text:
\section{Introduction}
\subsection{The Reducedness Conjecture}

 Let $ G $ be a complex semisimple group and consider the associated  thick affine Grassmannian $\mathbf{Gr}=G((t^{-1}))/G[t]$.  There is a Poisson structure on $G((t^{-1}))$ arising from the Manin triple $(\mathfrak{g}((t^{-1})),\mathfrak{g}[t],t^{-1}\mathfrak{g}[[t^{-1}]])$.  Hence $G[t]$ is a Poisson subgroup of $G((t^{-1}))$, and this coinduces a Poisson structure on $\mathbf{Gr}$.

 For a dominant coweight $\lambda$ of $G$ consider the $G[t]$-orbit given by $\mathbf{Gr}^\lambda=G[t]t^\lambda$.  Note that the thin affine Grassmannian $G[t,t^{-1}]/G[t]$ is isomorphic to  the union $\bigcup \mathbf{Gr}^\lambda$ over all dominant coweights.

Given a pair of dominant coweights such that $\mu \leq \lambda$ we have that $\mathbf{Gr}^\mu \subset \overline{\mathbf{Gr}^\lambda}$.  Our main objects of interest are transversal slices to $\mathbf{Gr}^\mu$ inside $\ol{\mathbf{Gr}^\lambda}$, which we denote $\mathbf{Gr}_\mu^\lambda$.  These slices arise in several contexts:
\begin{enumerate}
\item By the Geometric Satake Correspondence, the intersection homology of   $\mathbf{Gr}_\mu^\lambda$ can be identified with the associated graded of $V(\lambda)_\mu$, the $\mu$ weight space of the irreducible $^LG$ module of highest weight $\lambda$ \cite{L},\cite{G},\cite{MV}.
\item $\mathbf{Gr}_\mu^\lambda$ is a conical Poisson subvariety of the affine Grassmannian with respect to loop rotation \cite{KWWY}.

\item By recent work of Braverman, Finkelberg, and Nakajima \cite{BFN2}, these slices are the Coulomb branches of certain $3d$ $\mathcal{N}=4$ quiver qauge theories.
\item Closely related to the previous item, the slices in the affine Grassmannian are conjectured to be symplectic dual to corresponding Nakajima quiver varieties \cite{BLPW}.
\end{enumerate}

The transversal slice is constructed as an intersection $\mathbf{Gr}_\mu^\lambda=\mathbf{Gr}_\mu \cap \overline{\mathbf{Gr}^\lambda}$, where $\mathbf{Gr}_\mu$ is an orbit of the congruence subgroup of $G[[t^{-1}]]$ acting on $\mathbf{Gr}$.  The Reducedness Conjecture describes the ideal of $\mathbf{Gr}_\mu^\lambda$ inside $\mathbf{Gr}_\mu$.  More precisely, in \cite{KWWY} a Poisson ideal $J_\mu^\lambda \subset \mathcal{O}(\mathbf{Gr}_\mu)$ is explicitly defined via generators and it is shown that the vanishing of this ideal is  $\mathbf{Gr}_\mu^\lambda$.  Let $\mathcal{X}_\mu^\lambda$ be the (possibly non-reduced) scheme whose ideal is $J_\mu^\lambda$.

\begin{Conjecture}[Conjecture 2.20, \cite{KWWY}]
\label{MainConj}
The ideal of $\mathbf{Gr}_\mu^\lambda$ is $J_\mu^\lambda$.  Equivalently, $\mathcal{X}_\mu^\lambda$ is reduced.
\end{Conjecture}

Our aim is to prove this conjecture in type A:
\begin{Theorem}
\label{MainThm}
Let $G=SL_n$.  Then Conjecture \ref{MainConj} holds.
\end{Theorem}

For $G=SL_2,SL_3$, along with some special cases for general $n$, this conjecture is  proved in \cite{KMW}.  We will show how the main result of \cite{KMW} along with two additional ingredients proves Theorem \ref{MainThm}.  These ingredients are a) Weyman's description of the ideals defining nilpotent orbits in $\mathfrak{sl}_n$, and b) an isomorphism  motivated by \cite{BFN2} between certain $\mathcal{X}_\mu^\lambda$ for different $n$.

\subsection{Consequences of the Reducedness Conjecture}

Before describing the proof of Theorem \ref{MainThm}, we discuss some implications of Conjecture \ref{MainConj}.

\subsubsection{Truncated shifted Yangians}
\label{intro: truncated shifted Yangians}
The main aim of \cite{KWWY} is to introduce a quantum group, the so-called truncated shifted Yangian, in order to quantize the slice $\mathbf{Gr}_\mu^\lambda$.

The truncated shifted Yangian is defined in several steps.  Firstly, one constructs the shifted Yangian $Y_\mu$, a $\mathbb{C}[\hbar]$-algebra which quantizes $\mathbf{Gr}_\mu$ in the sense that
$$
Y_\mu/\hbar Y_\mu \cong \mathbb{C}[\mathbf{Gr}_\mu].
$$
Next, one defines the GKLO representation $\Psi_\mu^\lambda: Y_\mu \to D_\mu^\lambda$ which depends also on the parameter $\lambda$.  The target space $D_\mu^\lambda$ is an algebra of difference operators.

Finally, one defines the truncated shifted Yangian to be image $\im(\Psi_\mu^\lambda)$ of the shifted Yangian under the GKLO representation.

In \cite{KWWY} it is shown that the truncated shifted Yangian quantizes a scheme supported on $\mathbf{Gr}_\mu^\lambda$.  Furthermore it is shown that Conjecture \ref{MainConj} implies that this scheme is actually $\mathbf{Gr}_\mu^\lambda$.  Recently, in \cite[Appendix B]{BFN1}, the latter statement was proven for all simply-laced $G$ by identifying the truncated shifted Yangian with the quantized Coulomb branch $\mathcal{A}_\hbar$ of a $3d$ $\mathcal{N}=4$ quiver gauge theory.

By \cite[Theorem 4.10]{KWWY}, Conjecture \ref{MainConj} also implies that one can give a set of explicit generators for the kernel of $\Psi_\mu^\lambda$.  We thereby obtain a presentation for the truncated shifted Yangian, or equivalently for the quantized Coulomb branch. Denoting this explicitly presented algebra by $Y_\mu^\lambda$, in general there is a diagram
\begin{equation} \label{yangians diagram}
\begin{tikzcd}
Y_\mu \drar[two heads] \dar[two heads] &  \\ Y_\mu^\lambda \rar[two heads] & \im(\Psi_\mu^\lambda)  \cong \mathcal{A}_\hbar
\end{tikzcd}
\end{equation}
Theorem \ref{MainThm} implies that, in type A, the bottom row consists of isomorphisms.

We remark that in \cite{KTWWY} the highest weight theory of the truncated shifted Yangian is studied via the algebra $Y_\mu^\lambda$.  Therefore Theorem \ref{MainThm} implies that the results in \cite{KTWWY} actually also hold for the algebra $\im(\Psi_\mu^\lambda)$.

\subsubsection{Modular description of spherical Schubert varieties}

Another consequence of Conjecture \ref{MainConj} is a modular description of the spherical Schubert varieties.  We give a brief indication of this connection; for a thorough discussion  see Sections 1 and 2 of \cite{KMW} (cf. also \cite[Remark 2.1.7]{Z}).

The modular description of the thin affine Grassmannian (due to Beauville-Laszlo) is as follows: Let $X$ be a smooth curve, and $x\in X$ a closed point.  Then an $S$--point of $\mathbf{Gr}$ consists of a pair $(\mathcal{P},\phi)$, where $\mathcal{P}$ is a principal $G$--bundle on $S\times X$, and $\phi:\mathcal{P}_0|_{S\times (X\setminus x)} \to \mathcal{P}|_{S\times (X\setminus x)}$ is an isomorphism where $\mathcal{P}_0$ is the trivial bundle on $S\times X$.

Finkelberg and Mirkovi\'c propose a modular description of the spherical Schubert varieties $\ol{\mathbf{Gr}^\lambda}$ \cite[Proposition 10.2]{FM}.  They consider pairs $(\mathcal{P},\phi)$ as above, where the pole of $\phi$ at $x$ is controlled by $\lambda$ (see \cite[Section 2]{KMW} for a precise description).  While  this description is correct set-theoretically, it is not clear that the moduli space is reduced.

Conjecture \ref{MainConj} together with \cite[Proposition 5.1]{KMW} implies this modular description of $ \ol{\mathbf{Gr}^\lambda} $ is correct for $ G = SL_n $.

\subsection*{Acknowledgements}
We thank Alexander Braverman, Michael Finkelberg, and Anthony Henderson for helpful conversations.
J.K. is supported by NSERC and a Sloan fellowship. D.M. is supported by a PIMS Postdoctoral Fellowship. A.W. was supported in part by a Government of Ontario graduate scholarship.  O.Y. is supported by the Australian Research Council.
This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.

\section{Definitions and Overview}
\label{SecNot}
\subsection{Definitions} We recall some notations and results from \cite{KMW,KWWY}.  We work throughout over $\mathbb{C}$.

Let $G$ be a semisimple group, $\mathfrak{g}$ its Lie algbera, and let $I$ denote the nodes of the Dynkin diagram of $\mathfrak{g}$.
We write $\lambda,\mu,$ etc. for coweights of $G$ and $\lambda^\vee,\mu^\vee,$ etc. for weights of $G$.  Let $\varpi_i,\varpi_i^\vee$ (respectively $\alpha_i,\alpha_i^\vee$) be the fundamental coweights and fundamental weights of $\mathfrak{g}$ (resp. simple coroots and simple roots of $\mathfrak{g}$).  Let $w_0$ be the longest element of the Weyl group and set $\lambda^*=-w_0\lambda, \varpi_{i^*}=\varpi_i^*, \alpha_{i^*}=\alpha_i^*$. Let $(\cdot,\cdot)$ be the Killing form $\mathfrak{g}$, and for coweights $\mu,\lambda$ write $\mu \leq \lambda$ if $\lambda-\mu \in \bigoplus_i \mathbb{N}\alpha_i$.
\begin{Remark}
In Sections \ref{sec:iso} and \ref{sec:bfn} we will work with two semisimple groups simultaneously.  When writing $w_0, \lambda^\ast$, etc., the relevant group will be clear from context.
\end{Remark}

Let $G((t^{-1}))$ (respectively $G[t],G[[t^{-1}]]$) be the $\mathbb{C}((t^{-1}))$ points of $G$ (respectively the $\mathbb{C}[t],\mathbb{C}[[t^{-1}]]$ points of $G$).  Define also $G_1[[t^{-1}]] \subset G[[t^{-1}]]$ as the kernel of the evaluation $G[[t^{-1}]] \to G, t \mapsto \infty$.  A coweight $\lambda$ of $G$ can be considered as a $\mathbb{C}((t^{-1}))$ point of $G$, and we let $t^\lambda$ denote its image in $\mathbf{Gr}$. There is a corresponding orbit $\mathbf{Gr}^\lambda = G[t] t^\lambda$, and spherical Schubert variety $\ol{\mathbf{Gr}^\lambda}$. Recall that $\mathbf{Gr}^\mu \subset \ol{\mathbf{Gr}^\lambda}$ if and only if $\mu \leq \lambda$.

Consider a pair of dominant coweights $\mu,\lambda$ such that $\mu \leq \lambda$.  Define $\mathbf{Gr}_\mu^\lambda=\mathbf{Gr}_\mu \cap \ol{\mathbf{Gr}^\lambda}$, where
$$
\mathbf{Gr}_\mu=G_1[[t^{-1}]]t^{w_0\mu}.
$$
$\mathbf{Gr}_\mu^\lambda$ is a transversal slice to $\mathbf{Gr}_\mu$ in $\ol{\mathbf{Gr}^\lambda}$ at the point $t^{w_0\mu}$.

