^\dagger\right)_{ik} M_{kj} = \overline{M}_{ki}M_{kj}\\
&\stackrel{(\ref{Msymm})}{=} M_{k\overline{i}}M_{kj},
\end{align*}
as required. Similarly for the second statement.
}

We extend $M$ to an $(m+n)\times(m+n)$ matrix in a trivial way by defining
$$
M_{\mu\nu} = \delta_{\mu\nu},\ \ M_{i\mu} = M_{\mu i} = 0.
$$
We now introduce the Cartan-Weyl generators
\begin{align*}
S^p_{\ q} &= \left(M^{-1} \sigma M\right)^p_{\ q}\\
&= \left(M^{-1}\right)^p_{\ r}\sigma^r_{\ s} M^s_{\ q}
\end{align*}
where $M^{-1}$ satisfies
$$
\left( M^{-1} \right)_{i\mu} = \left( M^{-1}\right)_{\mu i} = 0,\ \ \left(
M^{-1}\right)_{\mu\nu} = \delta_{\mu\nu}.
$$
In view of equation (\ref{Racahrels}) we have the graded commutation relations
\begin{align*}
[S^p_{\ q},S^r_{\ s}] &= \left( M^{-1} \right)^p_{\ p'}\left( M^{-1} \right)^r_{\
r'}[\sigma^{p'}_{\ q'},\sigma^{r'}_{\ s'}]M^{q'}_{\ q}M^{s'}_{\ s} \\
&=\left( M^{-1} \right)^p_{\ p'}\left( M^{-1} \right)^r_{\ r'}
\left[ \delta^{r'}_{\ q'}\sigma^{p'}_{\ s'} - (-1)^{((p)+(q))((r)+(s))} \delta^{p'}_{\
s'}\sigma^{r'}_{\ q'} \right. \\
&\ \ \left. -(-1)^{(p)((p)+(q))}\theta_p\theta_q\left\{ \delta^{r'}_{\ \tilde{p}'}
\sigma^{\tilde{q}'}_{\ s'} - (-1)^{((p)+(q))((r)+(s))}\delta^{\tilde{q}'}_{\
s'}\sigma^{r'}_{\ \tilde{p}'} \right\}  \right] M^{q'}_{\ q}M^{s'}_{\ s} \\
&= \delta^r_{\ q}S^p_{\ s} - (-1)^{((p)+(q))((r)+(s))}\delta^p_{\ s}S^r_{\ q}\\
&\ \ -(-1)^{(p)((p)+(q))} \theta_p\theta_q\left\{  \left( M^{-1} \right)^p_{\ p'}
\left( M^{-1} \right)^r_{\ \tilde{p}'}\sigma^{\tilde{q}'}_{\ s'}M^{q'}_{\ q}M^{s'}_{\ s}
\right. \\
& \ \ \ \ \left.-(-1)^{((p)+(q))((r)+(s))}\left( M^{-1} \right)^p_{\ p'}\left( M^{-1} \right)^r_{\
r'} \sigma^{r'}_{\ \tilde{p}'} M^{q'}_{\ q}M^{\tilde{q}'}_{\ s} \right\}.
\end{align*}
Using Lemma \ref{lemmap11} we have
$$
\left(M^{-1}\right)^p_{\ p'}  \left( M^{-1} \right)^r_{\ \tilde{p}} =
\left\{
	\begin{array}{rl}
		\delta_{r\overline{p}}, & p,r \mbox{ even,}\\
		\delta_{r\overline{p}}, & p,r \mbox{ odd.}
	\end{array}
\right.
$$
Thus in either case,
\begin{align}
	\left(M^{-1}\right)^p_{\ p'}  \left( M^{-1} \right)^r_{\ \tilde{p}} =
	\delta_{r\overline{p}}. \label{starp13}
\end{align}
Similarly using
$$
S = M^{-1}\sigma M\ \ \Rightarrow\ \ \sigma M = MS
$$
we have
\begin{align*}
\sigma^{\tilde{q}'}_{\ s'} M^{q'}_{\ q}M^{s'}_{\ s}
&= (\sigma M)^{\tilde{q}'}_{\ s}M^{q'}_{\ q}\\
&=  (MS)^{\tilde{q}'}_{\ s}M^{q'}_{\ q}\\
&=  M^{\tilde{q}'}_{\ s'}S^{s'}_{\ s}M^{q'}_{\ q}
\end{align*}
where now
$$
M^{\tilde{q}'}_{\ s'}M^{q'}_{\ q} = \delta_{\overline{q}s'}
$$
(c.f. equation (\ref{starp13}) above).
In a similar way we may show that
$$
\left( M^{-1} \right)^p_{\ p'}  \left( M^{-1} \right)^r_{\ r'}\sigma^{r'}_{\ \tilde{p}'}
=
\left( M^{-1}\right)^p_{\ p'} S^r_{\ r'} \left( M^{-1} \right)^{r'}_{\ \tilde{p}'}.
$$
{\em Remark}: For index $p$ we now define the opposite index $\overline{p}$ by
$$
\overline{p} = \left\{
	\begin{array}{rl}
		\overline{i} = m+1-i,& p=i \mbox{ even,}\\
		\overline{\mu} = n+1-\mu,& p=\mu \mbox{ odd.}
	\end{array}
\right.
$$
$\blacksquare$ \medskip

Substituting into the previous relation we arrive at the graded commutation relations
\begin{align}
[S^p_{\ q},S^r_{\ s}] &=
	\delta^r_{\ q}S^p_{\ s} - (-1)^{((p)+(q))((r)+(s))}\delta^p_{\ s}S^r_{\ q}\\
	&\ \ -(-1)^{(p)((p)+(q))} \theta_p\theta_q\left\{ \delta^r_{\
\overline{p}}S^{\overline{q}}_{\ s} - (-1)^{((p)+(q))((r)+(s))}\delta^{\overline{q}}_{\
s}S^r_{\ \overline{p}}  \right\}.
\label{secondstarp13}
\end{align}

Recall that $\overline{i}=m+1-i,$ $\overline{\mu}=n+1-\mu,$ $\theta_i=1,$
$$
\theta_\mu \ \left\{ \begin{array}{rl} 1,& \mu\leq k,\\ -1,& \mu>k \end{array}\right.
$$
and $\theta_p^2=1,$ $\theta_p\theta_{\overline{p}} = (-1)^{(p)}$.
By case splitting it is easily seen that the generators $S^p_{\ q}$ satisfy the symmetry
relation
$$
S^p_{\ q} = -(-1)^{(p)((p)+(q))} \theta_p \theta_q S^{\overline{q}}_{\ \overline{p}}.
$$
In particular, setting $p=q$ into equation (\ref{secondstarp13}) we obtain
\begin{align}
	[S^p_{\ p},S^r_{\ s}] = \left( \delta^r_{\ p} - \delta^p_{\ s} - \delta^r_{\
	\overline{p}} + \delta^{\overline{p}}_{\ s} \right) S^r_{\ s} \label{starp14}
\end{align}
$$
\Rightarrow \ \ [S^p_{\ p},S^r_{\ r}]=0,\ \ S^p_{\ p} = -S^{\overline{p}}_{\
\overline{p}}.
$$
Thus the diagonal generators commute and thus span a Cartan subalgebra, while relations
(\ref{starp14}) show that our generators are indeed in Cartan-Weyl form.

To be explicit, as a Cartan subalgebra, we take the diagonal generators
\begin{align*}
	S^i_{\ i} &= -S^{\overline{i}}_{\ \overline{i}},\ \ 1\leq i\leq h = \left\lfloor
	\frac{m}{2}\right\rfloor\\
	S^\mu_{\ \mu} &= -S^{\overline{\mu}}_{\ \overline{\mu}},\ \ 1\leq \mu\leq k.
\end{align*}
{\em Notes:}
\begin{itemize}
	\item[(1)] Thus for $m=2h+1$ (an odd number) we have for the index $i=h+1$,
		$\overline{h+1}=h+1$
		$$
		\Rightarrow \ \ S^{h+1}_{\ h+1} = -S^{h+1}_{\ h+1} = 0,
		$$
i.e., this generator is identically zero in that case.
\item[(2)]
	Further for even indices we always have
	$$
	S^i_{\ \overline{i}} = S^{\overline{i}}_{\ i} = 0,\ \ 1\leq i\leq h
	$$
	which is consistent with the fact that the weight $2\varepsilon_i$ is not a root
	for $o(m)$.
\end{itemize}

\section{Root system, positive roots and $\mathbb{Z}$-grading ($m\geq 2$)} \label{sec6}

In ungraded index notation it follows that the generator $S^r_{\ s}$ has weight
$\varepsilon_r-\varepsilon_s$, where we define weights $\varepsilon_p$ by
$$
\varepsilon_{\overline{i}} = -\varepsilon_i,\ \ 1\leq i\leq h,
$$
with $\varepsilon_{h+1}=0$ for $m=2h+1$ and
$$
\varepsilon_\mu = \delta_\mu,\ \ \delta_{\overline{\mu}} = -\delta_\mu,\ \ 1\leq \mu\leq
k.
$$
Above, $\varepsilon_i$ ($1\leq i\leq h$), $\delta_\mu$ ($1\leq\mu\leq k$) denote our
elementary even and odd weights with 1 in position $i$ (respectively $\mu$) and zeros
elsewhere.

Thus our roots are $\varepsilon_r-\varepsilon_s\neq0$ or, in graded index notation, we
have the {\em even} roots
\begin{align*}
&\pm\varepsilon_i\pm\varepsilon_j,\ \ 1\leq i<j\leq h,\\
&\pm2\delta_\mu,\ \ \pm\delta_\mu\pm\delta_\nu,\ \ 1\leq\mu<\nu\leq k,
\end{align*}
together with $\pm\varepsilon_i$ ($1\leq i\leq h$) for odd $m=2h+1$. Our {\em odd} roots
are
$$
\pm\varepsilon_i\pm\delta_\mu,\ \ 1\leq i\leq h,\ 1\leq \mu\leq k
$$
together with
$$
\pm\delta_\mu = \varepsilon_{h+1}\pm\delta_\mu
$$
when $m=2h+1$ is odd.

To obtain the corresponding {\em positive} roots we need to look at the
$\mathbb{Z}$-graded structure of $L=osp(m|n)$.
Here our {\em odd} positive roots are given by the weights
$$
\varepsilon_i+\delta_\mu,\ \ 1\leq i\leq m,\ 1\leq\mu\leq k.
$$
In the case $m=2h+1$ is odd, this includes the odd roots
$\delta_\mu=\varepsilon_{h+1}+\delta_\mu,$ $1\leq\mu\leq k$. Thus the set of odd positive
roots is given by
$$
\Phi_1^+ = \{\varepsilon_i+\delta_\mu\ |\ 1\leq i\leq m,\ 1\leq \mu\leq k\}.
$$
The even positive roots are given by $\Phi_{\overline{0}}^+$ which is the set of (even)
positive roots of the even subalgebra $L_{\overline{0}} = o(m)\oplus sp(n)$. Thus our even
positive roots are given by
\begin{align*}
&\delta_\mu\pm\delta_\nu\ \ (1\leq\mu<\nu\leq k),\ \ 2\delta_\mu\ \ (1\leq\mu\leq k)\\
&\varepsilon_i\pm\varepsilon_j,\ \ 1\leq i<j\leq h,
\end{align*}
together with $\varepsilon_i$, $1\leq i\leq h$ in the case $m=2h+1$ is odd.

