text:
\section{Introduction}

Let $U(\mathfrak g )$ be  the universal enveloping algebra of a simple Lie algebra $\mathfrak g $. The Harish-Chandra isomorphism  identifies the  center $Z(\mathfrak g )$ of the algebra $U(\mathfrak g )$  with  the    algebra  of polynomials over the  Cartan subalgebra $\mathfrak  h$ of $\mathfrak g $ that are  invariant under a certain  action of the corresponding Weyl group $W$.  The elements of  $Z(\mathfrak g )$  act in finite-dimensional irreducible representations of $\mathfrak g $ by multiplication by scalars, and these eigenvalues  can be found  from the images of central elements under the  Harish-Chandra isomorphism.

For  the even orthogonal Lie algebra $\mathfrak g ={\mathfrak o}_{2n} $  the center is generated by  $n$  elements  those images  under the  Harish-Chandra isomorphism are  $W$-invariant polynomials over $\mathfrak  h$ of degrees $2,4,6,\dots,  2n-2$ and $n$.
There are several  constructions  that define these generators explicitly (see, e.g. \cite{HU, MN, PP, AW}).  In particular,  the generator of degree $n$ can be realized as a non-commutative  Pfaffian of a certain matrix with coefficients in $U({\mathfrak o}_{2n} )$.
Namely,  in the notations of Section  \ref{ext}, if $F$ is the $2n\times 2n$ matrix  those  entries $\{F_{ij}\}$, $(i,j\in \{-n,\dots, -1,1,\dots n\})$ are  standard generators of $U({\mathfrak o}_{2n} )$, and $J_{2n}$  is the  matrix  of the form that defines  ${\mathfrak o}_{2n} $, the Pfaffian of $\tilde F= FJ_{2n}$  is the central element defined by the formula
\begin{align}\label{defpf}
\text{Pf}\,(\tilde F)=\frac{1}{2^n n!}\sum_{\sigma\in S_{[-n;n]} }\text{sgn}\,{(\sigma)} (FJ_{2n})_{\sigma(-n)\sigma(-n+1)}\dots  (FJ_{2n})_{\sigma(n-1)\sigma(n)}.
\end{align}
In \cite {TH}   a combinatorial  formula for $\text{Pf}\,\tilde F$  is proved,   which  (with some change of notations  to match the notations of  this note) can be formulated as follows:
\begin{align}\label{HP}
\text{Pf}\,(\tilde F)=\sum_{k=0}^{{\lfloor{n/2}\rfloor}}
\sum_{\substack {I,J\subset \{-n,\dots, -1 \}\\|I|=|J|=2k}} \text{sgn}({\bar I, I}) \text{sgn} ({\bar J, J})\text{det}\, (\tilde F_{\bar I \bar J} +\rho(|J|)) \text{Pf}\, \tilde F_{-J\,J}\text{Pf}\, \tilde F_{I\,-I}.
\end{align}
 Here  for a subset $I$ of $\{-n,\dots, -1\}$,  the set  $\bar I$ is the complement    of $I$ in $\{-n,\dots, -1 \}$;  a  submatrix $\tilde F_{IJ}$  is  defined by
$\tilde F_{IJ}=(\tilde F_{ij})_{i\in I, j\in J}$;    $\rho(k)$ stands for the diagonal matrix $\rho(k)=\text{diag}\,(k-1,k-2,\dots, 1,0)$; the expressions
 $\text{det}\, (\tilde F_{\bar I \bar J} +\rho(|J|))$, $\text{Pf}\, \tilde F_{-J\, J}$,  $\text{Pf}\, \tilde F_{I\, -I}$ are certain  noncommutative analogues of
 the determinant and  Pfaffian of a matrix;
  $\text{sgn}({\bar I, I})$,  $\text{sgn}({\bar J,J})$  are the signs of  certain permutations.
(We refer the reader for full details to {\cite{TH}}.)
The advantage of this formula is that it immediately provides  the eigenvalues of the central element $\text{Pf}\, \tilde F$ on the highest weight modules.

The purpose of this note is  to prove a  similar to (\ref{HP})  combinatorial  formula  for a Pfaffian of the matrix of generators  for the universal  enveloping algebra  $U(\hat{{\mathfrak o}}_{2n} )$ of the affine Kac--Moody algebra $\hat{{\mathfrak o}}_{2n} $.
using the techniques of \cite{IU} and \cite{TH}. The main results of this  paper are  Theorems \ref{thm1} and \ref{thm2}.

 In general, for a simple Lie algebra $\mathfrak g $ over ${\mathbb C}$,  the  vacuum module $V_{-h^\vee}(\mathfrak g )$ over the affine Kac--Moody algebra $\hat \mathfrak g $  at the critical level  ${-h^\vee}$ is defined
 as the quotient of the universal enveloping algebra  $U(\hat \mathfrak g )$ by the left ideal generated by $\mathfrak g [t]$ and $K+ h^\vee$, where $ h^\vee$ denotes the
 dual Coxeter number  for $\mathfrak g $. As a vertex algebra,  $V_{-h^\vee}(\mathfrak g )$ has a non-trivial center $\mathfrak{z}(\hat \mathfrak g )$ defined by
  $$\mathfrak{z}(\hat \mathfrak g )=\{ S\in V_{-h^\vee}(\mathfrak g ) \,|\, \mathfrak g [t]S=0\}.$$
  It structure is described  by the Feigin-Frenkel theorem (1992); see
\cite{F}   for detailed exposition.
    In \cite{MC, CT, M1}  complete sets of  explicit generators of the commutative algebra
    $\mathfrak{z}(\hat \mathfrak g )$ are constructed for  classical Lie algebras.  In the case of $\mathfrak g ={\mathfrak o}_{2n} $ the set of generators contains an element $\text{Pf\,}\tilde F[-1]$
defined below by  (\ref{defpf}), and this element  is the main focus of this note.

\smallskip

Our goal is to prove a combinatorial formula  (Theorem \ref{thm1} and Theorem \ref{thm2})  for   $\text{Pf}\,\tilde F[-1]$   in the  spirit of formulas for Pfaffians
proved in \cite{TH}.  The immediate application of this result  is that it  allows  to compute  easily the image of the  Pfaffian $\text{Pf}\,\tilde F[-1]$ under  the Harish-Chandra homomorphism (Section \ref{sec_HC}).
The algebra  $\mathfrak{z}( \hat{{\mathfrak o}}_{2n} )$ is  a  commutative subalgebra of  $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}])^\mathfrak  h$ --  the algebra of invariants under the action of the Cartan subalgebra $\mathfrak  h$. A restriction of a  Harish-Chandra homomorphism
 $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}])^\mathfrak  h\to U(t^{-1}\mathfrak  h[t^{-1}]) $ to the subalgebra $\mathfrak{z}(\hat{{\mathfrak o}}_{2n} )$ yields  an isomorphism of
 $\mathfrak{z}( \hat{{\mathfrak o}}_{2n} )$ to the classical $\mathcal{W}$-algebra $\mathcal{W}({\mathfrak o}_{2n} )$ (see \cite{F,MM}). The Harish-Chandra images of generators
 of $\mathfrak{z}(\hat \mathfrak g )$   for classical Lie algebras $\mathfrak g $ were computed  in \cite{MC, MM}.  Below  Corollary {\ref{HC}}   deduces  the Harish-Chandra  image of $\text{Pf\,}\tilde F[-1]$  from the Theorem  \ref{thm1}  (or  Theorem \ref{thm2}) of the paper.

 Moreover, we illustrate in Section \ref{fol} that the formulas of \cite{TH}  for Pfaffians in classical case can be deduced from the formulas  for affine Lie algebra $\hat{{\mathfrak o}}_{2n} $ proved  in this note.

\subsection*{Acknowledgements} The author is very grateful to A.\,I.\,Molev for brining this problem to her attention and for  valuable discussions. She also would like to thank
Max Planck Institute for Mathematics   for  warm hospitality.

\section{ Application of  exterior calculus }\label{ext}
Let $J_{2n}$ be a  $2n\times 2n$ symmetric matrix  with  ones  on the anti-diagonal and  zeros elsewhere.  Consider
 the corresponding  realization  of   ${\mathfrak o}_{2n} $ as a Lie algebra  of $2n\times 2n$ complex-valued  matrices $X$ that satisfy
$ X^TJ_{2n}+J_{2n}X=0$.
We will use the notation  $[-n;n] =\{-n,\dots, -1,1,\dots, n\}$ (the interval of integers  from $-n$ to $n$  with  the element $0$  being omitted).
Let
$\{E_{i,j}\}_{i,j\in [-n; n]}$  be the matrices that   have one in the $i$th row and $j$th column and zeros elsewhere.
Set for $i,j\in{[-n;n]}$
$$
F_{i,j}=E_{i,j}- E_{-j,-i},\quad  ( F_{i,j}=-F_{-j,-i}).
$$
 The   Cartan  subalgebra $\mathfrak  h$ of  ${\mathfrak o}_{2n} $ is generated by elements $F_{ii}$ ($i=-n,\dots, -1)$.
Using the  bilinear form
$$
\langle X,Y \rangle = \frac{1}{2} \operatorname{tr} (XY) \quad \text{for}~ X,Y\in {\mathfrak o}_{2n} ,
$$
one defines the extended affine Kac--Moody algebra
$\hat{{\mathfrak o}}_{2n}  \oplus {\mathbb C}\tau= {\mathfrak o}_{2n} [t, t^{-1}]\oplus {\mathbb C} K\oplus {\mathbb C}\tau$ and set
$X[r]=Xt^r$, $r\in {\mathbb Z}$. Then the commutation relations in $\hat{{\mathfrak o}}_{2n} $ are
\begin{align*}
[X[r], Y[s]]&=[X,Y][r+s] +r\delta_{r, -s} \langle X,Y \rangle K,\\
[\tau,X[r]]&=-r X[r-1],\quad [\hat{{\mathfrak o}}_{2n} ,K]=0.
\end{align*}
In particular,  for  $i,j,k,l\in[-n; n]$ one gets
\begin{align*}
[F_{i,j}[r],F_{k,l}[s] ]&= \delta_{j,k}F_{i,l}[r+s]+ \delta_{i,l}F_{-j,-k}[r+s]- \delta_{j,-l}F_{i,-k}[r+s]- \delta_{i,-k}F_{-j,l}[r+s]\\
&+ r\delta_{r,-s}(\delta_{j,k}\delta_{i,l}-\delta_{i,-k}\delta_{j,-l}) K.
\end{align*}
$U(\hat{{\mathfrak o}}_{2n} )$ stands for the universal enveloping algebra of $\hat{{\mathfrak o}}_{2n} $.
Define $F[r]$ as  $U(\hat{{\mathfrak o}}_{2n} )$-valued matrix    of size  $2n\times 2n$  with entries $(F[r])_{i,j}=F_{i,j}[r]$ for  $i,j\in{[-n;n]}$.
Then the  matrix   $\tilde F[-1]= F[-1]J_{2n}$  is skew-symmetric
and, similarly to  the classical case, one defines the Pfaffian of $\tilde F[-1]$ as
\begin{align}\label{defpf}
\text{Pf}\,\tilde F[-1]=\frac{1}{2^n n!}\sum_{\sigma\in S_{[-n;n]} }\text{sgn}\,{(\sigma)}\tilde F_{\sigma(-n),\sigma(-n+1)}[-1]\dots \tilde F_{\sigma(n-1),\sigma(n)}[-1],
\end{align}
where the sum is over all permutations of the elements of the set $[-n; n]$.
By Lemma  4.3 in \cite{M1},    the Pfaffian  of $\tilde F[-1]$
is an element of Feigin-Frenkel  center $\mathfrak{z}(\hat{{\mathfrak o}}_{2n} )$.
Our goal is to prove a combinatorial formula  for $\text{Pf}\,\tilde F[-1]$  that, in particular, allows  easily to compute the image of this Segal-Sugawara vector  under  the Harish-Chandra homomorphism.

