%% 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[submission]{FPSAC2026} %% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc \usepackage{graphicx, tikz, amsmath, youngtab, ytableau, bm} %% note that the class file already loads {amsmath, amsthm, amssymb} \newcommand{\SL}{\mathrm{SL}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Sym}{\mathrm{Sym}} \def\x{\mathbf x} \def\y{\mathbf y} \def\z{\mathbf z} \def\XX{\mathbf X} \def\YY{\mathbf Y} \def\deltaD{\delta} \def\sl{\mathfrak {sl}} \def\GL{\mathrm {GL}} \def\sgn{\mathrm {sgn}} \def\SSYT{\mathrm {SSYT}} \def\Par{\mathrm {Par}} \def\SymG{\mathfrak S} \DeclareMathOperator{\ev}{ev} \newcommand{\qbinom}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}_q} \newcommand{\kbinom}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} \newcommand{\ydiagramb}[1]{\scalebox{1.25}{\ydiagram{#1}}} % MkW: bigger diagrams in the matrix headings \newcommand{\sm}[1]{\scalebox{0.85}{$#1$}} \renewcommand{\ge}{\geqslant} \renewcommand{\geq}{\geqslant} \renewcommand{\le}{\leqslant} \renewcommand{\leq}{\leqslant} \newcommand{\bwedge}[1]{{\textstyle{\bigwedge\!\!^{#1}\, }}} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Example}[Theorem]{Example} \newtheorem{Conjecture}[Theorem]{Conjecture} \newtheorem{Remark}[Theorem]{Remark} \newtheorem{Fact}[Theorem]{Fact} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Definition}[Theorem]{Definition} \newtheorem*{Note}{Note} %Theorem enumeration environment \newcounter{thmlistcnt} \newenvironment{thmlist}% {\setcounter{thmlistcnt}{0}% \begin{list}{\emph{(\roman{thmlistcnt})}}{% \usecounter{thmlistcnt}% \setlength{\topsep}{0pt}% \setlength{\leftmargin}{0pt}% \setlength{\itemsep}{-6pt}% \setlength{\labelwidth}{17pt} \setlength{\itemindent}{30pt}}% }% {\end{list}}% \usepackage[backend=bibtex]{biblatex} \addbibresource{bibliography.bib} %% define your title in the usual way \title[Plethystic lifts of $q$-binomial identities]{Plethystic lifts of $q$-binomial identities} %% define your authors in the usual way %% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details %% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author \author[Á. Gutiérrez, Á. L. Mart\'inez, M. Szwej, M. Wildon]{Álvaro Gutiérrez\addressmark{1}, Álvaro L. Martínez\addressmark{2},\\ Michał Szwej\thanks{\href{mailto:michal.szwej@bristol.ac.uk}{michal.szwej@bristol.ac.uk}. ÁG is funded by a University of Bristol Research Training Support Grant and MS and MW are funded by the Heilbronn Institute for Mathematical Research. }\addressmark{1} and Mark Wildon\addressmark{1}} %% then use \addressmark to match authors to institutions here \address{\addressmark{1} School of Mathematics, University of Bristol, UK \\\addressmark{2} Columbia University, New York, USA} %% put the date of submission here \received{\today} %% 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 develop a framework to study plethysms of $\mathrm{SL}_2$-representations over fields of arbitrary characteristic. Each representation of the type $\mathrm{Sym}_n \mathrm{Sym}^m \mathbb{F}^2$ is identified with a space of symmetric polynomials. A certain map on polynomials (defined by substitution) gives morphisms between these representations. We study the map combinatorially, and use it to