%% The class cedram-ALCO is just a wrapper around amsart.cls (version 2)
%% implementing the layout of the journal, and some additionnal
%% administrative commands.
%% You can place one option:
%% * "Unicode" if the file is UTF-8 encoded.
\documentclass[ALCO,Unicode,published]{cedram}
%% Here you might want to add some standard packages if those
%% functionnalities are required.
%\usepackage[matrix,arrow,tips,curve]{xy}
% ...
%% The production will anyway use amsmath (all ams utilities except
%% amscd for commutative diagrams which you need to load explicilty if
%% required), hyperref, graphicx, mathtools, enumitem...
% \usepackage[colorlinks=true,citecolor=cyan,backref=page]{hyperref}
\usepackage{graphicx}
\usepackage[labelfont=normalfont,labelformat=simple]{subcaption}
\usepackage{mathrsfs}
\usepackage[vcentermath]{youngtab}
\usepackage{ytableau}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{patterns}
\usetikzlibrary{backgrounds}
\usetikzlibrary{fadings}
\tikzfading[name=fade out, inner color=transparent!0, outer color=transparent!90]
%% User definitions if necessary... Such definitions are forbidden
%% inside titles, abstracts or the bibliography.
\DeclarePairedDelimiter\abs{\lvert}{\rvert} %Something useful only for this sample's sake: you can erase this line in your file (or find it useful...)
\newcommand{\Z}{\mathbb Z}
\newcommand{\Styl}{Styl}
\newcommand{\K}{\mathbb K}
\newcommand{\CC}{\mathbb C}
\newcommand{\Plax}{Plax}
\def\J{\mathscr{J}}
\frenchspacing
%% Theorem environment equivalences
\equalenv{proposition}{prop}
\equalenv{corollary}{coro}
%% The title of the paper: amsart's syntax.
\title
%% The optionnal argument is the short version for headings.
% [A \textup{(}useful?\textup{)} guide for authors]
%% The mandatory argument is for the title page, summaries, headings
%% if optionnal void.
% {A guide for authors preparing their accepted manuscript for AlCo}
{Quivers of stylic algebras}
%% Authors according to amsart's syntax + distinction between Given
%% and Proper names:
\author[\initial{A.} Abram]{\firstname{Antoine} \lastname{Abram}}
\address{Universit\'e du Qu\'ebec \`a Montr\'eal\\
D\'epartement de math\'ematiques\\
C.P. 8888, Succursale Centre-ville\\
Montr\'eal, QC H3C 3P8}
\email{abram.antoine@courrier.uqam.ca}
\author[\initial{C.} Reutenauer]{\firstname{Christophe} \lastname{Reutenauer}}
\address{Universit\'e du Qu\'ebec \`a Montr\'eal\\
D\'epartement de math\'ematiques\\
C.P. 8888, Succursale Centre-ville\\
Montr\'eal, QC H3C 3P8}
\email{reutenauer.christophe@uqam.ca}
\author[\initial{F.} \middlename{V.} Saliola]{\firstname{Franco} \middlename{V.} \lastname{Saliola}}
\address{Universit\'e du Qu\'ebec \`a Montr\'eal\\
D\'epartement de math\'ematiques\\
C.P. 8888, Succursale Centre-ville\\
Montr\'eal, QC H3C 3P8}
\email{saliola.franco@uqam.ca}
%% Do not include any other information inside \author's argument!
%% Other author data have special commands for them:
%% Current address, if different from institutionnal address
% \curraddr{Coll\`ege Royal Henri-Le-Grand\\
% Prytan\'ee National Militaire\\
% 72000 La Fl\`eche, Sarthe (France)}
%% e-mail address
%% possibly home page URL (not encouraged by journal's style)
% \urladdr{https://en.wikipedia.org/wiki/Marin\_Mersenne}
%% Acknowledgements are not a footnote in
%% \author, but are given apart:
\thanks{We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).