As in \cite{IU}, it is convenient for us to express Pfaffian-like elements as coefficients of powers of certain two-forms.
Consider the algebra $\Lambda$  generated by elements $\{e_i\}_{i\in[-n;n]}$ with the defining relations  $e_ie_j=-e_je_i$ for all $i$ and $j$. Define  the $U(\hat{{\mathfrak o}}_{2n} )$-valued  two-form
 \begin{align*}
\Omega[s] =\sum_{i,j\in[-n; n]} e_i e_{-j} F_{i,j}[s]\in \Lambda \otimes U(\hat{{\mathfrak o}}_{2n} ).
 \end{align*}
\begin{lemma}\label{omn}
 \begin{align*}
(\Omega[-1])^n=e_{-n}e_{-n+1}\dots e_{-1}e_{1}\dots e_{n} 2^n n!\text{Pf} \tilde F[-1].
 \end{align*}
\end{lemma}
\begin{proof}  (cf. Proposition 1.1 in  \cite{IU} and  Lemma 4.1  in \cite{TH}):
 \begin{align*}
  \Omega[-1] ^n&=
  \sum_{i_1,j_1,\dots, i_n, j_n\in [-n;n]}
  e_{i_1} e_{-j_1}\dots e_{i_n}e_{-j_n}F_{i_1,j_1}[-1]\dots F_{i_n,j_n}[-1]\\
   &=
  \sum_{i_1,j_1,\dots, i_n, j_n\in [-n;n]}
  e_{i_1} e_{j_1}\dots e_{i_n}e_{j_n}\tilde F_{i_1,j_1}[-1]\dots\tilde F_{i_n,j_n}[-1]\\
   &= e_{-n} e_{-n+1}\dots e_{n-1}e_{n}
  \sum_{\sigma\in S_{[-n;n]}}
\text{sgn} (\sigma) \tilde F_{\sigma(-n),\sigma(-n+1)}[-1]\dots\tilde F_{\sigma(n-1),\sigma(n)}[-1]\\
&=e_{-n} e_{-n+1}\dots e_{n-1}e_{n} \,2^n n! \text{Pf}\,\tilde F[-1].
 \end{align*}
 \end{proof}
We also  introduce the following forms:
\begin{align*}
\Theta[r]=\sum_{ i=-n}^{-1}\sum_ {j=1}^{n} {e_i e_{-j}} F_{i,j}[r],&
\quad
\tilde \Theta [r]=\sum_ {i=1}^{n} \sum_{ j=-n}^{-1} {e_i e_{-j}} F_{i,j}[r],\\
\Xi[r]=\sum_{i,j=-n}^{-1} {e_i e_{-j}} F_{i,j}[r] ,&\quad \Psi=\sum_{j=1}^{n} e_{j}e_{-j},\\
\xi[-1]=\Xi[-1] +2\Psi\tau, &\quad \tilde \xi[-1]=\Xi[-1] -2\Psi\tau,
\\
X[-1] = \Theta[-1]+\xi[-1],&\quad
Y[-1] =\tilde \Theta[-1] +\tilde \xi[-1].
\end{align*}

The following properties of the forms hold:
\begin{lemma}
\begin{align}
 \notag
\Psi=\sum_{j=-n}^{-1} e_{-j}e_{j}&=-\sum_{j=-n}^{-1} e_{j}e_{-j}=-\sum_{j=1}^{n} e_{-j}e_{j}. \\
[\Xi[r],\Xi[s]]&=-r\delta_{r,-s} \Psi^2 K,\\
[\Theta[r], \Theta[s]]&=[\tilde\Theta[r], \tilde\Theta[s]]=0,\label{tha}\\
[\Theta[r], \tilde\Theta[s]]&=-4\Psi \Xi[r+s] +2r\delta_{r,-s} \Psi^2 K,\label{4t}\\
[\Xi[r],\Theta[s] ]&= 2 \Psi\Theta[r+s],\quad  [\Xi[r],\tilde\Theta [s]]=-2 \Psi \tilde\Theta[r+s],\label{t1}\\
[\Xi[-1]+2a\Psi\tau ,\Theta[s] ]&= 2 \Psi(1-as)\Theta[s-1], \label{t2}\\

[\Xi[-1]+2b\Psi\tau ,\tilde\Theta[s] ]&= 2 \Psi(-1-bs)\tilde \Theta[s-1], \label{t3}\\
[X[-1],Y[-1]] &= 0.\label{xy}
\end{align}
\end{lemma}
\begin{proof}
All of  these relations can be checked directly. We illustrate this with  calculations of (\ref{4t}). With $i,l\in\{-n,\dots, -1\}$ and $j,k\in\{1,\dots, n\}$,
\begin{align*}
[\Theta[r], \tilde\Theta[s]]&=
\sum_{ijkl}e_{i}e_{-j} e_k e_{-l}[F_{i,j}[r], F_{k,l}[s]]
=\sum_{ijkl}e_{i}e_{-j} e_k e_{-l}\\
&\times \big(
 \delta_{j,k}F_{i,l}[r+s]+\delta_{i,l}F_{-j,-k}[r+s]
-\delta_{j,-l}F_{i,-k}[r+s] -\delta_{i,-k}F_{-j,l}[r+s]\\
&+r\delta_{r,-s}(\delta_{j,k}\delta_{i,l}-\delta_{i,-k}\delta_{j,-l})K\big)
\\
&= -\Psi\big(
\sum_{i,l}e_{i}e_{-l}F_{i,l}[r+s]
+\sum_{j,k}e_{-j}e_{k}F_{-j,-k}[r+s]\\
&+\sum_{i,k}e_{i}e_{k}F_{i,-k}[r+s]
+\sum_{j,l}e_{-j}e_{-l}F_{-j,l}[r+s]
\big) + 2r\delta_{r,-s} \Psi^2 K
\\
&=-4\Psi \Xi[r+s] +2r\delta_{r,-s} \Psi^2 K.
\end{align*}

\end{proof}

Observe that
$\Omega[-1] =\tilde \Theta [-1]+2\Xi[-1]+ \tilde \Theta [-1]
=X[-1] + Y[-1]$,
 and since $X[-1]$ and $Y[-1]$ commute, we can write
\begin{align*}
\Omega[-1]^n &= (X{[-1]}  +Y{[-1]})^n=\sum_{m+k=n}{n\choose m} Y[-1]^mX[-1]^{k}.
\end{align*}
From now on we will use the following notation: for a non-negative integer $a$ and a partition $\alpha\vdash a$, $\alpha=(a_1\ge\dots\ge a_l\ge 0)$, denote as
$\tilde \Theta[-\alpha] =\tilde \Theta[-a_1]\dots \tilde \Theta[-a_l]$, and  as
$ \Theta[-\alpha] = \Theta[-a_1]\dots \Theta[-a_l] $.
We also set
$\tilde \Theta[-(0)] = \Theta[-(0)]=1$
\begin{proposition}\label{yx}
\begin{align*}
Y[-1]^m=\sum_{a=0}^{m}
\sum_{\alpha\vdash  a} \frac{m!}{(m-a)!m_1!m_2!\dots}
 (-2\Psi)^{ a-l(\alpha)}
\tilde \Theta[-\alpha]\, (\tilde \xi[-1])^{m- a},
\\
X[-1]^m=\sum_{a=0}^{m}
\sum_{\alpha\vdash  a} \frac{m!}{(m-a)!m_1!m_2!\dots}
 (-2\Psi)^{ a-l(\alpha)}
( \xi[-1])^{m- a} \Theta[-\alpha] ,
\end{align*}
 where the sum is taken over all partitions $\alpha=(1^{m_1}, 2^{m_2}\dots )\vdash a$,
 and  $l(\alpha)$
 is the length of $\alpha$:
$ l(\alpha)= m_1+m_2+\dots$\,.
\end{proposition}

\begin{proof}
We prove the statement in two steps. First, we show that $Y[-1]^m$ and $X[-1]^m$  can be expressed as  linear combinations of ordered terms  labeled by {\it compositions} of integer numbers $a$, $1\le a\le m$.
\begin{lemma}\label{compos_lemma}
\begin{align*}
Y[-1]^m=\sum_{a=1}^{m}\sum_{\bar a} C(\bar a,m) (-2\Psi)^{ a-l}
\tilde \Theta[-a_1] \tilde \Theta[-a_2]  \dots \tilde \Theta[-a_l] \, (\tilde \xi[-1])^{m- a},\\
X[-1]^m=\sum_{a=1}^{m}\sum_{\bar a} C(\bar a,m) (-2\Psi)^{ a-l}
( \xi[-1])^{m- a}
 \Theta[-a_1] \ \Theta[-a_2]  \dots  \Theta[-a_l] ,
\end{align*}
 where the sum is taken over all ordered $l$-tuples $\bar a=(a_1,\dots, a_l)$,  with $0\le l \le m$, $a_i\ge 1$
and $\sum_{i=1}^l a_i=a\le m$. Here
 \begin{align*}
 C(\bar a,m)=
 \frac{m!}{(m-a)!}\prod_{s=1}^{l}\frac{a_s}{(a_1+\dots +a_s)}.
  \end{align*}
\end{lemma}