 Let $V$ be a representation of $G$ and $v\in V,\beta\in V^*$.  We'll introduce functions $\Delta_{\beta,v}^{(s)}$ on  $G_1[[t^{-1}]]$.   Let $\Delta_{\beta,v} \in \mathcal{O}(G)$ be the matrix coefficient: $g\mapsto \beta(gv)$.  Then $G_1[[t^{-1}]]$ acts on $V[[t^{-1}]]$ and for $g\in G_1[[t^{-1}]]$ we have
$$
\Delta_{\beta,v}(g)=\sum_{s\geq0}\Delta_{\beta,v}^{(s)}(g)t^{-s}.
$$

For instance, consider $G=SL_n$ and the representation $\bigwedge^i\mathbb{C}^n$ (this case is sufficient for our purposes).  If we take $v,\beta$ to be standard basis and dual basis elements, then $\Delta_{\beta,v}(g)$ is a $i\times i$ minor of $g$, and $\Delta_{\beta,v}^{(s)}(g)$ extracts its $t^{-s}$ coefficient.  It will be convenient for us to sometimes use the notation $\Delta_{C,D}^{(s)}$ to denote this function, where the matrix minor is taken with respect to rows specified by $C$ and columns specified by $D$ (here $C,D \subset \{1,...,n\}$ both have cardinality $i$).

We refer to \cite[Section 2]{KWWY} for results concerning the Poisson structure on $\mathbf{Gr}$.  We recall that $G_1[[t^{-1}]]$ is a Poisson algebraic group, $\mathbf{Gr}_\mu$ is a Poisson homogenous space, and $\mathbf{Gr}_\mu^\lambda$ is a Poisson subvariety of $\mathbf{Gr}$.
The Poisson bracket on $\mathcal{O}(G_1[[t^{-1}]])$ is specified by the following equations:
$$
\{\Delta_{\beta_1,v_1}^{(r+1)},\Delta_{\beta_2,v_2}^{(s)} \}-\{\Delta_{\beta_1,v_1}^{(r)},\Delta_{\beta_2,v_2}^{(s+1)} \}=\sum_a \Big( \Delta_{J_a\beta_1,v_1}^{(r)}\Delta_{J^a\beta_2,v_2}^{(s)}- \Delta_{\beta_1,J_av_1}^{(r)}\Delta_{\beta_2,J^av_2}^{(s)} \Big)
$$
for all $r,s\geq0$, where $\{J_a\},\{J^a\}$ are dual bases of $\mathfrak{g}$ with respect to $(\cdot,\cdot)$ (see \cite[Section 2.6]{KWWY}).

Let $V(\varpi_i^\vee)$ be the irreducible representation of $G$ of highest weight $\varpi_i^\vee$.  Fix a highest weight vector $v_i\in V(\varpi_i^\vee)$ and a lowest weight dual vector $v_i^*\in V(\varpi_i^\vee)^*$.  Write $\lambda-\mu=\sum_im_i\alpha_{i^*}$ (recall that $\mu\leq\lambda$, so all $m_i \geq 0$).
$\mathcal{O}(\mathbf{Gr}_\mu)$ can be canonically identified with the functions on $G_1[[t^{-1}]]$ invariant under the stabilizer in $G_1[[t^{-1}]]$ of $t^{w_0\mu}$.  This is a Poisson subalgebra of $\mathcal{O}(G_1[[t^{-1}]])$.  It can be shown that the functions $\Delta_{v_i^*,v_i}^{(s)}$ for $i\in I$ and $s>m_i$ are invariant under this stabilizer (\cite[Lemma 2.19]{KWWY}), and hence can be considered as functions on $\mathbf{Gr}_\mu$.

Now we can define the central objects appearing in Conjecture \ref{MainConj}:
\begin{itemize}
\item let $J_\mu^\lambda \subset \mathcal{O}(\mathbf{Gr}_\mu)$ be the Poisson ideal generated by $\Delta_{v_i^*,v_i}^{(s)}$ for $i\in I$ and $s>m_i$, and
\item let $\mathcal{X}_\mu^\lambda={^G}\mathcal{X}_\mu^\lambda$ be the corresponding subscheme of $\mathbf{Gr}$.
\end{itemize}
Note that in \cite{KMW} $\mathcal{X}_\mu^\lambda$ is defined as the intersection $\ol{\mathcal{X}^\lambda}\cap \mathbf{Gr}_\mu$, where $\ol{\mathcal{X}^\lambda}$ is given by the modular description of the orbit closures due to Finkelberg-Mirkovi\'c.  \cite[Theorem 8.4]{KMW} proves that the ideal of $\ol{\mathcal{X}^\lambda}\cap \mathbf{Gr}_\mu$ is $J_\mu^\lambda$, and so here we take this as the definition.

\subsection{Overview of Proof}
\label{section: proof overview}
For the remainder of the paper, unless otherwise noted we let $G=SL_n$.
The proof of Theorem  \ref{MainThm} relies on the following three results.  \begin{Proposition}
\label{PropWeyman}
Suppose that $\lambda \leq n\varpi_1$.  Then $\mathcal{X}^\lambda_0$ is reduced.
\end{Proposition}
Next  let $\lambda$ be an arbitrary coweight of $G$, such that $\lambda \geq \mu = 0$, and write it as a sum of simple coroots and also of fundamental coweights: $$\lambda=\sum m_j\alpha_{j^*}=\sum\lambda_j\varpi_{j^*}.$$  Set $k:=m_1 = \langle \lambda, \varpi_{n-1}^\vee \rangle $, and let $\tau$ be the standard inclusion of Dynkin diagrams $A_{n-1} \to A_{kn-1}$.  Denote by $\tau$ also the map on coweights, which extends $\varpi_i \mapsto \varpi_{\tau(i)}$ by linearity; for our chosen $\tau$ this is simply $\varpi_i \mapsto \varpi_i$.
\begin{Proposition}
\label{PropCalc}
There is an isomorphism
\begin{equation}
\label{eq:bfniso}
^{SL_n}\mathcal{X}^\lambda_0 \cong {^{SL_{kn}}\mathcal{X}^{\tau(\lambda)}_{k\varpi_n}}.
\end{equation}
\end{Proposition}
\begin{Proposition}[\mbox{\cite[Theorem 1.6]{KMW}}]
\label{PropKMW}
Let $\lambda$ be a dominant coweight.  If $\mathcal{X}^\lambda_0$ is reduced then $\mathcal{X}_\mu^\lambda$ is reduced for all $\mu \leq \lambda$.
\end{Proposition}

Given these results, and  a simple computation showing that $\tau(\lambda) \leq kn\varpi_1$ (Lemma \ref{lem:simpcomp}(b)), the proof of Theorem \ref{MainThm} is immediate.

It remains then only to prove the first two propositions.  Proposition \ref{PropWeyman} will be proved in Section \ref{SectionWeyman}, and Proposition \ref{PropCalc} in Section \ref{sec:iso}.  In Section \ref{sec:bfn} we   discuss how (\ref{eq:bfniso}) is motivated by as isomorphism between affine Grassmannian slices that follows from  \cite{BFN2}.  In Section \ref{sec:truncatediso} we prove that this isomorphism can be quantized using truncated shifted Yangians.

\section{Weyman's equations}
\label{SectionWeyman}

\subsection{The nilpotent cone}

Since $\mathbf{Gr}_0\cong G_1[[t^{-1}]]$ we can view $\mathcal{X}^{n\varpi_1}_0$ as a subscheme of $G_1[[t^{-1}]]$.  Let $\lambda\geq 0$ be a dominant coweight of $G$ and as above set $\lambda=\sum m_i\alpha_{i^*}$.  Recall the following (which holds for arbitrary semisimple groups):
\begin{Proposition}[\mbox{\cite[Proposition 2.15]{KWWY}}]
\label{PropKWWY}
$J_0^\lambda$ is generated as an ordinary ideal  by $\Delta_{\beta,v}^{(s)}$ for $s>m_i$, where $i\in I$ and $\beta,v$ range over bases for $V(\varpi_i)^*$ and $V(\varpi_i)$.
\end{Proposition}
In the case when $\lambda=n\varpi_1$, then $m_i=i$, and we have in particular that $J_0^{n\varpi_1}$ contains $\Delta_{\beta,v}^{(s)}$, where $s>1$ and $\beta,v$ range over bases for $V(\varpi_1^\vee)^*$ and $V(\varpi_1^\vee)$.  It's easy to see that these elements are sufficient to generate the whole ideal.
Therefore we have
$$
\mathcal{X}^{n\varpi_1}_0 \subset I_n +t^{-1}\rm{Mat}_{n},
$$
where $\rm{Mat}_{n}$ denotes the affine space of $n\times n$ matrices.  We use this to define an embedding $\mathcal{X}^{n\varpi_1}_0 \hookrightarrow \rm{Mat}_{n}$, by $I_n+t^{-1}X \mapsto X$.  The condition $\det(I_n+t^{-1}X)=1$ implies that the image of this map is precisely $\mathcal{N}$, the nilpotent cone of $\mathfrak{g}$.  Therefore we have an isomorphism of schemes $\mathcal{X}^{n\varpi_1}_0 \cong \mathcal{N}$.  In particular, the reducedness conjecture is true and we have $ \mathcal{X}^{n\varpi_1}_0 = \mathbf{Gr}^{n\varpi_1}_0 $.

\subsection{Nilpotent orbit closure}
Let $\lambda$ be a dominant coweight such that $0 < \lambda \leq n \varpi_1$. Recall that we have the expansions $\lambda = \sum \lambda_j \varpi_{j^*}= \sum m_i \alpha_{i^*}
$. We form the following partition, written in exponential notation:
\begin{align}
  \label{eq:4}
  \mathbf{v} = 1^{\lambda_{n-1}} 2^{\lambda_{n-2}} \cdots (n-1)^{\lambda_{1}}
\end{align}
The condition $0 < \lambda \leq n \varpi_1$ implies that $\sum j \lambda_{n-j} = n$, i.e., that $\mathbf{v}$ is a partition of $n$.

Let $\mathbf{u} = \mathbf{v}^T$ be the conjugate partition. Because $\lambda$ lies in the coroot lattice, we can also expand:

\begin{Lemma}
Let us write $\mathbf{u} = (\mathbf{u}_1 \geq \mathbf{u}_2 \geq \cdots \geq \mathbf{u}_{n-1})$. Then:
\begin{align}
  \label{eq:6}
  m_{n-i} = \mathbf{u}_1 + \cdots + \mathbf{u}_i - i
\end{align}
\end{Lemma}

Using the partition $ \mathbf{u} $, let us consider the nilpotent orbit closure $\overline{\mathbb{O}_\mathbf{u}} \subseteq \mathcal{N}$, where $ \mathbb{O}_\mathbf{u} $ denotes the orbit of nilpotent matrices whose Jordan form is given by $ \mathbf{u}$.
\begin{Proposition} \label{prop:reduced-slice-and-nilpotent-orbit-closure}
The isomorphism $ \mathbf{Gr}^{n\varpi_1}_0 = \mathcal{X}^{n\varpi_1}_0 \cong \mathcal{N} $ identifies $ \mathbf{Gr}^\lambda_0 $ with $\overline{\mathbb{O}_\mathbf{u}} $.
\end{Proposition}

Using results of Weyman \cite{W}, we will now see that under the identification $ \mathcal{X}^{n\varpi_1}_0 \cong \mathcal{N}$, the subscheme $\mathcal{X}^\lambda_0$ is equal to nilpotent orbit closure $ \overline{\mathbb{O}_\mathbf{u}} $ with its {\it reduced} induced scheme structure.

\subsection{Appearance of Weyman's equations}
Under the isomorphism $ \mathcal{X}^{n\varpi_1}_0 \cong \mathcal{N} $, we can identify $\mathcal{X}^\lambda_0$ as a subscheme of the nilpotent cone.  More precisely, $\mathcal{X}_0^\lambda$ is identified with the subscheme of $\mathcal{N}$ given by the functions
$$
\{f_{C,D}^{(s)}\mid 1\leq k<n, |C|=|D|=k, \text{ and } s>m_k\},
$$
where $f_{C,D}^{(s)}(X)=\Delta_{C,D}^{(s)}(I_n+t^{-1}X)$.  As functions on $ \mathcal{N} $, $ f_{C,D}^{(s)} $ is a sum of $s \times s$ minors.

Note that the ideal of $ \mathcal{N} $ as a subscheme of $ \rm{Mat}_{n} $ is generated by (for each $ p $) the sum of all principal $ p \times p $ minors.

