%% if you are submitting an initial manuscript then you should have submission as an option here %% if you are submitting a revised manuscript then you should have revision as an option here %% otherwise options taken by the article class will be accepted \documentclass[final]{FPSAC2022} %% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc %% note that the class file already loads {amsmath, amsthm, amssymb} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} %\usepackage{lipsum} %%%%%%%%%%%%%%%My stuff \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\ex}{ex} 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\setlength\cellsize{12\unitlength} \savebox2{% \begin{picture}(12,12) \put(0,0){\line(1,0){12}} \put(0,0){\line(0,1){12}} \put(12,0){\line(0,1){12}} \put(0,12){\line(1,0){12}} \end{picture}} \newcommand\cellify[1]{\def\thearg{#1}\def\nothing{}% \ifx\thearg\nothing\vrule width0pt height\cellsize depth0pt% \else\hbox to 0pt{\usebox2\hss}\fi% \vbox to 12\unitlength{\vss\hbox to 12\unitlength{\hss$#1$\hss}\vss}} \newcommand\tableau[1]{\vtop{\let\\=\cr \setlength\baselineskip{-12000pt} \setlength\lineskiplimit{12000pt} \setlength\lineskip{0pt} \halign{&\cellify{##}\cr#1\crcr}}} \newcommand{\e}{\mbox{}} % Fancy comments \usepackage[colorinlistoftodos]{todonotes} \newcommand{\dominic}[1]{\todo[size=\tiny,color=blue!30]{#1 \\ \hfill --- Dominic}} \newcommand{\Dominic}[1]{\todo[size=\tiny,inline,color=blue!30]{#1 \\ \hfill --- Dominic}} %\ytableausetup{centertableaux} %%%%%%%%END my stuff %% define your title in the usual way \title{$0$-Hecke--Clifford Modules From Diagrams} %% define your authors in the usual way %% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details \author[D.~Searles]{Dominic Searles\thanks{\href{mailto:dominic.searles@otago.ac.nz}{dominic.searles@otago.ac.nz}. Supported by the Marsden Fund, administered by the Royal Society of New Zealand Te \={A}parangi.}\addressmark{1}} %% then use \addressmark to match authors to institutions here \address{\addressmark{1}Department of Mathematics and Statistics, University of Otago, Dunedin 9016, New Zealand } %% put the date of submission here \received{November 25, 2021} %% leave this blank until submitting a revised version %\revised{} %% put your English abstract here, or comment this out if you don't have one yet %% please don't use custom commands in your abstract / resume, as these will be displayed online %% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year) \abstract{We construct modules of $0$-Hecke algebras and $0$-Hecke--Clifford algebras from fillings of diagrams of boxes in the plane. We apply this general construction method to solve a problem posed in (N.~Jing and Y.~Li, 2015), to define a new basis of the peak algebra analogous to quasiymmetric Schur functions, and to manifest a new connection between quasiymmetric Schur functions and Schur $Q$-functions.} %% put your French abstract here, or comment this out if you don't have one %\resume{\lipsum[2]} %% put your keywords here, or comment this out if you don't have them yet \keywords{$0$-Hecke algebra, $0$-Hecke--Clifford algebra, quasisymmetric Schur functions, Schur $Q$-functions, quasisymmetric Schur $Q$-functions} %% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier %% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text %% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex %% but you can use bibtex if you want \usepackage[backend=bibtex]{biblatex} \addbibresource{0HC.bib} %% note the \printbibliography command at the end of the file which goes with these biblatex commands %\usepackage{showkeys} \begin{document} \maketitle %% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above \section{Introduction} \subsection{Overview} The Hecke algebra is a deformation of the group algebra of the symmetric group by a parameter $q$, where the specialization $q=0$ is known as the $0$-Hecke algebra. The Grothendieck group of the $0$-Hecke algebra is isomorphic \cite{DKLT} to the Hopf algebra $\QSym$ of quasisymmetric functions, an important and widely-studied algebra that contains the symmetric functions. Under this isomorphism, called the \emph{quasisymmetric characteristic}, the images of the simple $0$-Hecke modules are precisely the \emph{fundamental quasisymmetric functions}, a basis of $\QSym$ introduced in~\cite{Gessel} as enumerating functions of $P$-partitions. Recently, several authors have constructed modules of $0$-Hecke algebras whose quasisymmetric characteristics are interesting bases of $\QSym$. Such modules were constructed for the dual immaculate functions in \cite{BBSSZ}, for the quasisymmetric Schur functions in \cite{TvW:1}, for the extended Schur functions in \cite{Searles:0Hecke}, and for the Young row-strict quasisymmetric Schur functions in \cite{Bardwell.Searles}. There has also been recent interest in the structure of such modules, \textit{e.g.}, in \cite{JKLO} all four of these families of $0$-Hecke modules are interpreted in terms of $0$-Hecke modules defined on intervals in the weak Bruhat order. The Hecke--Clifford (super)algebra was introduced in \cite{Olshanski}, combining the Hecke algebra and the Clifford algebra. The specialization $q=0$ is called the $0$-Hecke--Clifford algebra. The Grothendieck group of $0$-Hecke--Clifford modules is isomorphic \cite{BHT} to the \emph{peak algebra} $\Peak$, a subalgebra of $\QSym$ introduced in \cite{Stembridge:enriched}. Under this isomorphism, which we call the \emph{peak characteristic}, the images of the simple $0$-Hecke--Clifford (super)modules are (certain scalings of) the \emph{peak functions}, a basis of $\Peak$ introduced in~\cite{Stembridge:enriched} as enumerating functions of enriched $P$-partitions. The representation theory of $0$-Hecke--Clifford algebras has been further developed in \cite{Li}. A motivation was the question of finding $0$-Hecke--Clifford modules whose peak characteristics are the \emph{quasisymmetric Schur $Q$-functions}, a basis of $\Peak$ introduced in~\cite{Jing.Li} analogous to the dual immaculate basis \cite{BBSSZ:2} of $\QSym$. In this extended abstract, we summarize results from \cite{Searles:0HC}. We introduce a general method for constructing $0$-Hecke modules and $0$-Hecke--Clifford modules from certain standard fillings of box diagrams in the plane. The images of these modules under the quasisymmetric (respectively, peak) characteristic map are a sum of fundamental quasisymmetric (respectively, peak) functions determined by the descent sets of the standard fillings. An advantage of this framework is that it can be applied to families of standard tableaux of various shapes (\textit{e.g.}, straight-shape, skew, shifted, \textit{etc.