We first observe that $L$ admits a $\mathbb{Z}_2$-grading
$L=L_{\overline{0}}\oplus L_{\overline{1}}$ where $L_{\overline{0}}=o(m)\oplus sp(n)$ is
the underlying even sub-Lie algebra and
\begin{align*}
L_{\overline{1}} &= \mbox{span}\left\{ S^i_{\ \mu}\ |\ 1\leq i\leq m,\ 1\leq\mu\leq n
\right\}\\
&= \mbox{span}\left\{ S^\mu_{\ i}\ |\ 1\leq i\leq m,\ 1\leq \mu\leq n \right\}.
\end{align*}
We moreover have a $\mathbb{Z}$-grading
\begin{align}
L = L_{-2}\oplus L_{-1}\oplus L_0\oplus L_1\oplus L_2
\label{starp17}
\end{align}
where
\begin{align*}
L_2=\mbox{span}\left\{ S^\mu_{\ \overline{\nu}}\ |\ 1\leq\mu,\nu\leq k \right\},
L_{-2}=\mbox{span}\left\{ S^{\overline{\mu}}_{\ {\nu}}\ |\ 1\leq\mu,\nu\leq k \right\}.
\end{align*}
Also,
$$
L_{\overline{0}} = L_{-2}\oplus L_0\oplus L_2
$$
where $L_0 = o(m)\oplus gl(k)$ and
$$
gl(k) = \mbox{span}\left\{ S^{\mu}_{\ \nu}\ |\ 1\leq\mu,\nu\leq k \right\}.
$$
Then
$$
L_{\overline{1}} = L_{-1}\oplus L_1
$$
with
\begin{align*}
L_1 &= \mbox{span}\left\{ S^\mu_{\ i}\ |\ 1\leq i\leq m,\ 1\leq \mu\leq k \right\}\\
L_{-1} &= \mbox{span}\left\{ S^i_{\ \mu}\ |\ 1\leq i\leq m,\ 1\leq \mu\leq k \right\}.
\end{align*}

Note that the set of even positive roots is given by
$$
\Phi_{\overline{0}}^+ = \Phi_0^+ \cup \left\{ \delta_\mu+\delta_\nu\ | \
1\leq\mu\leq\nu\leq k \right\}
$$
with $\Phi_0^+$ the (positive) roots of $L_0 = o(m)\oplus gl(k)$.

The result of this discussion is that the above choice of positive roots is consistent with the
$\mathbb{Z}$-grading (\ref{starp17}) on $L=osp(m|n)$.

\noindent
{\em Note}:
In the case $m=2$, corresponding to the case $L=C(k+1)$, the above $\mathbb{Z}$-grading and
hence the choice of simple roots -- particularly the odd positive roots -- is quite different to Kac's distinguished choice. See Appendix C for
details.

\section{Highest weights, irreducible $L$-modules and $\mathbb{Z}$-gradings} \label{sec8}

Consistent with the $\mathbb{Z}$-gradation (\ref{starp17}), every finite dimensional irreducible
$L$-module $V(\Lambda)$, with highest weight $\Lambda$, admits a $\mathbb{Z}$-gradation
$$
V(\Lambda) = \bigoplus_{i=-d}^0 V_i(\Lambda)
$$
whose maximal $\mathbb{Z}$-graded component $V_0(\Lambda)$ is necessarily a finite dimensional
irreducible $L_0$-module. It follows that $\Lambda$ must be dominant for $L_0=o(m)\oplus gl(k)$.
Furthermore, $\Lambda$ must be the highest weight of an irreducible $L_{\overline{0}}=o(m)\oplus
sp(n)$  module so in fact $\Lambda$ must be dominant for $L_{\overline{0}}$.

It follows that the components of
$$
\Lambda = \sum_{i=1}^h\Lambda_i\varepsilon_i +\sum_{\mu=1}^k\Lambda_\mu\delta_\mu
$$
must satisfy
$$
\Lambda_{\mu=k}\geq\Lambda_{k-1}\geq \cdots\geq\Lambda_{\mu=1}\geq 0,\ \ \Lambda_\mu\in\mathbb{Z}
$$
(dominant condition for $sp(n)$) and
\begin{align*}
&\Lambda_{i=1}\geq \Lambda_2\geq \cdots\geq \Lambda_{i=h}\geq 0,\ \ m=2h+1,\\
&\Lambda_1\geq \Lambda_2\geq \cdots\geq |\Lambda_h|,\ \ m=2h,\\
& 2\Lambda_i\in\mathbb{Z},\ \ \Lambda_i-\Lambda_j\in\mathbb{Z},\ \ i\neq j,
\end{align*}
so that the even components of $\Lambda$ are simultaneously either all integers (tensor
representations) or all $1/2$-odd integers (spinor representations). Thus, unlike type I Lie
superalgebras (in the sense of Kac), we do {\em not} in general obtain one-parameter families of irreducible modules.

The eigenvalues of the universal Casimir element $C_L$ on $V(\Lambda)$ are well known to be given by
\begin{align}
\chi_\Lambda(C_L) = (\Lambda,\Lambda+2\rho)
\label{p19star}
\end{align}
where $\rho = \rho_0-\rho_1$,
\begin{align}
\rho_0 = \frac12\sum_{\alpha\in\Phi^+_{\overline{0}}} \alpha,\ \ \rho_1 = \frac12
\sum_{\beta\in\Phi^+_1}\beta,
\label{rhos}
\end{align}
is the graded $1/2$-sum of the positive roots, and $(\ ,\ )$ is the bilinear form induced on our
weights defined by
\begin{align}
(\varepsilon_i,\varepsilon_j) = \delta_{ij},\ \ (\delta_\mu,\delta_\nu) = -\delta_{\mu\nu},\ \
(\varepsilon_i,\delta_\mu)=0,\ \ 1\leq i,j\leq h,\ 1\leq \mu,\nu\leq k.
\label{signs}
\end{align}
We have explicitly
\begin{align}
\rho_0 &= \frac12\sum_{i=1}^h(m-2i)\varepsilon_i + \frac12\sum_{\mu=1}^k(n-2\mu+2)\delta_\mu,
\nonumber\\
\rho_1 &= \frac{m}{2}\sum_{\mu=1}^k \delta_\mu
\label{p19twostar}
\end{align}
\begin{align}
\Rightarrow \ \ \rho = \frac12\sum_{i=1}^h(m-2i)\varepsilon_i +
\frac12\sum_{\mu=1}^k(n-m-2\mu+2)\delta_\mu.
\label{starp20}
\end{align}

To see how the sign in the definition of the bilinear form $(\ ,\ )$ arises, it is instructive to
consider the quadratic invariant
\begin{align*}
I_2 &= \mbox{str}(\hat{\sigma}^2)\\
    &= \sigma^p_{\ q}(-1)^{(q)}\sigma^q_{\ p}\\
    &= S^p_{\ q}(-1)^{(q)}S^q_{\ p}\\
    &= (-1)^{(p)}\left({\left(\gamma S\right)^2}\right)^p_{ \ p} = \mbox{str}\left( \hat{S}^2 \right),
\end{align*}
where
$$
\hat{S}=\gamma S \ \ \Rightarrow \ \ \hat{S}^p_{\ q} = (-1)^{(p)}S^p_{\ q}.
$$
Thus if $v^\Lambda_+$ is the maximal weight vector of $V(\Lambda)$, we have (summation over repeated
indices)
\begin{align*}
I_2v^\Lambda_+ &= \left\{ S^p_{\ i}S^i_{\ p} - S^p_{\ \mu}S^\mu_{\ p} \right\}v^\Lambda_+\\
&= S^i_{\ j}S^j_{\ i}v^\Lambda_+ - S^\mu_{\ \nu}S^\nu_{\ \mu}v^\Lambda_+ +S^\mu_{\ i}S^i_{\
\mu}v^\Lambda_+ - S^i_{\ \overline{\mu}} S^{\overline{\mu}}_{\ i}v^\Lambda_+,
\end{align*}
where we have used the fact that the raising generators vanish on $v^\Lambda_+$. Now
$$
S^i_{\ j}S^j_{\ i} = 2C_{L_0}
$$
is twice the Casimir element of $o(m)$, so
$$
S^i_{\ j}S^j_{\ i}v^\Lambda_+ = 2(\Lambda_0,\Lambda_0+2\rho_0)v^\Lambda_+
$$
where
$$
\Lambda_0 = \sum_{i=1}^h\Lambda_i\varepsilon_i
$$
is the projection of $\Lambda$ onto the space of even weights. Similarly setting
$$
\Lambda_1 = \sum_{\mu}^k\Lambda_\mu\delta_\mu
$$
we have
$$
S^\mu_{\ \nu}S^\nu_{\ \mu}v^\Lambda_+ = 2(\Lambda_1,\Lambda_1+2\rho_1)v^\Lambda_+
$$
since $S^\mu_{\ \nu}S^\nu_{\ \mu}$ is twice the universal Casimir element of $sp(n)$ and where the
negative sign needs to be incorporated in the definition of the form $(\ ,\ )$ as in (\ref{signs})
above.

For the remaining two terms we have
\begin{align*}
S^\mu_{\ i}S^i_{\ \mu}v^\Lambda_+
&= \sum{i,\mu}[S^\mu_{\ i},S^i_{\ \mu}]v^\Lambda_+\\
&= \sum_{i,\mu}\left( S^\mu_{\ \mu} + S^i_{\ i} \right) v^\Lambda_+\\
&= m\sum_{\mu=1}^kS^\mu_{\ \mu}v^\Lambda_+ = m\sum_{\mu=1}^k\Lambda_\mu v^\Lambda_+
\end{align*}
and similarly
\begin{align*}
S^i_{\ \overline{\mu}} S^{\overline{\mu}}_{\ i}
&= \sum_{i,\mu}\left( S^{\overline{\mu}}_{\ \overline{\mu}} + S^i_{\ i} \right) v^\Lambda_+\\
&= -m\sum_{\mu=1}^k\Lambda_\mu v^\Lambda_+.
\end{align*}
Thus the eigenvalue $\chi_\Lambda(I_2)$ of $I_2$ on $V(\Lambda)$ is given by
$$
\chi_\Lambda(I_2) = 2(\Lambda,\Lambda+2\rho_0) + 2m\sum_{\mu=1}^k\Lambda_\mu.
$$
By comparison with our expression for $\rho_1$ given in (\ref{rhos}), we may write
\begin{align*}
\chi_\Lambda(I_2) &= 2(\Lambda,\Lambda+2\rho_0) - 2(\Lambda,2\rho_1)\\
&= 2(\Lambda,\Lambda+2\rho)
\end{align*}
which is consistent with equation (\ref{p19star}). The important point here is that we in fact have
$$
I_2=2C_L,
$$
i.e. the second order invariant is actually {\em twice} the universal Casimir element.

