


\section{The Matrix Ansatz and Hecke algebra}
\label{MAHA}

In this section we sketch an alternative proof of \cref{ASEP-f} using 
the Matrix Ansatz.


We recall the construction of Prolhac, Evans and Mallick \cite{ProlhacEvansMallick09} to compute
the stationary distribution of a state of the $L$-ASEP.
For $1\le i\le n$ and $0\le J,M<L$, they define
 matrices $a^{(L)}_{JM}$  by
\begin{eqnarray*}
a^{(L)}_{JM}=0 &{\rm for}\ 0 < M < J\\
a^{(L)}_{JM}=A^{\otimes (J-1)}\otimes \delta
\otimes 1^{\otimes (M-J-1)}\otimes \epsilon \otimes 1^{\otimes (L-M-1)}&{\rm for}\ 0 < J<M < L\\
a^{(L)}_{0M}=1^{\otimes (M-1)}\otimes \epsilon \otimes 1^{\otimes (L-M-1)}&{\rm for}\ 0<M < L\\
a^{(L)}_{JJ}=A^{\otimes (J-1)} \otimes 1^{\otimes (L-J)}
&{\rm for}\ 0 < J < L\\
a^{(L)}_{00}= 1^{\otimes (L-1)}
&\\
a^{(L)}_{J0}=A^{\otimes (J-1)}\otimes \delta
\otimes  1^{\otimes (L-J-1)}
&{\rm for}\ 0 < J < L\\
a^{(L)}_{L0}=A^{\otimes (L-1)};
&
\end{eqnarray*}
with $A$, $\delta$ and $\epsilon$ semi-infinite matrices
(with rows and columns indexed by the non-negative integers)
defined as follows
\begin{itemize}
\item $A_{i,j}=t^i$ if $i=j$ and 0 otherwise;
\item $\epsilon_{i,j}=1$ if $i=j-1$ and 0 otherwise;
\item $\delta_{i,j}=1-t^j$ if $i=j+1$ and 0 otherwise.
\end{itemize}

Then for $J\le L$ and $L>1$ they define
\begin{equation}
X_J^{(L)}=\sum_M a_{JM}^{(L)}\otimes X_{M^-}^{(L-1)}
\end{equation}
with $M^-={\rm max}(0,M-1)$ and $X_1^{(1)}=X_0^{(1)}=1$.

Let $\mu=(\mu_1,\ldots ,\mu_n)$ be a composition into parts less than or equal to $L$.
We define
\begin{equation}
Y^{(L)}_\mu={\rm Tr}\left(\prod_{i=1}^n X_{\mu_i}^{(L)}\right).
\end{equation}
Let $\lambda=(\lambda_1,\ldots ,\lambda_n)$ be a partition into parts less than 
or equal to $L$.
\begin{equation*}
Z^{(L)}_\lambda=\sum_{\mu} Y^{(L)}_\mu
\end{equation*}
where the sum is over all the distinct compositions $\mu$   obtained by pernuting the parts of $\lambda$.
The main result of \cite{ProlhacEvansMallick09} is that:
\begin{theorem}
	Let $\lambda = (\lambda_1,\dots, \lambda_n)$ be a partition
	and let $\mu$ be a composition obtained by permuting the parts
	of $\lambda.$  
	Then 
$$
\frac{Y^{(L)}_\mu}{Z^{(L)}_\lambda}
$$ is equal to the steady state probability of the state $\mu$ 
in the $L$-ASEP. 
\end{theorem}

We now include inhomogeneity parameters $x_1,\ldots ,x_n$. Let 
$$
a^{(L)}_{JM}(x_i)=\left\{
\begin{array}{ll}
x_ia^{(L)}_{JM} &{\rm if}\ J>0\\
a^{(L)}_{JM}&{\rm otherwise}
\end{array}\right.
$$
and we define for $J\le L$ and $L>1$
\begin{equation}
X_J^{(L)}(x_i)=\sum_M a_{JM}^{(L)}(x_i)\otimes X_{M^-}^{(L-1)}(x_i)
\end{equation}
with $M^-={\rm max}(0,M-1)$ and $X_1^{(1)}(x_i)=x_i$ and  $X_0^{(1)}(x_i)=1$.
We now introduce the parameter $q$ as in \cite{CGW}. Let
\begin{equation}
S^{(L)}= s^{(L)}\otimes \ldots \otimes  s^{(1)}
\end{equation}
with $s^{(1)}=1$ and $ s^{(L)}=D(1)\otimes \ldots \otimes D(q^{L-1})$
with $D(q)_{i,j}=q^i$ if $i=j$ and 0 otherwise.

Let $\mu=(\mu_1,\ldots ,\mu_n)$ be a composition into parts less or equal to $L$.
We define
\begin{equation}
Y^{(L)}_\mu(x_1,\ldots , x_n;q)={\rm Tr}\left(\left(\prod_{i=1}^n X_{\mu_i}^{(L)}(x_i)\right) S^{(L)} \right).
\end{equation}


Thanks to these parameters, we can now check that:
\begin{theorem}
For all $i$
\begin{eqnarray*}
T_i Y_\mu^{(L)}(x_1,\ldots ,x_n)= t Y_\mu^{(L)}(x_1,\ldots ,x_n) &{\rm if} \ \mu_i=\mu_{i+1}\\
T_i Y_\mu^{(L)}(x_1,\ldots ,x_n)=  Y_\mu^{(L)}(x_1,\ldots ,x_n) &{\rm if} \ \mu_i>\mu_{i+1}\\
q^{\mu_n}Y_{\mu}^{(L)}(x_1,\ldots ,x_n)= Y_{\mu_n,\mu_1,\ldots ,\mu_{n-1}}^{(L)}(qx_n,x_1,\ldots ,x_{n-1}).
\end{eqnarray*}
\end{theorem}
\begin{proof}
We use induction on $L$ and results proven in Section 5 of \cite{ProlhacEvansMallick09}. 
This requires a case by case analysis and is very similar to the proof we gave in Section \ref{inductive}.
\end{proof}


This implies \cref{ASEP-f}.





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