Let $W_{k,p} = \text{span} \{ f_{C,D}^{(p)} \mid |C| = |D| = k \}$, and let $U_{0,p}$ be the one-dimensional space spanned by the sum of all principal $p\times p$ minors.  We  then have that the ideal of $\mathcal{X}^\lambda_0$ in $\mathcal{O}(\rm{Mat}_{n})$ is generated by
\begin{equation}
\label{eq:Yideal}
\bigoplus_{k,p>m_k} W_{k,p} \oplus \bigoplus_{p=1}^n U_{0,p}.
\end{equation}

Let $M_p \subset \mathcal{O}(\rm{Mat}_{n})$ be span of all $p\times p$ minors, and set $E=\mathbb{C}^n$.  We let $GL_n$ act on $\mathcal{O}(\rm{Mat}_{n})$ by $(g\cdot f)(A)=f(g^{-1}Ag)$.  Under this action $M_p \cong \Lambda^p E \otimes \Lambda^p E^*$.  By the Pieri rule, we have:
\begin{align} \label{eq:10}
  M_p = \bigoplus_{0\leq k \leq \min{(p,n-p)}} U_{k,p}
\end{align}
where $U_{k,p}\cong S_{\alpha_k} E$, the Schur module of highest weight $\alpha_k=(1^k,0^{n-2k},(-1)^{k})$.
Note that in the case where $k=0$ this notation agrees with the  definition of $U_{0,p}$ in (\ref{eq:Yideal}).

It is clear that $W_{k,p}$ is a subrepresentation of $\mathcal{O}(\rm{Mat}_{n})$. Moreover, each $f_{C,D}^{(p)}$ is a weight vector for the torus of diagonal matrices in $GL_n$ with weight:
\begin{align}
  \label{eq:13}
  \text{wt} f_{C,D}^{(p)} = -\sum_{i \in C} \varepsilon_i + \sum_{j \in D} \varepsilon_j
\end{align}
Suppose now that $0\leq k\leq \min{(p,n-p)}$, and set $C_0=\{k+1,...,n\}$ and $D_0=\{1,...,n-k\}$.  Then we have
$
   \text{wt} f_{C_0,D_0}^{(p)} = \alpha_k.
$
 Furthermore $f_{C_0,D_0}^{(p)}$ is fixed by the unipotent upper triangular matrices in $GL_n$. Therefore, it generates a copy of $S_{\alpha_k}E$. Since $W_{n-k,p} \subset M_p$, by \eqref{eq:10} we see that
\begin{equation}
\label{eq:inclusion}
U_{k,p} \subset W_{n-k,p}.
\end{equation}

 We recall Weyman's Theorem on nilpotent orbit closures, which by the above lemma can be stated as follows:

\begin{Theorem}[\mbox{\cite[Theorem 4.6]{W}}]
The ideal of  $\overline{\mathbb{O}_\mathbf{u}}$ is generated by the following:
\begin{align}
  \label{eq:11}
   \bigoplus_{1 \leq i \leq n; p > m_{n-i}}  U_{i,p} \oplus \bigoplus_{1 \leq p \leq n} U_{0,p}
\end{align}
\end{Theorem}

By Equations (\ref{eq:Yideal}) and (\ref{eq:inclusion}), we see that Weyman's Theorem implies that $ I(\overline{\mathbb{O}_\mathbf{u}}) \subset I(\mathcal{X}^{\lambda}_0) $. By \cite[Corollary 2.16]{KWWY} $V(J^{\lambda}_0)=\mathbf{Gr}_0^\lambda $ and $ \mathbf{Gr}_0^\lambda = \overline{\mathbb{O}_\mathbf{u}}$ by Proposition \ref{prop:reduced-slice-and-nilpotent-orbit-closure}.  Therefore these ideals have the same radical, and since $I(\overline{\mathbb{O}_\mathbf{u}})$ is radical we obtain the following isomorphism of subschemes:
\begin{Proposition}{\label{prop:isomorphism-of-slice-to-nilpotent-orbit-closure}}
  Let $\lambda$ be a dominant coweight for $SL_n$ with $\lambda \leq n \varpi_1$, and let $\mathbf{u}$ be the corresponding partition of $n$. Then we have
\begin{align}
  \label{eq:01}
  \mathcal{X}^{\lambda}_0 = \overline{\mathbb{O}_\mathbf{u}}
\end{align}
as subschemes of $n \times n$ matrices.
\end{Proposition}
 This shows that $\mathcal{X}^{\lambda}_0$ is reduced for any $\lambda<n\varpi_1$, proving Proposition \ref{PropWeyman}.

\section{An isomorphism of slices}
\label{sec:iso}
\subsection{Some varieties of interest}
For the moment consider  the general setting where $G$ is a semisimple group, and $\lambda, \mu$ are dominant coweights for $G$ with $\lambda \geq \mu$.  In this case we denote $\lambda_i  = \langle \lambda^\ast, \alpha_i^\vee\rangle$, $\mu_i = \langle \mu^\ast, \alpha_i^\vee\rangle$ and $m_i = \langle \lambda^\ast-\mu^\ast, \varpi_i^\vee\rangle$.

As in \cite[Appendix B]{BFN2}, consider the subgroup
\begin{equation}
\label{eq: Gmu}
G_\mu := \left\{ g\in G_1[[t^{-1}]] : t^{-w_0\mu} g t^{w_0 \mu} \in G_1[[t^{-1}]] \right\},
\end{equation}
which has the property that there is an isomorphism $G_\mu \stackrel{\sim}{\longrightarrow} \mathbf{Gr}_\mu$ defined by $g\mapsto g t^{w_0 \mu}$.  Hence the translated subset \begin{equation}
\label{eq: cosets for Grmu}
\mathcal{W}_\mu := G_\mu \cdot t^{w_0\mu} \subset G((t^{-1}))
\end{equation}
maps bijectively onto $\mathbf{Gr}_\mu$ under the quotient map $G((t^{-1})) \rightarrow \mathbf{Gr}_G$.  It is naturally an affine scheme of infinite type.

Consider a closed subscheme $\mathcal{W}_\mu^\lambda$ of $\mathcal{W}_\mu$, which is defined by imposing the following conditions on $g t^{w_0\mu} \in \mathcal{W}_\mu$:
\begin{quote}
For every dominant weight $\tau^\vee$ of $G$, the valuation of $g t^{w_0 \mu}$ acting on $V(\tau^\vee) (( t^{-1}))$ is greater than or equal to $\langle \lambda,  w_0 \tau^\vee \rangle $.
\end{quote}
The above can be understood in terms of matrix coefficients, like in Section \ref{SecNot}. Note that in \cite{KMW} $\overline{\mathcal{X}^\lambda}$ is defined by these same conditions on valuations, but applied to a coset representative $[ h]\in \mathbf{Gr}$ (i.e. with $h$ in place of $g t^{w_0 \mu}$ above).  Since $\mathcal{W}_\mu$ provides a choice of coset representatives for $\mathbf{Gr}_\mu$, this implies:
\begin{Proposition}
\label{prop: slice via coset reps}
The quotient map $G((t^{-1})) \rightarrow \mathbf{Gr}_G$ induces an isomorphism of schemes $\mathcal{W}_\mu^\lambda \cong \mathcal{X}_\mu^\lambda$.
\end{Proposition}
\begin{proof}
This follows since $\mathcal{W}_\mu^\lambda$ is the fibre product:
\[
\begin{tikzcd}
\mathcal{W}_\mu^\lambda \rar \dar & \overline{\mathcal{X}^\lambda} \dar[hook] \\
\mathcal{W}_\mu \rar[hook] & \mathbf{Gr}_G
\end{tikzcd}
\]
where the bottom arrow comes from the quotient map $G((t^{-1})) \rightarrow \mathbf{Gr}_G$.

In other words, the definition of $\mathcal{W}_\mu^\lambda$ is exactly a translation of the conditions defining the scheme-theoretic intersection $\mathbf{Gr}_\mu \cap \overline{\mathcal{X}^\lambda}$ under the isomorphism $\mathcal{W}_\mu \rightarrow \mathbf{Gr}_\mu$.
\end{proof}

\subsection{Explicit description of certain slices}

We return to the case where $G$ is a special linear group.  Fix a dominant $SL_n$-coweight $\lambda\geq 0$, and write $ \lambda = \sum \lambda_i \varpi_{i^\ast} = \sum m_i \alpha_{i^\ast} $.  Set $k = m_1$.  Then $\lambda \leq k n \varpi_1$, and this is the minimal value of $k$ such that this inequality holds.

As in Section \ref{section: proof overview}, we consider $SL_n \subset SL_{kn}$ corresponding to the inclusion $\tau$ of Dynkin diagrams $\{ 1,\ldots, n-1\} \subset \{1,\ldots, kn-1\}$. Define a map $\tau$ taking $SL_n$ coweights to $SL_{kn}$ coweights, which extends $\varpi_i \mapsto \varpi_i$ by linearity.

Our goal is to explicitly describe an isomorphism:
\begin{equation}
\label{eq: goal iso}
^{SL_n} \mathcal{X}_0^\lambda \ \ \cong \ \ ^{SL_{kn}} \mathcal{X}^{\tau(\lambda)}_{k \varpi_n}
\end{equation}
Using Proposition \ref{prop: slice via coset reps}, we will work exclusively with $\mathcal{W}_\mu^\lambda$.  For simplicity, we will abuse notation and write continue to write $\mathcal{X}_\mu^\lambda$.

\subsubsection{Case of $SL_n$} Since $^{SL_n} \mathcal{W}_0 = (SL_n)_1[[t^{-1}]]$ and $\langle \lambda, w_0 \varpi_i^\vee \rangle = - m_i $, by Proposition \ref{prop: slice via coset reps} we have that
\begin{equation}
\label{eq: n presentation}
^{SL_n} \mathcal{X}_0^\lambda = \left\{ g\in (SL_n)_1[[t^{-1}]] : \begin{matrix} \text{the valuation of any } i\times i \text{ minor}  \\ \text{of } g \text{ is } \geq - m_i\end{matrix} \right\}
\end{equation}
Indeed, it suffices to consider only $\tau^\vee = \varpi_i^\vee$ the fundamental weights for $SL_n$.

\subsubsection{Case of $SL_{kn}$}
To describe $^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)}$, it will be convenient to write elements of $SL_{kn}$ as block matrices
$$ \left( \begin{array}{c|c} a & b \\ \hline c & d \end{array} \right) $$
where $a$ is $(kn-n)\times (kn-n)$, $d$ is $n\times n$, etc. With this convention, we have
$$ t^{w_0 (k \varpi_n)} = \left( \begin{array}{c|c} t^{-1} I & 0 \\ \hline 0 & t^{k-1} I \end{array} \right), $$
and following (\ref{eq: Gmu}) and (\ref{eq: cosets for Grmu}) we find that
\begin{equation}
^{SL_{kn}} \mathcal{W}_{k\varpi_n} = \left\{ \left( \begin{array}{c|c} a & b \\ \hline c & d \end{array} \right) \in SL_{kn}((t^{-1})) : \begin{array}{l} a \in t^{-1} I + t^{-2} M_{(k-1)n\times (k-1)n}[[t^{-1}]], \\ b\in t^{-2} M_{(k-1)n\times n}[[t^{-1}]], \\ c\in t^{-2} M_{n\times (k-1)n}[[t^{-1}]], \\ d\in t^{k-1} I + t^{k-2} M_{n\times n}[[t^{-1}]] \end{array} \right\}
\end{equation}

We can also describe $\tau(\lambda)$ more explicitly:
\begin{Lemma}
\label{lem:simpcomp}
We have:
\begin{itemize}
\item[(a)] $\tau(\lambda) = \sum_{i=1}^{n-1} m_{n-i} \alpha_i + k \varpi_n $

\item[(b)] $\tau(\lambda) \leq k n \varpi_1$
\end{itemize}
\end{Lemma}
\begin{proof}
\mbox{}
\begin{itemize}
\item[(a)]  Observe that $\tau(\alpha_i) = \alpha_i$ for $1\leq i \leq n-2$, while $\tau(\alpha_{n-1}) = \alpha_{n-1}+\varpi_n$.  Since $k = m_1$ is the coefficient of $\alpha_{n-1}$ in $\lambda$, the claim follows.