\section{Characteristic identities} \label{sec9}

Let $U=U(L)$ be the universal enveloping algebra of $L=osp(m|n)$ and $\Delta:U \rightarrow U\otimes
U$ the usual co-product defined by
$$
\Delta(x)=x\otimes 1+1\otimes x,\  \ x\in L,
$$
where $1$ is the unit element of $U$. This implies that
\begin{align*}
\Delta(C_L) &= \frac12\Delta(I_2) \\
		   &= \frac12 \left( S^p_{\ q}\otimes 1 + 1\otimes S^p_{\ q} \right)(-1)^{(q)}
\left( S^q_{\ p}\otimes 1 + 1\otimes S^q_{\ p} \right)
\end{align*}
since
\begin{align*}
	I_2 &= S^p_{\ q}(-1)^{(q)}S^q_{\ p} = \sigma^p_{\ q}(-1)^{(q)}\sigma^q_{\ p}\\
\Rightarrow \ \ \Delta(C_L) &= \frac12\left( \sigma^p_{\ q}\otimes 1 + 1\otimes \sigma^p_{\ q} \right)
	(-1)^{(q)}\left( \sigma^q_{\ p}\otimes 1+1\otimes \sigma^q_{\ p} \right) \\
	&= C_L\otimes 1+1\otimes C_L + (-1)^{(q)}\sigma^p_{\ q}\otimes \sigma^q_{\ p}.
\end{align*}
Now let $\pi^*$ be the dual vector irreducible representation so that
$$
\pi^*(x) = -\pi^T(x),\ \ x\in L
$$
with $T$ the supertranspose. Then by definition,
\begin{align*}
	\pi^*(\sigma^p_{\ q}) &= -\left( E^p_{\ q} - (-1)^{(p)((p)+(q))}\theta_p\theta_qE^{\tilde{q}}_{\
\tilde{p}} \right)^T
\mbox{ (c.f. equation (\ref{Racahgen}) )}\\
&= -(-1)^{((p)+(q))(p)}E^q_{\ p} + \theta_p\theta_qE^{\tilde{p}}_{\ \tilde{q}}(-1)^{(p)+(q)}.
\end{align*}

We now consider the matrix
\begin{align*}
A &= -\frac12 \pi^*\otimes\mbox{id} \left( \Delta(C_L) - C_L\otimes 1 - 1\otimes C_L \right) \\
  &= \frac12 \sum_{p,q}\left\{ (-1)^{((p)+(q))(p)}E^q_{\ p} -
(-1)^{(p)+(q)}\theta_p\theta_qE^{\tilde{p}}_{\ \tilde{q}}  \right\}\otimes
(-1)^{(q)}\sigma^q_{\ p}\\
&= \frac12\sum_{p,q}(-1)^{(p)(q) + (p)+(q)} E^p_{\ q}\otimes \sigma^p_{\ q} -
\frac12\sum_{p,q}\theta_{\tilde{p}}\theta_{\tilde{q}} E^p_{\ q}(-1)^{(p)}\otimes
\sigma^{\tilde{q}}_{\ \tilde{p}}\\
&= \sum_{p,q}(-1)^{(p)(q)+(p)+(q)}E^p_{\ q}\otimes \sigma^p_{\ q},
\end{align*}
where we have utilised the symmetry
$$
\sigma^{\tilde{q}}_{\ \tilde{p}} = -(-1)^{(p)((p)+(q))}\theta_p\theta_q\sigma^p_{\ q}
$$
together with
$$
\theta_p\theta_{\tilde{p}} = (-1)^{(p)}.
$$
Thus we arrive at our characteristic matrix
$$
A = \sum_{p,q}(-1)^{(p)(q)}E^p_{\ q}\otimes \sigma^p_{\ q} (-1)^{(p)+(q)}
$$
with entries
\begin{equation}
	A_{pq} = (-1)^{(p)}\sigma^p_{\ q}.
\label{charmat}
\end{equation}
{\em Note}: The entries of this matrix are in Racah form.

It follows immediately that this matrix satisfies the polynomial identity
$$
\prod_{p=1}^{m+n}(A-\alpha_p)=0,
$$
where the characteristic roots are given by
\begin{align*}
\alpha_p &= -\frac12\left[ \chi_{\Lambda-\varepsilon_p}(C_L) - \chi_{\delta_1}(C_L) -
\chi_{\Lambda}(C_L) \right]\\
&= -\frac12\left[ (\Lambda-\varepsilon_p,\Lambda - \varepsilon_p+2\rho) -
(\delta_1,\delta_1+2\rho)-(\Lambda,\Lambda+2\rho) \right]\\
&= \frac12 \left[ 2(\Lambda+\rho,\varepsilon_p) + m-n-1-(-1)^{(p)} \right]\\
&= (\Lambda+\rho,\varepsilon_p) + \frac12\left(m-n-1-(-1)^{(p)} \right).
\end{align*}
{\em Note}: Here we have used the fact that for the vector module $V\cong V^*$, we have
the $\mathbb{Z}$-grading
$$
V = V_0\oplus V_1\oplus V_2,
$$
where $V_0$ is the vector module of $gl(k)$, $V_1$ the vector module of $o(m)$ and $V_2$
the dual vector module of $gl(k)$. The weights in $V$ are the $\varepsilon_p$ ($1\leq
p\leq m+n$) and the highest weight is actually $\delta_1$.
$\blacksquare$ \medskip

Thus we have the odd characteristic roots, with $1\leq\mu\leq k$,
\begin{align*}
\alpha_\mu &= (\Lambda+\rho,\delta_\mu) + \frac12(m-n)\\
&= -\Lambda_\mu + m-n+\mu-1,\\
\alpha_{\overline{\mu}} &= -(\Lambda+\rho,\delta_\mu) + \frac12(m-n)\\
&= \Lambda_\mu - \mu+1,
\end{align*}
together with the even characteristic roots, with $1\leq i\leq h$,
\begin{align*}
\alpha_i &= (\Lambda+\rho,\varepsilon_i)+\frac12(m-n-2)\\
&= \Lambda_i+m-i-\frac{n}{2}-1,\\
\alpha_{\overline{i}} &= -(\Lambda+\rho,\varepsilon_i) + \frac12(m-n-2)\\
&= -\Lambda_i + i-\frac{n}{2} - 1,
\end{align*}
and the additional even root
$$
\alpha_{h+1} = \frac12(m-n-2)
$$
when $m=2h+1$ is odd.

If we define odd weight labels $\Lambda_\mu$, for $\mu>k$, by
$$
\Lambda_{\overline{\mu}} = m-\Lambda_\mu+1,
$$
then the above characteristic roots can be written in the unified form
$$
\alpha_\mu = -\Lambda_\mu+m-n+\mu-1,\ \ 1\leq\mu\leq n.
$$
Similarly if we define even weight labels $\Lambda_i$, for $i>h$, according to
$$
\Lambda_{\overline{i}} = 1-\Lambda_i,
$$
which is consistent with
$$
\Lambda_{h+1}=\frac12
$$
for odd $m=2h+1$, the even characteristic roots can also be expressed in the unified form
$$
\alpha_i = \Lambda_i+m-i-\frac{n}{2}-1,\ \ 1\leq i\leq m.
$$

Thus with these conventions we arrive at the $osp(m|n)$ identity
$$
\prod_{i=1}^m(A-\alpha_i)\prod_{\mu=1}^n(A-\alpha_\mu) = 0,
$$
where we define labels $\Lambda_p$ for all $p=1,2,\ldots,m+n$ according to the above, and
our even and odd roots are
\begin{align*}
\alpha_i &= \Lambda_i+m-i-\frac{n}{2}-1,\ \ 1\leq i\leq m,\\
\alpha_\mu &= -\Lambda_\mu+m-n+\mu-1,\ \ 1\leq\mu\leq n.
\end{align*}

\section{$osp(m+1|n)\supset osp(m|n)$ branching condition} \label{branchingcondition}

Here we add an extra even index $i=0$, to give the Racah generators $\sigma^p_{\ q}$,
$0\leq p,q\leq m+n$ of $osp(m+1|n)=K$ containing the canonical subalgebra $L=osp(m|n)$.
We have, by analogy with equation (\ref{starp17}),
the $\mathbb{Z}$-grading
$$
K = K_{-2}\oplus K_{-1}\oplus K_0\oplus K_1\oplus K_2
$$
where $K_0=o(m+1)\oplus gl(k)$ and $K_{\pm 1} = L_{\pm 1}\oplus M_{\pm}$ with
\begin{align*}
M_+ &= \mbox{span}\left\{ \sigma^\mu_{\ i=0}\ |\ 1\leq\mu\leq k\right\},\\
M_- &= \mbox{span}\left\{ \sigma^{i=0}_{\ \mu}\ |\ 1\leq \mu\leq k\right\}
\end{align*}
and $K_{\pm 2} = L_{\pm 2}.$

We also find it convenient to set
$$
L_+ = L_1\oplus L_2,\ \ K_+ = K_1\oplus K_2  = L_+\oplus M_+
$$
and similarly for $L_-,K_-.$

Now let $\tilde{V}(\tilde{\Lambda})$ be an irreducible $K=osp(m+1|n)$ module with highest
$\mathbb{Z}$-graded component $\tilde{V}_0(\tilde{\Lambda})$, constituting an irreducible $K_0$-module, such
that
$$
K_+\tilde{V}_0(\tilde{\Lambda}) = (0).
$$
On the other hand, $\tilde{V}_0(\tilde{\Lambda})$ is an irreducible $K_0$-module and $K=K_-\oplus K_0\oplus
K_+$. By the PBW theorem  we have
\begin{align*}
\tilde{V}(\tilde{\Lambda}) &= U(K_-)\tilde{V}_0(\tilde{\Lambda})\\
&= U(L_-)U(M_-) \tilde{V}_0(\tilde{\Lambda}).
\end{align*}
Hence setting
$
W = U(M_-)\tilde{V}_0(\tilde{\Lambda}),
$
we have
\begin{align}
\tilde{V}(\tilde{\Lambda}) = U(L_-)W. \label{starp27}
\end{align}
{\em Note}: $L_-$ and $M_-$ both transform as $L_0$-modules under the adjoint action. Also
$\tilde{V}_0(\tilde{\Lambda})$ decomposes into irreducible $L_0$-modules according to the usual
$o(m+1)\supset o(m)$ branching rule.
$\blacksquare$ \medskip

If we set $\tilde{\Lambda}=\tilde{\Lambda}_0+\tilde{\Lambda}_1$, then we write the above decomposition into irreducible
$L_0$-modules according to
\begin{equation}
\tilde{V}_0(\tilde{\Lambda}) = \bigoplus_{\Lambda_0} V_0(\Lambda_0 + \tilde{\Lambda}_1) \label{starp28}
\end{equation}
where the components of $\Lambda_0,$ $\tilde{\Lambda}_0$ obey the usual betweenness conditions
\begin{align}
& \tilde{\Lambda}_{i=1}\geq \Lambda_1\geq \tilde{\Lambda}_2\geq \cdots\geq \tilde{\Lambda}_h\geq\Lambda_h\geq -\tilde{\Lambda}_h,\ \
m=2h, \label{doublestarp28i}\\
& \tilde{\Lambda}_{i=1} \geq \Lambda_1\geq \tilde{\Lambda}_2\geq \cdots\geq \tilde{\Lambda}_h\geq \Lambda_h\geq |\tilde{\Lambda}_{h+1}|,\
\ m=2h+1, \label{doublestarp28ii}
\end{align}
with each such module occurring exactly once. It follows that the irreducible $L_0$-modules
occurring in $W$ have highest weights of the form
$$
\Lambda - \delta_{\mu_1} - \delta_{\mu_2} - \cdots - \delta_{\mu_r},\ \
1\leq\mu_1\neq\mu_2\neq\cdots\neq \mu_r\leq k.
$$

We now note that every finite dimensional irreducible $K$-module admits a non-degenerate
sesquilinear form which is {\em invariant} in the sense
$$
\langle xv,w\rangle = (-1)^{(x)(v)}\langle v,x^\ddagger w\rangle
$$
for all $v,w\in \tilde{V}(\tilde{\Lambda})$, $x\in K$, where now $\ddagger$ is a {\em super-conjugation}
operation (see Appendix A). Such a form, which is induced by an inner product on the
maximal $\mathbb{Z}$-graded component $\tilde{V}_0(\tilde{\Lambda})$ in a natural way, has all the
properties of an inner product except it is not generally positive definite. Note that in comparison to the
$C(k+1)=osp(2|2k)$ case this inner product has a different definition. See, for example, the article
\cite{ZhaGou19902}.