\begin{proof}

$
Y[-1]^m=( \tilde \Theta[-1] +\tilde \xi[-1])^m
$
is the sum of monomials of the form
 \begin{align}\label{mon1}
  \tilde \xi[-1]^{p_1} \tilde \Theta[-1]\tilde \xi[-1]^{p_2}  \tilde \Theta[-1]\dots  \tilde \xi[-1]^{p_l} \tilde \Theta[-1]\tilde \xi[-1]^{p_{l+1}} ,
  \end{align}
where $p_i\ge 0$ and
$
p_1+\dots+ p_{l+1}=m-l.
$
From (\ref{t3}),
\begin{align}\label{comb}
 \tilde \xi[-1]^{p}  \tilde \Theta[-1]=\sum_{s=1}^{p+1}{p\choose {s-1}} (-2\Psi)^{s-1}  s! \tilde \Theta[-s] \tilde \xi[-1]^{p-s+1},
 \end{align}
 and we obtain  that   the terms of the  monomials   (\ref{mon1}) can be permuted to obtain that  $Y[-1]^m$ as a linear combination
\begin{align}\label{mon2}
Y[-1]^m=
\sum_{\bar a} c(\bar a, m)(-2\Psi)^{ a-l} a_1! a_2!\dots a_l!
\tilde \Theta[-a_1] \tilde \Theta[-a_2]  \dots \tilde \Theta[-a_l] \, (\tilde \xi[-1])^{m- a}
\end{align}
with certain coefficients  $c(\bar a, m)$  and the sum   taken over all ordered  $l$-tuples $\bar a=(a_1,\dots, a_l)$  that satisfy $a_i\ge 1$\, $(i=1,\dots, l)$,  $a=\sum_{i=1}^l  a_i$. Our goal is  to compute $c(\bar a, m)$.
Fix the ordered  $l$-tuple  $(a_1,\dots, a_l) $ and consider the corresponding term  in the sum (\ref{mon2}).
We use the following notations:
\begin{align*}
&A_1=a_1-1, \quad A_2=(a_1-1)+(a_2-1),\quad \dots,\quad   A_l=(a_1-1)+\dots + (a_l-1),\\
&P_1=p_1+\dots +p_{l+1}=m-l,\quad  P_2=p_2+\dots +p_{l+1},\quad \dots, \\
&P_{l+1}=p_{l+1},\quad P_{k}=0 \quad  (k>l+1).
\end{align*}
Then from (\ref{comb}),
\begin{align*}
c(\bar a, m)=&\sum_{p_i\ge 0, \, p_1+\dots + p_{l+1}=m-l} {p_1\choose {a_1-1}} {{p_1+p_2-A_1}\choose {a_2-1}} \dots {{\sum_{i=1}^l p_i- A_{l-1}}\choose {a_l-1}}\\
=&\sum_{p_i\ge 0, \, p_1+\dots + p_{l+1}=m-l} {{m-l -P_2}\choose {a_1-1}} {{ m-l -P_3 -A_1}\choose {a_2-1}} \dots {{m-l- P_{l+1}- A_{l-1}}\choose {a_l-1}}\\
=&
\sum_{p_{l+1}=0}^{m-l}
{{m-l- P_{l+1}- A_{l-1}}\choose {a_l-1}}
\dots
\sum_{p_{3}=0}^{m-l-P_{4}}
{{ m-l -P_3-A_1}\choose {a_2-1}}  \sum_{p_{2}=0}^{m-l-P_3}
 {{m-l -P_2}\choose {a_1-1}}.
 \end{align*}
We compute the sums of this expression  starting from the last one. Recall that
 $$\sum_{j=0}^{r}{j\choose k}={{r+1}\choose {k+1}} \quad {\text {and}} \quad
 {{j-b}\choose a}{j\choose b}={j\choose {a+b}}{{a+b}\choose a}.
 $$
We get
 $$
 \sum_{p_{2}=0}^{m-l-P_3}
 {{m-l -P_2}\choose {a_1-1}}={{m-l+1-P_3}\choose {a_1}}
 $$
and
 $$
  { {m-l -P_3- A_1} \choose {a_2-1}}  {{m-l+1-P_3}\choose {a_1}}
= {{m-l+1-P_3}\choose {a_1+a_2-1}} {{ a_1+a_2-1}\choose {a_2-1}}.
  $$
Similarly,
$$
\sum_{p_s=0}^{m-l-P_{s+1}}  {{m-l+s-2-P_s}\choose {a_1+\dots +a_{s-1}-1}}= {{m-l+s-1-P_{s+1}}\choose {a_1+\dots +a_{s-1}}}
$$
and
\begin{align*}
{{m-l-P_{s+1}-A_{s-1}}\choose {a_s-1}}{{m-l+s-1-P_{s+1}}\choose {a_1+\dots +a_{s-1}}}={{m-l+s-1-P_{s+1}}\choose {a_1+\dots +a_{s}-1}}
{{a_1+\dots +a_s-1}\choose{a_s-1}}.
\end{align*}
By induction,
\begin{align*}
c(\bar a, m)=& {{m}\choose {a_1+\dots +a_l}}{{a_1+\dots +a_l-1}\choose{a_l-1}}\dots {{ a_1+a_2-1}\choose {a_2-1}}\\
&=\frac{m!}{(m-a)!}\prod_{s=1}^{l}\frac{1}{(a_1+\dots +a_s) (a_s-1)!}.
\end{align*}
Set  $C(\bar a, m)=a_1! a_2!\dots a_l!\,c(\bar a, m)$, and
 the   statement of Lemma \ref{compos_lemma} follows from (\ref{mon2}).

For $X[-1]^m$ the same arguments apply since
\begin{align*}
 \Theta[-1] \xi[-1]^{p}  =\sum_{s=1}^{p+1}{p\choose {s-1}} (-2\Psi)^{s-1}  s! \xi[-1]^{p-s+1}
 \Theta[-s].
 \end{align*}
\end{proof}
 Observe now  that  $\tilde \Theta[-a]$ and  $\tilde \Theta[-b]$ commute for $a,b\ge 1$.
 Thus the   factors of  any  monomial $\tilde \Theta[-a_1] \tilde \Theta[-a_2]  \dots \tilde \Theta[-a_l]$
 can be rearranged into a monomial  $\tilde \Theta[-a_{i_1}] \tilde \Theta[-a_{i_2}]  \dots \tilde \Theta[-a_{i_l}]$ so that  $(a_{i_1}\ge\dots\ge  a_{i_l}\ge1)$ is a {\it partition}  of $a$, and
\begin{align*}
Y[-1]^m&=\sum_{a=1}^{m}\sum_{\alpha\vdash  a}  \frac{m!}{(m-a)!}C(\alpha)
\prod_{s=1}^{l}   a_s \,   (-2\Psi)^{ a-l}
\tilde \Theta[-\alpha] \, (\tilde \xi[-1])^{m- a},
\end{align*}
where the  sum is taken over all  partitions $\alpha=(a_1\ge a_2\ge \dots \ge  a_l\ge 1)$ of $a$, and
$$
 C(\alpha)=\sum  \frac {1 }  {(a_{i_1}) (a_{i_1}+a_{i_2})\dots ( k_1 a_1\dots +k_la_{l})},
$$
--   the sum is taken over all  distinct permutations $(a_{i_1},\dots, a_{i_l}) $    of the  parts  of $\alpha$.

 The following lemma allows to compute the coefficients  of that linear combination.
 \begin{lemma}\label{calpha}
 Let $\{a_1, a_2,\dots,  a_r)$ be a set of distinct positive numbers and let\\ $\alpha=(a_1,\dots a_1,  a_2,\dots,a_2,\,  \dots \, a_r,\dots, a_r)$
 be an ordered $l$-tuple, where the part $a_i$ appears with multiplicity $k_i$
(therefore,  $l= k_1+\dots + k_r$).

Then
\begin{align}\label{C1}
\sum \frac{1}{(a_{i_1}) (a_{i_1}+a_{i_2})\dots ( k_1 a_1+\dots +k_ra_{r})} =\frac{1}{k_1! a_1^{k_1}\dots k_r!a_r^{k_r}}
\end{align}
where the sum on the left-hand side is taken over  distinct permutations $(a_{i_1}, \dots, a_{i_{l}})$  of $\alpha=(a_1,\dots a_1,  a_2,\dots,a_2,\,  \dots \, a_r,\dots, a_r)$.
\end{lemma}

\begin{proof}
  Denote the sum on the left-hand side of (\ref{C1}) as $C(\alpha)$.
  Assume first that
  $k_i=1$ ($i=1,\dots, r$).
Then  for $r=2$,
\begin{align}\label{int}
C((a_1, a_2))=
\frac{1}{a_1(a_1+a_2)}+\frac{1}{a_2(a_1+a_2)}= \frac{1}{a_1a_2}.
\end{align}
 By induction for general $\alpha$ with all  distinct parts we show that
\begin{align}\label{C}
C(\alpha)&=\sum_{\sigma\in S_r } \prod_{s=1}^r\frac{1}{(a_{\sigma(1)}+\dots +a_{\sigma(s)})}\\
& =
\sum_{k=1}^r\,  \sum_{\sigma\in S_r, \sigma(r)=k }  \frac{1}{(a_{\sigma(1)}+\dots +a_{\sigma(r-1)}+a_k)} \prod_{s=1}^{r-1}\frac{1}{(a_{\sigma(1)}+\dots +a_{\sigma(s-1)})} \\
&=\sum_{k=1}^r \,\sum_{\sigma\in S_r, \sigma(r)=k }  \frac{1}{(a_{\sigma(1)}+\dots +a_{\sigma(r-1)}+a_k)}\frac{1}{a_1 \dots \hat{a}_k\dots a_r }
=\frac{1}{a_1 \dots a_r }.
\end{align}
($\hat{a}_k$ means that the factor ${a}_k$  is omitted).
Now  suppose that not all of $k_i=1$ in $\alpha$. For simplicity of the argument, let us assume that   ${k_1}\ne 1$, and the rest of  $k_i= 1$.
Then consider $\alpha_\varepsilon= (a_1+\varepsilon_1, a_1+\varepsilon_2,\dots, a_{1}+\varepsilon_{k_1}, a_{2},\dots, a_r)$ with such values of $\varepsilon_i$ that all the terms of $\alpha_\varepsilon$ are distinct.  $C(\alpha_\epsilon)$ is a rational function of $(\varepsilon_1,\dots \varepsilon_{k_1})$. When all $\varepsilon_i\to 0 $,
$$
\frac{1} {(a_1+\varepsilon_1)\dots (a_{k_1}+\varepsilon_{k_1}) a_{k_1+1}\dots a_r}\to \frac{1}{(a_1)^{k_1} a_{k_1+1}\dots a_r}.
$$
On the other hand,
$$
C(\alpha_\epsilon)
\to \sum_{\sigma\in S_{r-k_1+1} } \prod_{s=1}^r\frac{k_1!}{(a_{\sigma(1)}+\dots +a_{\sigma(s)})},
$$
where the sum is taken over all permutations of the set $\{1,2,\dots, r\} $.
By generalizing this argument one can show that if $\alpha$   has distinct parts $\{ a_1, \dots, a_r\}$ with the corresponding multiplicities
 $\{ k_1,\dots, k_r \}$,
the value of  $C(\alpha)$ is given by the formula
$
C(\alpha)={1}/({k_1! a_1^{k_1}\dots k_r!a_r^{k_r}}).
$
\end{proof}
Applying Lemma \ref{calpha} and  rewriting the partition $\alpha$ in the form  $\alpha=( 1^{m_1}, 2^{m_2},\dots)$, we get the statement of Proposition \ref{yx}.
\end{proof}
\section{Formula for  $\Omega[-1]^n$}
We use notation  $\text{ad}_\tau$ for the operator $f\mapsto [\tau,f]$  acting on  $U(\hat{{\mathfrak o}}_{2n} )$.
\begin{theorem}\label{thm1}
For  the Lie algebra ${\mathfrak o}_{2n} $,
\begin{align*}
\Omega[-1]^n &= \sum_{l=0}^{\lfloor n/2\rfloor} \sum_{\substack{a,b,\\ 2l\le a+b\le n}}\frac{2^{n-2l} n!}{(n-a-b)!}
 \left(\sum_{\alpha\vdash a, l(\alpha)=l}\frac{1}{m_1! m_2!\dots }
  \tilde  \Theta[-\alpha]   \right)\\
&\quad \times(-\Psi)^{ a+b-2l} \left( (\Xi[-1] -\Psi \text{ad}_\tau)^{n-a-b}\cdot 1 \right)\,
\left(\sum_{\beta\vdash b,l(\beta)=l}
\frac{1}{m^\prime_1!m^\prime_2!\dots }
  \Theta[-\beta]\right),
\end{align*}
where  we use the notations $\alpha=(1^{m_1}, 2^{m_2},\dots )\vdash a$ and  $\beta=(1^{m^\prime_1}, 2^{m^\prime_2},\dots )\vdash b$ .

