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\title{Non-symmetric Cauchy kernels, Demazure measures, and LPP}

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\author{Olga Azenhas\thanks{\href{mailto:hello@world.c}{oazenhas@mat.uc.pt}. O. A. was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.}\addressmark{1}, Thomas Gobet \thanks{\href{thomas.gobet@lmpt.univ-tours.fr}{thomas.gobet@lmpt.univ-tours.fr}. T. G. was partially supported by the Agence Nationale de la Recherche funding ANR CORTIPOM 21-CE40-001.}\addressmark{2}, \and Cédric  Lecouvey\thanks{\href{cedric.lecouvey@lmpt.univ-tours.fr}{cedric.lecouvey@lmpt.univ-tours.fr}. C. L.  was partially supported by the Agence Nationale de la Recherche funding ANR CORTIPOM 21-CE40-001.}\addressmark{2}}


%%{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant %%2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

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\address{\addressmark{1} University of Coimbra,  CMUC, Department of Mathematics, Portugal \\ \addressmark{2} Institut Denis Poisson Tours, Universit\'{e} de Tours, Parc de Grandmont, 37200 Tours, France}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We use non-symmetric Cauchy kernel identities to get  the laws of last passage
percolation (LPP) models in terms of Demazure characters. The construction is based
on the restrictions of the RSK correspondence to  augmented stair (Young) shape matrices    and rephrased in a unified way compatible with crystal bases. }

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\keywords{Non-symmetric Cauchy identity, Demazure character, crystal,  percolation.}

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\begin{document}

\maketitle
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\section{Introduction }
We introduce the Demazure measure on nonnegative vectors corresponding to the directed last passage percolation (LPP) model on  matrices of Young shape, that is, nonnegative integer matrices whose positive  entries fit a Young shape. A nonnegative integer vector is always in the Weyl orbit of some partition and therefore all nonnegative  vectors in a same Weyl orbit share the size of  a largest entry  which  is the length of a longest row of the unique partition in its  orbit.  When the Young shape is a rectangle, we recover  the Okounkov's Schur measure \cite[Chapter 4]{BDS}, \cite{OK} on the unique partition of each Weyl orbit,  corresponding to the LPP model on   nonnegative integer matrices.  Our main contribution is the use of Demazure characters,  in general, non symmetric polynomials, to study LPP problems: this has only been carried out for models with more symmetries using symmetric polynomials, in particular, Schur polynomials or Weyl characters or geometric analogues as incarnations of Whittaker
functions (\cite{BZ1,BZ,NPS, Ye} and references therein). Crystal
theory allows the compatibility of Robinson--Schensted--Knuth
(RSK) correspondence with non-symmetric Cauchy identities by Lascoux \cite{Las2} and thus, in particular,  the Cauchy identity \eqref{cauchy}. This
interpretation was discovered by Choi--Kwon  \cite{CK} for the
non-symmetric case on stair cases (\ref{NSCK}). We complete the picture with the truncated and augmented stair shape.
This extended abstract is organized in four  sections.
In \S \ref{Subsec_DemaModules} we gather relevant definitions on crystals, in \S \ref{nonsymetric}  present our contributions, and  in \S \ref{example} provide an  example for our  main result.
 We refer the reader to the full version \cite{AGL}, accepted for publication,  for  details and proofs, containing the results hereby presented.