From Appendix A, we note that
$$
L_{\pm}^\ddagger = L_{\mp},\ \ K_{\pm}^\ddagger = K_{\mp}.
$$
Following our $gl(m|n)$ approach \cite{GIW1} we have the following result.

\begin{lemma} \label{lemma2}
Suppose $v_+$ is an $L$-maximal weight vector in $\tilde{V}(\tilde{\Lambda})$. Then
$$
\langle v_+,W \rangle \neq (0).
$$
\end{lemma}
\proof{
Otherwise we would have
\begin{align*}
(0)
&= \langle U(L_+)v_+,W \rangle\\
&= \langle v_+,U(L_-)W \rangle\\
&\stackrel{(\ref{starp27})}{=} \langle v_+,\tilde{V}(\tilde{\Lambda})\rangle
\end{align*}
$$
\Rightarrow \ \ v_+ = (0).
$$
}

Finally following exactly the same approach as for $gl(m|n)$ \cite{GIW1} we arrive at the
following {\em branching condition}. The details of the proof are given in Appendix B.

\begin{thm} \label{branchingtheorem}
Let $\tilde{V}(\tilde{\Lambda})$ be a finite dimensional irreducible $osp(m+1|n)$ module with highest weight
$\tilde{\Lambda}$ and $v_+$ a maximal weight vector of $osp(m|n)$ of highest weight $\Lambda$. Then
\begin{itemize}
\item[(a)] $v_+$ is unique (up to scalar multiples),
\item[(b)] the components of $\Lambda$ must satisfy the betweenness conditions
(\ref{doublestarp28i})-(\ref{doublestarp28ii}) and
\begin{equation}
\tilde{\Lambda}_\mu\geq \Lambda_\mu\geq \tilde{\Lambda}_\mu -1
\ \ 1\leq\mu\leq k.
\label{braconthm1}
\end{equation}
\end{itemize}
\end{thm}

It follows that our previous approach for evaluating the reduced matrix elements and
certain Wigner coefficients based on the characteristic identities, should extend to
$osp(m|n)$.

\noindent
{\em Note}: The above also extends to $osp(1|n=2k)$ but with even index generator
$$
\sigma^{i=1}_{\ i=1}
$$
which vanishes identically. In this case the even subalgebra is simply
$L_{\overline{0}}=sp(n=2k).$

\section{$osp(m|n)$ vector operators} \label{sec11}

Recall that the $osp(m|n)$ matrix has entries given by equation (\ref{charmat}), i.e.
$A=\gamma\sigma$, with $\gamma$ the grading automorphism introduced in (\ref{gradaut}). Now let $V$
be the vector module and $\pi$ the irreducible representation of $L=osp(m|n)$ afforded by $V$. Then
a {\em vector operator} $\psi$ is equivalent to an intertwining operator on an irreducible
$L$-module $V(\Lambda)$,
$$
\psi:V\otimes V(\Lambda)\rightarrow W,
$$
with $W=$Im$\psi$ some $L$-module. Thus for $x\in L$ we have
\begin{align*}
x\psi(e_p\otimes v) &= \psi\Delta(x)(e_p\otimes v)\\
&=\psi(xe_p\otimes v) + (-1)^{(p)(x)}\psi(e_p\otimes xv)
\end{align*}
$\forall v\in V(\Lambda)$, with $\{e_p\}$ the usual basis for $V$.

Equivalently we have a collection of components $\psi^p$, defined by
$$
\psi^pv = \psi(e_p\otimes v), \ \ \forall v\in V(\Lambda)
$$
so that, for all $v\in V(\Lambda)$,
\begin{align*}
	& x\psi^p v = \psi(xe_p\otimes v) + (-1)^{(p)(x)} \psi^px v \\
	\Rightarrow & (x\psi^p - (-1)^{(p)(x)}\psi^px)v = \pi(x)_{qp}\psi^q v.
\end{align*}
Therefore, by abstraction, we arrive at the transformation law of $osp(m|n)$ vector operators:
$$
[x,\psi^p] = \pi(x)_{qp}\psi^q,\ \ \forall x\in L,
$$
with the bracket on the left hand side being the usual graded commutator. Thus a vector operator
satisfies
$$
[\sigma^p_{\ q},\psi^r] = \pi(\sigma^p_{\ q})_{sr} \psi^s.
$$
Using
$$
\pi(\sigma^p_{\ q}) = e_{pq} - (-1)^{(p)((p)+(q))}\theta_p\theta_qe_{\tilde{q}\tilde{p}},
$$
where now $e_{pq}$ is an elementary matrix, it follows that
\begin{align*}
\pi(\sigma^p_{\ q})_{sr} &= \delta_{qr}\delta^p_{\ s} -
	(-1)^{(p)((p)+(q))}\theta_p\theta_q\delta_{\tilde{q}s}\delta_{\tilde{p}r} \\
\Rightarrow \ \ [\sigma^p_{\ q},\psi^r] &= \delta^r_{\ q}\psi^p -
	(-1)^{(p)((p)+(q))}\theta_p\theta_q\delta^{\tilde{p}}_{\ r}\psi^{\tilde{q}}.
\end{align*}
this is the transformation law for a vector operator.

Similarly we say that the operators $\phi_p$ transform as a {\em contragredient vector operator} if
\begin{align*}
[\sigma^p_{\ q},\phi_r] &= \pi^*(\sigma^p_{\ q})_{sr}\phi_s \\
&= -\pi^T(\sigma^p_{\ q})_{sr}\phi_s \\
&= -\left\{ e_{pq}^T - (-1)^{(p)((p)+(q))} \theta_p\theta_qe_{\tilde{q}\tilde{p}}^T
\right\}_{sr}\phi_s\\
&= -(-1)^{(p)((p)+(q))}\left\{ \delta^p_{\ r}\phi_q - (-1)^{((p)+(q))(q)}\delta^{\tilde{q}}_{\
r}\phi_{\tilde{p}} \right\}.
\end{align*}

\ \\

\noindent
\underline{Motivating example}: $osp(m+1|n)\supset osp(m|n)$

\ \\

Here we have the additional index $i=0$ giving rise to the operators
$$
\psi^p = \sigma^p_{\ 0},\ \ \phi_p = \sigma^0_{\ p}\ \ (1\leq p\leq m+n)
$$
which are easily seen to transform as vector (respectively contragredient vector) operators with
respect to $osp(m|n)$.

\section{Characteristic identities and reduced matrix elements} \label{sec12}

We now let
$$
A^p_{\ q} = (-1)^{(p)}\sigma^p_{\ q},\ \ 1\leq p,q\leq m+n
$$
be the $osp(m|n)$ matrix with characteristic roots $\alpha_r$ and
$$
B^p_{\ q} = (-1)^{(p)}\sigma^p_{\ q},\ \ 0\leq p,q\leq m+n
$$
the corresponding $osp(m+1|n)$ matrix with characteristic roots $\beta_r$ ($0\leq r\leq m+n$). We
have also the associated projections
\begin{align}
P[r] &= \prod_{q\neq r,q=1}^{m+n} \left( \frac{A-\alpha_q}{\alpha_r-\alpha_q} \right),\label{projA} \\
Q[r] &= \prod_{q\neq r,q=0}^{m+n} \left( \frac{B-\beta_q}{\beta_r-\beta_q} \right), \label{projB}
\end{align}
which satisfy
$$
P[q]P[r] = \delta_{qr}P[q],\ \ Q[q]Q[r] = \delta_{qr}Q[q]
$$
and the identity resolutions
$$
\sum_{q=1}^{m+n}P[q]_{rs} = \delta_{rs},\ \
\sum_{q=0}^{m+n}Q[q]_{rs} = \delta_{rs}.
$$
{\em Remark}:
As we have pointed out in our previous work relating to $gl(m|n)$ \cite{GIW1}, there may exist
irreducible representations for which the characteristic roots coincide. In such a case, the
projections defined above in (\ref{projA}) or (\ref{projB}) would be undefined. To circumvent this
issue, as in the case of $gl(m|n)$, we point out that the set of highest weights to which such
characteristic roots correspond is closed in the Zariski topology \cite{Humphreys} on $H^*$, the Cartan subalgebra
dual. It follows that there is a dense subset of $H^*$ on which the characteristic roots are distinct.
Without loss of generality, we therefore make the assumption that all characterstic roots are
distinct. This means that in practice, when applying the final formulae for the invariants which
appear as rational polynomial
functions of the characteristic roots, we must always be mindful of the fact that we are relying on
analytic continuation to define the function in certain pathological cases, and that terms
in the numerator and denominator should be cancelled where required.
This can always be done since polynomial functions on $H^*$ are continuous in the Zariski topology.
$\blacksquare$ \medskip

Following our $gl(m|n)$ derivation \cite{GIW1} we have for $1\leq r\leq m+n$,
$$
B^r_{\ s} Q[q]^s_{\ 0} + B^r_{\ 0}C_q = \beta_q Q[q]^r_{\ 0},
$$
where $C_q$ denotes the $osp(m|n)$ invariant
$$
C_q = Q[q]^0_{ \ 0}
$$
which has eigenvalues that determine the squares of a certain Wigner coefficient.

\noindent
{\em Note}: Recall that for $osp(m|n)$,
\begin{align*}
\alpha_r &= -\frac12 \left[ \chi_{\Lambda-\varepsilon_r}(C_L) - \chi_{\delta_1}(C_L) -
\chi_\Lambda(C_L) \right]\\
&= (\Lambda+\rho,\varepsilon_r)+\frac12 (m-n-1-(-1)^{(r)}),
\end{align*}
with a similar expression for $\beta_r$. The important point is that $P[r],$ $Q[r]$ project onto
a submodule with highest weight $\Lambda-\varepsilon_r$ in the tensor product module $V\otimes
V(\Lambda)$ (with $V$ the vector module).
$\blacksquare$ \medskip

Thus rearranging we may write
\begin{equation}
	(-1)^{(r)}\psi^rC_q = (\beta_q-A)^r_{\ s}Q[q]^s_{\ 0},
\label{starp4II}
\end{equation}
where
$$
\psi^r = (-1)^{(r)}B^r_{\ 0} = \sigma^r_{\ 0},\ \ 1\leq r\leq m+n
$$
is a vector operator of $osp(m|n)$ as we have seen.