\end{theorem}
\begin{example}
For $n=2$ the partitions that contribute to the sum of the formula are $\alpha=\beta=\emptyset$ and $\alpha=\beta=(1)$. Accordingly, in ${\mathfrak o}_{4}$ we get
\begin{align*}
\Omega[-1]^2=4(\Xi[-1]-\Psi\text{ad}_\tau)^2\cdot 1 + 2 \tilde  \Theta[-1]  \Theta[-1].
\end{align*}
For $n=3$ the partitions that contribute to the sum of the formula are $\alpha=\beta=\emptyset$; $\alpha=\beta=(1)$;   $\alpha=(2), \beta=(1)$ or  $\alpha=(1), \beta=(2)$. Accordingly, in ${\mathfrak o}_{6}$ we get
\begin{align*}
\Omega[-1]^3&=8(\Xi[-1]-\Psi \text{ad}_\tau)^3\cdot 1 + 12 \tilde  \Theta[-1] \,((\Xi[-1]-\Psi\text{ad}_\tau)\cdot 1)\,  \Theta[-1]\\
 & \quad
-12 \Psi( \tilde  \Theta[-2]   \Theta[-1] +\tilde  \Theta[-1]   \Theta[-2] ).
\end{align*}

\end{example}
\begin{proof}
\begin{align*}
\Omega[-1]^n &= \sum_{m=0}^n{n\choose m} Y[-1]^mX[-1]^{n-m}\\
&=\sum_{m=0}^n \sum_{a=0}^{m}\sum_{b=0}^{n-m} \sum_{\alpha\vdash a\beta\vdash b}  {n\choose m}\frac{m!\,(n-m)!}{ (m-a)!(n-m-b)! m_1! m_2!\dots m^\prime_1!m^\prime_2!\dots }\\
&\quad  (-2\Psi)^{ b+a-l(\alpha)-l(\beta)}
\tilde \Theta[-\alpha] (\tilde \xi[-1])^{m- a}( \xi[-1])^{n-m- b}\Theta[-\beta]\\
&=\sum_{0\le a+b\le n}\sum_{\alpha, \beta}
\frac{n!\,}{ (n-a-b)!m_1! m_2!\dots m^\prime_1!m^\prime_2!\dots }(-2\Psi)^{ a+b-l(\alpha)-l(\beta)}\\
& \,\times \tilde \Theta[-\alpha]  \left(\sum_{m=a}^{n-b}  {{n-a-b}\choose {m-a}}(\tilde \xi[-1])^{m-a}(\xi[-1])^{n-m- b}\right)\Theta[-\beta].
\end{align*}
Denote as
$$
P_r=\sum_{k=0}^{r}  {{r}\choose {k}}(\tilde \xi[-1])^{k}(\xi[-1])^{r-k}, \quad P_0=1.
$$
One has:
\begin{align*}
P_r&= \tilde \xi [-1]P_{r-1}+P_{r-1}\xi[-1]
=(\Xi[-1] -2\Psi\tau)P_{r-1}+P_{r-1}(\Xi[-1] +2\Psi\tau)\\
&=\Xi[-1] P_{r-1}+P_{r-1}\Xi[-1] -2\Psi[\tau, P_{r-1}].
\end{align*}
If $P_{r-1}$ is a sum of  monomials in $\Psi$ and  $\Xi[s] $ with some negative values of  $s$, then $[\tau, P_{r-1}]$ is also a sum of monomials in $\Psi$ and  $\Xi[s]$ with some negative values of $s$. This,  together with the  fact that $P_1=2\Xi[-1]$,
implies that all $P_r$ are linear combinations  of monomials in $\Psi$ and  $\Xi[s]$ ($s<0$),  each of them  commutes with $\Xi[-1] $ and  we can write
\begin{align*}
P_r=2(\Xi[-1] P_{r-1}-\Psi[\tau, P_{r-1}])= 2(\Xi[-1] -\Psi\text{ad}_\tau)\, P_{r-1}=2^r(\Xi[-1] -\Psi\text{ad}_\tau)^r\cdot 1.
\end{align*}
Thus
\begin{align*}
\left(\sum_{m=a}^{n-b}  {{n-a-b}\choose {m-a}}(\tilde \xi[-1])^{m-a}(\xi[-1])^{n-m- b}\right)=
   \left(   2^{n-a-b}(\Xi[-1] -\Psi\text{ad}_\tau)^{n-a-b}\cdot 1 \right).
   \end{align*}
   Also  notice that since $\Omega[-1]^n$ is a  form of full degree,  the number of parts $\alpha$ and $\beta$ in non-zero terms is the same: $l(\alpha)=l(\beta)=l$.
   This completes the proof of  the theorem.
\end{proof}

\section{Explicit formula for Pfaffian  $\operatorname{Pf}  \tilde F[-1]$}

  Pfaffian  $\text{Pf}\,\tilde F[-1]$ is given by  the coefficient of the monomial  $e_{-n}e_{-n+1}\dots e_{n-1}e_{n}$
  in the  form  $\Omega[-1]^n$ .
 One can use Lemma \ref{pfdet} below to interpret through Pfaffian- and determinant-like elements the  expressions $\tilde \Theta[-\alpha]$,  $(\Xi[-1] -\Psi \text{ad}_\tau)^{r}$, $\Theta[-\beta]$
in the statement of  Theorem \ref{thm1}.

 For  subsets $I, J$ of $[-n;n]$ such that $|I|=|J|=l$ we denote as  $\Phi[-s]_{IJ}$  an $l\times l$-submatrix of the matrix $\Phi[-s]=(F[-s]+{\text{ad}_\tau}\cdot Id )$
\begin{align*}
\Phi_{IJ} [-s]= (F_{i,j}[-s]+ {\text{ad}_\tau} \delta_{ij})_{i\in I, j\in J}.
\end{align*}
and set
\begin{align*}
 \text{det} \, \Phi_{IJ} [-1]=\sum_{\sigma\in S_l}\text{sgn}\,(\sigma) \Phi_{i_{\sigma (1)}, j_1}\dots \Phi_{i_{\sigma(l)}, j_l}.
\end{align*}

Also we define  for  subsets $I=( i_1<\dots <i_l)$    of $\{-n,\dots, -1\}$ the elements
\begin{align}
\text{Pf}\, \Phi_{+I-I} [-\beta]&=\frac{1}{2^l l!}\sum_{\sigma\in S_{2l}} \text{sgn}\,(\sigma) F_{i_{\sigma(1)}, -i_{\sigma (2)}}[-\beta_1]\dots  F_{i_{\sigma(2l-1)}, -i_{\sigma (2l)}}[-\beta_l],\label{pf2}\\
\text{Pf}\, \Phi_{-I+I} [-\alpha]&=\frac{1}{2^l l!}\sum_{\sigma\in S_{2l}} \text{sgn}\,(\sigma) F_{-i_{\sigma(1)}, i_{\sigma (2)}}[-\alpha_1]\dots  F_{-i_{\sigma(2l-1)}, i_{\sigma (2l)}}[-\alpha_l]. \label{pf3}
\end{align}
\begin{lemma}\label{pfdet}
For  partitions $\alpha$, $\beta$ and for   $r\in {\mathbb Z}_+$,
\begin{align*}
\tilde \Theta[-\alpha] &=2^l l!\sum_{-n\le i_1<\dots <i_{2l}\le -1} e_{-i_1}\dots  e_{-i_{2l}}\,  \text{Pf}\, \Phi_{-I+I} [-\alpha],\\
 \Theta[-\beta] &=2^l l!\sum_{-n\le i_1<\dots <i_{2l}\le -1} e_{i_1}\dots  e_{i_{2l}} \, \text{Pf}\,  \Phi_{+I-I} [-\beta],\\
 (\Xi[-1] -\Psi \text{ad}_\tau)^{r}\cdot 1&= r! \sum_ {\substack{
  -n\le i_1<\dots<i_r\le -1\\
   -n\le j_1<\dots<j_r\le -1}}
 e_{i_1}\dots e_{i_r}e_{-j_r}\dots e_{-j_1}\, (\text{det}\, \Phi_{I,J} [-1])\cdot1.
\end{align*}

 \end{lemma}
\begin{proof}
The first and the second equalities follow from the definitions (\ref{pf2}-\ref{pf3}), and the proof  of the third equality  repeats  word-to-word the proof of  Proposition 4.5.  in \cite{TH} if we observe that
$\Xi[-1]-\Psi \text{ad}_\tau=\sum_{j=-n}^{-1}\eta_j e_{-j}$, where $\eta_{j}=\sum_{i=-n}^{-1}e_i(F_{i,j}[-1]+\text{ad}_\tau\delta_{ij})$ and satisfy  $\eta_j\eta_k+\eta_k \eta_j=0$.
\end{proof}
\begin{example}\label{ex_2}
 For $n=2$ one gets
$
4(\Xi[-1]-\Psi \text{ad}_\tau)^2\,=\, 8 e_{-2}e_{-1} e_{1}e_{2}\, \text{det}(\Phi[-1])\cdot 1
$
and $\Theta[-1]=2e_{-2}e_{-1}F_{-2,1}[-1]$,
 $\tilde \Theta[-1]=2e_{1}e_{2}F_{1,-2}[-1]$. Therefore,
\begin{align*}
& \operatorname{Pf}  \tilde F[-1]\,=\,\text{det}(\Phi[-1])\cdot 1-F_{1,-2}[-1]F_{-2,1}[-1]\\

&=\, (F_{-2,-2}[-1]F_{-1,-1}[-1] - F_{-1,-2}[-1]F_{-2,-1}[-1]  + F_{1,-2}[-1]F_{-2,1}[-1] + F_{-1,-1}[-2]).
\end{align*}
\end{example}
This example illustrates that Theorem \ref{thm1}  can be used  to compute the Pfaffian $\operatorname{Pf}  \tilde F[-1]$:   we find  by  that theorem the value of $\Omega[-1]^n$, and  then by Lemma \ref{omn},  the coefficient of the monomial  $e_{-n} e_{-n+1}\dots e_{n-1}e_{n}$  in  $\Omega[-1]^n$ is  $2^n n!\operatorname{Pf}  \tilde F[-1]$. For the sake of completeness we write explicitly  the general  formula for that coefficient (c.f. Theorem 4.7 in \cite {TH}), though it involves more technical notations.  The statements of Theorems \ref{thm1}  above and \ref{thm2}  below are equivalent.