Given two sets of indeterminates $ x=\{x_{1},\ldots,x_{m}\}$ and
$ y=\{y_{1},\ldots,y_{n}\}$ the Cauchy identity  asserts that
\begin{equation}\label{cauchy}
\prod_{i=1}^{m}\prod_{j=1}^{n}\frac{1}{1-x_{i}y_{j}}=\sum_{\lambda
\in\mathcal{P}_{\min(m,n)}}s_{\lambda}( x)s_{\lambda}( y)
\end{equation}
where $\mathcal{P}_{\min(m,n)}$ is the set of partitions with at most
$\min(m,n)$ parts and, for each such partition $\lambda$, $s_{\lambda}( x)$ and
$s_{\lambda}( y)$ are the Schur polynomials in the indeterminates $ x$ and $y$,
respectively.
 This identity has several interpretations, applications and generalizations (see \cite{FMO} and references therein). In particular,  one can understand the left hand side as the character of polynomial functions on the space $\mathcal{M}_{m\times n}$ of matrices  with $m$ rows, $n$ columns and entries in $\mathbb{Z}_{\geq0}$  and  decompose this space into a direct sum of $\mathfrak{gl}_{m}\times\mathfrak{gl}_{n}$ bimodules. The products of Schur functions $s_{\lambda}( x)$ and $s_{\lambda}( y)$ on  the righthand side show this approach as
the characters of the tensor product of irreducibles finite dimensional representations of highest weight $\lambda$ for the linear
Lie algebras $\mathfrak{gl}_{m}(\mathbb{C})$ and $\mathfrak{gl}_{n}%
(\mathbb{C})$. In fact $\mathcal{M}_{m,n}$ is a realization of the
the bicrystal  of the
symmetric space $S(\mathbb{C}^{m}\otimes\mathbb{C}^{n})$ as a  $(\mathfrak{gl}_{m},\mathfrak{gl}_{n})$-module
 (see \cite{CK} and references therein).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The identity \eqref{cauchy} can also be proved
 using the RSK correspondence \cite{Fu,Sta}. This
is a one-to-one map $\psi$ between the set $\mathcal{M}_{m,n}$  and the set $\bigsqcup_{\lambda
\in\mathcal{P}_{\min(m,n)}} SSYT(\lambda,m)\times SSYT(\lambda,n)$ of pairs
$(P,Q)$ of semistandard tableaux of the same shape $\lambda$,
 and  entries in $[m]:=\{1,\ldots,m\}$ and $[n]:=\{1,\ldots,n\}$, respectively. (The convention that we use agrees with that of Kashiwara \cite{kash} to
which we refer for another description of the RSK\ procedure and the
connection with biwords. See \S \ref{example} and \cite{Fu} for  variations on RSK.) Regarding $SSYT(\lambda,k)$ as the tableau realization   for the $\mathfrak{gl}_{k}$-crystal $B(\lambda,k)$ of highest weight $\lambda$, then
\begin{align}%\label{Th_K}
\psi:\mathcal{M}_{m,n}&\rightarrow%
{\textstyle\bigsqcup\limits_{\lambda\in\mathcal{P}_{\min(m,n)}}}
B(\lambda,m)\times B(\lambda,n)\nonumber\\
A&\mapsto \psi(A)=(P(A),Q(A))\label{Th_K2}
\end{align}
 is a $(\mathfrak{gl}_{m},\mathfrak{gl}_{n})$-bicrystal isomorphism where  the  bicrystal structure on $\mathcal{M}_{m,n}$ is  afforded from $B(\lambda,m)\times B(\lambda,n)$ by $\psi^{-1}$, that is, by reverse column Schensted insertion.
The RSK correspondence has  interesting properties.\ For
each matrix $A$ in $\mathcal{M}_{m,n},$ the greatest integer $p(A)$
obtained by summing up the entries in all the possible paths $\pi$ starting at
position $(1,n)$ and ending at position $(m,1)$ with steps $\longleftarrow$ or
$\downarrow$
\begin{align}\label{LPPtime} p(A):=\max_{\mbox{\scriptsize$\pi$ path in $A$}}
\sum_{
(i,j)\in\pi}
a_{ij}
\end{align}
coincides
 with the common largest row length  of the tableaux $P(A)$ and $Q(A)$ in \eqref{Th_K2}. (We consider the paths which are
compatible with the version of RSK that is used here. See \S \ref{example}.) It is then
natural to study percolation models based on the RSK correspondence where
random matrices whose entries follow independent geometric laws are
considered  \cite{BDS}.
This type of model,  in the case of identical and independent geometric distribution, has been deeply studied by Johansson in
\cite{joh}, who proved that the fluctuations of the previous last passage
percolation, once correctly normalized, are controlled by the Tracy-Widom
distribution (defined from the study of the largest eigenvalues of random
Hermitian matrices). The Schur measure,  introduced
by Okounkov
based on the { Cauchy kernel identity}, is an extension of the probability measure on the partitions
corresponding to the directed last passage percolation model with  the independent and identical
geometric distribution of Johansson in
\cite{joh}, \cite[Chapter 4]{BDS}.
Let $ u_i, v_j$ be real numbers in  $[0, 1)$, for $1\le i\le m$, $1\le j\le n$.
Considering
an array $\mathcal{W}=\{W_{ij} : 1 \le i\le m, 1\le j \le n\}$ of independent   random variables, with values in $\mathbb{Z}_{\ge 0}$, called weights, geometrically  distributed
as
\begin{align}\mathbb{P}(W_{ij} =k) = (1 -u_iv_j)(u_iv_j)^{k},
 \mbox{ for any $k\in\mathbb{Z}_{\ge 0}$}\label{geometricw(i,j)},\end{align}
with parameter $ u_iv_j$,  $\mathcal{W}$ is a random matrix with values in  $\mathcal{M}_{m,n}$.  We then get
\[
\mathbb{P}(\mathcal{W}=A)=\left(  \prod_{1\leq i\leq m,1\leq j\leq n}%
(1-u_{i}v_{j})\right)  (uv)^{A}%
\]
where $(uv)^{A}=\prod_{1\leq i\leq m,1\leq j\leq n}(u_{i}v_{j})^{a_{i,j}}.$
The Last Passage Percolation (LPP) time $G$ of $\mathcal{W}$ is defined to be  the random variable $G:=p\circ\mathcal{W}$.
  %the length of the longest row of the tableaux $P(\mathcal{W})$ and $Q(\mathcal{W})$ in the RSK correspondence \eqref{Th_K2}.
Applying the RSK correspondence,
  its properties and the Cauchy identity \eqref{cauchy}, one obtains the law of the random variable $G$, for  any $k \in \mathbb{Z}_{\ge 0}$, in terms of Schur polynomials,
$$ \mathbb{P}(G= k) =\prod_{1\le i,j\le n}(1 - u_iv_j)\sum_{\lambda\in\mathcal{P}_{\min(m,n)}\mid
\lambda_{1}=k}
s_\lambda(u_1,\dots , u_m)  s_\lambda(v_1,\dots , v_n)$$
 where the sum is over partitions $\lambda$ with largest part $k$.  Johansson  \cite{joh}  has established this result  in the special case of identical geometric distribution, $u_i=v_j=\sqrt q$, $1\le i, j\le n$, for a fixed $q\in]0,1[$,  a special case of the Schur measure on partitions (see \cite[Chapter 10]{BDS}).
The RSK correspondence admits various generalizations and geometric versions
which can also be used to get interesting last passage percolation models involving symmetric polynomials,  %\cite{Lam},
 in particular, characters of representations of Lie algebras other
than $\mathfrak{gl}_{n}$ (symmetric with respect to the
 Weyl group) and  geometric analogues \cite{BZ1,BZ,NPS, Ye}. %We refer the reader to \cite{BZ1,BZ, NPS, Ye}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystal and Demazure modules}\label{Subsec_DemaModules}
The finite dimensional
irreducible polynomial representations of $\mathfrak{gl}_{n}=\mathfrak{gl}_{n}(\mathbb{C})$ are parameterized
by the partitions $\lambda$ %=(\lambda_1\ge\cdots\ge\lambda_n\ge 0)$
in $\mathcal{P}_{n}$. To each partition $\lambda\in\mathcal{P}_{n}$ corresponds  a finite dimensional representation $V(\lambda)$ (or $\mathfrak{gl}_{n}$-module),
and  a crystal graph
$B(\lambda)$ which can be regarded as the combinatorial skeleton of the simple
module $V(\lambda)$. The vertices of $B(\lambda)$ label a distinguished basis
of $V(\lambda)$.  On the other hand, $B(\lambda)$ has various combinatorial realizations (\textit{i.e.}, vertex
labelings) in terms of semistandard tableaux, Littelmann's paths
\cite{Li} or semiskylines  \cite{Mas}.
 The (abstract) crystal $B(\lambda)$ is a graph
whose set of vertices is endowed with a weight function ${wt}%
:B(\lambda)\rightarrow \mathbb{Z}^n$ and with the structure of a coloured and oriented
graph given by the action of the crystal operators $\tilde{f}_{i}$ and
$\tilde{e}_{i}$ with $i\in I=[n-1]$. One has an
oriented arrow $b\overset{i}{\rightarrow}b^{\prime}$ between two vertices $b$
and $b^{\prime}$ in $B(\lambda)$ if and only if $ b^{\prime}=\tilde{f}_{i}(b)\Leftrightarrow
b=\tilde{e}_{i}(b^{\prime})$ in which case $wt(b')=wt(b)-\alpha_i$, with $\alpha_i$ a simple root of $\mathfrak{gl}_{n}$.
%$\tilde{f}_{i}(b)=0$ (resp. $\tilde{e}_{i}(b)=0$) when no arrow $i$ starts from $b$ (resp. ends at$b$). %The  $0$ is  auxiliary and  not  in
The crystal $B(\lambda)$ is generated by the actions of the lowering (resp. raising) operators $\tilde{f}_{i}$ (respect. $\tilde{e}_{i}$) on the unique highest   (resp. lowest) weight vertex $b_\lambda$ (resp. $b_{\sigma_0\lambda})$ where one has $wt(b_\lambda)=\lambda$, and $\sigma_0$ is the longest element of the Weyl group $W$ here the symmetric group $\mathfrak{S}_n=<s_1,\dots,s_{n-1}>$  (unless mentioned differently).


   For $\lambda\in{\cal{P}}_n$, $W_\lambda$ is  the stabilizer of $\lambda$ under the action of $W$,  and $W^\lambda$ collects the unique  minimal length representative  of each  coset in  $W/W_\lambda$.
Let  $\lambda\in{\cal{P}}_n$ and $\sigma
\in W$.\ Up to a scalar in $\mathbb{C}$, there exists a unique
vector $v_{\sigma\lambda}$ in $V(\lambda)$ of weight $\sigma\lambda$. Recall the triangular decomposition
$\mathfrak{gl}_{n}=\mathfrak{gl}_{n}^{+}\oplus\mathfrak{h}\oplus
\mathfrak{gl}_{n}^{-}$ of $\mathfrak{gl}_{n}$ into its upper, diagonal and
lower parts. The
Demazure module associated to $v_{\sigma\lambda}$ is the $U(\mathfrak{gl}%
_{n}^{+})$-module defined by
$
V_{\sigma}(\lambda):= U(\mathfrak{gl}_{n}^{+})\cdot v_{\sigma\lambda}.
$
Demazure  introduced the character ${\kappa}_{\sigma,\lambda
}$ of $V_{\sigma}(\lambda)$ and showed that it can be computed by applying to
$x^{\lambda}$ a sequence of divided difference operators $D_{i_{1}}\cdots
D_{i_{\ell}}$  given by any reduced
decomposition of $\sigma=s_{i_{1}}\cdots s_{i_{\ell}}\in W$ where $\ell$ is the length of $\sigma$.
% We call  to  $(i_{1},\ldots, i_{\ell})$  a reduced word of $\sigma$.
For  $i\in I$,
 $D_i$ is a certain linear operator on $\mathbb{Z}[x_{1},\ldots,x_{n}]$ (see \cite{AGL} and references therein)
%by%
%\[
%D_{i}(P)=\frac{x_{i}P-x_{i+1}(s_{i}\cdot P)}{x_{i}-x_{i+1}},\,s_i\in W.
%\]
%Demazure proved that such operators
satisfying the relations
\begin{align*}
D_{i}^{2}  &  =D_{i}\text{ for any }i=1,\ldots,n-1\text{,}\quad
D_{i}D_{i+1}D_{i}    =D_{i+1}D_{i}D_{i+1}\text{ for any }i=1,\ldots
,n-2\text{,}\\
D_{i}D_{j}  &  =D_{j}D_{i}\text{ for any }i,j=1,\ldots,n-1\text{ such that
}\left\vert i-j\right\vert >1\text{.}%
\end{align*}
Thus,  by Mastumoto's Lemma, the operator $D_{\sigma}=D_{i_{1}}\cdots
D_{i_{\ell}}$ only depends on $\sigma$ and not on the chosen reduced decomposition, and
${\kappa}_{\sigma,\lambda}=D_{\sigma}(x^{\lambda})\in\mathbb{Z}%
[x_{1},\ldots,x_{n}]
$
is the \textsf{(Demazure) character} of $V_{\sigma}(\lambda)$. In particular,
we have ${\kappa}_{id,{\lambda}}=x^{\lambda}$ and ${\kappa
}_{\sigma_{0,\lambda}}=s_{\lambda}$.
\iffalse
and%
\begin{equation}
D_{i}({\kappa}_{\sigma,\lambda})=\left\{
\begin{array}
[c]{l}%
{\kappa}_{s_{i}\sigma,\lambda}\text{ if }\ell(s_{i}\sigma)=\ell
(\sigma)+1,\\
{\kappa}_{\sigma,\lambda}\text{ otherwise.}%
\end{array}
\right.  \label{D_i(Demazure)}%
\end{equation}
\fi