At this point we note degeneracies between the odd roots of $osp(m+1|n)$ and those of $osp(m|n)$
which parallels the situation with $gl(m|n)$ (but with even roots in that case). In terms of the
$osp(m+1|n)$ representation labels $\tilde{\Lambda}_\mu$ ($1\leq \mu\leq k$) we have the characteristic roots
$$
\beta_\mu = -\tilde{\Lambda}_\mu + m-1-n+\mu-1,\ \ 1\leq \mu\leq n,
$$
where we define the representation labels $\tilde{\Lambda}_\mu$ for $\mu>k=n/2$ by
$$
\tilde{\Lambda}_{\overline{\mu}} = m-\tilde{\Lambda}_\mu+1
$$
as before. If $\Lambda_\mu$ are the $osp(m|n)$ representation labels so
$$
\alpha_\mu = -\Lambda_\mu + m-n+\mu-1,\ \ 1\leq \mu\leq n,
$$
we have from the branching condition (\ref{braconthm1}) of Theorem \ref{branchingtheorem} that
$$
\beta_\mu =
\left\{
\begin{array}{rl}
1+\alpha_\mu, & \Lambda_\mu = \tilde{\Lambda}_\mu,\\
\alpha_\mu, & \Lambda_\mu = \tilde{\Lambda}_\mu - 1,
\end{array}
\right.
$$
while
$$
\beta_{\overline{\mu}} = \tilde{\Lambda}_\mu - \mu - \frac12 =
\left\{
\begin{array}{rl}
\alpha_{\overline{\mu}}, & \Lambda_\mu = \tilde{\Lambda}_\mu,\\
1+\alpha_{\overline{\mu}}, & \Lambda_\mu = \tilde{\Lambda}_\mu - 1.
\end{array}
\right.
$$
We now introduce the odd index set $I_1$ as follows. For $1\leq \mu\leq k$ we take
$$
\mu\in I_1 \ \ \Leftrightarrow\ \ \tilde{\Lambda}_\mu = \Lambda_\mu+1 \ (\mbox{or }\beta_\mu=\alpha_\mu),
$$
otherwise we take $\overline{\mu}\in I_1$, i.e.
$$
\overline{\mu}\in I_1\ \ \Leftrightarrow\ \ \tilde{\Lambda}_\mu = \Lambda_\mu.
$$
We define $\overline{I}_1$ by
$$
\overline{I}_1 = \{ 1\leq\mu\leq n\ |\ \mu \notin I_1\},
$$
so for $1\leq \mu\leq k$ we have
\begin{align*}
\mu\in\overline{I}_1 \mbox{ if } \tilde{\Lambda}_\mu = \Lambda_\mu,\\
\overline{\mu}\in\overline{I}_1 \mbox{ if } \tilde{\Lambda}_\mu = 1+\Lambda_\mu.
\end{align*}
Therefore, for $1\leq\mu\leq k,$
$$
\mu\in\overline{I}_1\ \ \Rightarrow \\ C_{\mu}=0
$$
since $\Lambda_\mu=\tilde{\Lambda}_\mu$ already takes its maximum value, whilst if $\mu\in I_1$ we have
$C_{\overline{\mu}}=0.$ Thus for every odd index, $\mu$, we have
$$
C_\mu=0, \mbox{ if } \mu\in\overline{I}_1
$$
and
$$
C_{\overline{\mu}} = 0, \mbox{ if } \mu\in I_1.
$$
{\em Note}: As for $gl(m|n)$, we have also the full index sets
$$
I = I_0\cup I_1,\ \ \tilde{I} = I\cup \{i=0\},
$$
where $I_0 = \{ i=1,2,\ldots,m\}$ is the set of even indices.
$\blacksquare$ \medskip

We shall also consider the contragredient vector operator
$$
\phi_r = \sigma^0_{\ r},\ \ 1\leq r\leq m+n.
$$
Then $\phi_r,$ $\psi^r$ may be resolved into shift components in the usual way:
$$
\phi_r = \sum_{q=1}^{m+n}\phi[q]_r,\ \ \psi^r = \sum_{q=1}^{m+n}\psi[q]^r
$$
where
\begin{align}
	\psi[q]^r &= (-1)^{(r)} P[q]^r_{\ s} (-1)^{(s)}\psi^s, \label{starp7IIi}\\
\phi[q]_r &= \phi_sP[q]^s_{\ r}, \label{starp7IIii}
\end{align}
with $P[q]$ the projection introduced earlier. Then we clearly have that if $q$ is an odd index then
$$
\psi[q]=0 \mbox{ if } q\in \overline{I}_1,
$$
while
$$
\phi[q]=0 \mbox{ if } q\in I_1.
$$
{\em Note}: Some care needs to be taken here. Actually $\phi[q]_r$ will {\em decrease} $\Lambda$ by
$-\varepsilon_q$, i.e. $\phi[q]$ affects the shift $\Lambda \rightarrow \Lambda-\varepsilon_q$.
Therefore for $q=\mu$ an odd index, $1\leq \mu\leq k,$ we see that $\phi[\mu]$ affects the shift
$\Lambda\rightarrow \Lambda-\delta_\mu$ so that $\phi[\mu]$ will {\em decrease} label $\Lambda_\mu$
by 1 unit. If $\Lambda_\mu = \tilde{\Lambda}_\mu - 1,$ i.e. $\mu\in I_1,$ then $\phi[\mu]$ must  vanish, and
vice-versa for $\mu\in\overline{I}_1$.
$\blacksquare$ \medskip

Inverting equation (\ref{starp4II}) and resolving $\psi^r$ into its shift components allows us to
write
\begin{equation}
	Q[q]^r_{\ 0} = (-1)^{(r)}\sum_{s\in I} (\beta_q-\alpha_s)^{-1}\psi[s]^rC_q,\ \ q\in \tilde{I},
\label{aboveequ}
\end{equation}
where we have used the fact that $\psi[s]^r,$ $Q[q]^r_{\ 0}$ and $C_q$ all vanish for
$s,q\in\overline{I}_1$. Using the easily established relation
$$
(\beta_q-\alpha_s)^{-1}\psi[s] = \psi[s](\beta_q-\alpha_s-(-1)^{(s)})^{-1}
$$
except for odd $m=2h+1$ where we have
$$
(\beta_q-\alpha_i)^{-1}\psi[i] = \psi[i](\beta_q - \alpha_i)^{-1},\ \ i=h+1,
$$
equation (\ref{aboveequ}) may be expressed in the form
\begin{equation}
	Q[q]^r_{\ 0} = (-1)^{(r)}\sum_{s\in I} \psi[s]^r (\beta_q-\alpha_s-(-1)^{(s)} + \delta_{s,h+1})^{-1}C_q,
\label{starp8II}
\end{equation}
where the term $\delta_{s,h+1}$ only contributes when $m=2h+1$ is odd.

\noindent
{\em Note}: Using the relation
$$
\left( \sigma^p_{\ q} \right)^\ddagger = (-1)^{((p)+(q))(p)}\sigma^q_{\ p}
$$
we have the easily established relations
$$
\left( \psi^p \right)^\ddagger = (-1)^{(p)}\phi_p,\ \ \phi_p^\ddagger = \psi^p.
$$
Thus resolving into shift components
$$
\Rightarrow \ \ \sum_r\phi[r]_p^\ddagger = \sum_r\psi[r]^p
$$
so we arrive at
$$
\phi[r]_p^\ddagger = \psi[r]^p.
$$
This can also be seen by noting for our $osp(m|n)$ matrix $A$ that
$$
A_{pq}^\ddagger = (-1)^{(q)((p)+(q))}A_{qp}
$$
and more generally or matrix powers
$$
\left( A_{pq}^N \right)^\ddagger = (-1)^{((p)+(q))(q)}A_{qp}^N.
$$

The matrix $A$ acts naturally on the right of $\phi_r = B^0_{\ r} = \sigma^0_{\ r}$ so that
\begin{align*}
\left( \phi[q]_r \right)^\ddagger &= \left( \phi_s P[q]_{sr} \right)^\ddagger \\
&= (-1)^{(s)((s)+(r))}\left( P[q]_{sr} \right)^\ddagger \phi_s^\ddagger \\
&= (-1)^{(s)((s)+(r))}(-1)^{((r)+(s))(r)} P[q]_{rs}\psi^s \\
&= (-1)^{(r)}P[q]_{rs}(-1)^{(s)}\psi^s \\
&\stackrel{(\ref{starp7IIi})}{=} \psi[q]^r
\end{align*}
as expected.

\noindent
{\em Note}: For odd index $p=\mu$, $1\leq\mu\leq k$, or even index $p=i$, $1\leq i\leq h$,
$\phi[p]_r$ {\em decreases} the representation label $\Lambda_p$ by 1 unit so $\psi[p]^r$ must
{\em increase} it.
$\blacksquare$ \medskip

Our analysis now follows closely that of $gl(m|n)$ \cite{GIW1}. Summing equation (\ref{starp8II})
over $q$ gives
$$
(-1)^{(r)}\sum_{s\in I}\psi[s]^r\sum_{q\in\tilde{I}}(\beta_q-\alpha_s-(-1)^{(s)} +
\delta_{s,h+1})^{-1} C_q = 0
$$
\begin{equation}
\Rightarrow \ \ \sum_{q\in\tilde{I}} (\beta_q-\alpha_s - (-1)^{(s)}+\delta_{s,h+1}) C_q=0,\ \ s\in
I.
\label{starp10II}
\end{equation}
Also using the resolution
\begin{equation}
\sum_{q\in\tilde{I}} C_q = 1,
\label{doublestarp10II}
\end{equation}
we obtain $|I|+1$ equations in $|\tilde{I}| = |I|+1$ unknowns which uniquely determines the
$osp(m|n)$ invariants $C_q$. By this means we arrive at the formula
\begin{equation}
C_q = \prod_{k\neq q}^{\tilde{I}} (\beta_q-\beta_k)^{-1}\prod_{r\in I}
(\beta_q-\alpha_r-(-1)^{(r)}+\delta_{r,h+1}).
\label{triplestarp10II}
\end{equation}
{\em Note}: Again, the term $\delta_{r,h+1}$ only applies when $m=2h+1$ is odd. Thus, to be precise, we have the
following formulae:
$$
C_q = \left\{
\begin{array}{rl}
\displaystyle{\prod_{k\neq q}^{\tilde{I}} (\beta_q-\beta_k)^{-1}\prod_{r\in I}(\beta_q -
\alpha_r-(-1)^{(r)})}, &
m=2h,\\
\displaystyle{\prod_{k\neq q}^{\tilde{I}} (\beta_q-\beta_k)^{-1}\prod_{r\in I}(\beta_q -
\alpha_r-(-1)^{(r)}+\delta_{r,h+1})}, & m=2h+1.
\end{array}
\right.
$$
However, it is convenient to use the unified formula of equation (\ref{triplestarp10II}).
$\blacksquare$ \medskip

We now invert equation (\ref{starp8II}) by writing
\begin{equation}
	\psi[p]^r = (-1)^{(r)}\sum_{s\in\tilde{I}} Q[s]^r_{\ 0} \gamma_{ps},\ \ p\in I
\label{starp11II}
\end{equation}
for suitable coefficients $\gamma_{ps},$ $p\in I$, $s\in \tilde{I}$. This leads us to consider the
unique solutions $\gamma_{ps}$ to the set of equations
\begin{align}
	\sum_{s\in\tilde{I}}\gamma_{ps}(\beta_s - \alpha_r - (-1)^{(r)}+\delta_{r,h+1})^{-1}C_s &=
\delta_{pr} \label{doublestar1p11II}\\
\sum_{s\in\tilde{I}}\gamma_{ps} C_s &=0,\ \ p,r\in I. \label{doublestar2p11II}
\end{align}
Then for each $p\in I$ this yields $|\tilde{I}| = |I|+1$ equations in $|\tilde{I}|$ unknowns
$\gamma_{ps}$, $s\in \tilde{I}.$