\item[(b)] The difference $kn \varpi_1 - \lambda$ is a linear combination of $\alpha_1,\ldots, \alpha_{n-2}$ with non-negative coefficients: $\alpha_{n-1}$ does not appear.  Since $\tau(\alpha_i) = \alpha_i$ for $1\leq i \leq n-2$,
$$ \tau( kn \varpi_1 - \lambda) = kn \varpi_1 - \tau(\lambda)$$
is also a linear combination of $\alpha_1,\ldots, \alpha_{n-2}$ with non-negative coefficients.
\end{itemize}
\end{proof}

To apply Proposition \ref{prop: slice via coset reps} to $^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)}$, we must compute the pairings $\langle \tau(\lambda), w_0 \varpi_j^\vee \rangle$ for the fundamental weights $\varpi_j^\vee$ of $SL_{kn}$.  We do so by using Lemma \ref{lem:simpcomp}(a), together with the expansion
$$ k \varpi_n = \sum_{j=1}^{n-1} j (k-1) \alpha_j  + \sum_{j=n}^{kn} (kn-j) \alpha_j $$
Altogether, we find that $^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)} \subset ^{SL_{kn}} \mathcal{W}_{k\varpi_n}$ is the closed subscheme defined by the conditions
\begin{equation}
\label{eq: kn presentation}
^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)} = \left\{ \left( \begin{array}{c|c} a & b \\ \hline c & d \end{array} \right) \in ^{SL_{kn}} \mathcal{W}_{k\varpi_n} :  \begin{array}{l} \ \text{the valuation of any } j \times j \text{ minor is:} \\ \ (a)\ \geq -j ,  \text{ for } 1\leq j \leq kn-n, \\  \begin{array}{l}(b)\  \geq -m_{j-kn+n} - (kn -j)(k-1), \\ \ \ \text{ for } j > kn-n\end{array} \end{array}\right\}
\end{equation}

\subsection{The isomorphism}
We begin with the following observation regarding $^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)}$:
\begin{Lemma}
The matrix coefficients of $a - t^{-1} I, b$ and $c$ are zero in $\mathbb{C}[^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)}]$.
\end{Lemma}
\begin{proof}
The $^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)}$ conditions tell us in particular that the valuation of any $1\times 1$ minor must be $\geq -1$.  Since $a \in t^{-1} I + t^{-2} M_{(k-1)n \times (k-1)n}[[t^{-1}]]$, this $1\times 1$ condition is only satisfied if $a = t^{-1} I$.  Similarly for $b, c$.
\end{proof}

The next result establishes Proposition \ref{PropCalc}:
\begin{Proposition}
\label{main prop}
There is an isomorphism of schemes
$$ ^{SL_n} \mathcal{X}_0^\lambda \ \ \stackrel{\sim}{\longrightarrow} \ \ ^{SL_{kn}} \mathcal{X}^{\tau(\lambda)}_{k \varpi_n} $$
defined by
$$ g \mapsto \left( \begin{array}{c|c} t^{-1}I & 0 \\ \hline 0 & t^{k-1} g \end{array} \right) $$
\end{Proposition}
\begin{proof}
We will show that the $^{SL_n} \mathcal{X}_0^\lambda$ conditions on $g$ imply the $^{SL_{kn}}\mathcal{X}_{k\varpi_n}^{\tau(\lambda)}$ conditions on its image.  The converse is similar.

Consider a $j\times j$ minor of the image of $g$.  To be nonzero, it must correspond to an $i\times i$ minor of $t^{k-1} g$ times a $(j-i)\times(j-i)$ minor of $t^{-1} I$. In other words, if nonzero, its valuation has the form
$$ i(k-1) + \operatorname{val}(\Delta) - (j-i) = \operatorname{val}(\Delta) + ki-j$$
where $\Delta$ is an $i\times i$ minor of $g$. By the $^{SL_n} \mathcal{X}_0^\lambda$ condition on $\operatorname{val}(\Delta)$, this is greater than or equal to
\begin{equation}
\label{eq: minor 1}
- m_i + ki -j
\end{equation}
We consider the cases $j\leq kn-n$ and $j > kn-n$ separately, as in (\ref{eq: kn presentation}).

If $j\leq kn-n$, then we must show that
$$ -m_i + ki - j \geq -j $$
Recalling that $kn \varpi_1 \geq \lambda$, the above follows from the inequality
$$ki = \langle kn \varpi_1, \varpi_{n-i}^\vee \rangle \geq \langle \lambda, \varpi_{n-i}^\vee\rangle = m_i$$

If $j > kn -n$, then we must show that
$$ -m_i  +ki -j \geq - m_{j-kn+n} - (kn-j)(k-1) $$
or equivalently
$$ m_i - m_{j-kn+n} \leq k \big( i - (j-kn+n)\big) $$
Note that $j-i \leq kn-n$, so $j -kn +n \leq i$.  Since $\lambda$ is dominant, it follows that $m_{\ell +1} - m_\ell \leq m_1$ for all $\ell$ (e.g. if we think of $\lambda = (p_1 \geq \ldots \geq p_n)$ with $\sum p_i = 0$, then $p_{n-\ell} = m_\ell - m_{\ell+1}$ and $p_n = -m_1$).  Since $m_i - m_{j-kn+n}$ is a telescoping sum of terms $m_{\ell+1} - m_\ell$, and since $m_1 = k$, the inequality follows.

In either case, we see that the $^{SL_{kn}} \mathcal{X}_{k\varpi_n}^{\tau(\lambda)}$ conditions hold on the image of $g$, as claimed.
\end{proof}

\section{Connection to quiver gauge theories}
\subsection{A general isomorphism between slices}
\label{sec:bfn}
We've now shown that $\mathcal{X}_\mu^\lambda$ is reduced in type A and hence is isomorphic to $\mathbf{Gr}_\mu^\lambda$.  In particular Proposition \ref{PropCalc} now says that we have an isomorphsm
\begin{align}
  \label{eq:19}
  \leftexp{SL_n}{\mathbf{Gr}^\lambda_0} \cong \leftexp{SL_{kn}}{\mathbf{Gr}^{\tau(\lambda)}_{k\varpi_n}}
\end{align}
In this section we show that this isomorphism has a natural interpretation in the context of Coulomb branches of quiver gauge theories, based on their description by Braverman, Finkelberg and Nakajima \cite{BFN2}.

We'll work more generally, for $G$ an arbitrary simply-laced semisimple group of Dynkin type $I$.  Consider $\lambda \geq \mu$ dominant $G$-coweights, and as per usual we denote $\lambda_i = \langle \lambda, \alpha_i^\ast\rangle$, $m_i = \langle \lambda-\mu, \varpi_i^\ast\rangle$.   Consider the vector spaces $ W_i = \mathbb{C}^{\lambda_i}$ and $V_i = \mathbb{C}^{m_i}$ for $i\in I$, and the group $\mathbf{G} := \prod_{i\in I } GL(V_i)$. Fix an orientation $\Omega$ of the Dynkin diagram $I$, and define
\begin{align}
  \label{eq:21}
  \mathbf{N} := \bigoplus_{i\rightarrow j\in \Omega} \hom(V_i, V_j) \oplus \bigoplus_{i\in I} \hom(W_i, V_i),
\end{align}
which is naturally a representation of $\mathbf{G}$.

To this data there is an associated commutative algebra $\mathcal{A} = \mathcal{A}_{\mathbf{G}, \mathbf{N}}$, which is a graded Poisson algebra, arising as a special case of the general construction \cite[Section 3(iv)]{BFN1} (see also \cite[Section 3(iii)]{BFN2}).  Consider
\begin{align}
  \label{eq:22}
   \mathcal{M}_C := \Spec  \mathcal{A}
\end{align}
This is proposed as a mathematical definition of the Coulomb branch associated to a $3d$ $\mathcal{N} =4 $ quiver gauge theory.

\begin{Theorem}[\mbox{\cite[Theorem 3.10]{BFN2}}]
\label{Theorem: BFN theorem}
For any dominant coweights $\lambda \geq \mu$ there is an isomorphism of Poisson varieties
\begin{align}
	\label{eq:23}
\mathbf{Gr}_{\mu}^{\lambda} \cong \mathcal{M}_C
\end{align}

\end{Theorem}

Let $\tau: I \hookrightarrow J$ be an inclusion of (simply-laced) Dynkin diagrams. For $j\in J$, consider the vector spaces
\begin{align}
	\label{eq:24}
\widetilde{W}_j = \left\{ \begin{matrix} W_i, & j = \tau(i), \\ 0, & \text{else} \end{matrix} \right., \qquad \widetilde{V}_j = \left\{ \begin{matrix} V_i, & j = \tau(i), \\ 0, & \text{else} \end{matrix} \right.
\end{align}
as well as the associated coweights $ \widetilde{\lambda}, \widetilde{\mu}$.

Fix an orientation of $J$ extending that of $I$.  Define the group $\widetilde{\mathbf{G}} := \prod_{j\in J} GL(\widetilde{V}_j) $, its representation $\widetilde{\mathbf{N}}$ analogous to $\mathbf{N}$ above, and the corresponding algebra $\widetilde{\mathcal{A}}$.  In other words, we are extending our data from the quiver of $I$ to that of $J$ by ``padding by zero''.

Clearly, there are compatible isomorphisms $\mathbf{G} \cong \widetilde{\mathbf{G}}$ and $\mathbf{N} \cong \widetilde{\mathbf{N}}$. From the definitions, there is therefore an isomorphism of graded algebras
\begin{align}
	\label{eq:25}
\mathcal{A} \cong \widetilde{\mathcal{A}}
\end{align}
In fact this is an isomorphism of graded Poisson algebras, since it lifts to an isomorphism of their deformations $\mathcal{A}_\hbar \cong \widetilde{\mathcal{A}}_\hbar$ (these deformations are defined as in \cite[Section 3(vi)]{BFN1}).  By applying Theorem \ref{Theorem: BFN theorem}, we get:

\begin{Proposition}
\label{Proposition: Isomorphism via BFN}
The isomorphism (\ref{eq:25}) induces a Poisson isomorphism
\begin{align}
	\label{eq:26}
\leftexp{I}{\mathbf{Gr}^\lambda_\mu} \cong \leftexp{J}{\mathbf{Gr}^{\widetilde{\lambda}}}_{\widetilde{\mu}}
\end{align}
\end{Proposition}

\begin{Remark}
\label{rmk: comparing with bfn}
Consider the inclusion $\tau$ of $I= \{1,\ldots, n-1\}$ into $J =\{1,\ldots, kn-1\}$, defined by $\tau(i) = kn-n+i$ (this agrees with our previous conventions, up to twisting by the longest elements of the symmetric groups $S_n$ and $S_{kn}$). For the dimension vectors on $I$ corresponding to $\lambda \geq \mu =0$, we recover the isomorphism (\ref{eq:19}): this follows from the decription of Theorem \ref{Theorem: BFN theorem} in terms of generalized minors given in \cite[Appendix B]{BFN2}.
\end{Remark}

\begin{eg}
As a variation on this construction, consider a slice of the form $\mathbf{Gr}_{\lambda - \alpha_i}^\lambda$.  On the one hand, it was shown in \cite[Example 2.2]{KWWY} that this variety is Poisson isomorphic to the Kleinian singularity $\mathbb{C}^2 / (\mathbb{Z} / n) $, where $n = \langle \lambda, \alpha_{i^\ast}^\vee \rangle $.   On the other hand, the quiver corresponding to $\mathbf{Gr}_{\lambda - \alpha_i}^\lambda$ has $V_{i^\ast} = \mathbb{C}$ and $V_{j^\ast} = 0$ for $j\neq i$, so we have $\mathbf{G} = GL(1)$ and $ \mathbf{N} = \hom(\mathbb{C}^n, \mathbb{C}) $.  In particular, there is an isomorphism to data corresponding to the Dynkin diagram of $SL(2)$, explaining the above ubiquitous appearance of Kleinian singularities.
\end{eg}

\subsection{An isomorphism of truncated shifted Yangians}
\label{sec:truncatediso}
In this section we'll explicitly describe the quantum analog of the isomorphism of Proposition \ref{Proposition: Isomorphism via BFN}, via algebras $^I Y_\mu^\lambda$ defined from Yangians (c.f. Section \ref{intro: truncated shifted Yangians}).