Kashiwara \cite{kash} and Littelmann \cite{Li} defined a relevant
notion of crystals for the Demazure modules. Recall $ O(\lambda)=\{\sigma\cdot b_{\lambda}=b_{\sigma\lambda}\mid\sigma
\in W/W_\lambda\}$ the orbit of the highest weight vertex $b_{\lambda}$ of $B(\lambda
)$. Its elements,  uniquely determined by their weight, are called the keys of $B(\lambda)$. (In this sense we may identify $ O(\lambda)$ with $W\lambda$.) Given $\sigma, \sigma'\in W/W_\lambda$, we write $\sigma\le \sigma'$ for  the Bruhat order on the cosets in $W/W_\lambda$ to mean that their  unique minimal (maximal) coset representatives satisfy the same relation in the strong Bruhat order restricted to $W^\lambda$. We also write $b_{\sigma\lambda}\leq b_{\sigma^{\prime}\lambda}$ when
$\sigma\leq\sigma^{\prime}$ in $W/W_\lambda$.
From the dilatation of crystals \cite{kash} each vertex $b$ of $B(\lambda)$ carries a pair of keys $K^+(b)\ge K^-(b)$, right, respectively, left key of $b$, in  $
O(\lambda)$. For any
$\sigma\in W$, consider the \textsf{Demazure atom}
\begin{align}
\overline{{B}}_{\sigma}(\lambda)=\{b\in B(\lambda)\mid K^{+}%
(b)=b_{\sigma\lambda}\}, \mbox{ where $\overline{{B}}_{id}(\lambda)=\{b_{\lambda}\}$} .\label{DemazAtom}
\end{align}
%In particular, $\overline{{B}}_{id}(\lambda)=\{b_{\lambda}\}$.
 For any $\sigma\in W$, the
\textsf{opposite Demazure module},  is defined to be
 $V^{\sigma}(\lambda):=U_{q}(\mathfrak{gl}_{n}^{-})\cdot v_{\sigma\lambda}$,
for which we define the \textsf{opposite Demazure atom}%
\begin{align}
\overline{{B}}^{\sigma}(\lambda)=\{b\in B(\lambda)\mid K^{-}%
(b)=b_{\sigma\lambda}\}, \mbox{ where $\overline{{B}}^{\sigma_{0}}(\lambda)=\{b_{\sigma
_{0}\lambda}\}$}. \label{OpDemazAtom}
\end{align}
 By definition we have $\overline{{B}}_{\sigma}(\lambda)=\overline
{{B}}_{\sigma^{\prime}}(\lambda)$ and $\overline{{B}}^{\sigma}(\lambda)=\overline
{{B}}^{\sigma^{\prime}}(\lambda)$ whenever $\sigma$ and $\sigma
^{\prime}$ belong to the same left coset of $W/W_{\lambda}$.
We then get $B(\lambda)=%
{\textstyle\bigsqcup\limits_{\sigma\in W^\lambda}}
\overline{{B}}_{\sigma}(\lambda)={\textstyle\bigsqcup\limits_{\sigma\in W^\lambda}}
\overline{{B}}^{\sigma}(\lambda)$.
%Given $\sigma$ and $\sigma^{\prime}$ in $\mathfrak{S}_{n}^{\lambda}$, we shall
%write $b_{\sigma\lambda}\leq b_{\sigma^{\prime}\lambda}$ when
%$\sigma\leq\sigma^{\prime}$ (recall that $\leq$ denotes the strong Bruhat
%order on $\mathfrak{S}_{n}$).
\label{keys} The \textsf{Demazure crystal} ${B}_{\sigma}(\lambda)$ and its \textsf{opposite
Demazure crystal} ${B}^{\sigma}(\lambda)$ are then defined by
\begin{align}
{B}_{\sigma}(\lambda)  &  =%
{\textstyle\bigsqcup\limits_{ \sigma'\in W^\lambda,\,
\sigma^{\prime}W_\lambda\leq\sigma W_\lambda}}
\overline{{B}}_{\sigma^{\prime}}(\lambda)=\{b\in B(\lambda)\mid
K^{+}(b)\leq b_{\sigma\lambda}\}, \mbox{  ${B}_{id}(\lambda)=\{b_{\lambda}\}$}\label{B_w(lambda)}\\
{B}^{\sigma}(\lambda)  &  =%
{\textstyle\bigsqcup\limits_{\sigma^{\prime}\in W^\lambda
,\,\sigma W_\lambda\leq\sigma^{\prime}W_\lambda}}
\overline{{B}}^{\sigma^{\prime}}(\lambda)=\{b\in B(\lambda)\mid
K^{-}(b)\geq b_{\sigma\lambda}\}, \mbox{  ${B}%
^{\sigma_{0}}(\lambda)=\{b_{\sigma_{0}\lambda}\}$}.\label{B^ww(lambda)}
\end{align}
{In particular, we have   ${B}_{\sigma_{0}%
}(\lambda)={B}(\lambda)={B}^{id}(\lambda)$.} We then note that for a given $\lambda\in {\cal{P}}_n$,
\begin{align}\label{DemazCartesian}
{\textstyle\bigsqcup\limits_{\sigma\in W^\lambda}}\overline{{B}}^{\sigma}(\lambda)\times {B}_{\sigma}(\lambda)=\{(b,b')\in B(\lambda)\times B(\lambda):K^-(b)\ge K^+(b')\}\simeq B(2\lambda).
\end{align}
 We refer to \cite{CK}, for the translation of \eqref{DemazCartesian} to the crystal  of Lakshmibai-Seshadri paths.
The Demazure  crystals respectively atoms and  their opposite,  are connected
via the Lusztig-Sch\"{u}tzenberger involution $\iota$ on the crystal $B(\lambda)$, a realization of the action of the longest element of W on finite irreducible representations.
 The map ${\iota}$ is a set involution on $B(\lambda)$ reversing
the arrows, flipping the labels $i$ and $n-i$, {and reversing the weight}.
We then have $K^{-}(b)=\sigma_{0}.K^{+}({\iota}(b))$ and
we get
\begin{align}
{B}^{\sigma}(\lambda)   ={\iota}({B}_{\sigma_{0}\sigma
}(\lambda)),\mbox{ or equivalently }  {B}^{\sigma_0\sigma}(\lambda)=\iota {B}_{\sigma}(\lambda),
%={\{\tilde{e}_{n-i_{1}}^{k_{1}}\cdots\tilde{e}_{n-i_{\ell}%}^{k_{\ell}}(b_{\sigma_{0}\lambda})\mid(k_{1},\ldots,k_{\ell})\in
%\mathbb{Z}_{\geq0}^{\ell}\}\setminus\{0\}}\text{ and }\label{Demaz_Updow1}
\quad\overline{{B}}^{\sigma}(\lambda)    ={\iota}(\overline
{{B}}_{\sigma_{0}\sigma}(\lambda)). \label{Demaz_Updow2}%
\end{align}
 Demazure (resp. opposite) crystals can also be generated by the actions of the lowering (resp. raising) operators given
by the reduced words in $W^\lambda$ (resp. $\sigma_0W^\lambda\sigma_0$)  on the highest (resp. lowest)  vertex of $B(\lambda)$.
The Demazure character ${\kappa}_{\sigma,\lambda}(x)$ of the Demazure module
$V^{\sigma}(\lambda)$  satisfies ${\kappa}_{\sigma,\lambda}(x)=\sum_{b\in\mathrm{B}_{\sigma
}(\lambda)}x^{{wt}(b)}$, and
the \textsf{opposite Demazure character}
${\kappa}_{\lambda}^{\sigma}(x)$ for the opposite Demazure module
$V^{\sigma}(\lambda)$ satisfies ${\kappa}_{\lambda}^{\sigma}(x)%
=\sum_{b\in{B}^{\sigma}(\lambda)}x^{{wt}(b)}$. Using the
involution ${\iota}${ and \eqref{Demaz_Updow2}}, we have %