\noindent
{\em Remark}: Since then we obtain
\begin{align*}
\sum_{s\in\tilde{I}} Q[s]^r_{\ 0}\gamma_{ps}
& \stackrel{(\ref{starp8II})}{=} (-1)^{(s)}\sum_{s\in \tilde{I}}\sum_{q\in I}\psi[q]^r
(\beta_s - \alpha_q-(-1)^{(q)} + \delta_{q,h+1})^{-1}C_s\gamma_{ps} \\
& \stackrel{(\ref{doublestar1p11II})}{=} (-1)^{(r)} \sum_{q\in I} \psi[q]^r
\delta_{pq}\\
& = (-1)^{(r)}\psi[p]^r
\end{align*}
as required.
$\blacksquare$ \medskip

The above equations (\ref{doublestar1p11II}) and (\ref{doublestar2p11II}) are easily solved using
matrix methods and yield the unique solution
$$
\gamma_{ps} = \gamma_p\left( \beta_s - \alpha_p - (-1)^{(p)} + \delta_{p,h+1} \right)^{-1},\ \ p\in
I,\ s\in\tilde{I}
$$
where
\begin{align}
	\gamma_p &= (-1)^{|I|} \prod_{r\neq p}^I\left( \alpha_p-\alpha_r+(-1)^{(p)} - (-1)^{(r)} +
\delta_{r,h+1} - \delta_{p,h+1} \right)^{-1}
\prod_{q\in \tilde{I}}\left( \beta_q - \alpha_p - (-1)^{(p)} + \delta_{p,h+1} \right).
\label{starp12II}
\end{align}
As for the case of $gl(m|n)$ these invariants have a natural interpretation. From the remarks above
we have
$$
\psi[p]^r = (-1)^{(r)} \sum_{s\in\tilde{I}} Q[s]^r_{\ 0}\gamma_{ps}
$$
so that (summation over repeated indices here and below)
\begin{align*}
	\phi[p]_{r} (-1)^{(r)} \psi[p]^r &= \phi_r (-1)^{(r)}
\psi[p]^r\\
&= \phi_r\sum_{s\in \tilde{I}}Q[s]^r_{\ 0} \gamma_{ps} \\
&= \sum_{s\in \tilde{I}} B^0_{\ r} Q[s]^r_{\ 0} \gamma_{ps} \\
&= \sum_{s\in\tilde{I}}(\beta_s - B^0_{\ 0})C_s\gamma_{ps}
\end{align*}
where we have utilised the $osp(m+1,n)$ characteristic identity. Also using
$$
B^{i=0}_{\ i=0} = \sigma^0_{\ 0} = 0,
$$
we obtain
\begin{align*}
	\phi[p]_{r}(-1)^{(r)}\psi[p]^r &= \sum_{s\in \tilde{I}}\beta_s
C_s\gamma_{ps}\\
& \stackrel{(\ref{starp12II})}{=} \gamma_p\sum_{s\in\tilde{I}}\beta_sC_s
\left( \beta_s - \alpha_p - (-1)^{(p)}+\delta_{p,h+1} \right)^{-1}\\
&= \gamma_p\left\{ \sum_{s\in\tilde{I}}C_s + (\alpha_p + (-1)^{(p)}-
\delta_{p,h+1})\sum_{s\in\tilde{I}} C_s\left( \beta_s - \alpha_p - (-1)^{(p)} + \delta_{p,h+1}
\right)^{-1} \right\} \\
& \stackrel{(\ref{starp10II}),(\ref{doublestarp10II})}{=} \gamma_p,\ \ p\in I.
\end{align*}
{\em Remarks}: It follows that the invariants $\gamma_p$, given explicitly by formula
(\ref{starp12II}) above, are closely related to the {\em reduced matrix elements} of the vector
operator $\psi^r = \sigma^r_{\ 0}$ (c.f. the case of $gl(m|n)$ in \cite{GIW1}).

\section{Reduced matrix elements and reduced Wigner coefficients} \label{sec13}

Since $P[r]$ and $Q[r]$ both determine projections, it can be shown that
\begin{equation}
	(P[r]Q[s]P[r])_{p q} = \lambda P[r]_{pq},\ \ 1\leq
	p,q\leq m+n
	\label{starp13II}
\end{equation}
for some $osp(m|n)$-invariant $\lambda$ commuting with both $Q[s]$ and $P[r]$ (generalised
angle operator). On the other hand from the $osp(m+1|n)$ characteristic identity we have
$$
(BQ[s])^p_{\ q} = \beta_sQ[s]^p_{\ q},\ \ 0\leq p,q\leq m+n
$$
which may be expressed
$$
B^p_{\ 0}Q[s]^0_{\ q} = (\beta_s - A)^p_{r}Q[s]^r_q
$$
(summation on $r$ from 1 to $m+n$), or
$$
(-1)^{(p)}\psi^p Q[s]^0_{\ q} = (\beta_s - A)^p_{\ r}Q[s]^r_{\ q}.
$$
Multiplication on the left by $P[t]$ we obtain in view of equation (\ref{starp7IIi}) and
the $osp(m|n)$ characteristic identity
$$
\psi[t]^p Q[s]^0_{\ q} = (-1)^{(p)}(\beta_s - \alpha_t) P[t]^p_{\
r} Q[s]^r_{\ q},
$$
which may be rearranged to give
\begin{equation}
	(P[t]Q[s])^p_{\ q} = (-1)^{(p)}(\beta_s -
	\alpha_t)^{-1}\psi[t]^p Q[s]^0_{\ q} \label{starp14II}
\end{equation}
with $1\leq p\leq m+n$, $0\leq q\leq m+n$, $t\in I$, $s\in \tilde{I}$.

On the other hand the $osp(m+1|n)$ characteristic identity also implies
$$
(Q[s]B)^0_{\ q} = Q[s]^0_{\ q}\beta_s
$$
or
\begin{align*}
	Q[s]^0_{\ 0} B^0_{ \ q} &= Q[s]^0_{\ r}(\beta_s - A)^r_{\ q} \\
	\Rightarrow \ \ C_s \phi_s &= Q[s]^0_{ \ r}(\beta_s - A)^r_{\ q},\ \
	1\leq q\leq m+n.
\end{align*}
In this case we multiply on the right by $P[t]$ to give
\begin{align}
	C_s\phi[t]_r &= Q[s]^0_{\ r} (\beta_s - \alpha_t) P[t]^r_{\ q}
	\nonumber\\
	\Rightarrow \ \ (Q[s]P[t])^0_{\ r} &= C_s \phi[t]_q(\beta_s -
	\alpha_t)^{-1} \nonumber \\
	&= C_s(\beta_s-\alpha_t - (-1)^{(t)} + \delta_{t,h+1})^{-1} \phi[t]_q,
	\label{doublestarp14II}
\end{align}
for $t\in I,$ $s\in\tilde{I}$.

Thus multiplying equation (\ref{starp14II}) on the right by $P[t]$ gives
\begin{align}
	(P[t]Q[s]P[t])^p_{\ q} &= (-1)^{(p)}(\beta_s -
\alpha_t)^{-1}\psi[t]^p (Q[s]P[t])^0_{\ q} \nonumber \\
&\stackrel{(\ref{doublestarp14II})}{=} (-1)^{(p)} (\beta_s - \alpha_t)^{-1}
\psi[t]^p C_s (\beta_s - \alpha_t - (-1)^{(t)}+\delta_{t,h+1})^{-1} \phi[t]_q.
\label{triplestarp14II}
\end{align}
{\em Note}: Since $C_s$ is expressible in terms of the $\beta_q$, $\alpha_r$ in accordance
with equation (\ref{triplestarp10II}), it follows that we must have
$$
(P[t]Q[s]P[t])^p_{\ q} = \lambda' \psi[t]^p\phi[t]_q(-1)^{(p)}
$$
for some function $\lambda'$ of the $\beta_q$, $\alpha_r$.
$\blacksquare$ \medskip

It then follows from equation
(\ref{starp13II}) that we may write
\begin{equation}
	(-1)^{(p)}\psi[t]^p\phi[t]_q = \mu_tP[t]_{pq}, \ \ 1\leq
	p,q\leq m+n \label{starp15II}
\end{equation}
where $\mu_t$ is some invariant. On the other hand we note that
$$
X_{pq} \equiv (-1)^{(p)} \psi[t]^p\gamma_t^{-1}\phi[t]_q = \nu
P[t]_{pq}
$$
with $\nu$ some function of the $\beta_t$, $\alpha_t$ (c.f. equation (\ref{starp12II}). Since
$$
\gamma_t = \phi[t]_\beta(-1)^{(q)}\psi[t]^q,\ \ t\in I
$$
we notice that $X$ is {\em idempotent}, i.e. $X^2=X$, which implies that $\nu^2=\nu$ or
$\nu(\nu-1)=0$. Thus for a non-zero contribution we must have $\nu=1$ which gives
\begin{equation}
	(-1)^{(p)}\psi[t]^p \gamma_t^{-1}\phi[t]_q = P[t]_{pq}.
	\label{doublestarp15II}
\end{equation}
By comparison with equation (\ref{starp15II}) above, we must have
$$
\psi[t]^p\gamma_t^{-1} = \mu_t^{-1} \psi[t]^p
$$
from which we obtain
\begin{equation}
	\mu_t = (-1)^{|I|}\prod_{s\neq t\in I}(\alpha_t - \alpha_s - (-1)^{(s)} + \delta_{s,h+1} -
	\delta_{s,\overline{t}})^{-1}\prod_{r\in\tilde{I}}(\beta_r - \alpha_t).
	\label{triplestarp15II}
\end{equation}
{\em Note}: It is understood that the term $\delta_{s,\overline{t}}$ only contributes when $t=i$ is
{\em even}. When $t$ is odd and $t\in I$, clearly $\overline{t}\notin I$ so this term vanishes.
$\blacksquare$ \medskip

Now returning to equation (\ref{triplestarp14II}), we note, from formula (\ref{triplestarp10II})
that
\begin{align*}
	\psi[t]^p C_s (\beta_s - \alpha_t - (-1)^{(t)}+\delta_{t,h+1})^{-1}
	=&C_s(\beta_s-\alpha_t - (-1)^{(t)}+\delta_{t,h+1})^{-1}(\beta_s-\alpha_{\overline{t}}
	- (-1)^{(t)} + \delta_{t,h+1})^{-1} \\
	& \ \ \times (\beta_s-\alpha_{\overline{t}} - 2(-1)^{(t)} +
	2\delta_{t,h+1})\psi[t]^p,\ \ t\in I
\end{align*}
where it is understood that the last two terms only occur if also $\overline{t}\in I$. In other
words they only occur for $t=i$ even.