In fact there is a family $^I Y_\mu^\lambda(\mathbf{c})$ of such algebras, where $\mathbf{c}$ are certain parameters.  We refer the reader to \cite[Section 3]{KTWWY} for the precise definition of $^IY_\mu^\lambda(\mathbf{c})$; we'll only recall the parts of the definition we'll need.   The Yangian $^IY$ has a presentation with generators $F_i^{(r)},H_i^{(r)}$, and $E_i^{(r)}$, where $r=1,2...$ and $i\in I$.  The shifted Yangian $^IY_\mu \subset ^IY$ is the subalgebra generated by all $H_i^{(r)}$ and $E_i^{(r)}$, and the $F_i^{(s)}$ such that $s>\langle \mu^*,\alpha_i\rangle$.  Note that $^I Y$ and $^I Y_\mu$ can also be defined as $\mathbb{C}[\hbar]$--algebras, and we are working with the specialization $\hbar =1$.

Now we fix $\mathbf{c}=(\mathbf{c}_i)_{i\in I}$, where each $\mathbf{c}_i$ is a multiset of complex numbers with $|\mathbf{c}_i| = \lambda_i$.  From this data we define series $r_i(u)$ (see \cite[Section 3.2]{KTWWY}) and new Cartan generators $A_i^{(s)}$ using the formula
$$
H_i(u)=r_i(u)\frac{\prod_{j\sim i}A_j(u-\frac{1}{2})}{A_i(u)A_i(u-1)},
$$
where $H_i(u)=1+\sum_{s>0}H_i^{(s)}u^{-s}$ and $A_i(u)=1+\sum_{s>0}A_i^{(s)}u^{-s}$.  Then $^IY_\mu^\lambda(\mathbf{c})$ is the quotient of $Y_\mu$ by the two-sided ideal generated by $A_i^{(s)}$ where $i\in I$ and $s>m_i$.

Let $\tau(\mathbf{c})$ be the collection of multisets indexed by $J$, where $\tau(\mathbf{c})_{\tau(i)}=\mathbf{c}_i$ and otherwise $\tau(\mathbf{c})_{j}$ is empty if $j \notin \tau(I)$.

\begin{Proposition}
\label{Prop: isomorphism of Yangians}
We have an isomorphism of algebras
\[
^IY_\mu^\lambda(\mathbf{c}) \cong {^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))}.
\]
\end{Proposition}

\begin{proof}
Consider the inclusion of algebras $^IY \hookrightarrow ^JY$ defined by $X_i^{(r)} \mapsto X_{\tau(i)}^{(r)}$, for $X=F,H,E$.
For any $\ell \in I$
\begin{align*}
\langle \widetilde{\mu} , \alpha_{\tau(\ell)} \rangle &= \langle \tau(\lambda),\alpha_{\tau(\ell)} \rangle -\sum_{i\in I}m_i  \langle \alpha_{\tau(i)},\alpha_{\tau(\ell)} \rangle \\
&= \lambda_\ell -\sum_{i \in I} m_ia_{i\ell} \\ &= \langle \mu, \alpha_\ell \rangle.
\end{align*}
Therefore for any $F_\ell^{(r)} \in ^IY_\mu$ we have that $F_{\tau(\ell)}^{(r)} \in ^JY_{\widetilde{\mu}}$, and hence $^IY_\mu\hookrightarrow ^JY_{\widetilde{\mu}}$.

Define $\mathcal{I}_{\widetilde{\mu}}^{\tau(\lambda)}$ to be  the two sided ideal in $^JY_{\widetilde{\mu}}$ generated by
\begin{align*}
&A_{\tau(i)}^{(s)} \quad s>m_i, i\in I,\\ \quad &A_j^{(r)} \quad r>0, j\in J\setminus \tau(I).
\end{align*}
By construction we have the s.e.s.
$$
0 \longrightarrow \mathcal{I}_{\widetilde{\mu}}^{\tau(\lambda)} \longrightarrow ^JY_{\widetilde{\mu}} \longrightarrow ^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c})) \longrightarrow 0.
$$
Composing with the inclusion of the previous paragraph we obtain a map
\[
\varphi: ^IY_\mu\longrightarrow ^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c})).
\]

We make some observations about $\varphi$.  First, $\varphi(A_i^{(r)})=A_{\tau(i)}^{(r)}$.  Indeed, in $^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))$ we have $A_j(u)=1$ for $j\in J\setminus \tau(I)$.  Hence for any $i\in I$ the following equality in $^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))$ holds:
\[
H_{\tau(i)}(u)=r_{\tau(i)}(u)\frac{\prod_{\ell \sim i}A_{\tau(\ell)}(u-\frac{1}{2})}{A_{\tau(i)}(u)A_{\tau(i)}(u-1)}
\]
Since $r_{\tau(i)}(u)=r_i(u)$ and $\varphi(H_i(u))=H_{\tau(i)}(u)$, this implies that $\varphi(A_i(u))=A_{\tau(i)}(u)$.

Since $^IY_\mu^\lambda(\mathbf{c})$ is the quotient of $^IY_\mu$ by the ideal generated by
\[
A_{i}^{(s)} \quad s>m_i, i\in I,
\]
$\varphi$ factors through a map $\varphi':{^IY_\mu^\lambda(\mathbf{c})} \longrightarrow ^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))$.

Second, $\varphi$ is surjective.  To see this first note that in $^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))$, we have $X_j^{(r)}=0$ for any $r>0, j\in J\setminus I$, and $X=F,H,E$.  This follows since $A_j(u)=1$ and the relations
\begin{align*}
(u-v)[A_j(u),E_j(v)]&=A_j(u)(E_j(u)-E_j(v)), \\
(u-v)[A_j(u),F_j(v)]&=(F_j(u)-F_j(v))A_j(u), \\
[E_j^{(r)},F_j^{(s)}]&=H_j^{(r+s-1)}.
\end{align*}
Therefore $^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))$ is generated by the $X_{\tau(i)}^{(r)}$, $i\in I$, and these generators are all in the image of $\varphi$.

It follows that $\varphi'$ is also surjective.  We can define a map in the opposite direction by declaring $X_{\tau(i)}^{(r)} \mapsto X_i^{(r)}$ for any $i\in I, r>0$.  This defines an algebra homomorphism.  Indeed any relation only involving generators over $\tau(i)$'s gets mapped to the same relation only involving generators over $i$'s.  Moreover, any relation involving generators over $j$'s, where $j\in J\setminus \tau(I)$, is already zero in $^JY_{\widetilde{\mu}}^{\tau(\lambda)}(\tau(\mathbf{c}))$.  Finally, as we saw above, this identifies $A_{\tau(i)}(u)$ with $A_{i}(u)$.  Since this map is manifestly surjective it defines an inverse to $\varphi'$.
\end{proof}

\begin{Remark}
With notation as Section \ref{sec:bfn}, there is a deformation of the (quantum) Coulomb branch using the ``flavour symmetry'' group $\mathbf{G}_F := \prod_i GL(W_i)$ \cite[Section 3(v)]{BFN2}.  The data $\mathbf{c}$ defines a specialization $H^\ast_{\mathbf{G}_F\times \mathbb{C}^\times}(pt)\rightarrow \mathbb{C}$ (we also specialize $\hbar = 1$), and a corresponding specialized algebra $\mathcal{A}_{\hbar = 1,\mathbf{c}}$. There is a surjection $^I Y_\mu^\lambda(\mathbf{c}) \twoheadrightarrow \mathcal{A}_{\hbar = 1, \mathbf{c}}$ by \cite[Appendix B]{BFN2}. Appropriately twisting data by factors of $w_0$ as in Remark \ref{rmk: comparing with bfn}, one can verify that this surjection intertwines the isomorphism from Proposition \ref{Prop: isomorphism of Yangians} with an isomorphism  $\mathcal{A}_{\hbar=1,\mathbf{c}} \cong \widetilde{\mathcal{A}}_{\hbar=1, \tau(\mathbf{c})}$ analogous to Proposition \ref{Proposition: Isomorphism via BFN}
\end{Remark}

redpajama_set_name: RedPajamaArXiv


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text:
"Test debugger_r, coverage 30%."

from idlelib import debugger_r
import unittest
from test.support import requires
from tkinter import Tk

class Test(unittest.TestCase):

##    @classmethod
##    def setUpClass(cls):
##        requires('gui')
##        cls.root = Tk()
##
##    @classmethod
##    def tearDownClass(cls):
##        cls.root.destroy()
##        del cls.root

    def test_init(self):
        self.assertTrue(True)  # Get coverage of import

# Classes GUIProxy, IdbAdapter, FrameProxy, CodeProxy, DictProxy,
# GUIAdapter, IdbProxy plus 7 module functions.

if __name__ == '__main__':
    unittest.main(verbosity=2)

redpajama_set_name: RedPajamaGithub


Proposals for an extension of the planned Crossrail 2 line through Hackney have been postponed, leaving just one station in the borough at Dalston Junction, the latest consultation documents reveal.
Crossrail 2 aims to connect the north and south of Greater London in the same way as Crossrail, still under construction, plans to connect the east and west. But the proposed eastern branch via Hackney Central has been placed on hold as there is not a strong enough "business case" to support its funding.
If all of Crossrail 2 is granted permission, construction would start in 2020 with the new line becoming fully operational by 2030.
Crossrail 2 could provide more opportunities for local businesses due to the increased transport links.


Via Fox Business, Shell is selling its 8% interest in Australia's Wheatstone-Iago gas field to a state-owned Kuwaiti exploration firm. The Kuwait Foreign Petroleum Exploration Co. will also buy Shell's 6.4% interest in the Wheatstone LNG project. The deal is expected to be worth $1.14 billion.
Shell is in cost-cutting mode under the leadership of new CEO Ben van Beurden, who on Friday issued a profit warning ahead of the company's Q4 earnings announcement. Last week several media outlets reported that Shell is looking to sell $15 billion in global assets.


text:
Myanmar's Suu Kyi Appears Before Court As Supporters Continue Protests
Anti-junta protesters run from teargas fired by police during a demonstration in Yangon on Monday. At least 18 people were killed over the weekend as Myanmar police reportedly used live ammunition against protesters.
Myanmar's detained former leader, Aung San Suu Kyi, appeared for a court hearing Monday, a month after being ousted in a coup. Her supporters again staged protests calling for her release, despite a deadly crackdown by police.
After 18 people were reportedly killed and another 30 wounded by police in anti-junta demonstrations on Sunday in the bloodiest day since the Feb. 1 military takeover, supporters of Suu Kyi and her National League for Democracy, or NLD, party showed up on the streets once again to oppose the country's sudden relapse into authoritarianism.
Although scenes on Monday were less dramatic, police on Sunday reportedly fired rubber bullets, and in some cases live ammunition, along with tear gas to disperse protests in the major cities of Yangon, Mandalay and Dawei, where thousands have gathered in the weeks since the coup to oppose a return to military rule.
Social media have been flooded with images from Sunday's violence, showing bloodied protesters and footage of unarmed demonstrators retreating in the face of overwhelming police force. In one posted video, a man can be seen being beaten by several officers in full riot gear as gunfire echoes off the surrounding buildings.
The death toll from Sunday brings to 21 the number of people killed in protests in the past month, Reuters reports. There were no immediate reports of casualties from Monday's protests, but police fired tear gas and stun grenades in Yangon, the most populous city, witnesses told the news agency.
Junta leader Senior Gen. Min Aung Hlaing later went on state television to warn that "instigators" of the protests would be punished, Reuters reports.
Last month, the Biden administration announced that it was imposing sanctions against individual members of the junta. The United Kingdom has slapped similar sanctions and the European Union may do so as well.
The coup that toppled Suu Kyi, who held the title of state counselor and was the country's de facto leader, followed accusations by the military that November's elections were fraudulent. The polls, deemed largely free and fair by international observers, showed that the NLD trounced candidates allied to Myanmar's armed forces. Despite the military's poor showing, it is constitutionally guaranteed a minimum of one-quarter of all seats in the parliament.
Suu Kyi has not appeared in public since her arrest shortly after the coup. She was initially charged with illegally importing walkie-talkies, and charges of breaching coronavirus protocols were later added.
In Monday's hearing, Suu Kyi appeared via video conference before the court in the capital, Naypyitaw.
Lawyer Min Min Soe told Reuters that she "looks healthy" and that "she asked to meet her lawyer."
Suu Kyi spent a total of 15 years under house arrest during previous military regimes before winning the country's first free election in decades by a landslide in 2015.
During her years of house arrest, Suu Kyi became a cause célèbre and in 1991, while under house arrest, she was awarded the Nobel Peace Prize for her efforts to bring democracy to Myanmar, also known as Burma.
However, in recent years, she has come under harsh criticism for her seeming defense of a 2017 army crackdown in Myanmar's western Rakhine state directed at minority ethnic Rohingya Muslims. The attacks on civilians, which came in response to assaults on soldiers by Rohingya insurgents, has been described by the head of the United Nations Human Rights Council as a "textbook example of ethnic cleansing."