%\begin{align*} %{\kappa}_{\sigma,\lambda}(x)={\kappa}_{\mu}(x),\quad
${\kappa}_{\lambda}^{\sigma}(x_{1},\ldots,x_{n})%={\kappa}^{\mu}(x_{1},\ldots,x_{n})
$ $={\kappa}_{\sigma_{0}\sigma\lambda}(x_{n},\ldots,x_{1})$ and  %\label{character}
$\overline{{\kappa}}_{\lambda}^{\sigma}(x_{1},\ldots,x_{n}%
)%=\overline{{\kappa}}^{\mu}(x_{1},\ldots,x_{n})
$ $=\overline
{{\kappa}}_{\sigma_{0}\sigma\lambda}(x_{n},\ldots,x_{1})=\sum_{b\in
\overline{{B}}^{\sigma}(\lambda)}x^{{wt}(b)}.$ %\label{characteratom}$
%\end{align*}
Alternatively we may also label the Demazure crystals and the Demazure characters
of $B(\lambda)$  directly by the elements in  the orbit of $\lambda$, $W\lambda$. Given $\mu\in W\lambda$ where $\mu
=\sigma\lambda$ and $\sigma\in W^{\lambda}$, we  write
${B}_{\mu},$ ${B}^{\mu}=\iota{B}_{\sigma_0\mu}$ instead of ${B}_{\sigma}%
(\lambda),$ ${B}^{\sigma}(\lambda)$ respectively, and ${\kappa}_{\mu
}, {\kappa}^{\mu
}$ $={\kappa}_{\sigma_0\mu
}$, $ \overline{{\kappa}}_{\mu}, \overline{{\kappa}}^{\mu}=\overline{{\kappa}}_{\sigma_0\mu}$ instead of ${\kappa}_{\sigma
,\lambda}, {\kappa}^{\sigma}
_\lambda$ and $\overline{{\kappa}}_{\sigma,\lambda},\,\overline{{\kappa}}^{\sigma}_\lambda$ respectively.  The operators $D_{i}$ act on Demazure characters $\kappa_{\mu}$ and Demazure atoms $\overline \kappa_{\mu}$ as follows
\begin{eqnarray}\label{D_i(Demazure)}
D_{i}(\kappa_{\mu})=%
\begin{cases}
\kappa_{s_{i}\mu} & \mbox{if }\mu_{i}>\mu_{i+1}\\
\kappa_{\mu} & \mbox{if }\mu_{i}\leq\mu_{i+1},%
\end{cases}&&
D_{i}(\overline\kappa_{\mu})=%
\begin{cases}
\overline\kappa_{s_{i}\mu}+\overline\kappa_{\mu} & \mbox{if }\mu_{i}>\mu_{i+1}\\
\overline \kappa_{\mu} & \mbox{if }\mu_{i}=\mu_{i+1}\\%
 0,& \mbox{ else}.
\end{cases}
%\Leftrightarrow D_{i}(\kappa_{\mu})=\kappa_{\pi_{i}(\mu)}.
%\label{dempro}%
\end{eqnarray}
For $i\in [n-1]$, we define below $\Delta_{i}$ and $\dot{\Delta}_{i}$ as
operators on Demazure  respectively  Demazure atom crystals  to mimic the action of the operator $D_{i}$ on Demazure
 respectively on Demazure atom charaters \eqref{D_i(Demazure)}, and  we then always have
${char}({\Delta}_{i}({{B}}_{\mu}))=D_{i}%
({{\kappa}}_{\mu}), \,\mbox{ and }
{char}(\dot{\Delta}_{i}(\overline{{B}}_{\mu}))=D_{i}%
(\overline{{\kappa}}_{\mu}),$
\begin{eqnarray}
\Delta_{i}({B}_{\mu})=\left\{
\begin{array}
[c]{l}%
{B}_{s_{i}\mu}\text{ if }\mu_{i}>\mu_{i+1}\\
{B}_{\mu}\text{ otherwise},%
\end{array}
\right.&
\dot{\Delta}_{i}(\overline{{B}}_{\mu})=\left\{
\begin{array}
[c]{l}%
\Delta_{i}(\overline{{B}}_{\mu})=\overline{{B}}_{\mu}%
%TCIMACRO{\tbigsqcup }%
%BeginExpansion
{\textstyle\bigsqcup}
%EndExpansion
\overline{{B}}_{s_{i}\mu}\text{ if }\mu_{i}>\mu_{i+1}\\
\Delta_{i}(\overline{{B}}_{\mu})=\overline{{B}}_{\mu}\text{ if
}\mu_{i}=\mu_{i+1}\\
\emptyset\text{ if }\mu_{i}<\mu_{i+1}.
\end{array}
\right. \label{DeltaDot}%
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-symmetric Cauchy kernels, RSK on Young shapes and LPP}\label{nonsymetric} We now   consider last passage percolation
models based on the non-symmetric Cauchy kernel \eqref{NSCK} as  studied by
Lascoux in \cite{Las2} and its extensions to augmented stair shapes.  Demazure { crystals} with { their}  opposite Demazure  atoms, and  certain parabolic subcrystals  will describe the image of RSK,  as a bicrystal isomorphism,  restricted to stair shape, truncated stair shape and to augmented stair shape matrices. We detach the truncated case from the general Young shape  due to its more explicit as well interesting structure.
%See \S\ \ref{Subsec_DemaModules}.
\subsection{LPP, staircase and Demazure measure}
 \ The ordinary Cauchy identity \eqref{cauchy} is  then
replaced by its non-symmetric analogue%
\begin{equation}
\prod_{1\leq j\leq i\leq n}\frac{1}{1-x_{i}y_{j}}=\sum_{\mu\in\mathbb{Z}%
_{\geq0}^{n}}\overline{\kappa}^{\mu}( x)\kappa_{\mu}( y) \label{NSCK}%
\end{equation}
where $\overline{\kappa}^{\mu}( x)$ and $\kappa_{\mu}( y)$ are this time
(opposite) Demazure atoms and Demazure characters
 in the indeterminates $ x$ and $ y$ (with
$m=n$). These polynomials are not
symmetric in $ x$ and $ y$. They  correspond to characters of
representations for subalgebras of the enveloping algebra $U(\mathfrak{gl}%
_{n})$. It was proved in \cite{Las2} that the identity (\ref{NSCK}) can be
obtained by restricting the RSK correspondence $\psi$ to the set of lower
triangular matrices. (The convention of our paper differs from
that in \cite{Las2} which considers matrices with nonzero entries in positions
$(i,j)$ with $1\leq i+j\leq n+1$ rather than lower-triangular matrices.)%
\ Since then, other proofs have been proposed using combinatorial objects  which explicitly carry the pairs of right and left keys \cite{Fu}.
More precisely,
\cite[Theorem 3, Corollary 2]{AE} uses the combinatorics of Mason's semiskyline augmented fillings \cite{Mas}, and \cite{CK} uses
the combinatorics of crystal bases, in particular, the combinatorial model of Lakshmibai-Seshadri paths \cite{Li}.
Recently Assaf-Schilling provided an explicit tableau crystal for Mason’s semiskyline augmented fillings \cite{Mas}, { replacing the former objects by  equivalent ones, termed} semistandard key tableaux  (see \cite{AGL} and references therein).  Here we { stand in}   the tableau model for $\mathfrak{gl}_n$ crystals where  one has the effective Lascoux's jeu de taquin procedure \cite{Fu} to compute the right key $K^+(T)$ and left key $K^-(T)$  of a
tableau $T$.