\noindent
{\em Remark}: Thus for $t=\mu$ odd we have
$$
\psi[\mu]^p C_s(\beta_s - \alpha_\mu+1)^{-1} = C_s(\beta_s-\alpha_\mu+1)^{-1}\psi[\mu]^p
$$
but for $t=i$ even we have
\begin{align*}
	\psi[i]^p C_s (\beta_s - \alpha_i - 1+\delta_{i,h+1})^{-1}
	=&C_s(\beta_s-\alpha_i - 1+\delta_{i,h+1})^{-1}(\beta_s-\alpha_{\overline{i}}
	- 1 + \delta_{i,h+1})^{-1} \\
	& \ \ \times (\beta_s-\alpha_{\overline{i}} - 2 +
	2\delta_{i,h+1})\psi[i]^p.
\end{align*}
These two cases are summarised above.
$\blacksquare$ \medskip

Thus substituting into equation (\ref{triplestarp14II}) we arrive at
\begin{align*}
	(P[t]Q[s]P[t])^p_{\ q} =&
	(-1)^{(p)}(\beta_s-\alpha_t)^{-1}(\beta_s-\alpha_t-(-1)^{(t)}+\delta_{t,h+1})^{-1}
	(\beta_s - \alpha_{\overline{t}} - (-1)^{(t)} + \delta_{t,h+1})^{-1} \\
	&\ \ \times
	(\beta_s -\alpha_{\overline{t}} - 2(-1)^{(t)} + 2\delta_{t,h+1})  C_s  \psi[t]^p  \phi[t]_q.
\end{align*}
Finally, using equation (\ref{starp15II}), we arrive at
\begin{align}
	(P[t]Q[s]P[t])^p_{\ q} =&
	(-1)^{(p)}(\beta_s-\alpha_t)^{-1}(\beta_s-\alpha_t-(-1)^{(t)}+\delta_{t,h+1})^{-1}
	(\beta_s - \alpha_{\overline{t}} - (-1)^{(t)} + \delta_{t,h+1})^{-1} \nonumber\\
	&\ \ \times
	(\beta_s -\alpha_{\overline{t}} - 2(-1)^{(t)} + 2\delta_{t,h+1})  C_s  \mu_t P[t]^p_{\
	q}.
	\label{starp17II}
\end{align}
Thus we may write (c.f. equation (\ref{starp13II}))
\begin{equation}
	(P[t]Q[s]P[t])^p_{\ q} = \omega_{s,t}P[t]^p_{\ q}
	\label{doublestarp17II}
\end{equation}
where
\begin{align}
	\omega_{s,t} =& C_s \mu_t
	(\beta_s-\alpha_t)^{-1}(\beta_s-\alpha_t-(-1)^{(t)}+\delta_{t,h+1})^{-1}
	(\beta_s - \alpha_{\overline{t}} - (-1)^{(t)} + \delta_{t,h+1})^{-1} \nonumber\\
	&\ \ \times
	(\beta_s -\alpha_{\overline{t}} - 2(-1)^{(t)} + 2\delta_{t,h+1}).
	\label{triplestarp17II}
\end{align}

As for $gl(m|n)$ \cite{GIW1}, the invariants $\mu_t$ of equation (\ref{triplestarp15II}) determine squares of reduced matrix elements,
while the invariants $C_k$, $\omega_{s,t}$ of equations (\ref{triplestarp10II}),
(\ref{triplestarp17II}) determine squared reduced Wigner coefficients. Together they determine
matrix elements of the generators as for $gl(m|n)$.
This problem will be investigated in a future article. Of particular interest is the determination
of a full branching rule, considering the
occurrence of indecomposable representations, and interpreting the matrix element formulae that arise
in these situations.

\section*{Acknowledgements}

This work was supported by the Australian Research Council through Discovery Project DP140101492.

redpajama_set_name: RedPajamaArXiv


Vancouver, Canada – June 5, 2018 – Crop Infrastructure Corp. (CSE: CROP) ("Crop" or the "Company") announces that it is accelerating the expiry date of the share purchase warrants (the "Warrants") that were issued in connection with the March 2018 "fundamental change" transaction between Fortify Resources Corp. and DV Infrastructure Corp. ("DVI"). The Warrants were initially issued by DVI in connection with a private placement of units, with those shares and warrants becoming free trading on July 6, and with each whole share purchase warrant being exercisable into one additional common share at a price of $0.20 per share.
The Warrants were subject to the right of the Company to accelerate the expiry date of the Warrants if the Company's common shares close at or above $0.40 per share for a period of five (5) consecutive trading days. The Company is entitled to accelerate the expiry of the Warrants to that date that is not less than 20 days from the date of issuance of a news release announcing the exercise of the acceleration right. The news release is issued today, June 5, 2018.
The expiry date of the Warrants has now been accelerated to June 26, 2018. This news release constitutes notice to Warrant holders of the new expiry date.
In order to exercise your Warrants, you are required to complete and submit the Warrant Exercise Form attached to your respective warrant certificate on or before 5:00 pm (PST) on June 26 2018. For detailed exercise requirements, please refer to your Warrant Certificate.
After 5:00 p.m. (PST) on June 26, 2018, no Warrants may be exercised and all unexercised Warrants will be void and of no effect.
The Company announces a warrant exercise incentive program (the "Program") in connection with the acceleration of the Warrants to encourage the exercise of the Warrants.
Pursuant to the Program, the Company is offering an inducement to each warrant holder that exercises their Warrants on or before June 26, 2018 (the "Acceleration Exercise Period"), consisting of an additional one-half of one transferable share purchase warrant, with each whole warrant (the "Incentive Warrant") entitling the holder to purchase one additional common share for a period of two (2) years from the date of issuance of such Incentive Warrant at a price of $0.75 per common share. The Incentive Warrants, and any shares issued upon exercise thereof, will be subject to a four-month hold period.
CROP Infrastructure is engaged in the business of investing, constructing, owning, optimizing and branding light supplemented greenhouse projects for lease to cannabis producers and processors offering best-in-class operations. The Company's main focus is currently California and Washington.
The securities of the Company are considered highly speculative due to the nature of the Company's business. The Company is indirectly involved through its business in both the medical and recreational cannabis industry in the United State where local state law permits such activities. As a result of the conflicting views between state legislatures and the federal government regarding cannabis, investments in cannabis businesses in the United States are subject to inconsistent legislation and regulation and therefore there are risks of federal government enforcement. Marijuana-related practices or activities, including the cultivation, possession or distribution of marijuana, are illegal under U.S. federal law.
Certain statements included in this press release constitute forward-looking information or statements (collectively, "forward-looking statements"), including those identified by the expressions "anticipate", "believe", "plan", "estimate", "expect", "intend", "may", "should" and similar expressions to the extent they relate to the Company or its management. The forward-looking statements are not historical facts but reflect current expectations regarding future results or events. This press release contains forward looking statements. These forward-looking statements are based on current expectations and various estimates, factors and assumptions and involve known and unknown risks, uncertainties and other factors.
Statements about the Company's business, plans and other statements, are all forward-looking information. Forward-looking statements are not guarantees of future performance and involve risks, uncertainties and assumptions which are difficult to predict. Such statements and information are based on numerous assumptions regarding present and future business strategies and the environment in which the Company will operate in the future, anticipated costs and the ability to achieve goals, that general business and economic conditions will not change in a material adverse manner, that the U.S. federal government will not prosecute people and companies who are involved directly or indirectly in the legal state cannabis industry, that financing will be available if and when needed and on reasonable terms, and that third party contractors, equipment and supplies and governmental and other approvals required to conduct the Company's business will be available on reasonable terms and in a timely manner. While such estimates and assumptions are considered reasonable by the management of the Company, they are inherently subject to significant business, economic, competitive and regulatory uncertainties and risks.
Factors that could cause the actual results to differ materially from those in forward-looking statements include, failure to obtain regulatory approvals, increase in construction and operating costs, government regulations, changes in laws and the enforcement of such laws, in particular the laws of the Unites States, lack of tenants at the Company's joint-venture facilities, loss of key employees and consultants, the continued availability of capital and financing, general economic conditions, competition, litigation, market or business conditions. Forward-looking statements contained in this press release are expressly qualified by this cautionary statement. These statements should not be read as guarantees of future performance or results. Such statements involve known and unknown risks, uncertainties and other factors that may cause actual results, performance or achievements to be materially different from those implied by such statements. These assumptions, risks and uncertainties include, among other things, including, but not limited to: regulatory and political risks, in particular those of the United States; change in laws, regulations and guidelines; limited operating history; competition; risks inherit ant in an agricultural business; vulnerability to rising energy costs; environmental and employee health and safety regulations; lack of demand of the Company's services; dependence on suppliers and skilled labour; reliance on key inputs; operating risk and lack of initial insurance coverage; unfavourable publicity or consumer perception; operation permits and authorizations; risks associated with acquisitions; risks associated with loaned money; financial projections may prove materially inaccurate or incorrect; management of growth; difficulty implementing business strategy; costs of maintaining a public listing; conflicts of interest; litigation; share price fluctuations; limited market for securities; lack of dividends; environmental laws and regulations; economic environment; global economy risk; going-concern risk; and financial risk exposures.
The Company assumes no responsibility to update or revise forward-looking information to reflect new events or circumstances unless required by law. Readers should not place undue reliance on the Company's forward-looking statements. Neither the CSE nor its regulation services provider accepts responsibility for the adequacy or accuracy of this news release.