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: got bad config for $resource in angular js I am trying to run my angular code using rest service $resource function. I got bad configuration error in my code. I have included # in my base URL so I got that but If I remove its showing 404. please help me to render my page properly.
Here's app.js
'use strict';

angular.module('confusionApp', ['ui.router','ngResource'])
.config(function($stateProvider, $urlRouterProvider) {
        $stateProvider

            // route for the home page
            .state('app', {
                url:'/',
                views: {
                    'header': {
                        templateUrl : 'views/header.html',
                    },
                    'content': {
                        templateUrl : 'views/home.html',
                        controller  : 'IndexController'
                    },
                    'footer': {
                        templateUrl : 'views/footer.html',
                    }
                }

            })

            // route for the aboutus page
            .state('app.aboutus', {
                url:'aboutus',
                views: {
                    'content@': {
                        templateUrl : 'views/aboutus.html',
                        controller  : 'AboutController'
                    }
                }
            })

            // route for the contactus page
            .state('app.contactus', {
                url:'contactus',
                views: {
                    'content@': {
                        templateUrl : 'views/contactUs.html',
                        controller  : 'ContactController'
                    }
                }
            })

            // route for the menu page
            .state('app.menu', {
                url: 'dishes',
                views: {
                    'content@': {
                        templateUrl : 'views/menu.html',
                        controller  : 'MenuController'
                    }
                }
            })

            // route for the dishdetail page
            .state('app.dishdetails', {
                url: 'dishes/:id',
                views: {
                    'content@': {
                        templateUrl : 'views/dishdetail.html',
                        controller  : 'DishDetailController'
                   }
                }
            });

        $urlRouterProvider.otherwise('/');
    })
;

my services.js
'use strict';

angular.module('confusionApp')
        .constant("baseURL","http://localhost:8080/app/index.html#/")
        .service('menuFactory',['$resource','baseURL', function($resource,baseURL) {

            var dishes=[
                         {
                          _id:0,
                          name:'Uthapizza',
                          image: 'images/uthapizza.png',
                          category: 'mains',
                          label:'Hot',
                          price:'4.99',

                           comments: [
                               {
                                   rating:5,
                                   comment:"Imagine all the eatables, living in conFusion!",
                                   author:"John Lemon",
                                   date:"2012-10-16T17:57:28.556094Z"
                               },

                               {
                                   rating:4,
                                   comment:"Ultimate, Reaching for the stars!",
                                   author:"Ringo Starry",
                                   date:"2013-12-02T17:57:28.556094Z"
                               },
                               {
                                   rating:2,
                                   comment:"It's your birthday, we're gonna party!",
                                   author:"25 Cent",
                                   date:"2011-12-02T17:57:28.556094Z"
                               }

                           ]
                        },
                        {
                          _id:1,
                          name:'Zucchipakoda',
                          image: 'images/zucchipakoda.png',
                          category: 'appetizer',
                          label:'',
                          price:'1.99',
                          description:'Deep fried Zucchini coated with mildly spiced Chickpea flour batter accompanied with a sweet-tangy tamarind sauce',
                          comments: [

                               {
                                   rating:2,
                                   comment:"It's your birthday, we're gonna party!",
                                   author:"25 Cent",
                                   date:"2011-12-02T17:57:28.556094Z"
                               }

                           ]
                        },
                        {
                          _id:2,
                          name:'Vadonut',
                          image: 'images/vadonut.png',
                          category: 'appetizer',
                          label:'New',
                          price:'1.99',
                          description:'A quintessential ConFusion experience, is it a vada or is it a donut?',
                           comments: [
                               {
                                   rating:5,
                                   comment:"Imagine all the eatables, living in conFusion!",
                                   author:"John Lemon",
                                   date:"2012-10-16T17:57:28.556094Z"
                               },

                               {
                                   rating:2,
                                   comment:"It's your birthday, we're gonna party!",
                                   author:"25 Cent",
                                   date:"2011-12-02T17:57:28.556094Z"
                               }

                           ]
                        }
                        ];
            var promotions = [
                {
                          _id:0,
                          name:'Weekend Grand Buffet',
                          image: 'images/buffet.png',
                          label:'New',
                          price:'19.99',

                }

            ];

                this.getDishes = function(){

                    return $resource(baseURL+"dishes/:id",null,{'update':{method:'PUT'}});

                };

              /*  this.getDish = function (index) {

                    return $http.get(baseURL,"dishes/"+index;
                };*/

                    this.getPromotion=function(index){
                        return promotions[index];
                    };

        }])

        .factory('corporateFactory', function() {

            var corpfac = {};

            var leadership = [
                {
                    name: "Peter Pan",
                    image: 'images/alberto.png',
                    designation: "Chief Epicurious Officer",
                    abbr: "CEO",

                },
                {
                    name: "Dhanasekaran Witherspoon",
                    image: 'images/alberto.png',
                    designation: "Chief Food Officer",
                    abbr: "CFO",

                },
                {
                    name: "Alberto Somayya",
                    image: 'images/alberto.png',
                    designation: "Executive Chef",
                    abbr: "EC",

                }

            ];

        corpfac.getLeaders=function(){
            return leadership;
        }
        corpfac.getLeader=function(index){
            return leadership[index];
        }

        return corpfac;

        })

;

Here's my controller.js
'use strict';

angular.module('confusionApp')

        .controller('MenuController', ['$scope', 'menuFactory', function($scope, menuFactory) {
            $scope.showMenu=true;
            $scope.message="loading...";
            $scope.tab = 1;
            $scope.filtText = '';
            $scope.showDetails = true;
            /* $scope.dishes=[];*/
            $scope.dishes= menuFactory.getDishes().query(
             function(response) {
                    $scope.dishes = response;
                    $scope.showMenu = true;
                },
                function(response) {
                    $scope.message = "Error: "+response.status + " " + response.statusText+JSON.stringify(response);
                });

                        }])

        .controller('DishDetailController', ['$scope', '$stateParams', 'menuFactory', function($scope, $stateParams, menuFactory) {

              $scope.dish=  menuFactory.getDishes.get({id:parseInt($stateParams.id,10)})

            .$promise.then(
            function(response){
                $scope.dish=response.data;
                $scope.showDish=true;
            },
                   function(response) {
                                $scope.message = "Error: "+response.status + " " + response.statusText;
                            }
            );

        }])

        .controller('DishCommentController', ['$scope','menuFactory', function($scope,menuFactory) {

            $scope.mycomment = {rating:5, comment:"", author:"", date:""};

                        $scope.submitComment = function () {
                                $scope.mycomment.date = new Date().toISOString();
                console.log($scope.mycomment);
                                $scope.dish.comments.push($scope.mycomment);

                menuFactory.getDishes().update({id:$scope.dish.id},$scope.dish);
                                $scope.commentForm.$setPristine();
                                $scope.mycomment = {rating:5, comment:"", author:"", date:""};
            }
        }])

        .controller('IndexController',['$scope','menuFactory','corporateFactory',function($scope,menuFactory,corporateFactory){
            $scope.promotion=menuFactory.getPromotion(0);
           $scope.leader= corporateFactory.getLeader(3);

$scope.showDish = true;
                        $scope.message="Loading ...";
                        $scope.feturedDish = menuFactory.getDishes().get({id:0});

 console.log($scope.feturedDish)

        }])

        .controller('AboutController',['$scope','corporateFactory',function($scope,corporateFactory){
            $scope.leaders=corporateFactory.getLeaders();

        }])

;

redpajama_set_name: RedPajamaStackExchange


text:
Digging it: Indie band Any Anything explore looser sound at Dublin gig
News & Reviews	by Lazer Guided Reporter July 7, 2018
Directly across from the Tivoli Theatre, Francis Street, is (probably) the most yellow bar front in Dublin – Drop Dead Twice. It was here last week, where Any Anything played their first Dublin gig following their Galway debut and single premiere. They shared the night with jazz-fusion prog rock outfit, Auxiliary Phoenix Trio.
Any Anything's performance at Drop Dead Twice was a significant milestone in the band's musical odyssey – not only for the fact that they broke onto the scene in the Big Shmoke (Dublin) – but because they showed us that their sound can be fluid and loose. It might be something that evolves over time when musicians develop synergy with each other and the performance loses that linear mentality. Or it could have just been a noteworthy performance without any explanation needed. Whatever it was, there was a difference between this performance and their debut – and it feels promising.
Galway born Indie pop band Any Anything gave their debut performance at the iconic live music venue the Róisín Dubh in March, 2018. Their single, 'First Time', was picked up by Irish journalists such as Remy, and was premiered in April by The Thin Air.
The video below is from their debut at the Róisín Dubh. There is a very convoluted pun to be made about Any Anything performing their first single for the 'First Time' at their first gig. But I won't risk it.
Against a west Ireland backdrop of a city that has a reputation for hosting outstanding artists (Primal Scream, Yann Tiersen, Lisa Hannigan, Blondie, Mercury Rev, Panda Dub, Seasick Steve, Spiritualised) and has witnessed the beginnings of some of Ireland's most innovative and trailblazing artists, collectives and record labels (Rua Sound, Puzzle Events, Galway Street Club, My Fellow Sponges, Dead Horse Jive, New Pope, Elaine Mai) – Galway is a city where artists can tap into a creative atmosphere.
Any Anything proclaim themselves as "Hailing from the graveyard of ambition; although showing no obvious signs of its affliction". Robert, Christopher, Luke, and Morgan are the four faces of Any Anything, and while Galway has been nicknamed 'the graveyard of ambition' by corporate-minded lizard people, it certainly doesn't feel that way when you see people actively pursuing a creative passion.
The band's performance at Drop Dead Twice stood out as something memorable. About half-way through their setlist they came out with 'We Got Sumthin' Goin' On'
If I had to pick one moment from their performance it would be this minute and a half intro. This is what I was talking about when I said they had a looser, fluid sound. Standing in front of the stage listening to the music, it feels like everything flows naturally without any desire to push on to the next verse. It's that loose treatment of the music that does the sound justice.
The future looks good for Any Anything. Let the odyssey continue.
Opening the gig that Thursday evening in Drop Dead Twice was the three piece jazz-fusion group Auxiliary Phoenix Trio. This band has built a strong reputation for themselves in Ireland's music scene, and their performance was outstanding. When I hear an artist perform, a stream of musical influences automatically crisscross through my mind – during their set I was thinking about french electro-jazz and Air's 1997 EP Premiers Symptômes, and prog rock instrumentalist hooligans which included Deep Purple and Caravan. Joining them on flute was Tom, who by all means stole the show.
Any AnythingGalway musicIndie rock
A guide to feminist media research: Part one, introductory papers
New Made in Galway playlist celebrates local talent


I was never a fan of extracurricular activities in high school or college, so I surprised even myself when I recently formed the Mozilla Roomba Users Group – a club of home-cleaning-robot enthusiasts. The idea was that interested Roomba owners and spectators at Mozilla would get together and participate in events testing the various skills of the robots. Our first RUG meeting was today, with three of the four members in attendance: myself, osunick, and dolske.
The first event was "function properly", in which all participating robots were placed on a conference room table and released from slumber to interact with each other. Sadly, osunick's roomba did not function properly, and had to be disqualified from subsequent events. fligtar and dolske tied for winning the first event, as both of their roombas functioned properly.
The second event was a timed race from one end of the table to the other, knocking a styrofoam section completely off the table. Both robots started off with a straight path, tying at 12 seconds each. The meeting was heating up, with both dolske and fligtar tied with 2 events won.
The last event of this meeting was a synchronized demo race. The first roomba to pass others by more than a second in the recorded demo would emerge victorious. fligtar's roomba was able to pull ahead of dolske's midway through the demo to clinch the event and meeting, with 3 events won to dolske's 2.
Events planned for the next meet-up include a more challenging timed race with obstacles and turns, as well as several varieties of cleaning competitions.
My brief search seems to indicate that this may be the only Roomba Users Group/club, competitive or otherwise, so I'll continue documenting our adventures if we continue to have them.