Let $D$ be any subset of $[n]\times [m]$ and write
$\mathcal{M}_{m,n}^{D}$ for the subset of $\mathcal{M}_{m,n}$ containing the
matrices $A$ such that $a_{i,j}\neq0$ only if $(i,j)\in D$.  For $D$ in general, the
set $ \mathcal{M}_{m,n}^{D}$ is not stable for the $\mathfrak{gl}%
_{m}\times\mathfrak{gl}_{n}$-crystals operators. Nevertheless, when $D$
corresponds to the Young diagram of a fixed partition $\Lambda$, see \eqref{picture},  $D=D_{\Lambda}$ is
stable under the action of the crystal raising operators. When $m=n$ and $\varrho
=(n,n-1,\ldots,1)$, we get in matrix coordinates
$
D_{\varrho}=\{(i,j)\mid1\leq j\leq i\leq n\}.
$
Then the bijection $\psi$ \eqref{Th_K2} restricts to a bijection
from the set $\mathcal{M}_{m,n}^{D_\varrho}$ of $n\times n$ lower triangular matrices to the set of pairs $(P, Q)$ of semistandard Young tableaux of the same shape on the alphabet $[n]$ such that  $ K^-(P)\ge K^+(Q)$ (entrywise comparison). (See also \cite[Corollary 2]{AE} for the Knuth version of RSK.)   This means that the image of this restriction, for a fixed $\lambda\in {\cal{P}}_n $, is ${\textstyle\bigsqcup\limits_{\sigma\in W^\lambda}}
\overline{{B}}^{\sigma}(\lambda)\times{B}_{\sigma}(\lambda)$ \eqref{DemazCartesian}.
Thus the restriction of RSK correspondence $\psi$ to $D_\varrho$ gives
\begin{align}\label{Th_LKO}
\psi:\mathcal{M}_{n,n}^{D_{\varrho}}&\rightarrow%
%TCIMACRO{\tbigsqcup \limits_{\lambda\in\mathcal{P}_{n}}}%
%BeginExpansion
{\textstyle\bigsqcup\limits_{\lambda\in\mathcal{P}_{n}}}
{\textstyle\bigsqcup\limits_{\sigma\in W^\lambda}}
\overline{{B}}^{\sigma}(\lambda)\times{B}_{\sigma}(\lambda)\\
A&\mapsto \psi(A)=(P(A),Q(A)): K^+(Q(A))\le K^-(P(A)),\label{StaircaseKernel}
\end{align}
where $B_\sigma(\lambda)$ is a Demazure crystal \eqref{B_w(lambda)} and  $\overline{{B}}^{\sigma}(\lambda)$  its opposite Demazure atom \eqref{OpDemazAtom}.
This time, we
only consider independent random variables $W_{i,j}$ when $1\leq j\leq i\leq
n$ with geometric distributions as in (\ref{geometricw(i,j)}).\ This defines a
lower triangular random square matrix $\mathcal{L}$ with nonnegative integer entries. In this model we
consider paths from position $(1,n)$
to position $(n,1)$ where only the entries in the lower part of $A$ contribute
to the length of the paths.\ We  define the random variable
$L={p}\circ\mathcal{L}$ and  determine its law. Since
\eqref{Th_LKO} gives a bijective correspondence obtained as the
restriction of the RSK map $\psi$  \eqref{Th_K2} to lower triangular matrices, the value of $L$ still corresponds to the length of the
largest part of the partitions on the right hand side of
(\ref{StaircaseKernel}).
\begin{thm}For any  $k\in\mathbb{Z}_{\ge 0}$, we have the law%
\begin{align}\label{stair}
\mathbb{P}(L=k)&=\prod_{1\leq j\leq i\leq n}(1-u_{i}v_{j})\sum_{\mu\in\mathbb{Z}_{\geq0}^{n}\mid\max(\mu
)=k}{\overline{\kappa}}^{\mu}(u_1,\dots,u_n){\kappa}_{\mu}(v_1,\dots,v_n).
\end{align}
\end{thm}
This law was also obtained by Baik-Rains \cite[Section 4]{BR,BZ1} when $u_i=v_i$.  In this case, \eqref{DemazCartesian}, and \eqref{NSCK} with $x_i=y_i$, together give  a refinement of  a Littlewood identity: $\prod_{1\leq j\leq i\leq n}(1-x_{i}x_{j})^{-1}=$ $\sum_{\mu\in\mathbb{Z}%
_{\geq0}^{n}}\overline{\kappa}^{\mu}( x)\kappa_{\mu}( x)=$ $\sum_{\lambda\in\mathcal{P}_n}s_{2\lambda}(x)$.  In \cite{BZ1} it is called a law in the point-to-line last passage percolation in zero temperature limit. However this formula is not produced in  \cite{BZ1} by the geometric RSK  but rather one in terms of a symplectic Cauchy like identity.
\subsection{Main results: LPP on Young shapes and Demazure measure}
Lascoux \cite{Las2} also
established generalizations of the formula (\ref{NSCK}) where positions with
nonzero entries are allowed in the matrices outside their lower triangular
part.\ These  augmented staircase formulas below $(*)$ were then obtained just by
computations on polynomials and thus not related to the RSK
correspondence. This connection was partially done in \cite{AO2} where
certain truncated staircases formulas are proved to be
compatible with the RSK correspondence using the combinatorics of semiskyline augmented fillings \cite{Mas}. 
More precisely, this applies to the case where  nonzero entries are authorized only in
positions $(i,j)$ with $n-p\leq i\leq j\leq q$, for $p$ and $q$
two nonnegative integers such that $n\geq q\geq p\geq1$.
 We consider the Young
diagram
 $
D_{p,q}=\{(i,j)\mid n-p+1\leq i\leq n,1\leq j\leq q\}\cap D_{\varrho}%
$
defined by using the matrix coordinates $(i,j)$. It is the intersection
of $D_{\varrho}$ with a quarter of plane defined by the lines $i=p$ and $j=q$
(in Cartesian coordinates). When $~n-p+1\leq q$, we get the Young diagram
$
D_{p,q}=D_{\Lambda(p,q)}\text{ with }\Lambda(p,q)=(q^{n-q+1},q-1,\dots,n-p+1)$,  and $D_{n,n}=D_{\Lambda(n,n)}=D_{\varrho}$. Below one illustrates the truncated Young shape $D_{\Lambda(p,q)}$, in green, fitting the $p$
by $q$ rectangle so that the staircase $D_{\varrho}$ of size $n$, in red, is the
smallest one containing $D_{\Lambda(p,q)}$. If $p\leq q$, $D_{(p,p-1,\ldots,1)}$ is
the biggest staircase inside $D_{\Lambda(p,q)}$.