categorify numerous $q$-binomial identities including the $q$-analogues of the team-and-leader identity $(n+1)\binom{n+d+1}{n+1} = \binom{n+d}{n}(n+d+1)$ and its generalisations. We conclude with a potential application to the generalised Foulkes Conjecture for $\mathrm{SL}_2(\mathbb{C})$, and some further conjectures and directions raised by this work. } % The $k$-fold plethystic substitution is a map on symmetric functions given by replacing each variable with $k$ of its copies. % We introduce a polynomial interpretation of plethystic representations of $\SL_2(\F)$, where the $k$-fold map gives rise to an automorphism of $\Sym\Sym \F^2$. % %\MS{Q: how can we prove it's an automorphism when $\F\neq \C$? If we can't, the sentence will be rephrased}. % Unlike the traditional character-theoretic approach, our framework identifies a representation with an entire algebra rather than a single polynomial, allowing us to work in arbitrary characteristic. When $\F=\C$, we compute the structure constants of the $k$-fold substitution and reveal numerous remarkable symmetries among them. We also provide evidence that the $k$-fold map is a candidate for the $q$-Foulkes conjecture in a large class of parameters. Finally, we present applications of the $2$-fold substitution in modular plethystic $\SL_2(\F)$-representations, leading to a conjectural filtration of the general plethysm $\Delta^\lambda \Sym^d \F^2$.} %% put your French abstract here, or comment this out if you don't have one %% put your keywords here, or comment this out if you don't have them yet \keywords{Symmetric polynomials, plethysm, modular representation, $q$-analogues} %% 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; i.e., [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{sample.bib} %% note the \printbibliography command at the end of the file which goes with these biblatex commands \articlenumber{65} \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}\label{sec:intro} %Since their introduction only a few decades ago, $q$-analogues have played a central role in combinatorics. A powerful `metatheorem' in enumerative combinatorics says that every interesting identity has a $q$-analogue which refines it. Let $q$ be an invertible variable. The $q$-analogue of the integer $n$ (the \emph{$q$-integer $n$}) is %MkW: no 'or' for stylistic consistency \[ [n]_q = \frac{q^n - q^{-n}}{q - q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{1-n}, \] and the $q$-analogue of the binomial coefficient (the \emph{$q$-binomial}) is \[ \qbinom{n}{k} = \frac{[n]_q[n-1]_q\cdots[n-k+1]_q}{[k]_q[k-1]_q\cdots[1]_q}\,. \] The $q$-binomials are omnipresent %MkW: maybe we might try this word? Revert to 'ever present' if you prefer. in the vast landscape of $q$-hypergeometric identities~\cite{Koepf}. They can be interpreted combinatorially as the generating functions for $k$-subsets of~$\N_0$ by the sum of their elements, partitions in an $(n-k)\times k$ rectangle, and inversions in permutations of $n$ objects \cite[Ch.