This research was facilitated by computer exploration using the open-source
mathematical software system \textsc{SageMath}~\cite{SageMath} and its algebraic
combinatorics features developed by the \textsc{Sage-Combinat} community
\cite{sage-combinat}.}
%% If co-authors exist, add them the same way, in alaphabetical order
%\author{\firstname{Joseph} \lastname{Fourier}}
%\address{Universit\'e de Grenoble\\
% Institut Moi-m\^eme\\
% BP74, 38402 SMH Cedex (France)}
%\email{fourier@fourier.edu.fr}
% Key words and phrases:
\keywords{representation theory, quiver with relations, quiver presentation,
complete system of primitive orthogonal idempotents, $\J$-trivial monoid,
stylic monoid, plactic monoid}
%% Mathematical classification (2010)
%% This will not be printed but can be useful for database search
\subjclass{05E10, 16G20, 20M25, 20M30}
\datepublished{2024-01-08}
\begin{document}
%% Abstracts must be placed before \maketitle
\begin{abstract}
We construct a complete system of primitive orthogonal idempotents and
give an explicit quiver presentation of the monoid algebra of the stylic
monoid introduced by Abram and Reutenauer.
\end{abstract}
\maketitle
%\tableofcontents
\section{Introduction}
We study the monoid algebra of the stylic monoid $\Styl(A)$ introduced by the
first two authors in \cite{AR}. We begin by recalling its definition.
Let $A$ be a totally ordered finite alphabet
and $A^*$ the free monoid that it generates.
The \emph{Robinson--Schensted--Knuth (RSK) correspondence} associates
with each word $w \in A^*$ a semistandard tableau $P(w)$ with entries in $A$
called its \emph{$P$-symbol}.
If $w$ is a decreasing word, then its $P$-symbol $P(w)$ is a column, which
allows us to identify the set of decreasing words on $A$ with the set
$\Gamma(A)$ of column-shaped tableaux with entries in $A$.
This induces a left action of $A^*$ on $\Gamma(A)$:
for a word $x \in A^*$ and a column $\gamma \in \Gamma(A)$,
take $x \cdot \gamma$ to be the first column of the tableau $P(x w)$, where
$w$ is the decreasing word corresponding to the column $\gamma$.
(This action can be defined using the Schensted column insertion procedure;
see \S\ref{Schensted-column-insertion}.)
The finite monoid of endofunctions of $\Gamma(A)$ obtained by this action is
the \emph{stylic monoid $\Styl(A)$}.
It turns out that $\Styl(A)$ is canonically isomorphic to a quotient of the
celebrated plactic monoid. Recall that the plactic monoid has appeared in many
contexts in algebraic combinatorics and was used to give the first rigorous
proof of the \emph{Littlewood--Richardson rule} \cite{Schutzenberger1977,
Lascoux-Schutzenberger-1981, Lothaire2002}.
The monoid algebra $\K\Styl(A)$, where $\K$ is any field, is the
first example of a finite dimensional representation of the plactic monoid that
does not pass through the abelianisation (to our knowledge).
This article is a first step towards understanding the structure of this
representation.
Stylic monoids are examples of $\J$-trivial monoids \cite{AR}, which are
a ubiquitous class of monoids that arise naturally in algebraic combinatorics.
Other examples include the $0$-Hecke monoids associated with finite Coxeter
groups, and the monoids of regressive order-preserving functions on a poset;
see \cite{DHST} for many more examples.
It follows that the monoid algebra $\K\Styl(A)$ admits
a \emph{quiver presentation}: that is, $\K\Styl(A)$ is isomorphic to a quotient
of the path algebra $\K Q(A)$ of a canonical quiver $Q(A)$.
Obtaining a quiver presentation is an essential step towards applying the tools
and techniques from the modern representation theory of finite dimensional
algebras~\cite{ASSVol1}.
One of our main
results is an explicit presentation of $\K\Styl(A)$ as a \emph{quiver
with relations}.