 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\label{fig:trunc}
\begin{tikzpicture} [scale=0.18] \filldraw[color=green!25] (-1,0)
rectangle (3,5); \filldraw[color=green!25] (3,0)
rectangle (5,2);  \filldraw[color=green!25] (-1,0)
rectangle (3,5); \filldraw[color=green!25] (3,1) rectangle
(5,3);\filldraw[color=green!25] (3,3) rectangle (4,4);
\draw[line width=0.5pt](-1,0)-- (-1,5);
\draw[line width=0.5pt, red](-1,6)-- (-1,8);
\draw[line width=0.5pt, red](-1,5)-- (-1,6);
\draw[line width=0.5pt](0,0)-- (0,5);
\draw[line width=0.5pt, red](0,8)-- (0,7);
\draw[line width=0.5pt, red](1,7)-- (1,6);
\draw[line width=0.5pt](1,0)-- (1,5);
\draw[line width=0.5pt](2,0)-- (2,5);
\draw[line width=0.5pt](3,0)-- (3,5);
\draw[line width=0.5pt](4,0)-- (4,4);
\draw[line width=0.5pt](-1,0)-- (5,0);
\draw[line width=0.5pt, red](4,0)-- (7,0);
\draw[line width=0.5pt](-1,1)-- (5,1);
\draw[line width=0.5pt](4,0) rectangle  (5,3);
\draw[line width=0.5pt, red](7,1)-- (6,1);
\draw[line width=0.5pt](-1,2)-- (5,2);
\draw[line width=0.5pt, red](6,2)-- (5,2);
\draw[line width=0.5pt](-1,3)-- (4,3);
\draw[line width=0.5pt](-1,4)-- (4,4);
\draw[line width=0.5pt](-1,5)-- (3,5);
\draw[line width=0.5pt,red](1,6)-- (2,6);
\draw[line width=0.5pt,red](2,5)-- (2,6);
\draw[line width=0.5pt, red](7,0)-- (7,1);
\draw[line width=0.5pt, red](6,2)-- (6,1);
\draw[line width=0.5pt, red](5,3)-- (5,2);
\draw[line width=0.5pt, red](5,3)-- (4,3);
\draw[line width=0.5pt](4,2)rectangle (5,3);
\draw[line width=0.5pt, red](-1,8)-- (0,8);
\draw[line width=0.5pt, red](1,7)-- (0,7);
\draw[line width=0.5pt] (-4,0) -- (-4,8);
\draw[line width=0.5pt] (-4,0) -- (-3.8,0);
\draw[line width=0.5pt] (-4,8) -- (-3.8,8);
\draw[line width=0.5pt] (-2,0) -- (-2,5);
\draw[line width=0.5pt] (-2,0) -- (-1.8,0);
\draw[line width=0.5pt] (-2,5) -- (-1.8,5);
\node at (-2.7,2.5){$\scriptstyle p$}; \node at (-4.7,4){$\scriptstyle n$};
\draw[line width=0.5pt] (-1,-1) -- (5,-1);
\draw[line width=0.5pt] (-1,-1) -- (-1,-0.8);
\draw[line width=0.5pt] (5,-1) -- (5,-0.8);\node at (2.5,-1.8){$\scriptstyle q$};
\end{tikzpicture}$