~1]{StanleyEC2}. Each $q$-binomial can be written uniquely as a linear combination of $q$-integers, \begin{equation} \label{eq:q-bin and plet} \qbinom{n+m}{m} = \sum_{k} a_{n,m}^k [k]_q\,. \end{equation} The coefficients in this expansion are the \emph{$\SL_2(\C)$-plethysm coefficients}: a \emph{plethysm} is a representation of $\SL_2(\C)$ of the form $\Sym^n\Sym^m\C^2$ and there is an $\SL_2(\C)$-isomorphism \vspace*{-6pt} \begin{equation} \label{eq:plet} \Sym^n\Sym^m\C^2 \cong \bigoplus\nolimits_k (\Sym^{k-1}\C^2)^{\oplus a_{n,m}^k}\,. \end{equation} \smallskip\noindent One recovers \eqref{eq:q-bin and plet} by taking characters. We say that \eqref{eq:plet} \emph{lifts} or \emph{categorifies} \eqref{eq:q-bin and plet} over~$\C$. Thus the fundamental bridge between combinatorics and algebra is that \smash{$\qbinom{n+m}{m}$} is the character of \smash{$\Sym^n \Sym^m \C^2$} on the complex diagonal matrix $\mathrm{diag}(q, q^{-1})$. %\scalebox{0.85}{$\left(\begin{matrix} q & 0 \\ 0 & q^{-1}\end{matrix}\right)$}. Despite the number of existing $q$-binomial identities, understanding $\SL_N(\C)$-plethysm coefficients for general $N$ is an elusive and hard problem. The problem dates to Littlewood in the 1930s and was identified by Stanley as one of the major problems in algebraic combinatorics \cite[Problem 9]{StanleyList}. It has deep connections with computational complexity \cite{FischerIkenmeyer}. The recent FPSAC abstracts~\cite{Seamus,GutCrystals-FPSAC,MingYean} show that it is a highly active research topic. Our research is motivated by the following overarching question: \begin{quote} \itshape Which $q$-binomial identities reflect a deeper property of plethysms? \end{quote} Let us proceed by example. The \emph{team-and-leader identity} mentioned in the abstract draws its name from the combinatorial proof, in which teams of size $n+1$ with a single leader are counted in two ways, firstly by choosing $n+1$ of the $n+d+1$ people and having them elect a leader; and secondly by choosing a leader in $n+d+1$ ways and allowing the leader to pick the rest of the team. %MkW I think we could have the proof, but delete it you prefer More generally, we count teams of size $n+m$ with~$m$ leaders and $n$ followers. %Using $\qbinom{n+m}{m} = \qbinom{n+m}{n}$, one One $q$-analogue of the team-and-leader identity is then \begin{equation} \label{t&l} \qbinom{n+m}{n}\qbinom{n+m+d}{n+m} % MkW: note (1.1) and (1.2) give Sym^n Sym^m C^2 <-> \qbinom{n+m}{m}, % i.e. inner m appears twice, and here to generalize T&L as stated in abstract with m leaders, we need to choose % m leaders in \qbinom{n+m}{m}. But this doesn't work later on in the modular case, so I won't change it here. = \qbinom{n+d}{n}\qbinom{n+m+d}{m}. \end{equation} % Consider the $q$-analogue of the \emph{team-and-leader identity}: % \begin{equation} % \label{t&l} % \qbinom{n+m}{n}\qbinom{n+m+d}{n+m} % = % \qbinom{n+d}{n}\qbinom{n+m+d}{m}. % \end{equation} % When $q=1$, the left-hand side counts the number of ways of electing $n+m$ lawmakers from a population of $n+m+d$, and $n$ ministers among the lawmakers; the right-hand side counts the number of ways of first electing $m$ lawmakers (which are not ministers), and $n$ ministers from the remaining population. Automatically from~\eqref{eq:q-bin and plet} and~\eqref{eq:plet}, %MkW we obtain an $\SL_2(\C)$-isomorphism \[ \Sym^m\hskip1pt\Sym^n\hskip1pt\C^2 \otimes %MkW: swapped n and m here to reflect earlier swap of \qbinom{n+m}{n} to \qbinom{n+m}{m} \Sym^{m+n}\hskip1pt\Sym^d\hskip1pt\C^2 \cong \Sym^n\hskip1pt\Sym^d\hskip1pt\C^2 \otimes \Sym^m\hskip1pt\Sym^{d+n}\hskip1pt\C^2 \] % \[ % \Sym^n\C^2\otimes\Sym^{n+1}\Sym^d\C^2\cong\Sym^n\Sym^d\C^2\otimes\Sym^{n+d}\C^2 % \] lifting \eqref{t&l} over~$\C$. However, we will shortly see that there is a much more general result. Let $\F$ be a field. There are two different ways of constructing a symmetric power in the category of $\SL_2(\F)$-representations, by taking invariants or coinvariants: \[ \Sym_n V = (V^{\otimes n})^{\SymG_n} \quad \text{and} \quad \Sym^n V = V^{\otimes n}/\langle \cdots\otimes x\otimes y\otimes \cdots - \cdots\otimes y\otimes x\otimes \cdots \rangle. \] When $\F=\C$ these are isomorphic $\SL_2(\C)$-representations, but they are typically not isomorphic as $\SL_2(\F)$-representations when the field has positive characteristic. This makes the following theorem from~\cite{GMSW} notable. %MkW added citation \begin{Proposition} \label{prop:t&l} There is a characteristic-independent $\SL_2(\F)$-isomorphism \[ \Sym_m\Sym^n\hskip1pt\F^2 \otimes \Sym_{m+n}\Sym^d\hskip1pt\F^2 \cong \Sym_n\Sym^d\hskip1pt\F^2 \otimes \Sym_m\Sym^{d+n}\hskip1pt\F^2 \] lifting \eqref{t&l} over $\F$. \end{Proposition} Isomorphisms that exist over $\C$ do not in general hold over $\F$: see for instance \cite[Th.~1.6]{McDW} for many counterexamples. This makes characteristic-independent isomorphisms a rarity. Their study is fast becoming a vibrant field. %MkW In \cite{McDW}, McDowell and the fourth author lifted \emph{Hermite reciprocity} \begin{equation}\label{eq:hermite} \Sym_m\Sym^n\hskip1pt\F^2 \cong \Sym^n\hskip1pt\Sym_m\hskip1pt\F^2 \end{equation} and the \emph{Wronskian isomorphism} \begin{equation} \label{eq:Wronskian iso} \bwedge{n}\Sym^{n+m-1}\hskip1pt\F^2 \cong \Sym_n\Sym^{m}\hskip1pt\F^2. %MkW: inner sym powers were wrong way round \end{equation} The second and fourth authors lifted a special case of Stanley's $q$-Hook-Content Formula \cite{MW}. We generalise this in Proposition~\ref{prop:hooks}. Recently, Ikenmeyer, Omar, and Tsintsilidas lifted a correspondence of certain Kronecker and plethysm coefficients \cite{IOT}. %The second and fourth authors lifted a special case of Stanley's $q$-Hook-Content Formula \cite{MW}. We generalise this in Proposition~\ref{prop:hooks} below. %Recently, Ikenmeyer, Omar, and Tsintsilidas lifted an isomorphism between a plethysm and a %space of $\SL_2(\F)$-invariants motivated by the Kronecker coefficients \cite{IOT}. %\medskip % MkW deleted to save space and for uniformity We construct a new framework for lifting $q$-binomial identities over~$\F$. In \S\ref{sec:poly inter} we~give an isomorphism between each plethysm and a certain space of symmetric~polynomials: % MkW: it isn't a ring unless we quotient out, which takes time to explain. \vspace*{-9pt} \begin{equation} \label{eq: main iso} \Sym_m\Sym^n\hskip1pt\F^2 \cong \Lambda_{\le n}[x_1, \ldots, x_m]. \end{equation} This brings symmetric polynomials into representation theory \emph{without taking characters}, but instead by using symmetric polynomials as the \emph{elements} of the representation. With the exception of \eqref{eq:hermite}, all of the above isomorphisms (both old and new) become simple maps between polynomial rings, defined by specialisations