If  $I=( i_1<\dots <i_k)$  is  a  string   of elements of  subset  of    of $\{-n,\dots, -1\}$ written in increasing order,  denote as  $- I= ( -i_1>\dots >-i_k)$.
Consider subsets  $J, I_1,I_2,J_1,J_2$  of  $\{-n,\dots,-1\}$ that satisfy the properties:
\begin{align}
I_1 \sqcup J_1 \sqcup J= J \sqcup J_2 \sqcup I_2=\{-n,\dots,-1\},\label{subsets1}\\
|I_1|=|I_2|=2l, \quad  |J_1|=|J_2|=n-a-b.\label{subsets2}
\end{align}

Write the elements  of  strings    $[J,J_2, I_2]$   exactly   in that order.   We denote  by
$ \text{sgn}\, ({ J,J_2, I_2})$  the sign of the  permutation of   the elements of this string that puts  them in the  increasing order  $( -n,\dots, -1 )$.
Similarly,  $\text{sgn}\, ( {-I_1,-J_1,-J})$  is the sign of the  permutation of   the elements of this string that puts  them in the  increasing order  $( 1,\dots, n )$.

 \begin{theorem}\label{thm2}
  \begin{align}
 \operatorname{Pf}  \tilde F[-1]&=\sum_{l=0}^{\lfloor n/2\rfloor}
\sum_{\substack{ a,b\\0\le a+b\le n }}\sum_{I_1,J_1,I_2,J_2, J }
\text{sgn}\, ( {J,J_2,I_2})\text{sgn}\, ( {-I_1,-J_1,-J})\notag\\
&\quad \times \left(
\sum_{\alpha\vdash a, l(\alpha)=l}\frac{l!}{m_1! m_2!\dots}
\text{Pf}\, \Phi_{-I_1+I_1} [-\alpha]\right)
\notag\\
&\quad \times
\left(
{(a+b-2l)! }
 (\text{det}\, \Phi_{J_2,J_1} [-1]\cdot1)
 \right)
 \left(  \sum_{\beta\vdash b, l(\beta)=l}\frac{l!}{m^\prime_1! m^\prime_2!\dots}\Phi_{+I_2-I_2} [-\beta]\right),\label{uzhas}
 \end{align}
the third sum  is taken over subsets $J, I_1,I_2,J_1,J_2$  of  $\{-n,\dots,-1\}$ that satisfy the properties (\ref{subsets1}, \ref{subsets2}).
 \end{theorem}
\begin{proof}
Using  for a subset  $I=( i_1<\dots <i_k)$    of $\{-n,\dots, -1\}$ the  notations
$
{e_{I}}=  e_{i_1}\dots  e_{i_{k}}$,
${e_{-I}}= e_{-i_1}\dots  e_{-i_{k}}$, $
{e_{-\tilde I}}=e_{-i_k}\dots  e_{-i_{1}},
$
combine Theorem  \ref{thm1}   and Lemma \ref{pfdet}  to obtain
\begin{align*}
\Omega[-1]^n &= 2^{n} n! \sum_{l=0}^{\lfloor n/2\rfloor}
\sum_{0\le a+b\le n} \sum_{I_1,J_1,I_2,J_2}
(-\Psi)^{ a+b-2l}
{e_{-I_1}} {e_{J_2}} {e_{-\tilde {J}_1}} {e_{I_2}}\\
&\quad \times \sum_{\alpha\vdash a, l(\alpha)=l}\frac{l!}{m_1! m_2!\dots}
\text{Pf}\, \Phi_ {-I_1 I_1}[-\alpha]  ( \text{det}\, \Phi_ {J_2 J_1}[-1]  \cdot 1 )\\
&\quad \times
\sum_{\beta \vdash b, l(\beta)=l}\frac{l!}{m^\prime_1! m^\prime_2!\dots}
\text{Pf}\, \Phi_ {I_2 -I_2}[-\beta]
\end{align*}
Note that
\begin{align*}
(-\Psi)^{ r}&= (r)!\sum_{ J\subset\{-n\dots -1\},\, |J|=r} (-1)^{\frac{r(r-1)}{2}} e_{J} e_{-J},
\end{align*}
and that
$$ (-1)^{\frac{{(a+b-2l)}(a+b-2l-1)}{2}} e_{J} e_{-J}{e_{-I_1}} {e_{J_2}} {e_{-\tilde {J}_1}} {e_{I_2}}
= {e_{J}} {e_{{J}_2}} {e_{I_2}}e_{-I_1} e_{-J_1} {e_{-J}}.$$

By comparing the  order of  the  terms of this monomial    with the monomial  $e_{-n}e_{-n+1}\dots\\ e_{-1}e_1\dots e_{n-1}e_{n}$ we get  formula (\ref{uzhas}).
\end{proof}

\section{Harish-Chandra image of the Pfaffian }\label{sec_HC}
The  Feigin-Frenkel center $\mathfrak{z}(\hat{{\mathfrak o}}_{2n} )$ is a commutative subalgebra of $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}]) ^\mathfrak  h$. Consider  a left ideal  $ I$    of $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}])$ that is  generated by the elements $F_{ij}[r]$ for $-n\le i<j\le n$ and $r<0$. The quotient of   $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}]) ^\mathfrak  h$ by the  two-sided ideal  $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}]) ^\mathfrak  h\cap  I$ is  isomorphic to  the commutative algebra  that is freely generated  by the images of the elements
 $F_{ii}[r]$ ($i=-n,\dots, -1$,  $r<0$). Denote these images as $\mu_i[r]=F_{ii}[r]$
 We get an analogue of  classical Harish-Chandra homomorphism
$$\chi: U(t^{-1}{\mathfrak o}_{2n} [t^{-1}]) ^\mathfrak  h\to U(t^{-1}\mathfrak  h[t^{-1}]), $$
and  $U(t^{-1}\mathfrak  h[t^{-1}]) ^\mathfrak  h$ can be viewed as a polynomial algebra in variables $\{\mu_i[r], i=-n, \dots,-1, r<0\}$.
The  restriction  of $\chi$ to  $\mathfrak{z}(\hat{{\mathfrak o}}_{2n} )$ yields an isomorphism
 $\mathfrak{z}(\hat{{\mathfrak o}}_{2n} )\to \mathcal W({\mathfrak o}_{2n} )$, where $\mathcal W({\mathfrak o}_{2n} )$ is the classical $\mathcal{W}$-algebra of ${\mathfrak o}_{2n} $. It is a subalgebra of $U(t^{-1}{\mathfrak o}_{2n} [t^{-1}]) ^\mathfrak  h$ annihilated by screening operators (\cite{F}, Chapter 7).
The elements of $\mathcal W({\mathfrak o}_{2n} )$ are polynomials in $\mu_i[r]$ ($r<0$, $i=-n,\dots, -1$).
We give a new direct proof of the following corollary established in \cite{MM}.
\begin{corollary}\label{HC}
\begin{align*}
\chi( \operatorname{Pf} \tilde F[-1])= (\mu_{-n}[-1] + \text{ad}_\tau)\dots(\mu_{-1}[-1] +\text{ad}_\tau)\cdot1.
\end{align*}
\end{corollary}
\begin{proof}
 Only  the terms with $a=b=0$ will survive under Harish-Chandra homomorphism in the formula  of Theorem \ref{thm1} for $\Omega[-1]^n$:
\begin{align*}
\chi(\Omega [-1]^n)&=\chi (
2^n (\Xi[-1] -T\text{ad}_\tau)^{n}\cdot 1)=
2^nn!\, e_{1}\dots e_n e_{-1}\dots e_{-n}\,   \chi (\text{det}\,(F [-1]+\text{ad}_\tau)\cdot1)\\
&=2^n n! \,  e_{1}\dots e_n e_{-1}\dots e_{-n}\, (\mu_{-n}[-1]+\text{ad}_\tau )\dots(\mu_{-1}[-1]+\text{ad}_\tau )\cdot 1
\end{align*}
\end{proof}

\section{Deducing  the  formula for  Pfaffian for  $U({\mathfrak o}_{2n} )$}\label{fol}
In this section we would like to illustrate that the application of evaluation homomorphism to the statement of Theorem \ref{thm1}  implies the formulas for Pfaffian of the universal enveloping algebra $U({\mathfrak o}_{2n} )$  in \cite{TH} (namely, Proposition 4.4.  there).

 Let $z$ be another variable and consider a homomorphism of algebras  $\varphi:\,  U(t^{-1}{\mathfrak o}_{2n} [t^{-1}])\to   U({\mathfrak o}_{2n} )\otimes z^{-1} {\mathbb C}\langle z^{-1} \rangle $, defined  by  $\varphi(F_{ij}[r])= F_{ij}\otimes z^r$,  $\varphi(K)= 0$.
  Then the action of $\text{ad}_\tau$ is identified with the operator $-\partial_z$ by the rule
  $\varphi([\tau, f])=-\partial_z(\varphi(f))$.
Let us apply $\varphi$ to the formula of the Theorem \ref{thm1}.

We have:
\begin{align*}
\varphi(\Omega [-1]^n)=\Omega ^n z^{-n}, \quad \text{where}\quad  \Omega=\sum_{i,j\in[-n;n]}e_ie_{-j} F_{i,j}\in \Lambda\otimes U({\mathfrak o}_{2n} ).
\end{align*}
Note that  for  a  partition $\alpha\vdash a$ of length $l(\alpha)=l$, the  evaluation map gives
$\varphi(\tilde \Theta [-\alpha])= \tilde \Theta^l z^{-a}\quad \text{and}\quad \varphi(\Theta [-\alpha])= \Theta^l z^{-a},$ where $ \tilde \Theta=\sum_{i=1}^{n}\sum_{j=-n}^{-1}e_ie_{-j} F_{i,j}$,
$\Theta=\sum_{i=-n}^{1}\sum_{j=1}^{n}e_ie_{-j} F_{i,j}$.
Also
$$ \varphi ((\Xi[-1] -\Psi\, \text{ad}_\tau)^{k}\cdot 1)=
((\Xi z^{-1} +\Psi\, \partial_z)^{k}\cdot 1), $$
where $
\Xi=\sum_{i,j=-n}^{-1} {e_i e_{-j}} F_{i,j}
$.
Using that  in the notations $\alpha =(1^{m_1}, 2^{m_2},\dots)$,
\begin{align}
\sum_{\alpha\vdash a, l(\alpha)=l}\frac{l!}{m_1! m_2!\dots}={{a-1} \choose {l-1}}
\end{align}
(see e.g. Corollary 6 in \cite{CW}),
and that
\begin{align}
\sum_{a+b=p}{{a-1} \choose {l-1}}{{b-1} \choose {l-1}}= {{p-1} \choose {2l-1}},
\end{align}
we obtain from Theorem \ref{thm1},
\begin{align}\label {z-p}
\Omega^n z^{-n} &= \sum_{l=0}^{\lfloor n/2\rfloor}  \frac{2^{n-2l} n!}{l!l!}
  \tilde  \Theta^l \quad\Big( \sum_{2l\le p\le n} \frac{1}{(n-p)!}{{p-1}
\choose {2l-1}}\\
&\quad \times (-\Psi)^{ p-2l} \,(\Xi z^{-1} +\Psi\partial_z)^{n-p}\cdot 1\,
\Big)  \Theta^l z^{-p}.
\end{align}
Let us multiply both parts of (\ref{z-p})  by $z^{-n}$.
Observe that
$$
((\Xi z^{-1} +\Psi\partial_z)^{n-p}\cdot 1) z^{n-p}=  (\Xi)(\Xi-\Psi)(\Xi -2\Psi)\dots (\Xi-(n-p+1)\Psi),
$$
and
\begin{align*}
\frac{1}{(n-p)!}{{p-1}
\choose {2l-1}}=
\frac{1}{(n-2l)!}{{n-2l}\choose{n-p}}\frac{(p-1)!}{(2l-1)!}.
\end{align*}
Therefore, we can rewrite the  central part expression of  (\ref{z-p})  as
\begin{align}\label{xyz}
&\sum_{2l\le p\le n} \frac{1}{(n-p)!}{{p-1}
\choose {2l-1}}
 (-\Psi)^{ p-2l} \,((\Xi z^{-1} +\Psi \partial_z)^{n-p}\cdot 1)\, z^{n-p}\notag
\\
=
& \frac{1}{(n-2l)!} \sum_{p=0}^{n}{{n-2l}\choose{n-p}}y^{(p-2l)}x^{(n-p)}
\end{align}
with
\begin{align*}
 y=-2l\Psi,&\quad y^{(p-2l)}=  (-2l \Psi)\dots  (-(p-1)\Psi),\\
x=\Xi, &\quad x^{(n-p)}=\Xi(\Xi-\Psi)\dots (\Xi-(n-p+1)\Psi).
\end{align*}
Since $\Xi$  and $\Psi$ commute, (\ref{xyz}) can be rewritten using binomial theorem for falling factorial powers  as
\begin{align*}
 \frac{1}{(n-2l)!} (x+y)^{(n-2l)}= \frac{1}{(n-2l)!} (\Xi-2l\Psi)^{(n-2l)}= \frac{1}{(n-2l)!}(\Xi-2l\Psi)\dots (\Xi-(n+1)\Psi).
\end{align*}
Hence, from  (\ref{z-p}),
\begin{align}
\Omega^n=\sum_{l=0}^{\lfloor n/2\rfloor}  \frac{2^{n-2l} n!}{l!l!}
  \tilde  \Theta^l
\frac{1}{(n-2l)!}(\Xi-2l\Psi)\dots (\Xi-(-n+1)\Psi)
 \Theta^l .
\end{align}
Finally, using the commotion relation
\begin{align*}
 \tilde  \Theta^l (\Xi-2l\Psi)\dots (\Xi+(n-1)\Psi)=\Xi (\Xi-\Psi)\dots (\Xi-(n-2l-1)\Psi)\tilde \Theta^l,
\end{align*}
we get
\begin{align}\label{imp}
\Omega^n=\sum_{l=0}^{\lfloor n/2\rfloor}  \frac{2^{n-2l} n!}{(n-2l)! l!l!}
\Xi(\Xi-\Psi)\dots (\Xi-(n-2l-1)\Psi)\
 \tilde  \Theta^l  \Theta^l,
\end{align}
which is the statement of Proposition 4.4. in \cite{TH}.
\begin{remark} (a) As  it is discussed  in the proof of Theorem 4.17 of \cite{TH}, necessarily  $p=q$ in Proposition 4.4 of \cite{TH}.\\
(b) Our notation  $\Psi$ stands for $-\tau$ in \cite{TH}.\\
(c)  As it was mentioned before,  Theorem \ref{thm1} and Theorem \ref{thm2} of this note are equivalent statements. Similarly, the results of \cite{TH} cited here in  (\ref{imp}) and (\ref {HP})
are equivalent.  We chose to illustrate how  (\ref{imp}) is deduced from Theorem \ref{thm1} rather then  (\ref {HP}) from Theorem \ref{thm2} in order to avoid technically involved notations of the other two statements.
\end{remark}