We write $B_{p}(\lambda)$ for the
subcrystal of the $\mathfrak{gl}_{n}$-crystal $
B(\lambda,0^{n-p})$ with $\lambda\in \mathcal{P}_p$, obtained by keeping only the vertices connected to its
highest weight vertex  by $i$-arrows with $i\in [p-1]$. Given
$u\in {\mathfrak{S}}_p$,  ${B}_{p,u}(\lambda
),{B}_{p}^{u}(\lambda),\overline{{B}}_{p,u}(\lambda)$ and
$\overline{{B}}_{p}^{u}(\lambda)$ denote the Demazure, its opposite, respectively, atom and its opposite crystals
 associated to $u$ in the $\mathfrak{gl}_{p}$-crystal $B_{p}(\lambda)$. See Example \ref{example} and  \eqref{P}, \eqref{Q}.
The restriction of the map
$\psi$ from $\mathcal{M}_{n,n}^{D\varrho}$  \eqref{StaircaseKernel} to $\mathcal{M}_{n,n}%
^{D_{\Lambda(p,q)}}$ gives %
\begin{align*}
\psi(\mathcal{M}_{n,n}^{D_{\Lambda(p,q)}})=%
%TCIMACRO{\tbigsqcup \limits_{\lambda\in\mathcal{P}_{n}}}%
%BeginExpansion
{\textstyle\bigsqcup\limits_{\lambda\in\mathcal{P}_{n}}}
%EndExpansion%
%TCIMACRO{\tbigsqcup \limits_{\sigma\in\mathfrak{S}_{n}^{\lambda}}}%
%BeginExpansion
{\textstyle\bigsqcup\limits_{\sigma\in W^{\lambda}}}
%EndExpansion
\overline{{B}}^{\sigma}(\lambda)\cap B^{p}(\lambda)\times
{B}_{\sigma}(\lambda)\cap B_{q}(\lambda)=%
%TCIMACRO{\tbigsqcup \limits_{\mu\in\mathbb{Z}_{\geq0}^{n}}}%
%BeginExpansion
{\textstyle\bigsqcup\limits_{\mu\in\mathbb{Z}_{\geq0}^{n}}}
%EndExpansion
\overline{{B}}^{\mu}\cap B^{p}(\lambda)\times{B}_{\mu}\cap
B_{q}(\lambda).
\end{align*}
By the Borel-Weil theorem,  Demazure crystals and  Schubert varieties are in natural correspondence.
 Let $\sigma\in \mathfrak{S}_{n}$ and $\sigma_0^{[q]}$ be the longest element of $\mathfrak{S}_{q}$. From the  Billey--Fan--Losonczy parabolic map (see \cite[Algorithm 3.1, Proposition 3.4]{AGL}  and references therein) the set $\{v\in\mathfrak{S}_{q}\mid v\leq\sigma\}$ has
a unique maximal element $\sigma^{I_{q}}$ for the Bruhat order $\le$ in $W$.  For $\sigma\in W^\lambda$, the  intersections
$S_{\sigma_0^{[q]}}\cap S_\sigma=S_{\sigma^{I_{q}}}$ and  $  {B}_{\sigma}(\lambda)\cap B_{q}(\lambda)={B}_{\sigma}(\lambda)\cap B_{\sigma_0^{[q]}}(\lambda)=\mathrm{B}_{q,\sigma^{I_{q}}}(\lambda)$ translate into each other, \label{parabolic}
  where $S_\sigma$ is a Schubert variety of the flag variety $G/B$ 
  with $B$ a Borel subgroup  of the reductive group $G$ with  Weyl group $W$  \cite[Chapter III]{Fu}.
However  $\overline{{B}}^{\sigma
}(\lambda)\cap B^{p}(\lambda)=\emptyset$ unless $\sigma\in\sigma
_{0}\mathfrak{S}_{p}^{\lambda}$, $\lambda\in\mathcal{P}_{p}$ and then
$\overline{{B}}^{\sigma}(\lambda)\cap B^{p}(\lambda)={\iota
}\overline{{B}}_{p,\sigma_{0}\sigma}(\lambda)$ \cite{AGL}. In
this case  ${B}_{\sigma}(\lambda)\cap B_{q}(\lambda)={B}%
_{q,\sigma^{I_{q}}}(\lambda)$.
The restriction of the
RSK correspondence $\psi$ to $\mathcal{M}_{n,n}^{D_{\Lambda(p,q)}}$ then gives  a
one-to-one correspondence%
\begin{align}\label{Th_truncatedRectangle}
\psi:\mathcal{M}_{n,n}^{D_{\Lambda(p,q)}}&\rightarrow%
{\textstyle\bigsqcup\limits_{\mu\in\mathbb{Z}_{\geq0}^{p}}}
%EndExpansion
{\iota}(\overline{{B}}_{p,\mu})\times{B}_{q,\widetilde
{\mu}}, \mbox{
and} \\
\prod_{(i,j)\in D_{\Lambda(p,q)}}\frac{1}{1-x_{i}y_{j}}&=\sum_{(\mu_{1}%
,\ldots,\mu_{p})\in\mathbb{Z}_{\geq0}^{p}}\overline{{\kappa}}_{(\mu
_{p},\ldots,\mu_{1})}(x_{n},\ldots,x_{n-p+1}){\kappa}_{\widetilde{\mu}%
}(y_{1},\ldots,y_{q}),
\end{align}
where for each $\mu\in \mathbb{Z}_{\geq0}^{p}$, the vector $\tilde \mu=(\sigma_{0}\tau)^{I_{q}}(\lambda,0^{q-p},0^{n-q})$ with $\tau\in\mathfrak{S}_{p}%
^{\lambda}$  is such that $\mu=\tau\lambda$. It can also   be  explicitly computed by a simple algorithm in \cite[Theorem 3.20]{ AE,AGL} (see also examples in \cite[Section 3.1]{AGL}).
 One can then similarly
use  \eqref{Th_truncatedRectangle} to study the percolation model on
random matrices $\mathcal{T}_{p,q}$ with nonnegative random integer
coefficients having zero entries in each position $(i,j)$ such that $i\leq
n-p$ and $j>q$. Each random variable $W_{i,j}$ with $i\geq n-p+1$ and $j\leq
q$ follows a geometric distribution of parameter $u_{i}v_{j}$. Using the same
arguments as before, we obtain the law of the random
variable $T_{p,q}={p}\circ\mathcal{T}_{p,q}$.
\begin{thm}For any nonnegative integer $k$, we have for $v=(v_{1},\ldots,v_{q})$%
\[
\mathbb{P}(T_{p,q}=k)=\prod_{(i,j)\in D_{\Lambda(p,q)}}(1-u_{i}v_{j})\sum_{(\mu_{1},\ldots,\mu_{p})\in\mathbb{Z}%
_{\geq0}^{p}\mid\max(\mu)=k}\overline{{\kappa}}_{(\mu_{p},\ldots,\mu_{1}%
)}(u_{n},\ldots,u_{n-p+1}){\kappa}_{\widetilde{\mu}}(v).
\]
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsubsection{LPP, augmented staircases and Demazure measure}
In {\cite{Las2}} Lascoux gave other non-symmetric Cauchy type identities for any
partition $\Lambda\in\mathcal{P}_{n}$. One considers the largest
staircase $\rho_{\Lambda}=(m,m-1,\ldots,1)$ contained in the Young diagram of
$\Lambda$. Then one  chooses a box  at position $(i_{0},j_{0})$, {in
Cartesian coordinates}, in the augmented staircase $(m+1,m,\ldots,1)$ which is
not in $\Lambda$. The diagonal $L_{i,j}:j-i=j_{0}-i_{0}$, {in Cartesian
coordinates}, cuts $\Lambda$ in a northwest part and a southeast part
corresponding to the boxes above and below $L_{i,j}$, respectively.\ Now fill
the boxes $(i,j)$, {in the $n\times n$ matrix convention}, of the $NW$ part of
$\Lambda$ by $ n-i$ (\textit{i.e.}, by the { $n\times n$ matrix { reverse row index (equivalently counting rows from bottom to top)} minus one)},
and the boxes $(i,j)$ of the $SE$ part by $j-1$ (\textit{i.e.}, {by the index of the
column minus one}). Let $\sigma(\Lambda,NW)=s_{i_{1}}\cdots s_{i_{a}}$ be the
element of $W$ {where the word $i_{1}\cdots{i_{a}}$ is obtained
from} right to left column reading of the $NW$ part of $\Lambda,$ each column
being {read} from top to bottom. Similarly, let $\sigma(\Lambda,SE)=s_{j_{1}%
}\cdots s_{j_{b}}$ be the element of $W$ {where the word
${j_{1}}\cdots j_{b}$ is obtained from} {top to bottom row reading} of the
$SE$ part of $\Lambda,$ each row being {read} from right to left.
For instance, let $n=8$ and $\Lambda=(7,4,2,2,2)$. Take $(i_{0},j_{0})=(3,3)$ (the box with $\blacktriangle$). Hence
$m=4$, $\rho_{\Lambda}=(4,3,2,1)$,   and $\sigma(\Lambda,NW)= s_{\color{blue}4}s_{\color{blue}3}s_{\color{blue}4}$,  $\sigma(\Lambda,SE)=s_{\color{magenta}3}s_{\color{magenta}6}s_{\color{magenta}5}s_{\color{magenta}4}$,
\begin{align}\label{picture}
\YT{0.14in}{}{
			{{\color{blue}\scriptstyle 4},{\color{blue}\scriptstyle 4}},
			{{\blacksquare},{\color{blue}\scriptstyle 3}},
			{{\blacksquare},{\blacksquare},{\blacktriangle }},
            {{\blacksquare }, {\blacksquare}, {\blacksquare}, {\color{magenta}\scriptstyle 3}},
            {{\blacksquare}, {\blacksquare}, {\blacksquare}, {\blacksquare}, {\color{magenta}\scriptstyle 4},
{\color{magenta}\scriptstyle 5},{\color{magenta}\scriptstyle 6}}}
\end{align}