of certain variables to other variables. The canonical example of such a map is the $k$-fold plethystic substitution (or simply, the \emph{$k$-fold map}) of symmetric functions \begin{align} \Lambda[\x_\infty] &\to \Lambda[\y_\infty] \nonumber\\ f(x_1,x_2, \ldots) = f(\x_\infty) &\mapsto f[k\y_\infty] = f(y_1, \stackrel{k}{\ldots}, y_1, y_2, \stackrel{k}{\ldots}, y_2, \ldots) \end{align} defined in~\S\ref{sec:k-fold}. In this abstract we give combinatorial interpretations for the structure constants of the $k$-fold map on the Schur and the monomial bases. We sketch a proof of Proposition~\ref{prop:t&l} (lifting the $q$-team-and-leader identity) in \S\ref{sec:k-fold} and Proposition~\ref{prop:hooks} (lifting the $q$-Hook-Content Formula) in \S\ref{hooks}.%\medskip We finish by giving two conjectures. First, that a certain $q$-binomial identity implied by the $q$-Pfaff--Saalschütz identity \cite{GSMW-Pfaff} lifts to a filtration of $\SL_2(\F)$-plethysms. This result would complete a crucial step in the categorification of Lusztig's integral form of the Cartan subalgebra of the quantum group $U_q(\mathfrak{sl}_2)$. See \cite{Martinez} for details on this programme, which predicted Propositions~\ref{prop:t&l} and \ref{prop:hooks}, and is heuristic evidence for Conjecture~\ref{con:filtration}. The second is the generalised Foulkes Conjecture for $\SL_2(\C)$: if $a = \min\{a,b,c,d\}$ and $ab=cd$ then \[ \qbinom{c+d}{c} \ge_q \qbinom{a+b}{a}\,, \] where $f \ge_q g$ if $f-g$ is a non-negative combination of $q$-integers. This conjecture was posed by Doran, Vessenes, Abdesselam and Chipalkatti, and Bergeron \cite{Doran,Vessenes,AC,Bergeron}. %Chronological order, 98, 04, 07, 17 We conjecture, moreover, that the $k$-fold map lifts this inequality (over~$\C$) when $a$ divides $c$. \section{Plethysms as spaces of symmetric polynomials} \label{sec:poly inter} Let $\F_d[X,Y]$ be the ring of homogeneous polynomials in $X$ and $Y$ of degree $d$. The group $\SL_2(\F)$ acts on $\F_d[X,Y]$ by \begin{align*}\label{action-of-SL_2} \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}\cdot P(X,Y)=P(\alpha X+\gamma Y,~\beta X+\delta Y). \end{align*} A classical result is that $\C_d[X,Y]$ is the irreducible $\SL_2(\C)$-representation of dimension $d+1$. Moreover $\C_d[X,Y] \cong \Sym^d\hskip0.5pt\C^2$, and more generally $\F_d[X,Y] \cong \Sym^d\F^2$. Consider now the plethysm $\Sym_n\Sym^d\hskip1pt\F^2$. By definition, \begin{align*} \Sym_n\Sym^d\hskip1pt\F^2 = \bigl\{P\in\F[X_1,\ldots,X_n,Y_1,\ldots,Y_n]\mid %\deg_{X_i}\!P + \deg_{Y_i}\!P = d ~\text{for all $i$} \text{degree}\,d\,\text{multi-homogenous in pairs} \bigr\}^{\SymG_n}, \end{align*} where $\sigma\in\SymG_n$ acts by sending $X_i$ to $X_{\sigma(i)}$ and $Y_i$ to $Y_{\sigma(i)}$ for all $i$. We remark that, using tensor products of algebras, the right-hand side becomes \smash{$(\F_d[X,Y]^{\otimes n})^{\SymG_n}$}. Let $\XX = (X_1, X_2, \ldots, X_n)$. For ease of notation, we specialise $Y_i = 1$ for all $i$, to obtain polynomials in $X_1, \ldots, X_n$ of \emph{degree at most $d$ in each variable} and invariant in the $\SymG_n$ action. This is a subspace of the space $\Lambda[\XX]$ of symmetric polynomials in $\XX$, which we shall call $\Lambda_{\leq d}[\XX]$. We obtain an isomorphism \[ \Sym_n\Sym^d\hskip1pt\F^2 \cong \Lambda_{\le d}[\XX]. \] The action of an element $U=\left(\begin{smallmatrix} \alpha&\beta\\\gamma&\delta \end{smallmatrix}\right)\in\SL_2(\F)$ inherited by $\Lambda_{\leq d}[\XX]$ is given by \begin{equation}\label{eq:SL2action} Uf(X_1,\ldots,X_n)=\prod_{i=1}^n(\beta X_i+\delta)^d\cdot f\!