P-ROL Process Control Inc. is a company involved in process design and control. We are capable of providing PLC and HMI control of various processes in chemical,food,pharmaceutical,water treatment, drinking water and related industries. We are stationed in the vicinity of Toronto and provide services in Canada, USA as well as in other parts of the world. These are some projects completed in the past: WATER PUMPING STATION The project included PLC control of two separate pumping stations and radio communication to the central office, where control was done using Wonderware HMI system. It also involves gathering historical data to satisfy government requirements using MS SQL system with InSQL. ZOOK CANADA Inc. in Burlington is manufacturing rupture disks for industry. Rupture disk are made of different materials and are of different sizes. In order to maintain consistency of production, we were requested to provide computer control of their presses. SHERWIN WILLIAMS CANADA Inc. contracted us to install storage tanks level monitoring system. The project included AB equipment and programming, two panel view local displays and link via internet to head office in Cleveland USA. AVENTIS PASTEUR pharmaceutical company from Toronto performed some modifications on their Waste Water Treatment System. On their behalf we performed Pre-Start Health and Safety Inspection as required by Ministry of Labor in Ontario. COLORTECH Inc. in Brampton. Programming changes to the extruder to incorporate VFD to feed and chopper assembly. Extruder was imported from Germany and the modification was required to accommodate to the process. GM Corporation in Chicago. We were programming the automatic bolting machine for the engine assembly line. The project involved programming, installation, start up and debugging of the machine without production interruption. PARISCO Inc. in Toronto. An exact amount of warm water at precise temperature was required to be delivered to the mixers in cookie production for every batch. AB PLC controlled process. LATEX CORPORATION in Illinois. This 1.5 Million-Dollar project included design, installation of production storage tanks and reactor. The installation was required to increase the capacity and improve operation safety. Siemens PLC was chosen for control of the process. TENNECO AUTOMOTIVE. Retrofit of the existing chemical treatment facilities involved implementation of HMI and AB PLC to control quality and flow of water before it is released into the community. For DOMTAR in Hamilton we performed design and installation of a pilot size (500 L) chemical reaction vessel. The unit was used for heat treatment of heavy aromatic oils. The purpose of the installation was to study the progress of the reaction and derive the parameters required to build an industrial size reactor. DOMTAR in Mississauga. The vacuum drum filter from Komline-Sanderson needed modifications to perform filtration of heated carcinogenic material. Hazardous fumes must be kept sealed within the filter. The entire operation was upgraded to computer (PLC) control. Design and installation of a stearate production line. The project-included design of raw material storage tanks, three production kettles, two semi automatic continuous centrifuges, a continuous drying unit as well as grinding and packaging equipment. The total capacity of the operation was 500 tons annually. The installation increased capacity, improved product quality and reduced manpower. Design of a batch reactor for pharmaceutical grade Lithium Carbonate. The project included design of delivery system to feed the reactor and gas dispersers to introduce gas into the liquid phase. A centrifuge system was selected to separate particles for final drying of the material in ovens. These projects represent only a small portion of work done in the past, and hopefully illustrate our experience and strength in the field of process industry.


text:
All posts in category Avigdor Liberman
Israel Political Brief April 4, 2012: PM Benjamin Netanyahu, Defense Minister Ehud Barak under fire over Machpelah house eviction
Netanyahu, Barak under fire over Machpelah house eviction
Source: YNet News, 4-4-12
'This policy undermines the coalition's stability,' FM Lieberman says Prime Minister Benjamin Netanyahu and Defense Minister Ehud Barak are the focus of fierce criticism following the eviction of the Machpelah house in Hebron on Wednesday….READ MORE
Ministers slam PM, defense minister for misguided handling of Hebron house affair, suggest decision motivated by personal, political considerations. 'This policy undermines the coalition's stability,' FM Lieberman says
Prime Minister Benjamin Netanyahu and Defense Minister Ehud Barak are the focus of fierce criticism following the eviction of the Machpelah house in Hebron on Wednesday.
Foreign Minister Avigdor Lieberman said: "The handing of the issue from start to finish was bad. One cannot use the right of property claim to evict en entire community established 13 years ago and at the same time not honor the right of property when it comes to the purchase of a Hebron house by Jews."
Police evacuate Machpelah house
Palestinians deny sale of Machpelah house
Op-ed: Arab racism prevents peace
He accused the Defense Ministry of trying to stir up a controversy and failing to provide security to Israelis living in Judea and Samaria. "This policy undermines the coalition's stability. The only solution which will restore stability will be to transfer all authorities on Jewish settlement in Judea and Samaria to a special ministerial committee that will aptly reflect the opinion of the majority of the public and government."
Sources close to Netanyahu said in response: "The coalition is stable, the eviction was coordinated." Netanyahu himself reportedly expressed his commitment to the continuation of the settlement in Judea and Samaria, specifically in Hebron.
Machpelah house eviction (Photo: Gil Yohanan)
Ministers from the Right reproached Defense Minister Barak, who authorized the eviction. Habayit Hayehudi Chairman Minister Daniel Hershkowitz compared the move to "poking fingers in the eyes of the people.
"This is not a battle between law-abiders and lawbreakers, but rather between those who cherish the settlement project and those who seek to destroy it," Hershkowitz noted.
by bonniekgoodman on April 4, 2012 • Permalink
Posted in Avigdor Liberman, Benjamin Netanyahu, Ehud Barak, Israel News, Israeli Politics, West Bank Communities, West Bank Settlements
Tagged Benjamin Netanyahu, Defense Minister, Ehud Barak, Hebron home eviction, Jewish Settlers, Machpelah house
Posted by bonniekgoodman on April 4, 2012
https://togetherwithisrael.wordpress.com/2012/04/04/israel-political-brief-april-4-2012-pm-benjamin-netanyahu-defense-minister-ehud-barak-under-fire-over-machpelah-house-eviction/
Israel Political Brief March 23, 2012: Israeli Foreign Minister Avigdor Liberman: Israel might withdraw from UN Human Rights Council
Liberman: Israel might withdraw from U.N. Human Rights Council
Source: JTA, 3-23-12
Israeli Foreign Minister Avigdor Liberman reportedly said he might recall Israel's ambassador to the U.N. Human Rights Council after the council voted 36-1 to investigate the effects of Jewish settlements on Palestinians.
Liberman also said Israel would not cooperate with the fact-finding mission established by the council to probe settlements, the Jerusalem Post reported.
On Thursday, the council passed a resolution, with 10 abstentions, to investigate how Israeli settlement construction affects Palestinian human rights. The United States was the only country to vote against the resolution.
"This is a hypocritical council with an automatic majority against Israel," Israeli Prime Minister Benjamin Netanyahu said on Thursday. "This council ought to be ashamed of itself."
The Israeli leader noted that the council has made 91 decisions, 39 of which dealt with Israel, three with Syria and one with Iran.
"One only had to hear the Syrian representative speak today about human rights in order to understand how detached from reality the council is," he said.
The decision requires the council to "dispatch an independent international fact-finding mission, to be appointed by the President of the Human Rights Council, to investigate the implications of the Israeli settlements on the civil, political, economic, social and cultural rights of the Palestinian people throughout the Occupied Palestinian Territory, including East Jerusalem." The mission will generate a report for the council.
The council on Thursday approved five resolutions critical of Israel, including implementing the Goldstone report on the Gaza war and criticizing Israel's occupation of the Golan Heights.
The resolution on the settlements, which calls on Israel to cooperate in the investigation, also called on Israel to prevent settler violence against Palestinians in the West Bank.
A U.S. representative to the council said the U.S. is "deeply troubled by this council's bias against Israel."
by bonniekgoodman on March 23, 2012 • Permalink
Posted in Avigdor Liberman, Benjamin Netanyahu, Israeli Politics, Palestinians -- UN, United Nations
Tagged Avigdor Liberman, Benjamin Netanyahu, Golan Heights, Goldstone report, Israeli Foreign Minister, Jewish Settlements, Palestinians, UN Human Rights Council, West Bank
Posted by bonniekgoodman on March 23, 2012
https://togetherwithisrael.wordpress.com/2012/03/23/israel-political-brief-march-23-2012-israeli-foreign-minister-avigdor-liberman-israel-might-withdraw-from-un-human-rights-council/