The Mongoose Scan R70 BMX Bike is a strong BMX with a chromoly main triangle, making it perfect for showing off those stunts down the skate park. It boasts 25/9 gearing with an alloy chainring, a semi integrated headset and alloy anodised rims. Topped off with a front calliper brake and rear U-brake for reliable stopping power, the R70 is a perfect choice for the budding BMXer.
The Mongoose Scan R70 BMX bike, Helmet & Gloves Bundle will fit your _VEHICLEYEAR_ _VEHICLEMAKE_ _VEHICLEMODEL_ Click here to Change vehicle and no additional parts are required for fitting.
Once you've added the Mongoose Scan R70 BMX bike, Helmet & Gloves Bundle to your basket, the items you've selected above will also be added to your order.
£244 This bundle includes the Mongoose Scan R70 BMX bike, Mongoose Skull Kids' Bike Helmet and the X Rated Kids' Fingerless BMX Gloves.


text:
D: Deniz Gamze Ergüven
Ad Vitam (Charles Gillibert)
France/Turkey 2015
W: Deniz Gamze Ergüven & Alice Winocaur
DP: David Chizaliet & Ersin Gok
Ed: Mathilde van de Moortel
Mus: Warren Ellis
Günes Sensoy (Lale), Doga Doguslu (Nur), Elit Ìscan (Ece), Tugba Sunguroglu (Selma)
This French produced Turkish film takes place in a small rural village, where traditional values are still of the utmost importance. The plot follows a group of orphaned schoolgirls, under the guardianship of their strict uncle who turns their home into a prison after the girls are seen innocently playing with some teenage boys on the beach. The girls rebel against the authority, sneaking out to watch football matches and experience life, but one-by-one, they are forced into arranged marriages by their patriarch.
The story is quite a slow mover, though it builds to a nail-biting and bittersweet conclusion, though there are some scenes in the build up which feel a little contrived.
Feminists championed the film for their own social justice arguments, but it's more a story about the clash between Eastern and Western cultures.
The film was deservedly nominated for Best Foreign Language Film at the Oscars.


text:
Take me home to die.
GEORGIO SAW FEAR in his father's eyes as he sat beside the campfire on Houston Creek, or maybe it was sadness. The old man had come to Texas searching for a home. But he had only found a place to stay for awhile, and his time was running out. His eyes had turned as gray as his hair, and a thick white beard hid the wrinkles on his face, and he didn't laugh much anymore.
Georgio knew that his father had seen the end time and had suddenly felt like a stranger upon the land that bore his crops and fed his children. The old man wasn't afraid of dying. Georgio was sure of it. There was a time to be born and a time to die. His father had known it all his life. The old man was not afraid of the devil himself and had always said hell was just a place to cool off from one of those Texas summers that sapped the soil and fried his vegetables on the vine.
A month earlier, he had come to Georgio, his first born, about sundown as the frogs began to complain down along the Elm Fork of the Trinity River. "I'm dying," he said softly.
Georgio had tried to protest, but his father wouldn't listen. It's all right, the old man said, He had expected to die someday anyway. His years had already been long enough, and they had been good and bad but mostly better than he probably deserved.
Georgio nodded.
His father looked out across the gently rolling plains as he had done for the past forty-six years. He knew them well. But they didn't belong to him. Not really. They had kept him alive and their dirt had stuck beneath his fingernails, and their rocks had dulled his plow. But they were foreign to him, and he wanted to go back home. Lately the old man had been thinking a lot about the home of his childhood.
"I want you make me a promise, Georgio," he said firmly.
"What, papa?"
"Don't let me die here. I want to be buried in the earth of my fathers. Carry me back to Italy, to sunny Italy, before I die.'
"It's a long way, papa."
"Promise me, Georgio."
"Yes, papa."
The old man kissed his children and grandchildren goodbye, placed a handful of wildflowers on the grave of his wife who had left him so many years ago, and climbed into the oxcart. For a moment, Georgio thought he saw a faint smile tug at the corners of his father's mouth, and he heard the old man whisper, "Italy." He seemed stronger than before, and the gray cloud had faded from his eyes. "Italy." The old man could not wait to touch the soil of homeland and let it hold him close forever.
The journey turned southward, and the oxen were headed toward the port of Galveston. There, Georgio knew, he would a find a ship to carry his father home. For the first few days, the old man spoke loudly and often of his years on a farm outside of Rome, of his family and his mother who wept and refused to see him the morning he sailed away to seek his fortune in a new land of promise. He wondered if he would be able to find any of his kin and if they remembered him or had even heard of him. Maybe he was just an outcast and would come home and no one would care.
He had told his mother that someday he would return as a wealthy man to walk with dignity and respect down the road outside their farm home. The old man sighed. He was only returning to die. But maybe someone would carry his coffin with dignity and respect down the old road. He suddenly felt very tired. The miles were exacting a heavy price. He grew weak, then silent, and the gray cloud found its way back to his eyes.
He sat staring at the campfire on Houston Creek, chilled even though the night was warm, alone even though Georgio sat beside him. "Are we going to get there?" he asked his son.
The old man closed his eyes, and Georgio thought the old man had drifted off to sleep. He pulled the blanket tighter around his father's shoulders and threw another dried oak limb onto the fire. He, too, was tired. It had rained early that morning, warm and gentle but enough to turn the black land into a quagmire. It had slowed the old cart to a crawl and drained the strength from his oxen.
Beyond the trees, Robert Aycock sat in the dim candlelight of a house that served as both a grocery store and a post office for the settlers scattered along Houston Creek. He was a perplexed man. Around him were the makings of a good town, and all it needed was a name. He was having a hard time finding one.
Most of his neighbors wanted to call the hamlet Houston since they all believed that old Sam Houston had once camped out along the creek that wound among their farms. So Aycock, being the postmaster, had applied for the name, but, alas, the post office rejected it. Texas already had a Houston, the letter said, and one was plenty.
Aycock was too frustrated to sleep, and he hard someone knocking on his door. He opened it and saw a frightened Georgio.
"My father's extremely ill," the young man said. "Can I bring him inside?"
Robert Aycock rushed with him back to the oxcart and helped carry the old man into the darkness of a bedroom. The hot air was stifling.
For days, Georgio sat beside his father and held his hand, waiting to see those gray eyes open again. On a Thursday morning, Robert Aycock walked into the room as the old man eased out of his coma and turned his head toward the window, watching as sunlight streaked across the prairie. His voice was faint and drifting. "Italy," he said softly. "Sunny Italy."
Aycock saw Georgio close his eyes and wrestle with his conscience before speaking. Then the young man answered, "Yes, papa. It is Italy."
The old man smiled broadly and tightly grasped Georgio's hand. "Thank you, son," he whispered.
That afternoon, Georgio borrowed a shovel and dug his father's grave beneath a live oak on the banks of Houston Creek. He grieved, but not for his father's death. "I lied to him," the young man told Robert Aycock. The old man had depended on his son to return him to Italy, and Georgio had failed. Then he had lied. It would be a burden and a sin that he would have to shoulder for the rest of his life.
Aycock shook his head. "You did your best," he said, "and I'll make sure you didn't lie to your father." That night in March of 1880, he sat down and wrote the Post Office Department a letter. His hamlet at last had a name. It was then, as it is now, the only town in America called Italy.
Please click the book cover image to read more about Caleb Pirtle III and his books.
author, burden, Caleb Pirtle III, dying, foreign, Hamlet, homeland, ill, Italy, outcast, promise, sin, sundown, Texas
Build a Fan Base and Sell More Books
Wonder Weapons: Release the Magic


Looking for Dallas Mavericks vs. San Antonio Spurs free NBA picks & NBA odds? NBA betting sees the Mavericks taking on the Spurs on Saturday December 16, 2017 at the AT&T Center in San Antonio, Texas. Cappers Picks provides complimentary expert handicapping picks on all NBA basketball matchups so stay tuned for more FREE daily NBA hoops predictions like this Mavericks Spurs matchup.
This is the fourth and final time these two teams will play one another this season. Dallas has so far lost two of the three games that have already been played, by a combined 13 points. The Mavericks won the last matchup earlier this week in San Antonio when Kawhi Leonard made his season debut.
Yogi Ferrell continued his good play as a starter with Dennis Smith Jr. out. He had 16 points and eight rebounds — one of five Mavericks in double-figures. Salah Mejiri caught the attention of the Internet when he tried to trip Patty Mills as the two teams were leaving the court at the end of the game. We'll see if that situation is over.
Dallas' last game on Thursday was a 15-point loss at Golden State. Most teams are in action on Friday night, including the Spurs, but the Mavs get the night off. They'll use that time wisely. They'll have to if they want any shot at beating San Antonio twice in a week.
The Spurs haven't played since that game on Tuesday. Kawhi Leonard played 16 minutes and scored 12 points on 6-for-12 shooting. He was particularly aggressive but that isn't out of the ordinary. It's unknown how Gregg Popovich will use him in the back-to-back the Spurs have coming.
San Antonio is in Houston on Friday night for a showdown with the best team in the NBA. Leonard's presence will bring some excitement to that game. He's had some fun battles with James Harden in recent years. It may be that Pop plays Leonard more minutes on Friday night and takes it easy on him in this game (if he even plays at all). But outsiders never know what Popovich is apt to do.
What I do know is that San Antonio's rotation looks a lot deeper with Leonard in it. The Spurs used 13 players in Tuesday's game because there are that many bodies competing for minutes. That's a good problem to have and Popovich will take full advantage of the competition within the team to get the most out of everybody in these regular season games.
Under is 6-1-1 in DAL last 8 vs. NBA Southwest.
Under is 13-3-1 in DAL last 17 Sat. games.
Under is 8-2 in DAL last 10 vs. Western Conference.
SA are 8-3-1 ATS in their last 12 home games.
SA are 3-7 ATS in their last 10 Sat. games.
Head to Head: Underdog is 22-6 ATS in the last 28 meetings.
Free Atlanta Hawks vs. San Antonio Spurs NBA Prediction: San Antonio gets its revenge and takes the game and season series.
It won't be an easy spot flying back from Houston to play this game the next day but the Spurs won't lost twice in a week to Dallas. It's very possible this could be the game standing between them and an 0–3 week, too.
The Spurs will put their best foot forward.
If you're looking for more NBA picks like this Dallas Mavericks vs. San Antonio Spurs matchup we'll be providing 4* free NBA predictions on the blog during the entire 2017/18 NBA season, and check out our Experts for guaranteed premium & free NBA predictions!


text:
Q: Rails 4 Nested Form I have three models:
class Item < ActiveRecord::Base
   has_many :item_points
   has_many :groups,  through: :item_points
   accepts_nested_attributes_for :item_points
end

class ItemPoint < ActiveRecord::Base
  belongs_to :group
  belongs_to :item
  accepts_nested_attributes_for :group
end

class Group < ActiveRecord::Base
  has_many :item_points
  has_many :items, through: :item_points
end

The schema for items
create_table "items", force: :cascade do |t|
  t.string   "name",       null: false
  t.datetime "created_at", null: false
  t.datetime "updated_at", null: false
end

The schema for item_points
create_table "item_points", force: :cascade do |t|
  t.decimal  "points",          null: false
  t.integer  "item_id",         null: false
  t.integer  "group_id",   null: false
  t.datetime "created_at",      null: false
  t.datetime "updated_at",      null: false
end

The schema for groups
create_table "groups", force: :cascade do |t|
    t.string   "name",       null: false
    t.datetime "created_at", null: false
    t.datetime "updated_at", null: false
end

In my groups table, I've created a number of rows, e.g. group 1 and group 2.
In the form for creating items I'd really like to see a field each for group 1 and group 2, so that I might be able to enter the points for that item.  e.g. In item X, group 1 is worth 10 points, and group 2 is worth 5 points.
EDIT  Added the form
The form:
 <%= form_for(@item) do |item_form| %>

      <div class="form-group">
        <%= item_form.label :name %>
        <%= item_form.text_field :name, :class => 'form-control' %>
      </div>

      <%= item_form.fields_for(:groups) do |groups_form| %>
        <% group = groups_form.object_id.to_s%>
        <%= groups_form.hidden_field :id %>
        <%= groups_form.fields_for(:item_point) do |entity_form| %>
          <%= entity_form.text_field :points %>
        <% end %>
      <% end %>

This provides me with a form, which contains one extra entry box, called item[groups][item_point][points], and has no label.
What I'd like to know is how do I get all of the rows I've added into groups as fields into a Rails Form? And when I do, how do I save the associated item_points data using strong parameters?
I've spent quite some time looking for an answer, and I can't seem to find anything other than a series of StackOverflow questions, which don't quite seem to have the same problem as me.
All help is wonderfully appreciated.