\left( \frac{\alpha X_1+\gamma}{\beta X_1+\delta}, \ldots, \frac{\alpha X_n+\gamma}{\beta X_n+\delta} \right). \end{equation} Alternatively, as we show in \cite{GMSW}, the $\SL_2(\F)$ action is completely determined by the action of the element $\mathbf{f} = \left(\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix}\right)$ of $\mathfrak{sl}_2(\C)$ as the operator $\mathrm{D}_{\XX} = \frac{\partial}{\partial X_1} + \cdots + \frac{\partial}{\partial X_n}$.\medskip %Following this isomorphism, the element %$\mathbf{f} = \left(\begin{smallmatrix} % 0 & 0 \\ 1 & 0 %\end{smallmatrix}\right)$ %of $\mathfrak{sl}_2(\C)$ acts on $\Lambda_{\le d}[\XX]$ as the~operator $\mathrm{D}_{\XX} = %\frac{\partial}{\partial X_1} + \cdots + \frac{\partial}{\partial X_n}$. This suffices to determine the %entire $\SL_2(\F)$-action~\cite{GMSW}. To illustrate the strength of this identification, we sketch a brief proof of \eqref{eq:Wronskian iso}. \begin{Proposition}[Wronskian isomorphism \cite{McDW, Grinberg, IOT}]\label{prop:Wronskian} Let $n, d \in \N_0$. %such that $N-1\le d$. Let $V(\x)$ be the Vandermonde determinant. The map \begin{align*} \zeta_{n,d} : \Lambda_{\le d}[\x] &\to V(\x)\Lambda_{\leq d}[\x]\\ P(\x) &\mapsto V(\x) P(\x) \end{align*} realises an isomorphism of $\SL_2(\F)$-representations \( \Sym_n\Sym^{d} \F^2\cong \bwedge{n}\Sym^{d+n-1} \F^2. \) \end{Proposition} \begin{proof}[Sketch.] Since polynomial rings are integral domains, the map $\zeta_{n,d}$ is injective. Since $V(\x)$ is a determinant, every element of $\SL_2(\F)$ acts on it trivially, so $\zeta_{n,d}$ is $\SL_2(\F)$-equivariant. Conclude by noting each side has dimension \smash{$\binom{d+n}{d}$}. \end{proof} Observe that \( \{s_\lambda(\XX) \mid \lambda\in L(n,d)\} \) and \( %\quad\text{and}\quad \{m_\lambda(\XX) \mid \lambda\in L(n,d)\} \) are bases of $\Lambda_{\le d}[\XX]$, where $L(n,d)$ is the set $\{\lambda\in\Par\mid \lambda_1\le d,~\ell(\lambda)\le n\}$ of partitions in an $n\times d$ rectangle and $s_\lambda$ and $m_\lambda$ are the Schur and monomial symmetric polynomials. \section{The $k$-fold map} \label{sec:k-fold} % I think it's a good idea to start with the k-fold map in C, as many results use some nice symmetric-functionology, which is perhaps more in line with FPSAC philosophy. Also our modular maps can then be viewed as two-fold substitutions. % The product of an alphabet $\mathbf{x} = (x_1, x_2, \ldots)$ with a scalar $k\in\N$ is the alphabet % \[ % k\mathbf{x} = (x_1, \stackrel{k}{\ldots}, x_1, x_2, \stackrel{k}{\ldots}, x_2, \ldots). % \] \begin{Definition}\label{de:k-fold map} Suppose either $|\x| = k|\y|$ or $|\x| = \infty = |\y|$. The \emph{$k$-fold map} is defined by \begin{align*} \Lambda[\x] &\to \Lambda[\y] \\ f(x_1,x_2,\ldots ) = f(\x) &\mapsto f[k\y] = f(y_1, \stackrel{k}{\ldots}, y_1, y_2, \stackrel{k}{\ldots}, y_2, \ldots). \end{align*} \end{Definition} \noindent Equivalently, the $k$-fold