Our approach is constructive in the sense that we explicitly identify
a complete system of primitive orthogonal idempotents in $\K\Styl(A)$
(Theorem~\ref{system}) that we use to define a quiver $Q(A)$ together with
a surjective map $\varphi: \K Q(A) \xrightarrow{} \K\Styl(A)$
(Corollary~\ref{quiver-map-is-surjective}) whose kernel is an admissible
ideal (Theorem~\ref{span}). General theory then implies that $Q(A)$ is
the quiver of $\K\Styl(A)$ (Theorem~\ref{quiver}).
We remark that the representation theory of finite monoids naturally occurring
in algebraic combinatorics, especially in connection with Markov chains,
has been investigated by many authors:
\cite{BD1998,
BHR1999,
BBD1999,
Brown2000,
Brown2004,
Saliola2007,
Saliola2009,
BBBS2011,
GanyushkinMazorchuk2011,
MargolisSteinberg2011,
MargolisSteinberg2012,
MazorchukSteinberg2012,
HST2013,
GrensingMazorchuk2014,
ASST2015,
MargolisSteinbergSaliola2015,
MargolisSteinberg2018,
Stein2020,
MargolisSteinbergSaliola2021};
see especially Steinberg's recent book and the references therein \cite{SteinbergMonoidRepTheory}.
Those most closely related to the present work are \cite{DHST},
\cite[Chapter~17]{SteinbergMonoidRepTheory} and \cite{MargolisSteinberg2018},
which describe the quiver of the algebra of a $\J$-trivial monoid.
While guided by this work, our approach is complementary and completely
self-contained as their techniques do not involve constructing primitive
orthogonal idempotents or a quiver presentation. In fact, in
\cite{MargolisSteinberg2018} one reads \emph{``It is notoriously difficult to write
down explicit primitive idempotents for monoids algebras (c.f. \cite{BBBS2011,
Denton2011}) and often they have complicated expressions in terms of the monoid
basis, making it virtually impossible to determine even the dimension of the
corresponding projective indecomposable module let alone construct a matrix
representation out of it.''}
\section{Stylic monoid and algebra}
We consider a totally ordered finite set $A$, whose elements are called {\it letters}, and the free monoid $A^*$ that it generates. Its elements are called {\it words}. The {\it alphabet} of a word~$x$ is the set of letters $Alph(x)$ appearing in $x$.
\subsection{Tableaux}
We call a {\it tableau} what is usually called a {\it semistandard Young tableau}: a finite lower order ideal
of the poset $\mathbb N^2$, ordered naturally (that is, a finite subset $E\subset \mathbb N^2$ such that $x\leq y$ and $y\in E$ implies $x\in E$),
together with a weakly increasing mapping into $A$, such that the restriction of this mapping to
each subset with given $x$-coordinate is injective. A tableau is usually represented as in Figure \ref{tab}. The
conditions may be expressed by saying that the letters in $A$ are weakly increasing from left to right in each row, and
strictly
increasing from the bottom to top in each~column.
\begin{figure}
\begin{ytableau}
d\\
b& b\\
a&a&c \end{ytableau}
\caption{A tableau}\label{tab}
\end{figure}
A {\it column} is a tableau with only one column. The set of columns on $A$ is denoted by $\Gamma(A)$. A column is identified naturally with a subset of $A$, and also with the word in $A^*$ that is the decreasing product of its elements.
\subsection{Schensted's column insertion procedure}
\label{Schensted-column-insertion}
Let us recall the Schensted {\it column insertion} algorithm. Let $\gamma$ be a column, viewed here as a subset
of $A$, and let $x\in A$. There are two cases: if $\forall y\in\gamma, x>y$, then define $\gamma'=
\gamma\cup x$. Otherwise, let $y$ be the smallest element in $\gamma$ with $y\geq x$; then define $
\gamma'=(\gamma\setminus y)\cup x$. Then $\gamma'$ is the column obtained by {\it column insertion of $x$ into} $
\gamma$, and in the second case, $y$ is said to be {\it bumped}.