The following identity  is established in {\cite{Las2}} and reproved { for near stair shapes} in
\cite{AO2},
\begin{align*}(\ast)\;
\prod_{(i,j)\in\Lambda}\frac{1}{1-x_{i}y_{j}}=\sum_{(\mu_{1},\ldots,\mu
_{m})\in\mathbb{Z}^{m}}D_{\sigma(\Lambda,NW)}\overline{{\kappa}}%
_{(\mu_{m},\ldots,\mu_{1})}(x_{n},\ldots,x_{n-m+1})D_{\sigma(\Lambda
,SE)}{\kappa}_{(\mu_{1},\ldots,\mu_{m})}(y),\label{Th_ASC}
\end{align*}
where $y=(y_{1},\ldots,y_{m})$, and $D_{\sigma(\Lambda,NW)}=D_{\color{blue}{i_{1}}}\cdots D_{\color{blue}{i_{a}}}$, $D_{\sigma
(\Lambda,SE)}=D_{\color{magenta}{j_{1}}}\cdots D_{\color{magenta}{j_{b}}}$  are compositions of Demazure operators \eqref{D_i(Demazure)}.
 \begin{thm}The restriction of
the RSK correspondence $\psi$ to $\mathcal{M}_{n,n}^{D_{\Lambda}}$ gives the
one-to-one correspondence
\begin{equation}
\psi:\mathcal{M}_{n,n}^{D_{\Lambda}}\rightarrow%
%TCIMACRO{\tbigsqcup \limits_{(\mu_{1},\ldots,\mu_{m})\in\mathbb{Z}_{\geq0}%
%^{m}}}%
%BeginExpansion
{\textstyle\bigsqcup\limits_{(\mu_{1},\ldots,\mu_{m})\in\mathbb{Z}_{\geq0}%
^{m}}}
%EndExpansion
{\iota}\left(  \dot{\Delta}_{\sigma(\Lambda,NW)}%
(\overline{{B}}_{(\mu_{m},\ldots,\mu_{1})})\right)  \times
\Delta_{\sigma(\Lambda,SE)}\left(  {B}_{(\mu_{1},\ldots,\mu_{m}%
)}\right)  \label{Image}%
\end{equation}
where $\Delta_{\sigma(\Lambda,SE)}=\Delta_{j_{1}}\cdots\Delta_{j_{b}}$ and
$\dot{\Delta}_{\sigma(\Lambda,NW)}=\dot {\Delta}_{i_{1}%
}\cdots\dot{\Delta}_{i_{a}}$ \eqref{DeltaDot}. (As usual $\emptyset\times U$
$=\emptyset$.)
\end{thm}
Now, for a fixed partition
$\Lambda$ in $\mathcal{P}_{n}$, we consider random matrices $\mathcal{A}%
_{\Lambda}$ with nonnegative random integer coefficients having zero entries
in each position $(i,j)$ such that $(i,j)\notin\Lambda$. Here again each
random variable $W_{i,j}$ for $(i,j)\in\Lambda$ follows a geometric
distribution of parameter $u_{i}v_{j}$. Define the random variable
$A_{\Lambda}={p}\circ\mathcal{A}_{\Lambda}$. Then, by
$(\ast)$ and \eqref{Image}, we get the law of $A_{\Lambda}$.
\begin{thm}
For any nonnegative integer $k$,
\begin{align*}
&\mathbb{P}(A_{\Lambda}=k)=\prod_{(i,j)\in D_{\Lambda}}(1-u_{i}v_{j}).\\
&.\sum_{(\mu_{1},\ldots,\mu_{m})\in\mathbb{Z}^{m}\mid\max(\mu
)=k}D_{\sigma(\Lambda,NW)}\overline{{\kappa}}_{(\mu_{m},\ldots,\mu_{1}%
)}(u_{n},\ldots,u_{n-m+1})D_{\sigma(\Lambda,SE)}{\kappa}_{(\mu_{1}%
,\ldots,\mu_{m})}(v_{1},\ldots,v_{m}).
\end{align*}
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An example for RSK on augmented stair shapes}\label{example}
 Let us resume to the setting of \eqref{picture} with
 $ n=8$,  $ \Lambda=(7,4,2,2,2)$,  %$m=4$, and  %%$\varrho_{\Lambda}=(4,3,2,1)$ as in \eqref{picture}
and $\sigma(\Lambda,NW)=s_{\color{blue}4}s_{\color{blue}3}s_{\color{blue}4}$, $\sigma(\Lambda,SE)=s_{\color{magenta}3}%
s_{\color{magenta}6}s_{\color{magenta}5}s_{\color{magenta}4}$.
Let $\psi$ be the RSK { restricted} to $\mathcal{M}_{8,8}^{D_{\Lambda}}$.
Then \eqref{Image} gives for $m=4$
\begin{align*}\psi:\mathcal{M}_{8,8}^{D_{(7,4,2,2,2)}}&\rightarrow%
{\textstyle\bigsqcup\limits_{(\mu_{1},\ldots,\mu_{4})\in\mathbb{Z}_{\geq0}%
^{4}}}
%EndExpansion
{\iota}\left(  \dot{\Delta}_4 \dot{\Delta}_3 \dot{\Delta}_4 (\overline{{B}}_{(\mu_{4},\ldots,\mu_{1})})\right)  \times
{\Delta}_3{\Delta}_6{\Delta}_5{\Delta}_4\left(  {B}_{(\mu_{1},\ldots,\mu_{4}%
)}\right)\\
A&\mapsto \psi(A)=(P,Q).
\end{align*}
Let
$\scriptstyle
A=\left(\begin{smallmatrix}
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0\\
 \textbf{\color{blue}0}&\textbf{\color{blue}1}&0&0&0&0&0&0\\
\textbf{1}&\textbf{\color{blue}1}&0&0&0&0&0&0\\
\textbf{0}&\textbf{0}&0&0&0&0&0&0\\
\textbf{2}&\textbf{0}&\textbf{1}&\textbf{\color{magenta}1}&0&0&0&0\\
\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{\color{magenta}1}&\textbf{\color{magenta}0}&\textbf{\color{magenta}2}&0\\
\end{smallmatrix}\right)  \in \mathcal{M}_{8,8}^{D_{\Lambda}}$ encoded  as a tensor product of row tableaux $577\otimes 45\otimes 7\otimes 7\otimes  8\otimes\emptyset\otimes 88\otimes\emptyset$ on the
alphabet $[8]$ where  $a_{i,j}$ is the number of letters $i$ in the tensor
$j$-th component.
 One then applies the column insertion procedure from left to right. This means
that we begin by reading the first column (of $A$)  $775$  and  compute the column
insertions $5\rightarrow 7\rightarrow 7$ to get $577$ then read the   second column $54$ and compute the column insertion
$4\rightarrow 5\rightarrow 577$ to get $45577$, then  $7\rightarrow 45577$ to get
$\scriptstyle\begin{smallmatrix}7&&&&\\
4&5&5&7&7
\end{smallmatrix}$,
and  eventually get the tableau $P$ below. The  "recording tableau" $Q$ is obtained by filling {with} letters
$j$ the new boxes appearing during the insertion of column $j$ of $A$,
\begin{align}\label{P}
&\small P=\YT{0.13in}{}{
			{{\scriptstyle 8},{\scriptstyle 8}},
			{{\scriptstyle 7},{\scriptstyle 7},{\scriptstyle 8}},
			{{\scriptstyle 4},{\scriptstyle 5},\scriptstyle 5,{\scriptstyle 7},{\scriptstyle 7}}}
&\small
K^-(\small P)=\YT{0.13in}{}{
			{{\scriptstyle 8},{\scriptstyle 8}},
			{{\scriptstyle 7},{\scriptstyle 7},{\scriptstyle 8}},
			{{\scriptstyle 4},{\scriptstyle 4},{\scriptstyle 4},{\scriptstyle4},\scriptstyle 4}}
=\small K(0^3,5,0^2,2,3)\\\label{Q}
&\small Q=\YT{0.13in}{}{
			{{\scriptstyle 5},{\scriptstyle 7}},
			{{\scriptstyle 3},{\scriptstyle 4},{\scriptstyle 7}},
			{{\scriptstyle 1},{\scriptstyle 1},\scriptstyle 1,{\scriptstyle 2},{\scriptstyle 2}}}
 &\quad \small K^+(\small Q)=\YT{0.13in}{}{
			{{\scriptstyle7},{\scriptstyle 7}},
			{{\scriptstyle 4},{\scriptstyle 4},{\scriptstyle 7}},
			{{\scriptstyle 2},{\scriptstyle 2},\scriptstyle 2,{\scriptstyle 2},{\scriptstyle 2}}}
=\small K(0,5,0,2,0^2,3,0).
 \end{align}
We show that there exists $\mu=(\mu_{1},\mu_{2},\mu_{3},\mu_{4})\in\mathbb{Z}^4_{\geq0}$ such that  $\psi(A)=(P,Q)\in \iota(\dot{\Delta}_4\dot{\Delta}_3\dot{\Delta}_4%
\overline{{B}}_{(\sigma_0\mu,0^4)})\times {\Delta}_3{\Delta}_6{\Delta}_5{\Delta}_4B_{(\mu,0^4)},
$
 where  $\sigma_0\in \mathfrak{S}_4$ and $\iota $ is the Sch\"utzenberger (evacuation) involution. 
From \eqref{DeltaDot} one has
$\iota (\dot{\Delta}_4\dot{\Delta}_3\dot{\Delta}_4%
\overline{{B}}_{(\mu_{4},\ldots,\mu_{1},0^4)})=$
\begin{align}&=\begin{cases}\iota\overline{{B}}_{(\mu_{4},\mu_3,\mu_2,0,0^4)} \bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,0,\mu_2,0^4)}\bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,0^2,\mu_2,0^3)},\mbox{ if $\mu_2>\mu_1=0$ \quad $(\ast\ast)$}\nonumber\\
\iota\overline{B}_{(\mu_{4},\mu_3,0,0,0^4)}, \mbox{ if $\mu_1=\mu_2=0$}\nonumber\\
\iota\overline{B}_{(\mu_{4},\mu_3,\mu_2,\mu_2,0^4)} \bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,\mu_2,0,\mu_2, 0^3)}\bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,0,\mu_2,\mu_2,0^3)},\mbox{ if $\mu_1=\mu_2>0$}\nonumber\\
 \emptyset,\; \mbox{ if  $\mu_1>\mu_2\ge 0$}\\
\iota\overline{{B}}_{(\mu_{4},\dots,\mu_{1},0^4)}\bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,\mu_1,\mu_{2},0^4)}\bigsqcup  \iota\overline{{B}}_{(\mu_{4},\mu_3,\mu_2,0,\mu_{1},0^3)}\bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,0,\mu_2,\mu_{1},0^3)}
\bigsqcup \iota\overline{{B}}_{(\mu_{4},\mu_3,0,\mu_1,\mu_{2},0^3)} \nonumber\\ \bigsqcup\iota\overline{{B}}_{(\mu_{4},\mu_3,\mu_1,0,\mu_{2},0^3)}, \mbox{  if $\mu_2>\mu_1>0$}.
\end{cases}
\end{align}
Then, by \eqref{OpDemazAtom},
 $K^-(P)=K(0^3,5,0^2,2,3)\Leftrightarrow
P\in \overline{{B}}^{(0^3,5,0^2,2,3)}=
\iota
\overline{{B}}_{(3,2,0^2,5,0^3)}$, and we are in case $(\ast\ast)$,  where $\mu_2=5>\mu_1=0,\,\mu_3=2,\,\mu_4=3$.
Hence, $\mu=(0,5,2,3)$ and $\iota(\dot{\Delta}_4\dot{\Delta}_3\dot{\Delta}_4%
\overline{{B}}_{(3,2,5,0,0^4)})=\iota\overline{{B}}_{(3,2,5,0,0^4)} \bigsqcup \iota\overline{{B}}_{(3,2,0,5,0^4)}\bigsqcup \iota\overline{{B}}_{(3,2,0^2,5,0^3)}$. Therefore, by the LHS of \eqref{DeltaDot},
     $ {\Delta}_3{\Delta}_6{\Delta}_5{\Delta}_4B_{(\mu,0^4)}= B_{(0,5,0,2,0,0,3,0)} $.
Indeed
$K_+(Q)\le K(0,5,0,2,0^2,3,0)$ and from \eqref{B^ww(lambda)}, $Q\in B_{(0,5,0,2,0^2,3,0)}.$ Hence,
$(P,Q)\in \overline{{B}}^{(0^3,5,0^2,2,3)}\times {B}_{(0,5,0,2,0^2,3,0)}$ and $\psi(A)\in $ $\iota(\dot{\Delta}_4\dot{\Delta}_3\dot{\Delta}_4%
\overline{{B}}_{(3,2,5,0,0^4)})\times {\Delta}_3{\Delta}_6{\Delta}_5{\Delta}_4B_{(0,5,2,3,0^4)}.
$

  




\acknowledgements{ Authors thank to S. Assaf, E. Feigin, D. Orr and T. Scrimshaw for valuable discussions.}


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