map sends each $x_i$ to $y_{\lceil i/k\rceil}$. %It is a $\C$-algebra homomorphism. Letting $\x_n$ denote an alphabet of size~$n$, note that $\Lambda_{\le m}[\x_{kn}]$ gets mapped to $\Lambda_{\le km}[\y_n]$. \begin{Proposition}\label{prop:k-fold-sl2} The $k$-fold map $\Lambda_{\le m}[\x_{kn}] \to \Lambda_{\le km}[\y_n]$ is $\SL_2(\F)$-equivariant. \end{Proposition} % \begin{proof} % Let $U=\left(\begin{smallmatrix} % \alpha &\beta\\ \gamma& \delta % \end{smallmatrix}\right)\in\SL_2(\F)$. For an alphabet $\z$, let $(a\z+b)$ denote $\prod_{z\in\z}(az+b)$. The explicit computation % \begin{align*} % (Uf)[k\y_{n}]=(\beta[k\y_{n}]+\delta)^m\cdot f\left(\frac{\alpha[k\y_n]+\gamma}{\beta[k\y_n]+\delta}\right)=(\beta\y_n+\delta)^{km}\cdot f\left[k\frac{\alpha\y_n+\gamma}{\beta\y_n+\delta}\right]=U\left(f[k\y_n]\right) % \end{align*} % shows that the action of $\SL_2(\F)$ intertwines with the $k$-fold map. % \end{proof} %attempt 2 \begin{proof} Let $\kappa$ denote the $k$-fold map defined above, and let $U=\left(\begin{smallmatrix} \alpha &\beta\\ \gamma& \delta \end{smallmatrix}\right)\in\SL_2(\F)$. Following \eqref{eq:SL2action}, we compute explicitly that both $\kappa(Uf(\x_{kn}))$ and $U\kappa(f(\x_{kn}))$ equal \[ \prod_{i=1}^n\left((\beta y_i+\delta)^m\right)^k\cdot f\!\left( \frac{\alpha y_1+\gamma}{\beta y_1+\delta}, \stackrel{k}{\ldots}, \frac{\alpha y_1+\gamma}{\beta y_n+\delta}, \ldots, \frac{\alpha y_n+\gamma}{\beta y_n+\delta}, \stackrel{k}{\ldots}, \frac{\alpha y_n+\gamma}{\beta y_n+\delta} \right)\,.\qedhere \] \end{proof} %\begin{proof}[Sketch] % Since the $\SL_2(\F)$-action is determined by the derivative operators from \S~\ref{sec:poly inter}, % it suffices to show that % \[ % (\mathrm{D}_{\mathbf{x}_{kn}} f)[k\mathbf{y}_n] = \mathrm{D}_{\mathbf{y}_n}(f[k\mathbf{y}_n]) %% (\mathrm{D}_{\mathbf{x}_{kn}} f)[k\mathbf{y}_n] = \mathrm{D}_{\mathbf{y}_n}(f(\x_{kn})) % \] % for $f\in\Lambda_{\le m}[\x_{kn}]$. By linearity of the derivative, it suffices to check the above identity on a single monomial $\prod_{i=1}^{kn} x_i^{m_i}$ appearing in $f$. This is a simple direct computation. %\end{proof} \subsection{The $k$-fold map on the Schur basis} \begin{Definition} For each $\lambda, \mu \in \Par$, define $d_\lambda^\mu(k)$ by $s_\lambda[k \mathbf{y}_\infty] = \sum_\mu d_\lambda^\mu(k) s_\mu(\mathbf{y}_\infty)$. \end{Definition} We study $D(k) = \big( d_\lambda^\mu(k)\big)_{\lambda,\mu\in\Par}$. Observe first that this matrix is block-diagonal. \begin{Lemma} If $|\lambda|\ne|\mu|$ then $d_\lambda^\mu(k) = 0$. \end{Lemma} % In fact, since $p_\lambda[k\mathbf{y}_\infty] = k^{\ell(\lambda)}p_\lambda(\mathbf{y}_\infty)$, the matrix $D(k)$ is diagonalised by the matrix $X = (\chi^\lambda(\mu))_{\lambda,\mu\in\Par}$ of character values of the symmetric groups. More precisely, $X^{-1}\cdot D(k)\cdot X = \mathrm{diag}(1,k,k,k^2,\ldots) = \mathrm{diag}(k^{\ell(\lambda)})_{\lambda\in\Par}$.