redpajama_set_name: RedPajamaCommonCrawl


text:
California Protects Food Workers Where The Federal Government Does Not
The State of California Legislature, Executive Branch, and administrative agencies continue to step up to protect food workers where the federal government cannot or will not.
Here is a straightforward explanation of the most recent laws put on the books that are aimed at protecting food workers in the Golden State.
Paid Sick Leave (California Assembly Bill 1867)
California State Assembly Bill 1867 ("AB 1867") combines into the same bill COVID-19 supplemental paid sick leave and mandated hand washing requirements for food workers.
COVID-19 Supplemental Paid Sick Leave
AB 1867 operates as a stop-gap by mandating COVID-19 paid sick leave for workers who were excluded from this benefit under the federal FFCRA or California's prior food worker paid sick leave Executive Order.
The federal Families First Coronavirus Response Act ("FFCRA") — which provides for COVID-19 paid sick leave to workers of small employers — excludes workers employed by private businesses with 500 or more employees in the United States and by health care provider or emergency responders. AB 1867 covers these workers.
California Governor Newsom's Executive Order N-51-20 came with enforceability concerns, which AB 1867 addressed by codifying most of the Executive Order and making it retroactive.
Now, under AB 1867, California full-time employees are entitled to 80 hours of paid time off (and part-time employees a proportionate amount), for any of the following reasons:
The worker is subject to a federal, state, or local quarantine or isolation order related to COVID-19.
The worker is advised by a health care provider to self-quarantine or self-isolate due to concerns related to COVID-19.
The worker is prohibited from working by the worker's hiring entity due to health concerns related to the potential transmission of COVID-19.
The worker's rate of pay is the highest of the regular rate of pay for the last pay period or the state or local minimum wage, and may not exceed $511 per day, with a maximum cap of $5,110 total.
Unlike the FFCRA and other California paid sick leave ordinances, AB 1867 does not provide leave for those caring for a family member who is quarantined or sick or caring for a minor child whose school or childcare has closed due to COVID-19.
Like the FFCRA, AB 1867 expires on December 31, 2020, or upon the expiration of any federal extension of the FFCRA, whichever is later.
Hand-washing
AB 1867 requires a food sector worker working in any food facility to be permitted to wash their hands every 30 minutes and more as needed. This codifies a similar requirement set forth in the Executive Order.
Unpaid Medical/Family Leave (California Senate Bill 1383)
California State Senate Bill 1383 ("SB 1383") expands the breadth of the California Family Rights Act ("CFRA"), which already provides California employees with up to 12 weeks of unpaid leave during each 12-month period for purposes of family and medical leave. SB 1383 bolsters this by covering more employers and more employee family members.
SB 1383 mandates California employers with as few as five employees to provide CFRA protections. The threshold had been private employers with 50 or more employees within 75 miles of the worksite.
SB 1383 expands the list of employee family members that receive protection to include siblings, grandparents, grandchildren, and domestic partners; and, the definition of "child" now covers all adult children (regardless of whether they are dependent) and children of a domestic partner.
Workers' Compensation (California Senate Bill 1159)
California Senate Bill 1159 ("SB 1159") expands the definition of worker injury under the workers' compensation system to include illness or death resulting from COVID-19.
Like the prior Executive Order that expired in July, SB 1159 creates a disputable presumption that COVID-19 injuries arose out of and in the course of employment and are compensable.
SB 1159 makes a workers' compensation claim relating to a COVID-19 illness presumptively compensable after 30 days or 45 days, rather than 90 days. It allows for the same presumption of injury for employees whose coworkers test positive for COVID-19.
The presumption will remain in effect until January 1, 2023, and it goes into effect immediately.
Notice of Exposure (California Assembly Bill 685)
California State Assembly Bill 685 ("AB 685") increases COVID-19 exposure notice and reporting requirements on employers.
Notice to Workers
AB 685 requires employers having notice of a potential COVID-19 exposure (e.g., individual testing positive for COVID-19 was in the workplace) provide a written notice to employees and subcontractor employees who were at the worksite when a potentially infected individual was there and may have been exposed to COVID-19 as a result.
This notice must be provided within one business day of the employer being notified of a potential exposure and may be done in "a manner that the employer normally uses to communicate employment-related information," such as personal service, mail, or text message.
Notice to Public Health Department
AB 685 requires that upon an outbreak, the employer must notify its local public health department within 48 hours and be prepared to provide information on the number of COVID-19 cases at the worksite, their names, occupation, and other pertinent information. And, employers must keep working with the local health department and provide updates on new laboratory-confirmed COVID-19 cases.
Notice to Cal/OSHA
AB 685 confirms employers must report COVID-19 cases the California Division of Occupational Safety and Health ("Cal/OSHA"), as it meets the agency's definition of a serious injury or illness. It authorizes Cal/OSHA to issue citations for serious violations related to COVID-19 without requiring the agency to comply with pre-citation requirements.
Health And Safety Administration Enforcement
The Federal Safety and Health Administration has been criticized for lackluster enforcement of worker protections. In this vacuum, California is starting to act aggressively.
On September 4, Cal/OSHA cited 11 California employers for not protecting employees from COVID-19 exposure during inspections of industries where workers have an elevated risk of exposure. Nine of the 11 industries are food processing, meatpacking, or agriculture.
The employers were cited for various violations including some classified as serious, with proposed penalties ranging from $2,025 to $51,190.
D.L. Poultry, Inc. in Monterey Park and Olson Meat Company in Orland, California, "put their workers at risk for serious illness because they did not ensure their workers were physically distanced at least six feet apart in the processing area, nor did they install Plexiglas or other barriers between the workers."
Uni-Kool Partners in Salinas, California, "put their workers at risk as they did not implement procedures to screen employees and visitors arriving at the facility, and failed to take appropriate measures for employees who exhibited COVID-19 symptoms at the facility."
The hands-off federal approach has left states like California responsible for passing and enforcing emergency legislation like the above, to protect workers while keeping the economy somewhat open.


text:
161-й пехотный Александропольский полк — пехотная воинская часть Русской императорской армии.

Полковой праздник — 21 апреля.

Формирование полка
Сформирован 1 августа 1874 г. из 4-х батальонов Кавказской гренадерской дивизии, благодаря чему в полку сразу же оказались многочисленные знаки отличия, унаследованные от 4-х батальонов полков: 13-го лейб-гренадерского Эриванского Его Величества полка; 14-го гренадерского Грузинского полка; 15-го гренадерского Тифлисского полка, кроме того, были сохранены и боевые отличия 41-го Егерского полка (расформированного в 1834 г.); 16-го гренадерского Мингрельского полка, а также и переданные последнему от 42-го Егерского (расформированного в 1834 г.) полка. Таким образом, молодому по имени Александропольскому полку, получившему при сформировании высшие отличия, пожалованные за боевую 75-летнюю службу его славным предшественникам, предстояло на деле доказать, что Александропольцы равняются кавказцам-героям.

Кампании полка

Русско-турецкая война 1877—1878 гг.
Полк, носивший с 16 декабря 1875 г. наименование 161-го пехотного Александропольского Его Императорского Высочества великого князя Алексея Михаиловича, перед началом русско-турецкой войны находился на границе Гурии и в апреле 1877 г. весь был сосредоточен в Николаевске, будучи предназначен в состав Кобулетского отряда.

12 апреля Александропольцы выступили в поход, причём 1-й и 4-й батальоны вошли в авангард, которым командовал командир полка полковник князь Абашидзе. С первых же шагов проявились кавказцы: в тот же день, после тяжелого перехода, они без выстрела заняли Лельский пост. 29 апреля Александропольцы, участвуя в деле у высот при р. Ачхуа, заняли их, получив боевое крещение после обстрела с турецких броненосцев. По занятии Хуцубанских высот, батальонам приходилось заниматься землекопными работами, устраивая позицию и прокладывая дороги в дикой гористой стране, покрытой девственными лесами, а также производить разведки по р. Кинтриши, через которую отряд переправился 16 мая и утвердился на высотах Сальба.

Жители Кобулетии, отрезанные от турок, хотя внешне и были покорны, но лишь временно. С 20-х чисел мая появились местные враждебные шайки, поэтому войскам Кобулетского отряда пришлось подвижными колоннами действовать и против местного населения, вскоре сдавшее розданное турками оружие и через почётных лиц изъявившее покорность.

11 июня Александропольцы участвовали в атаке турецкого укрепленного лагеря на высотах: Квириках, Дзегва, Цихидзири. При этом полк находился в 14-часовом бою под палящим солнцем. Выяснившаяся невозможность сбить турок, оказавшихся в значительном численном превосходстве, вынудила оставить занятые нами позиции, что и предполагалось совершить 12-го июня, но в действительности в этот день отряд геройски вынес 9-часовой бой, отбиваясь штыками и картечью от перешедших в наступление турок. Стойко и мужественно бились и в этот день Александропольцы. Из рядов полка выбыл тяжело раненым в живот флигель-адъютант подполковник Териев, кавалер одного офицерского и четырёх солдатских георгиевских крестов, в сильных страданиях скончавшийся через два дня. «Под Цихидзири ни днем ни ночью нет покоя; в бою ни своих не видать, ни его (турка), а бьют из леса и не видим откуда» — вот меткая солдатская оценка тяжёлых дней 11 июня и последующих.

К 18 июня отряд сосредоточился на прежней позиции Мухаэстатэ, где Александропольцы и оставались до ноября. В общем, они за этот 2-месячный боевой период действий в отряде, помимо постоянных перестрелок с кобулетцами, соорудили 70 вёрст дороги в малодоступной местности, построили 4 укрепленных позиции, совершили 4 блестящих горных экспедиции, приняв участие в 2 крупных делах и одном 2-дневном сражении. 15 ноября Александропольцы, в числе прочих, вновь заняли позицию Хуцубани, а затем, изредка участвуя в незначительных перестрелках, завершили свою боевую службу 18 января 1878 г. атакою высот Цихидзири. 22 января было заключено перемирие и военные действия прекратились.

За отличия в войну 1877—1878 гг. полку было пожаловано: 1-му и 4-му батальонам — поход за военное отличие, а 2-му и 3 батальонам — дополнительные надписи на знаменах: «за отличие в Турецкую войну 1877 и 1878 гг.».

В Закаспийской области
В 1879 г. 4-й батальон Александропольского полка принял участие в неудачной Ахал-Текинской экспедиции, но ему пришлось быть не в бою, а находиться в гарнизонах Терсакана и Бендессена и нести службу по охране тыла главных сил генерала Ломакина. Несмотря на это, в батальоне оказалась убыль, так как среди нижних чинов развилась сильная цынга. К концу года батальон возвратился обратно на Кавказ.

В 1883 г. весь полк перевели в Европейскую Россию, где для него наступил 20-летний мирный период службы. По случаю кончины шефа великого князя Алексея Михайловича, полк с 1 марта 1895 г. именуется 161-м пехотным Александропольским.

Русско-японская война
В русско-японскую войну полк также принял участие, но лишь в последний период военных действий. 30 октября 1904 г. император Николай II напутствовал полк, а в январе 1905 г. началась его боевая служба, в формуляр которой занесён и 15-дневный Мукденский бой. В период январских и февральских боёв, полк понес значительные потери, причём убыль в офицерском составе дошла до 50 %. Особенное отличие в эту войну оказала 15-я рота, получившая 5 июля 1910 г. особую боевую награду — георгиевский серебряный рожок, с надписью: «за отличие 23 февраля 1905 г.».

Первая мировая война
Перед Первой мировой войной был расквартирован в Саратове. Участвовал в боевых действиях на Юго-Западном (1915 г.) и Западном (1916 г.) фронтах. Входил в 41-ю пехотную дивизию (16 армейский корпус)

Расформирован в январе 1918 года.