We define a left action of $A^*$ on $\Gamma(A)$, denoted $u\cdot \gamma$, for each $u\in A^*$ and each
column $\gamma$. Since $A^*$ is the free monoid on $A$, it is enough to define the action for each letter $a\in A$.
Define
$$a\cdot \gamma=\gamma'$$
if $\gamma'$ is obtained from $\gamma$ by column insertion of $a$ into $\gamma$.
For further use, we note that if $\gamma$ is a column, then we have
$$
\gamma\cdot \emptyset=\gamma,
$$
where on the left-hand side, $\gamma$ is viewed as a decreasing word.
\subsection{Stylic monoid}
\label{Stylic-monoid}
We denote by $\Styl(A)$ the monoid of endofunctions of the set $\Gamma(A)$ of columns obtained by the action defined above. Thus, a typical element of $\Styl(A)$ is a function
\begin{eqnarray*}
\mu_w : \Gamma(A) & \to & \Gamma(A) \\
\gamma & \mapsto & w\cdot \gamma
\end{eqnarray*}
for some word $w\in A^*$. Since $\Gamma(A)$ is finite, $\Styl(A)$ is finite. Let $\mu:A^*\to \Styl(A)$ be the canonical monoid homomorphism defined by $\mu(w) = \mu_w$.
We denote by $\equiv_{styl}$ the
corresponding monoid congruence of $A^*$, called the {\it stylic congruence}:
\begin{equation*}
u\equiv_{styl} v \quad \Longleftrightarrow \quad \mu(u)=\mu(v)
\quad \Longleftrightarrow \quad u\cdot \gamma=v\cdot \gamma \text{~for all columns $\gamma$.}
\end{equation*}
The monoid $\Styl(A)$ acts naturally on the set of columns, and we
take the same notation: $m\cdot \gamma=w\cdot \gamma$ if $m=\mu(w)$.
\subsection{Relationship with the plactic monoid}
\label{Relationship-with-the-plactic-monoid}
The {\it Schensted $P$-symbol} is a mapping that associates with each word $w$ on $A$ a tableau $P(w)$, see \cite{Sagan, Lothaire2002}.
The relation~$\equiv_{plax}$ on $A^*$, defined by
$$
u\equiv_{plax} v \quad \Longleftrightarrow \quad P(u)=P(v),
$$
is a congruence of the monoid $A^*$, called the {\it plactic congruence}. The quotient monoid~$A^*/{\equiv_{plax}}$ is called the {\it plactic monoid}.
The {\it column-reading word} of a tableau is the word obtained by reading the columns from left to right, each column being read as a decreasing word. For example, the column reading word of the tableau from Figure~\ref{tab} is the word $dbabac$. If $T$ is a tableau, with column-reading word $w$, then
\begin{equation}\label{Pw=T}
P(w)=T
\end{equation}
by a theorem of Schensted.