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Home › Meetings & Events › 2017 › HR's Role in Complying with Nevada's CARE ACT
HR's Role in Complying with Nevada's CARE ACT
Date: July 11, 2017, 7:30am – 9:30am
1385 E Pama Ln #111
$25 Regular | $15 Student
iCal link
Hundreds of thousands of Nevadans care for older parents, spouses or loved ones, helping them live independently in their own homes. These family caregivers have a huge responsibility, and now a new law will make life a little bit easier for them. The CARE ACT helps family caregivers when their loved ones go into the hospital and as they transition home. This presentation will highlight HR's role in supporting employees who are family caregivers.
What attendees will learn:
· What rights do the CARE ACT create for caregivers?
· What role does HR have in guaranteeing the rights of its employees who are caregivers?
· How does the CARE ACT compare to the Family Medical Leave Act?
· What community resources are available to which HR can refer employees?
Lynda Hascheff - Lynda is a native Nevadan and has lived most of her life in Reno. She relocated to the Las Vegas area last year to work for AARP as their Director of Community Outreach and Volunteer Program Manager.
Prior to joining AARP, Lynda was the Executive Director of a local non-profit in Reno called Moms on the Run, which assists patients in treatment for breast cancer. She also worked for the Governor's Office of Economic Development as a Project Manager managing public boards, commissions and taking legislation live once it was passed in the legislature.
Her background is in law practice management, public official management and campaign management. She managed a private law firm and local elected official for 24 years prior to her segue into the non-profit sector. She has two grown children; her daughter Giuliana is a senior at Long Beach State and plays for their basketball team. Her son, Anthony, is a Peace Corp volunteer in Paraguay, South America.
Barry Gold, Director of Government Relations AARP Nevada - Barry Gold is director of government relations for AARP Nevada, working on state and national legislative issues on behalf of all 50+ Nevadans and their families. For more than 30 years, he has fought for the issues most important to older Americans. From on-the-ground case work for the Older Americans Act, to serving as executive director of the Governor's Advisory Council on Aging for the State of Arizona, his career has been dedicated to improving individuals' quality of life, and helping those who are underrepresented to have a voice. His breadth of experience in the aging network includes work with senior centers, residential care facilities, and state and municipal agencies, bringing a unique range of perspectives on the complex issues facing our aging population today.
This meeting has been approved for 1 Re certification credit for HRCI and SHRM
Chapter Membership Designation
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J. Ryan Fuller, Ph.D. is a licensed clinical psychologist and has practiced for more than 10 years. He has presented scientific studies in the United States, Canada, Russia, and India on a number of subjects including Weight Loss, Aggression in the Schools, Satisfaction with Life, and Anger Management.
New York Behavioral Health's mission is to provide the highest level of expert care to our clients in a comfortable setting. Cognitive Behavioral Therapists and psychologists use scientifically tested techniques to bring an end to emotional pain and to increase coping skills and other helpful behaviors. The goal is to provide New Yorkers and other clients with tools they can use on their own to cope with intense emotional experiences, stressful situations, and tough interpersonal problems in the future. We move beyond symptom relief and often target productivity, life satisfaction, goal achievement, and improved relationships- once clients determine what is personally meaningful in the midst of demanding New York City living.
New York Behavioral Health is conveniently located in midtown Manhattan, a block away from Grand Central Station. For directions, see our Contact Us page.
The psychologists of New York Behavioral Health strongly emphasize Cognitive Behavior Therapy (CBT) as well as traditional behavioral therapy to address depression, anger management, anxiety, addiction, and relationship issues. Our cognitive behavior therapists (CBT) utilize Rational Emotive Behavior Therapy (REBT), Dialectical Behavior Therapy (DBT), Acceptance and Commitment Therapy (ACT), and Behavior Therapy (BT) techniques specifically suited to each individual client. This cognitive behavioral approach helps clients efficiently overcome current problems, whether they are acute or long-standing anxiety, anger, addiction, depression symptoms, or work, weight or relationship problems.
Addiction Treatment - Cognitive Behavior Therapy (CBT) has been shown to effectively treat drug addiction and alcohol abuse. Our warm, caring professionals offer CBT in a comfortable setting at New York Behavioral Health.
Anger Management - Anger management problems can be very serious. If you have anger issues, you may be having trouble with relationships at home or at work. Our therapists can offer CBT for anger that is targeted and effective. Anger management clients set concrete goals and are given cognitive behavioral tools (CBT) to manage anger episodes.
Compulsions - Cognitive Behavior Therapy (CBT) can be used to address compulsive behaviors of Obsessive-compulsive Disorder (OCD) or other compulsions by teaching new ways of thinking and developing coping tools to manage emotions and urges. While OCD and compulsions are incredibly distressing, they are typically very responsive to cognitive behavioral treatment, as are many of the anxiety disorders related to OCD.
Depression - Symptoms can vary greatly, but can include apathy, hopelessness, difficulty sleeping or wanting to sleep all of the time, guilt, anxiety, loss of appetite or overeating, lethargy, changes in libido, and loss of pleasure. Cognitive Behavior Therapy (CBT) has been shown to effectively treat depression. Major Depressive Disorder responds very well to a number of depression treatments we utilize, e.g., cognitive therapy, Rational Emotive Behavior Therapy (REBT), Acceptance and Commitment Therapy (ACT), and Behavioral Activation Therapy.
Work Issues - Anyone living in New York City understands difficult commutes, expensive housing costs, and long work hours. There is even evidence that contact with crowds in areas of dense population can be stressful. Simply put, New Yorkers know stress. Cognitive Behavior Therapy (CBT) provides clients with tools for managing stress. Cognitive Behavioral Therapy techniques can efficiently provide clients with methods for changing their emotions, physiology, and ways of thinking.
Dysthymia - Dysthymia is a mood disorder similar to depression. The symptoms of dysthymia may be a little less intense, but unfortunately they tend to last much longer than those of clinical depression, i.e., major depressive disorder. Cognitive Behavior Therapy (CBT) can reduce the pain of dysthymic disorder, i.e., dysthymia.
Bipolar Disorder - Suffering from Bipolar Disorder can feel like a never-ending rollercoaster-ride emotionally for both you and your loved ones. Cognitive Behavior Therapy (CBT) has been shown to help those with depression and bipolar disorder, especially when combined with the proper medication.
Anxiety Disorders – Anxiety treatment can be very effective. Anxiety has many forms and many related disorders, e.g., Generalized Anxiety Disorder (GAD), Panic Disorder, Obsessive-Compulsive Disorder (OCD). Since New York Behavioral Health professionals use scientifically supported treatments for anxiety, there is good reason to believe your (or your loved one's) anxiety symptoms can begin to improve very soon.
Weight Loss – As a New Yorker, chances are you have either thought about losing weight or have tried one or many diets to help you lose weight. Cognitive Behavior Therapy has been shown to be effective for weight loss. Our behavioral health approach emphasizes developing life-long healthy habits that allow clients to not only lose weight, but to maintain the weight loss.
Grief - If you find yourself unable to resume your life after losing someone, if you experience depression, hopelessness, lack of focus, sadness, isolation, sleeplessness for an extended period of time, finding someone who can help you move on and find a way to live a fully functioning life can be crucial (while still honoring and remembering your loved one). Cognitive Behavior Therapy (CBT) has been shown to effectively treat complicated grief.
Procrastination - Whatever the underlying cause, you can learn how to act in line with your goals and values despite the tendency to procrastinate. CBT can also help you figure out the underlying reasons for your procrastination and develop effective strategies to achieve your goals moving forward.
Couples Therapy - If you are in a relationship that is affected negatively by arguments over finances, jealousy, parenting, infidelity, sexual satisfaction, goals and values, schedules, division of labor, or other family members, couples therapy can help. Couples therapy can help both individuals with depression, anger, and anxiety symptoms, as well as improving communication, intimacy, and overall relationship satisfaction.
Family Therapy - Behavioral Family Therapy has been proven effective in the treatment of family issues. In therapy, you can learn how to identify the behaviors that maintain the problems in your family and choose more effective behaviors that are in line with your goals and values.
Child Behavioral Problems - Children can experience a range of psychological and behavioral problems including ADHD, anxiety, depression, OCD, defiance, school refusal, autism, aggression, social and performance anxiety, eating disorders, academic problems, and trauma. Cognitive Behavioral Therapy has demonstrated efficacy in the treatment of child behavioral issues.
ADHD (ADD) Treatment - Attention Deficit Hyperactivity Disorder (ADHD) is characterized by behaviors that lead to academic and work difficulties, as well as social problems. Cognitive Behavioral Therapy (CBT) can be used to treat both child and adult ADHD.
Binge Eating - Binge Eating Disorder is now an official psychiatric diagnosis and it is reimbursable by insurance companies. If you consume a large amount of food in a short amount of time beyond the point of fullness, we can help. Binge eating results in weight gain, and can put you at risk for many medical problems. Fortunately, Behavior Therapy and CBT are highly effective treatments for this disorder.
Obsessive-Compulsive (OCD) - OCD is an anxiety disorder, in which obsessions and compulsions cause significant emotional pain and/or impair an individual's ability to function in different areas of his or her life. While OCD is an incredibly difficult disorder to experience, the good news is anxiety treatments are highly effective for OCD. Our therapists can help.
Panic Disorder - Even though panic attacks can be incredibly painful, they are also one of the most treatable of all psychological disorders. Our CBT therapists at New York Behavioral Health uses evidence-based anxiety treatments and can help you improve your symptoms very soon.
Insomnia - Insomnia, which is difficulty obtaining a good night's sleep, involves difficulty falling asleep, staying asleep through the night, or early wakenings. Cognitive Behavioral Therapy (CBT) is more effective than sleep medications for the treatment of insomnia.
Post-Traumatic Stress Disorder (PTSD) - The symptoms of Post-Traumatic Stress Disorder (PTSD) often develop weeks after the actual trauma and can surprise the sufferer who thought that they have moved on. If you are suffering from PTSD, getting help in very important as these symptoms can severely interfere with your quality of life and well-being. While PTSD can invade every area of your life and experience, there is hope. PTSD is very responsive to CBT and Behavioral Therapy. If you, or a loved one, are suffering in silence or coping in unhealthy ways, please consider calling us.
Social Anxiety - Social anxiety can cause extreme discomfort in social situations and can interfere with one's ability to make friends, maintain relationships, get to know people at work or one's personal life. Anxiety treatment of social anxiety can help you not only reduce the anxiety, but learn to be more confident in your communication skills. You can learn to be more effective socially and in work settings.
Stress Management - Because our current New York lifestyle produces a lot of stress, learning stress management is crucial in NYC to anyone craving health and a satisfying life. We are here to help you with that. CBT along with relaxation, problem-solving and time management skills can effectively reduce your stress.
Worries - Excessive worrying, also known as Generalized Anxiety Disorder, can be accompanied by depression, OCD, social anxiety, and phobias. Since our CBT therapists and psychologists use scientifically supported anxiety treatments for excessive worrying, there is good reason to believe your problems can begin to improve very soon.
Click here for a step to step guide in finding a therapist in New York City. If you need to ask questions, call us. We will be happy to help.
Group Therapy forms and guidelines can be found by selecting this link.


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La circonscription électorale de Málaga est l'une des cinquante-deux circonscriptions électorales espagnoles utilisées comme divisions électorales pour les élections générales espagnoles.

Elle correspond géographiquement à la province de Malaga.

Historique
Elle est instaurée en 1977 par la loi pour la réforme politique puis confirmée avec la promulgation de la Constitution espagnole qui précise à l'article 68 alinéa 2 que chaque province constitue une circonscription électorale.

Congrès des députés

Synthèse

1977

1979

1982

1986

1989

 José María Ruiz Povedano est remplacé en  par Juan Carlos Lomeña Villalobos.

1993

 Antonio Romero Ruiz est remplacé en  par José Luis Centella Gómez.
 Celia Villalobos est remplacée en  par Pablo Marcial Izquierdo Juárez.
 María Esperanza Oña Sevilla est remplacée en  par Luis Casaseca Navas.

1996

2000

2004

 Manuel Atencia Robledo est remplacé en  par Juan Manuel Moreno.

2008

 Magdalena Álvarez (PSOE) est remplacée en  par Daniel Pérez Morales.

2011

 Juan Manuel Moreno (PP) est remplacé en  par Paloma García Gálvez.
 Paloma García (PP) est remplacée en  par María del Mar Baena Azuaga.
 Carmen del Cid (PP) est remplacée en  par Jesús Ignacio Vázquez García.

2015

2016

Avril 2019

Novembre 2019

 Pablo Montesinos (PP) est remplacé en  par Ángel Luis González Muñoz.
 Carolina España (PP) est remplacée en  par Isabel Gema Pérez Recuerda.

Sénat

Synthèse

1977

1979

1982

1986

1989

1993

1996

2000

 María del Pilar Jurado de Miguel est remplacée en  par Rosa María Pérez Hernández.

2004

2008

 Emelina Fernández (PSOE) est remplacée en  par Beatriz Álvarez Urda.

2011

 Francisco de la Torre (PP) est remplacé en  par Avelino de Barrionuevo.
 Patricia Navarro (PP) est remplacée en  par María Teresa Otero Cobos.

2015

2016

Avril 2019

Novembre 2019

 Josele Aguilar (PSOE) est remplacé en  par Paloma Alonso Sahagún.

Notes et références

Voir aussi

Articles connexes
 Circonscription électorale

Liens externes
Qu'est-ce qu'une circonscription ?

Circonscription électorale espagnole du Congrès des députés
Circonscription électorale espagnole du Sénat

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