Знаки отличия полка
 Георгиевское знамя, с надписью: «За отличные подвиги при поражении 36-тыс. турецкого корпуса на Баш-Кадыкларских высотах 19 ноября 1853 г., в сражении 17 июля 1854 г. на Чингальских высотах, за войну 1854 и 1855 годов, за Кавказскую войну и за отличие в Турецкую войну 1877—1878 годов»
 Георгиевския трубы с надписью «За Кавказскую войну»

При формировании в батальонах полка сразу появились знаки отличия:
 В 1-м батальоне: Георгиевское знамя с надписью «За отличия, оказанные в войну 1854 и 1855 годов и за Кавказскую войну» и Знаки на головные уборы с надписью «За отличие», пожалованные за войну с Персией в 1826—1828 гг. — перешли от 4-го батальона 13-го лейб-гренадерского Эриванского Его Величества полка;
 во 2-м батальоне: Георгиевское знамя с надписью «За отличные подвиги при поражении 36-тысячного турецкого корпуса на Баш-Кадыкларских высотах 19 Ноября 1855 г., за отличие в войну 1854 и 1855 годов и за Кавказскую войну» и Знаки на головные уборы с надписью «За отличие» за русско-турецкую войну 1828—1829 гг. — перешли от 4-го батальона 14-го гренадерского Грузинского полка;
 в 3-м батальоне: Георгиевское знамя с надписью «За отличие в сражении 17-го Июля 1854 г. на Чингальских высотах» и Знаки на головные уборы с надписью «За отличие» за русско-персидскую войну 1826—1828 гг. и Георгиевская серебряная труба с надписью «За Кавказскую войну» — перешли от 4-го батальона 15-го гренадерского Тифлисского полка; кроме того, батальону сохранены и боевые отличия 41-го егерского полка, расформированного в 1834 г. и имевшего «Поход за военное отличие» за Итальянские походы А. В. Суворова и Знаки на головные уборы «За отличие» за русско-турецкую войну 1828—1829 гг.;
 в 4-м батальоне: те же знаки отличия, как и в 3-м батальоне, полученные от 16-го гренадерского Мингрельского полка, а также Знаки на головные уборы за русско-персидскую войну 1826—1828 гг., полученные 4-м батальоном 16-го гренадерского Мингрельского полка от 42-го егерского полка, расформированного в 1834 г.

Самостоятельно полк заслужил следующие награды:
 За русско-турецкую войну 1877—1878 гг. — 1-му и 4-му батальону пожалован «Поход за военное отличие», 2-му и 3-му батальонам — дополнительные надписи на знамёна: «За отличие в Турецкую войну 1877 и 1878 годов».
 5 июля 1910 г. 15-я рота полка за отличие в русско-японскую войну пожалована Георгиевским серебряным рожком с надписью «За отличие 23 Февраля 1905 г.».

Командиры полка
 1874—1878 — полковник Абашидзе, Ростом Николаевич
 в 1881—1883 — полковник Нельдихин, Евгений Альбертович
 11.08.1883 — 19.02.1890 — полковник Озаровский, Александр Федорович
 07.03.1890 — 18.01.1895 — полковник Пневский, Вячеслав Иванович
 20.01.1895 — 03.04.1897 — полковник Орлов, Владимир Александрович
 12.02.1907 — 07.07.1914 гг. — полковник Туров, Пётр Николаевич
 31.10.1915 — 18.07.1916 гг. — полковник Ростковский, Георгий Николаевич

Шефы полка
 16.12.1875 — 1.03.1895 — великий князь Алексей Михайлович

Известные чины полка

 с марта по май 1916 года обязанности полкового священника полка исполнял отец Феодор.
 с декабря 1916 года по январь 1918 года в полку служил будущий советский военачальник  Чехарин, Андрей Евсеевич.

Примечания

Литература

 Е. Р. Краткий очерк военных действий 1877—1878 гг. Киев, 1885.
 Гизетти А. Л. Хроника Кавказских войск 1896 г. Тифлис, 1896.
 Гродеков Н. И. Война в Туркмении. Т. 1. СПб., 1883.
 Климан Ф. Описание боевой жизни 41 артиллерийской бригады в минувшую войну 1877 и 1878 годов. Тифлис, 1883.
 Колюбакин Б. М. Воспоминания офицера Кобулетского отряда в кампанию 1877—78 гг. СПб., 1897.
 Летопись войны с Японией 1904 и 1905 гг., № 34, 44, 55 и другие номера.
 Туган-Мирза-Барановский Б. А. Русские в Ахал-Теке в 1879 г. СПб., 1881.
 Энциклопедия военных и морских наук под редакцией Г. А. Леера. Т. I.
 Краткая историческая памятка 161-го пехотного Александропольского полка. Казань, 1911.

Пехотные полки Российской империи
Воинские части и формирования, созданные в 1874 году
Воинские части и формирования, расформированные в 1918 году

redpajama_set_name: RedPajamaWikipedia


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Egypt expects to export garlic, a crop it has typically had in surplus, to Indonesia soon, Agriculture Minister Ezz el-Din Abu-Steit announced on Thursday.
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1. Trends in Older Women's and Men's Time Use in the United States, 1965-2012 • Sarah Flood , University of Minnesota; Phyllis Moen, University of Minnesota; Renada Goldberg, University of Minnesota; Katie Genadek, University of Colorado.
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Conversations/Conversas
Department of Art
Maria Magdalena Campos-Pons
Professor of Fine Arts
Cornelius Vanderbilt Endowed Chair of Fine Arts
Drawing, Performance, Installation
Professor Maria Magdalena Campos-Pons with artist, Brandon Donahue in Space 204.
E. Broson Ingram Studio Arts Center
email: maria.m.campos-pons@vanderbilt.edu
"María Magdalena Campos-Pons was born in in 1959 in the province of Matanzas, in the town of La Vega, Cuba. She grew up on a sugar plantation in a family with Nigerian, Hispanic and Chinese roots. Her Nigerian ancestors were brought to Cuba as slaves in the 19th century and passed on traditions, rituals, and beliefs. Her polyglot heritage profoundly influences Campos-Pons' artistic practice, which combines diverse media including photography, performance, painting, sculpture, film, and video. Her work is autobiographical, investigating themes of history, memory, gender and religion and how they inform identity. Through deeply poetic and haunting imagery, Campos-Pons evokes stories of the Trans-Atlantic slave trade, indigo, and sugar plantations, Catholic and Santeria religious practices, and revolutionary uprisings.
In the late 1980s, Campos-Pons taught at the prestigious Instituto Superior de Arte in Havana and gained an international reputation as an exponent of the New Cuban Art movement that arose in opposition to Communist repression on the island. In 1991, she emigrated to Boston, where she continues to live and work. She has had solo exhibitions at the Museum of Modern Art in New York, the Indianapolis Museum of Art, the Peabody Essex Museum, and the National Gallery of Canada, among other distinguished institutions. She has presented over 30 solo performances commissioned by institutions including the Guggenheim and The Smithsonian's National Portrait Gallery. She has participated in the Venice Biennale, the Dakar Biennale, Johannesburg Biennial, Documenta 14, the Guangzhou Triennial and is included in Pacific Standard Time: LA/LA and Prospect.4 Triennial. In October 2017 she will receive the endowed Cornelius Vanderbilt Chair at Vanderbilt University in Nashville, Tennessee.
Campos-Pons' works are in over 30 museum collections including the Smithsonian Institution, The Whitney, the Art Institute of Chicago, the National Gallery of Canada, the Victoria and Albert Museum, the Museum of Modern Art, New York, The Museum of Fine Arts, Boston, the Perez Art Museum, Miami and the Fogg Art Museum."
– Gallery Wendi Norris
* Brooklyn Museum
* Gallery Wendi Norris
* Spotlight on Matanzas: María Magdalena Campos-Pons and "Ríos Intermitentes." Cuban Art News, May 1, 2019.
* Campos-Pons strengthens Vanderbilt-Cuban cultural ties through art exhibition, Ann Marie Deer Owens, Vanderbilt News, April 22, 2019.
* Art, Democracy and Justice Series with Campos-Pons to Launch, Ann Marie Deer Owens, Vanderbilt News, November 10, 2018.
* Gibbes Museum's Society 1858 announces finalists for southern art award, Katie Lyons, Charleston City Paper, June 2018.
* Notes on Sugar/Like the Lonely Traveler, a two-part exhibition curated by Neon Queen Collective, January 25 – May 5, 2018 and September 21 – December 8, 2018.
* "Through the Fire: International Acclaimed Artist Puts Down Roots in Nashville", Sara Lee Burd, Nashville Arts Magazine, February 2018.
* "María Magdalena Campos-Pons at Gallery Wendi Norris, San Francisco," The Editors of ARTNews, Art News, January 12, 2018.
* "Must-See Art Guide: San Francisco," Tatiana Berg, Art Net News, January 11, 2018.
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Department News! Faculty appointed to positions with University Arts Council
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text:
Be prepared - confused...
By ParkMan, May 22, 2018 in Forum Support & Announcements
ParkMan 801
Moderators,
Got to admit. I'm not sure why you locked and then hid the thread.
Was it because of the "adult nature" of the content? Was it because of the debate over values?
The comment and you'll lock future threads just has me wondering what to stay away from.
Not complaining or questioning, just looking for clarity.
David CO 979
I was under the impression that it was locked because it contained criticism of the WOSM.
Edited May 22, 2018 by David CO
Ohh - that never even crossed my mind.
FormerProfessional 11
I thought the United States Congress established The Boy Scouts of America in 1910. How does the WOSM have authority of a "scout license" and power to revoke it from the Boy Scouts of America?
Peregrinator 85
24 minutes ago, FormerProfessional said:
WOSM doesn't have a "scout license" but they could, in theory at least, revoke the BSA's membership in that association.
From the other thread that is locked
BSA is a national member society of WOSM. WOSM is the ultimate grantor of the license for Scouting.
The he thread will stay locked until we've had a chance to talk. Future threads will be deleted out of hand.
JoeBob 519
FormerProfessional:
Welcome to the forum?
(quoting someone else) "BSA is a national member society of WOSM. WOSM is the ultimate grantor of the license for Scouting."
That's an odd statement (about WOSM being the grantor of a license for scouting). I'm sure WOSM would like things to be that way, but as Baden-Powell put it many years ago, "Scouting is a movement, not an organization." There are many scouting and guiding associations that have nothing to do with WOSM or WAGGGS.
Saltface 164
Does his own stunts.
49 minutes ago, Peregrinator said:
That doesn't sound so bad. Between the proclivity towards promiscuity, antipathy towards knives, and uninspiring coed programs of other countries...
I did not make the
BSA is a national member society of WOSM. WOSM is the ultimate grantor of the license for Scouting.
Just curious when this happened since BSA was founded in 1910 and WOSA in 1922?
My understanding is that the WOSM is like the United Nations. National Scouting organizations can be members and work collectively together, but they don't grant the license for a Scouting association.
qwazse 3010
Just one more beggar ...
All I know is there was a link to an article from quite a few years ago in the original thread, an I would like to reference it on my cousin Robbie's wall. But now I have to go fish.
http://www.wnd.com/2018/05/condoms-required-at-scouts-24th-world-jamboree/
It's been a long conversation with the moderators.
ianwilkins 338
8 hours ago, Saltface said:
Ha ha, very good.
Ian (UK degenerate)
P.s. moderators, seriously, if it was anything I said, or stirred someone else to say from my comments, my heartfelt apologies.
Edited May 23, 2018 by ianwilkins
added a PS
RememberSchiff 1962
old fogey nostalgic for the good old days
7 hours ago, JoeBob said:
A response to above link. Note in this response, Vice Chairperson of the World Scout Committee Andy Chapman is American. What remains unanswered is how
the hosts Scouts Canada, the Scout Association of Mexico, and the Boy Scouts of America will implement the requirements. Will it be by hosts, nations, units,...? How will parents be notified (this was news to me!)?
https://www.snopes.com/fact-check/boy-scouts-condoms-jamboree/
And a link to WOSM Guidelines for Hosting World Jamborees
Original topic is still under moderator review. Please keep discussion professional and respectful. Thank you.
Edited May 23, 2018 by RememberSchiff
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