}), from which many important families of symmetric and quasisymmetric functions are naturally generated. We apply this construction method to produce $0$-Hecke--Clifford modules whose peak characteristics are the quasisymmetric Schur $Q$-functions, answering the question raised in \cite{Jing.Li}. We moreover use this method to construct modules whose peak characteristics form a new basis of $\Peak$ analogous to the quasisymmetric Schur basis \cite{HLMvW11:QS} of $\QSym$. As a further application, we establish a new connection between the quasisymmetric Schur functions and the \emph{Schur $Q$-functions}: an important family of symmetric functions related to projective representations of symmetric groups. \subsection{Quasisymmetric functions and peak functions} A \emph{composition} is a finite sequence $\alpha = (\alpha_1, \ldots , \alpha_k)$ of positive integers. The integers $\alpha_i$ are called the \emph{parts} of $\alpha$. When the parts of $\alpha$ sum to $n$, $\alpha$ is called a \emph{composition of $n$} and we write $\alpha \vDash n$. Let $[n]$ denote the set $\{1, \ldots , n\}$. The \emph{descent set} $\Des(\alpha)$ of $\alpha= (\alpha_1, \ldots , \alpha_k)\vDash n$ is the subset $\{\alpha_1, \alpha_1+\alpha_2, \ldots , \alpha_1+\alpha_2 + \cdots + \alpha_{k-1}\}$ of $[n-1]$. The map $\Des$ taking a composition to its descent set is a bijection between compositions of $n$ and subsets of $[n-1]$. Given $X\subseteq [n-1]$, let $\comp_n(X)$ denote the composition of $n$ whose descent set is $X$. For example, $\Des(2,2,1,2) = \{2,4,5\}\subseteq [7]$ and $\comp_8(\{1,3,6\}) = (1,2,3,2)\vDash 8$. Denote by $\C[[x_1, x_2, \ldots ]]$ the Hopf algebra of formal power series of bounded degree in infinitely many commuting variables. The Hopf algebras $\Sym$ of symmetric functions and $\QSym$ of quasisymmetric functions are subalgebras of $\C[[x_1, x_2, \ldots ]]$, and $\Sym$ is a subalgebra of $\QSym$. For $\alpha\vDash n$, the \emph{fundamental quasisymmetric function} $F_\alpha$ is defined by \[F_\alpha = \sum x_{i_1}x_{i_2} \cdots x_{i_n},\] where the sum is over sequences $i_1, \ldots , i_n$ of integers satisfying $1\le i_1 \le \cdots \le i_n$ and $i_j < i_{j+1}$ whenever $j\in \Des(\alpha)$. The functions $F_\alpha$ form a basis for $\QSym$. \begin{example} If $\alpha = (1,2,1)$ then $\Des(\alpha) = \{1,3\}$, and we have \[F_{(1,2,1)} = \sum_{1\le i < j < k} x_ix_j^2x_k + \sum_{1\le i 1, i\in X \mbox{ and } i-1 \notin X\}$. For any composition $\alpha$, the \emph{peak set of $\alpha$} is the set $\Peak(\Des(\alpha))$, which we write as $\Peak(\alpha)$ for short. The \emph{peak functions} $K_\alpha$ \cite{Stembridge:enriched} are defined by \[K_\alpha = 2^{|\Peak(\alpha)|+1}\sum_{\beta: \Peak(\alpha)\subseteq \Des(\beta)\Delta(\Des(\beta)+1)} F_\beta,\] where $X\Delta Y$ denotes the symmetric difference of sets $X$ and $Y$. Since $K_\alpha$ depends only on $\Peak(\alpha)$, the peak functions can be indexed by peak compositions. As $\alpha$ ranges over peak compositions, the $K_\alpha$ form a basis for $\Peak$. \subsection{0-Hecke algebras and 0-Hecke--Clifford algebras} The $0$-Hecke algebra $H_n(0)$ is the $\C$-algebra with generators $\pi_1, \ldots , \pi_{n-1}$ and relations \begin{align}\label{eqn:0Heckeq} \pi_i^2 & = -\pi_i & & \mbox{ for all } 1 \le i \le n-1, \nonumber \\ \pi_i\pi_j & = \pi_j\pi_i & &\mbox{ for all } i, j \mbox{ such that } |i-j|\ge 2, \\ \pi_i\pi_{i+1}\pi_i & = \pi_{i+1}\pi_i\pi_{i+1} & & \mbox{ for all } 1\le i \le n-2. \nonumber \end{align} For an $H_n(0)$-module ${\bf N}$, let $[{\bf N}]$ denote its isomorphism class. Let $G_0(H_n(0))$ denote the Grothendieck group of the category of finite-dimensional $H_n(0)$-modules, and let $\mathcal{G} = \bigoplus_{n\ge 0} \mathcal{G}_0(H_n(0))$. There are $2^{n-1}$ simple $H_n(0)$-modules; these may be indexed by the compositions of $n$. Let $\mathbf{F}_\alpha$ denote the simple $H_n(0)$-module corresponding to the composition $\alpha$. In \cite{DKLT}, an algebra isomorphism $ch \colon \mathcal{G}\rightarrow \QSym$ is defined by \[ch([\mathbf{F}_\alpha]) = F_\alpha.