The {\it plactic relations}, due to Knuth, are the following relations:
\begin{equation}\label{plactic-abc}
bac\equiv_{plax} bca, \qquad acb\equiv_{plax}cab
\end{equation}
for any choice of letters $a**=latex,line join=bevel,framed, baseline=-50pt, scale=0.8]
\node (node_0) at (3.5bp,73.5bp) [draw] {$\epsilon$};
\node (node_1) at (31.5bp,73.5bp) [draw] {$a$};
\node (node_2) at (31.5bp,6.5bp) [draw] {$b$};
\node (node_3) at (64.5bp,73.5bp) [draw] {$ba$};
\draw [black,->] (node_1) to (node_2);
\draw (40bp,40.0bp) node [font=\small] {$b$};
\end{tikzpicture}
\qquad
\qquad
\begin{tikzpicture}[>=latex,line join=bevel,framed, scale=0.8]
\node (node_0) at (-10.5bp,145.5bp) [draw] {$\epsilon$};
\node (node_1) at (31.5bp,145.5bp) [draw] {$a$};
\node (node_2) at (9.5bp,76.0bp) [draw] {$b$};
\node (node_3) at (33.5bp,6.5bp) [draw] {$c$};
\node (node_4) at (100.5bp,145.5bp) [draw] {$ba$};
\node (node_5) at (100.5bp,76.0bp) [draw] {$ca$};
\node (node_6) at (100.5bp,6.5bp) [draw] {$cb$};
\node (node_7) at (138.5bp,145.5bp) [draw] {$cba$};
\draw [black,->] (node_1) ..controls (22.792bp,134.24bp) and (18.048bp,127.64bp) .. (15.5bp,121.0bp) .. controls (12.042bp,111.99bp) and (10.531bp,101.28bp) .. (node_2);
\definecolor{strokecol}{rgb}{0.0,0.0,0.0};
\pgfsetstrokecolor{strokecol}
\draw (20bp,112.0bp) node [font=\small] {$b$};
\draw [black,->] (node_1) ..controls (34.839bp,134.07bp) and (36.723bp,127.18bp) .. (37.5bp,121.0bp) .. controls (42.487bp,81.312bp) and (40.868bp,70.858bp) .. (37.5bp,31.0bp) .. controls (37.29bp,28.521bp) and (36.959bp,25.909bp) .. (node_3);
\draw (48.5bp,76.0bp) node [font=\small] {$c$};
\draw [black,->] (node_2) ..controls (9.349bp,60.728bp) and (10.106bp,43.953bp) .. (15.5bp,31.0bp) .. controls (16.967bp,27.477bp) and (19.097bp,24.021bp) .. (node_3);
\draw (20.5bp,40.0bp) node [font=\small] {$c$};
\draw [black,->] (node_4) ..controls (100.5bp,128.68bp) and (100.5bp,107.56bp) .. (node_5);
\draw (110.0bp,112.0bp) node [font=\small] {$c$};
\draw [black,->] (node_5) ..controls (100.5bp,59.182bp) and (100.5bp,38.062bp) .. (node_6);
\draw (110.0bp,40.0bp) node [font=\small] {$b$};
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[>=latex,line join=bevel, scale=0.8, framed]
\node (node_0) at (-16.084bp,289.5bp) [draw] {$\epsilon$};
\node (node_1) at (52.084bp,289.5bp) [draw] {$a$};
\node (node_2) at (10.084bp,220.0bp) [draw] {$b$};
\node (node_3) at (40.084bp,148.0bp) [draw] {$c$};
\node (node_4) at (40.084bp,76.0bp) [draw] {$d$};
\node (node_5) at (197.08bp,289.5bp) [draw] {$ba$};
\node (node_6) at (174.08bp,220.0bp) [draw] {$ca$};
\node (node_8) at (198.08bp,148.0bp) [draw] {$da$};
\node (node_7) at (127.08bp,148.0bp) [draw] {$cb$};
\node (node_9) at (174.08bp,76.0bp) [draw] {$db$};
\node (node_10) at (197.08bp,6.5bp) [draw] {$dc$};
\node (node_11) at (285.08bp,289.5bp) [draw] {$cba$};
\node (node_12) at (285.08bp,220.0bp) [draw] {$dba$};
\node (node_13) at (285.08bp,148.0bp) [draw] {$dca$};
\node (node_14) at (285.08bp,76.0bp) [draw] {$dcb$};