A: There's a helpful post with some examples at: https://robots.thoughtbot.com/accepts-nested-attributes-for-with-has-many-through. It specifically talks about adding inverse_of on your associations.
In your view you'll need to use fields_for. You could (for example) put that in a table or a list and have each entry be a row.
If you share what you've tried in your view so far you may be able to get a more detailed response if you need it.
As for permitted_params in your controller, you can nest them something like:
def permitted_params
  params.permit(item: [
    :name,
    item_points_attributes: [
      :id,
      :points,
      :group_id,
    ]
  )
end

Update:
Rather than fields_for(:groups) I think you want your controller to build the models for all the item_points (@item_points = Group.all.collect {|group| ItemPoint.new({group_id: group.id, item_id: @item.id}).
Then you can use a fields_for(:item_points, @item_points).
You can add a label for the field so it's not just an unlabeled field using the HTML label tag.


We had a great time finding out about our favourite books and even dressing up as our favourite characters.
Have you ever wondered what would happen if Harry Potter met the Big Bad Wolf, or Alice became best friends with Hermione Granger, or Denis the Menace found Wally? Well now is your chance to find out!
On the day we were very lucky to work with pupils from Carres Grammar School who helped us come up with a brand new adventure for our favourite characters. Using inspiration from our fancy dress, we created a brand new adventure for the characters, wrote the script and even acted it out.


This would be usefull to know as I would like to test the motherboard before I order the complete top body with keyboard.
They are usually just right above the very leftmost edge of the keyboard ribbon cable. If it doesn't work, try disconnecting the ribbon cable, and then try again. Good luck!
I used your video today as I replaced the keyboard on a recently purchased MacBook Pro Unibody 13" 2009 model. The computer worked but had problems with the keyboard and trackpad, so those were the things I was replacing. I didn't have immediate success with the keyboard replacement, so I was troubleshooting by trying to jump start from the logic board. I touched the two metal spots you indicated with tweezers and got a "zap" followed by electrical smell. So... before I started I could check the battery charge tested full and now it does not.
Do you think I completely destroyed the logic board... or is there something more that could be done. As far as I can tell everything has been installed back to correct places - time consuming, but painstakingly done and rechecked. However, the computer will not start up or do anything at all since the "zap". My $500 investment in a new used laptop looks like it went up in smoke. I'm hoping there's something else that can be done.
Just an FYI, I am not a novice at computer repair or parts replacement and I followed various videos for all the aspects of this job today.
Yikes, sorry to hear that. My guess is that you may not have been grounded and were carrying around a buildup of static electricity, and then when you jumped the board, it received the full impact of this. If you noticed a smell, most likely the board is damaged, but what I would do before coming to conclusions is remove all sources of power (AC, battery) and let the machine sit overnight to make sure any kind of charge it has taken is 100% discharged. Sometimes a "bad charge" will cause a computer to get stuck in a specific behavior, and letting that charge dissipate over time eliminates the behavior. I would also disconnect the DC-in connector from the board, because it is a power-related component and also has the ability to maintain a charge.
If after sitting overnight disconnected it still will not power on, I'd test the RAM slots by attempting to power on with one slot empty, and then the other empty. I'd also try different RAM if you have any, because a memory module might have been damaged. If none of that works, I'd use a magnifying glass to examine the board and see if you can find any burn marks, as those might present clues. You can also test powering on with battery vs AC adapter, to see if one or the other works.
Anyway, good luck, and let me know how it goes.


text:
Les élections législatives de Gibraltar en 1996 se sont tenues le  pour élire les 15 membres du parlement pour un mandat de quatre ans.

Partis en présence

Résultats

Par partis

Par candidat

Notes et références

Gibraltar
1996
1996 à Gibraltar
Politique en mai 1996

redpajama_set_name: RedPajamaWikipedia


text:
Posted on May 30, 2017 August 18, 2018 by Grimmedian
Fantasy Flashback: Farscape 1999-2002
In 1999, one of the best space operas ever written was produced by Nine Network. A SciFi Fantasy labor of love.
A genuine SciFi Fantasy series, Australian and American, conceived by Rockne S. O'Bannon and produced by the Jim Henson Company and Hallmark Entertainment. The show was the flagship of the then titled The Sci FI Channel now known only as SYFY.
The Jim Henson Company was responsible for the various alien make-up and prosthetics, and two regular characters(the animatronic puppets Rygel and Pilot) are entirely Creature Shop creations.
The Jim Henson Company, which had been known up to that point for the making of The Dark Crystal and Labyrinth, as well as the iconic Muppets and Sesame Street, crossed over brilliantly with this foray into a very adult themed series.
Farscape centers it's story around John Crichton, an IASA astronaut on a test flight in the Farscape module ship, who is suddenly sucked into a wormhole and flung to the farthest ends of the universe. He finds himself taken aboard the leviathan Moya, a race of bio-mechanoid creatures that function as living ships. He's joins a very unusual group of aliens who, along with the ship itself, is on the run from the Peacekeepers, a race of Nazi-like beings called Sebacceans, who are startlingly human in appearance. Moya was being used by the Peasekeepers as a prison ship before the escape of both ship and prisoners.
The Peacekeepers have an ongoing war with the Scarrans and soon John finds himself the focus of a battle to get the information in his head. Information that resides in his subconscious and was a result of his travel through the wormhole phenomenon. This wormhole technology would allow either faction to control the egress to our galaxy. John Crichton's wry humor as the only human in this part of the universe, and his constant use of pop culture quips added a punch to the storyline as he desperately tries to return to earth. Saying as he contemplates his predicament, "Boy was Speilberg ever wrong…close encounters my ass."
Awkward ending:
The show was an international co-production between the United States and Australia. The flagship series of SyFy, then known as The SciFi Channel, the show ran four seasons, then was abrubtly cancelled in September 2002 after ending with its standard MO, a cliffhanger. Fans of the show went on mass to voice their disappointment. At great expense to himself, Co-producer Brian Henson later secured the rights to Farscape, paving the way to produce a three-hour miniseries to wrap up the cliffhanger, titled Farscape: The Peacekeeper Wars, which was aired on two consecutive nights on SyFy in March of 2004.
CategoriesFantasy Flashback TagsFantasy Flashback, Farscape, Henson, SciFi
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text:
Q: How to fixed second row in sorttable javascript? Here is sorttable.js: https://kryogenix.org/code/browser/sorttable/
and usage as below:

        Str
        Int

        a
        1

        b
        2

now I want to create a filter below first row:

        Str
        Int

        filter
        <select>
        <option>a</option>
        <option>b</option>
        </select>

        a
        1

        b
        2

but it will sorted, how can I edit this sorttable.js and can fixed the second row?
(my sorttable.js is save in local file)
very thaks!

A: in

you just need to remove class="sortable" and sortable.js won't work in the second row

A: maybe this work, remove filter part into

        Person
        Monthly pay

        filter
        <select>
        <option>a</option>
        <option>b</option>
        </select>

Jan Molby	£12,000

Steve Nicol	£8,500

Steve McMahon	£9,200

John Barnes	£15,300

TOTAL	£45,000


text:
<?php

class Google_Service_Plus_CommentPlusoners extends Google_Model
{
  public $totalItems;

  public function setTotalItems($totalItems)
  {
    $this->totalItems = $totalItems;
  }
  public function getTotalItems()
  {
    return $this->totalItems;
  }
}


Grizzly 3-blade Broadheads have a reputation for being tough and dependable.
Now you can have the same tough, dependable performance you're used to with their glue-on broadheads on any carbon or aluminum arrow shaft with the Grizzly Instinct 3-Blade Screw-In Broadhead!
Grizzly Instinct 3-Blade Screw-In Broadheads are made of multiple layers of 1075 carbon steel, which are spot welded together and then copper brazed. The core of the Instinct is the triple laminated core, which gives it incredible strength.
The Instinct's sold steel tip reinforcement gives it even more structural integrity, all but eliminating tip curl. Grizzly Instinct Screw-In Broadheads are heat treated to 50 Rockwell, and Teflon® coated. The Broadheads are then ground to a burr, making them extremely easy to sharpen to a razor's edge.
With a long, lean design, the Instinct is the perfect predator, penetrating with a cutting diameter of 1-1/8". The Grizzly Instinct Screw-In Broadhead will mount on all carbon and aluminum arrows as they have an industry standard 8-32 thread.
Available in 200 grains only. Sold by the 3-pack.


text:
NUBCEconomic Data/News
Initial Jobless Claims Rise by 5,000 as One-Month Average Hits November 2007 Level
Andrew Klips Follow
| Thursday, 08 August 2013 09:02 (EST)
The number of Americans filing for first-time unemployment benefits rose less than expected in the week ended August 3, according to a weekly report from the Labor Department, to remain close to five-year lows and keep speculation alive about the Federal Reserve tapering its third version of quantitative easing.
Last week, initial jobless claims rose by 5,000 to 333,000 from the prior week's revised figure of 328,000 (up from an originally estimated 326,000). In the week ended July 27, the number of claims hit the lowest level since January 2008.
Economists were expecting a slightly higher rise in claims for last week, forecasting claims at 338,000.
The four-week moving average, a less volatile barometer of labor trends because it irons out weekly volatility, dropped by 6,250 to 335,500, marking the lowest level since November 2007. Generally speaking, economists consider claims at 350,000 per week to indicated modest growth in the jobs market.
July is known to be a volatile month because of the Independence Day holiday and auto factories shutting down for maintenance, but with that month now behind us, economists start paying even closer attention to new unemployment filings, especially as to how it can relate to the Fed starting to unwind QE3. On Wednesday, Sandra Pianalto, president of the Cleveland Federal Reserve Bank, reiterated the sentiment of many in saying that the Fed could start scaling back its asset purchases as long as the labor market holds its recent gains. Pianalto did not offer a timeframe, but Thursday's report lends some credence to the strengthening jobs market.
Continuing claims, or people that are already receiving unemployment benefits, increased by 67,000 to a seasonally adjusted 3.02 million in the week ended July 27. Analysts were pegging a slight decline to 2.95 million. Continuing claims are reported at a one-week lag to initial claims and do not include people receiving extended benefits under Federal programs.
The total number of Americans collecting benefits in all programs for the week ended July 20 decreased by 174,418 to 4.52 million. Total claims are reported at a two-week lag. Compared to the same time in 2012, total claims were down 21 percent from 5.75 million.
The largest increases in initial claims for the week ending July 27 were in Idaho (+264), Wisconsin (+179), and South Dakota (+115). 12 states reported declines, with the largest decreases in California (-21,479), Michigan (-8,647) and Missouri (-3,208).
The slightly better-than-expected report is helping keep futures in the green ahead of the opening bell. Dow futures are up 58 points, S&P 500 futures are ahead 6 points and Nasdaq futures are up by 14 points. Equities have been beaten-up this week so far with the blue chip indexes all down about 1 percent or more.
Follow	 NUBC Northumberland Bancorp 36.00 0.00 0.00 0
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