\] The quasisymmetric function $ch([{\bf N}])$ is called the \emph{quasisymmetric characteristic} of ${\bf N}$. The \emph{$0$-Hecke--Clifford algebra} $HCl_n(0)$ is the algebra generated by $\pi_1, \ldots , \pi_{n-1}$ and $c_1, \ldots , c_n$, where the $\pi_i$ generate the $0$-Hecke algebra $H_n(0)$ and the $c_j$ generate the \emph{Clifford algebra} $Cl_n$, \textit{i.e.}, satisfy the relations \begin{equation} c_i^2 = -1 \,\,\, \mbox{ for all } 1 \le i \le n \qquad \mbox{ and } \qquad c_ic_j = -c_jc_i \,\,\, \mbox{ for all } i, j \mbox{ such that } i\neq j \end{equation} and the $\pi_i$ and $c_j$ additionally satisfy the cross-relations \begin{align}\label{eqn:crossrelations} \pi_ic_j & = c_j\pi_i & & \mbox{ for } j\neq i, i+1, \nonumber \\ \pi_ic_i & = c_{i+1}\pi_i & &\mbox{ for all } 1\le i \le n-1, \\ (\pi_i+1)c_{i+1} & = c_i(\pi_i+1) & & \mbox{ for all } 1\le i \le n-1. \nonumber \end{align} The $0$-Hecke--Clifford algebra is moreover a superalgebra: it is graded by $\mathbb{Z}_2$ with each $\pi_i$ having degree $0$ and each $c_j$ having degree $1$. Bergeron, Hivert and Thibon \cite{BHT} defined a family of $HCl_n(0)$-modules ${\bf M}_\alpha$, and extracted the simple $HCl_n(0)$-modules from these. Letting $\widetilde{\mathcal{G}}$ denote $\bigoplus_{n\ge 0} \mathcal{G}_0(HCl_n(0))$, they defined an algebra isomorphism $\widetilde{ch} \colon \mathcal{\widetilde{G}}\rightarrow \Peak$, which we call the \emph{peak characteristic}. Under this map, one has \begin{equation}\label{eqn:peakchartheta} \widetilde{ch}([{\bf M}_\alpha]) = K_{\comp_n(\Peak(\alpha))}. \end{equation} \section{Diagram modules}\label{sec:modules} A \emph{diagram} $D$ is a collection of $n$ boxes in the coordinate plane. A \emph{standard filling} of $D$ is a bijective assignment of the numbers $1, \ldots, n$ to the boxes of $D$, typically depicted by writing each number inside its corresponding box. The numbers in the boxes are called the \emph{entries} of a standard filling. Let $\StdFill(D)$ denote the set of all $n!$ standard fillings of $D$. Fix any total ordering on the boxes of $D$ and call this the \emph{reading order}. For $T\in \StdFill(D)$, define the \emph{reading word} $\rw(T)$ to be the word obtained by listing the entries of $T$ in reading order from left to right. Let $\StdTab(D)$ denote an arbitrary subset of $\StdFill(D)$. Next we define a function $\Des \colon \StdTab(D) \rightarrow [n-1]$ by \[\Des(T) = \{i \in T : i \mbox{ is to the right of } i+1 \mbox{ in } \rw(T)\}.\] We call $\Des(T)$ the \emph{descent set} of $T$, and call its elements the \emph{descents} of $T$. The elements of $[n-1]$ not in $\Des(T)$ are called the \emph{ascents} of $T$. Given $1\le i \le n-1$ and $T\in \StdFill(D)$, let $s_iT$ be the standard filling of $D$ obtained by exchanging the entries $i$ and $i+1$ of $T$. We define an ascent of $T$ to be \emph{attacking} if $s_iT\notin \StdTab(D)$, and \emph{nonattacking} otherwise. We also say $T$ has an \emph{ascent in positions $(r,s)$} if $1\le r