\node (node_15) at (360.08bp,289.5bp) [draw] {$dcba$};
\draw [black,->] (node_1) ..controls (37.816bp,283.2bp) and (23.186bp,276.23bp) .. (16.084bp,265.0bp) .. controls (10.811bp,256.66bp) and (9.3099bp,245.67bp) .. (node_2);
\draw (19.084bp,256.0bp) node [font=\small] {$b$};
\draw [black,->] (node_1) ..controls (49.887bp,262.97bp) and (44.096bp,195.65bp) .. (node_3);
\draw (54.084bp,220.0bp) node [font=\small] {$c$};
\draw [black,->] (node_1) ..controls (58.066bp,278.21bp) and (61.705bp,271.35bp) .. (64.084bp,265.0bp) .. controls (69.871bp,249.56bp) and (71.069bp,245.37bp) .. (73.084bp,229.0bp) .. controls (77.966bp,189.34bp) and (85.587bp,146.09bp) .. (66.084bp,103.0bp) .. controls (63.451bp,97.184bp) and (58.908bp,91.945bp) .. (node_4);
\draw (86.084bp,184.0bp) node [font=\small] {$d$};
\draw [black,->] (node_2) ..controls (9.5925bp,204.58bp) and (10.101bp,187.69bp) .. (16.084bp,175.0bp) .. controls (18.51bp,169.85bp) and (22.322bp,165.03bp) .. (node_3);
\draw (21.084bp,184.0bp) node [font=\small] {$c$};
\draw [black,->] (node_2) ..controls (4.2842bp,200.26bp) and (-4.4016bp,166.15bp) .. (3.0837bp,139.0bp) .. controls (8.0786bp,120.88bp) and (19.913bp,102.72bp) .. (node_4);
\draw (9.084bp,148.0bp) node [font=\small] {$d$};
\draw [black,->] (node_3) ..controls (40.084bp,130.49bp) and (40.084bp,108.25bp) .. (node_4);
\draw (47.084bp,112.0bp) node [font=\small] {$d$};
\draw [black,->] (node_5) ..controls (186.58bp,278.56bp) and (180.92bp,272.0bp) .. (178.08bp,265.0bp) .. controls (174.48bp,256.11bp) and (173.44bp,245.39bp) .. (node_6);
\draw (181.58bp,256.0bp) node [font=\small] {$c$};
\draw [black,->] (node_5) ..controls (201.53bp,278.16bp) and (204.04bp,271.29bp) .. (205.08bp,265.0bp) .. controls (211.62bp,225.54bp) and (210.33bp,214.65bp) .. (205.08bp,175.0bp) .. controls (204.63bp,171.59bp) and (203.86bp,167.99bp) .. (node_8);
\draw (218.58bp,220.0bp) node [font=\small] {$d$};
\draw [black,->] (node_6) ..controls (158.48bp,208.88bp) and (149.05bp,201.52bp) .. (143.08bp,193.0bp) .. controls (137.05bp,184.39bp) and (133.03bp,173.24bp) .. (node_7);
\draw (147.0bp,184.0bp) node [font=\small] {$b$};
\draw [black,->] (node_6) ..controls (173.12bp,204.75bp) and (172.99bp,188.0bp) .. (178.08bp,175.0bp) .. controls (179.81bp,170.59bp) and (182.53bp,166.29bp) .. (node_8);
\draw (183.58bp,184.0bp) node [font=\small] {$d$};
\draw [black,->] (node_7) ..controls (129.99bp,132.62bp) and (134.51bp,115.25bp) .. (143.08bp,103.0bp) .. controls (146.9bp,97.544bp) and (152.15bp,92.564bp) .. (node_9);
\draw (147.58bp,112.0bp) node [font=\small] {$d$};
\draw [black,->] (node_8) ..controls (187.36bp,136.31bp) and (181.16bp,128.85bp) .. (178.08bp,121.0bp) .. controls (174.58bp,112.06bp) and (173.55bp,101.35bp) .. (node_9);
\draw (183.58bp,112.0bp) node [font=\small] {$b$};
\draw [black,->] (node_8) ..controls (201.86bp,135.89bp) and (204.15bp,128.04bp) .. (205.08bp,121.0bp) .. controls (210.33bp,81.345bp) and (211.62bp,70.463bp) .. (205.08bp,31.0bp) .. controls (204.63bp,28.247bp) and (203.89bp,25.383bp) .. (node_10);
\draw (218.58bp,76.0bp) node [font=\small] {$c$};
\draw [black,->] (node_9) ..controls (173.03bp,60.718bp) and (172.84bp,43.937bp) .. (178.08bp,31.0bp) .. controls (179.55bp,27.39bp) and (181.76bp,23.897bp) .. (node_10);
\draw (184.58bp,40.0bp) node [font=\small] {$c$};
\draw [black,->] (node_11) ..controls (285.08bp,272.68bp) and (285.08bp,251.56bp) .. (node_12);
\draw (296.08bp,256.0bp) node [font=\small] {$d$};
\draw [black,->] (node_12) ..controls (285.08bp,202.49bp) and (285.08bp,180.25bp) .. (node_13);
\draw (296.08bp,184.0bp) node [font=\small] {$c$};
\draw [black,->] (node_13) ..controls (285.08bp,130.49bp) and (285.08bp,108.25bp) .. (node_14);
\draw (296.08bp,112.0bp) node [font=\small] {$b$};
\end{tikzpicture}
\end{center}
\caption{The quivers for alphabets of cardinality $2,3,4$; the columns are
represented by decreasing words and the empty word is denoted $\epsilon$.}
\label{3quivers}
\end{figure}
For later use, we note the following result relating the left and right actions. The proof is left to the reader.
\begin{lemma}\label{left-right-action}
For two columns of the same height $\gamma,\delta$, and two letters $b,c$, the two following conditions are equivalent:
\begin{enumerate}[label=(\roman*)]
\item $b\cdot \delta=\gamma$ and $c$ is bumped;
\item $\gamma \cdot c=\delta$, and $b$ is bumped.
\end{enumerate}
\end{lemma}
\subsection{A lemma on edges and idempotents}
We give a technical, but important, result on the idempotents of the stylic algebra and the quiver introduced previously.
\begin{lemma}\label{identity} Let $\gamma\xrightarrow{c} \delta$ be an edge in the quiver $Q(A)$, and denote by $b$ the bumped letter, so that
$\delta=c\cup (\gamma\setminus b)$, and $b\in \gamma, c\notin\gamma, b\notin \delta, c\in \delta$. Then in $\Z\Styl(A)$
\begin{equation*}
be_\gamma c=bce_\delta
\qquad\text{and}\qquad
e_\gamma c e_\delta=e_\gamma c.
\end{equation*}
\end{lemma}
\begin{proof} I. We prove the first identity. Let $a=\min(A)$. As in the proof of Theorem \ref{system}, denote by $e'_{\gamma'}$ the idempotents (\ref{egamma}) relative to the alphabet $A\setminus a$.
1. Suppose that $a\in \gamma\cap \delta$. Since the action $\gamma\cdot c$ is frank, and since $a$ cannot be bumped, we have $a****c$)
$=y\prod_{k=1}^{t}(1 -
z_k)a\delta$, by the same identity in Lemma \ref{superplax}.
It follows that $ae_\gamma c=a(1-c)\prod_{k=1}^{t}(1 - z_k)a\delta=a(1-c)\prod_{k=1}^{t}(1 - z_k)ac\delta
$ (since $c\delta=\delta$) $
=a(1-c)a\prod_{k=1}^{t}(1 - z_k)c\delta$ (by the first identity in Lemma \ref{superplax}) $=a(1-c)ac\prod_{k=1}^{t}(1 - z_k)c\delta$
(by Lemma~\ref{axa}\ref{it:221})
$ =(ac-ca)\prod_{k=1}^{t}(1 - z_k)\delta$.
On the other hand, we have $ace_\delta=ac(1 - a) \prod_{j=1}^{s}(1 - y_j) \prod_{k=1}^{t}(1 - z_k) \delta=(ac-ca)\prod_{j=1}^{s}(1 - y_j) \prod_{k=1}^{t}(1 -
z_k) \delta$. Note that $\prod_{j=1}^{s}(1 - y_j) $ is equal to $1$ plus a linear combination of $yu$, with $a**