\medskip Our first formula expresses $d_\lambda^\mu(k)$ in terms of $k$-multi-Littlewood--Richardson coefficients, defined by \smash{$c_{\bm{\nu}}^\lambda = \langle s_{\nu^{(1)}}\cdots s_{\nu^{(k)}}, s_\lambda\rangle$}, where \smash{$\bm{\nu}=(\nu^{(1)},\ldots,\nu^{(k)})\in\Par^k$}. The $k$-multi-Littlewood--Richardson coefficient $c_{\bm\nu}^\lambda$ counts the number of lattice $k$-multi-tableaux of multi-shape $\bm{\nu}$ and content $\lambda$ \cite[\S3 and Lemma~6.1]{SXP}. \begin{Proposition}\label{p:coefs} For all $\lambda, \mu\in\Par$, we have \( d_\lambda^\mu(k) = \sum_{\bm{\nu} \in \Par^k} c_{\bm{\nu}}^\lambda c_{\bm{\nu}}^\mu \). \end{Proposition} The symmetries of Littlewood--Richardson coefficients \cite{BriandRosas} endow $D(k)$ with a rich combinatorial structure. \begin{Lemma}{\ }\label{lem:symmetries} \begin{thmlist} \item The matrix $D(k)$ is symmetric: $d_\lambda^\mu(k) = d_\mu^\lambda(k)$. \item The blocks of $D(k)$ are conjugation invariant: $d_\lambda^\mu(k) = d_{\lambda'}^{\mu'}(k)$. %MkW: changed from antisymmetric \item We have $d_\lambda^\mu(k) = d_{\lambda^\square}^{\mu^\square}(k)$, where the box-complement of $\lambda$ (resp.~$\mu$) is taken inside $L(b,a)$ (resp.~$L(d,c)$), for all choices of $a, b, c, d$ such that $c/a = b/d = k$. \end{thmlist} \end{Lemma} \begin{Proposition}[Cauchy formula] For all $\lambda, \mu\in\Par$, we have \[d_\lambda^\mu(k) = \Big\langle s_\lambda(\mathbf{y}_\infty)s_\mu(\mathbf{z}_\infty), ~ \prod_{i,j} (1-y_jz_i)^{-k} \Big\rangle.\] \end{Proposition} % The third formula for $d_\lambda^\mu(k)$ involves Kronecker products. Let $f*g$ denote the Kronecker product of symmetric polynomials. The Kronecker coefficients are defined by \( g_{\lambda\mu\nu} = \langle s_\lambda, ~s_\mu*s_\nu\rangle. \) %denote the \emph{Kronecker coefficients}. \begin{Proposition}[Kronecker formula]\label{p:Kronecker1} For all $\lambda,\mu\in\Par$, we have \(d_\lambda^\mu(k) = (s_\lambda * s_\mu)(1, \stackrel{k}{\ldots}, 1).\) \end{Proposition} \begin{Corollary}\label{cor:d-expresion-ssyt} For all $\lambda,\mu\in\Par$, we have $d_\lambda^\mu(k) = \sum_{\nu} g_{\lambda\mu\nu} ~\#\hskip1pt\mathrm{SSYT}_k(\nu)$. \end{Corollary} \begin{proof} Expand the Kronecker product to obtain \[ s_\lambda * s_\mu(1, \stackrel{k}{\ldots}, 1) = \sum_\nu g_{\lambda\mu\nu} ~ s_{\nu}(1, \stackrel{k}{\ldots}, 1) = \sum_\nu g_{\lambda\mu\nu} ~\#\hskip1pt\mathrm{SSYT}_k(\nu). \qedhere \] \end{proof} \vspace*{-3pt} We believe this new connection between Kronecker coefficients and the structure constants of the $k$-fold map will repay further investigation. % We can now deduce necessary conditions for some entries in the blocks of $D(k)$ to be zero or non-zero. % \begin{Corollary} % If $k<\max\left(\frac{\ell(\lambda)}{\ell(\mu)}, \frac{\ell(\mu)}{\ell(\lambda)}\right)$ then $d_\lambda^\mu(k)=0$. If $k>|\lambda\cap\mu'|$ then $d_\lambda^\mu(k)\neq0$. % \end{Corollary} %\vspace{-6pt} \begin{Example}\label{eg:D(2)} The following is the top-left corner of the infinite matrix $D(2)$. \end{Example} %Moved matrix outside Example environment to avoid italics for numbers \vspace*{-24pt} \[ \ytableausetup{boxsize=.2em,centertableaux} \begin{tikzpicture} \node[matrix, row sep={1.2em,between origins}, column sep={1.5em,between origins}] (*) at (0,0) { & \node{$\scriptstyle \varnothing$}; & \node{\ydiagramb{1}}; &\node{\ydiagramb{2}}; &\node{\ydiagramb{1,1}}; &\node{\ydiagramb{3}}; &\node{\ydiagramb{2,1}}; &\node{\ydiagramb{1,1,1}}; &\node{\ydiagramb{4}}; &\node{\ydiagramb{3,1}}; &\node{\ydiagramb{2,2}}; &\node{\ydiagramb{2,1,1}}; &\node{\ydiagramb{1,1,1,1}}; \\% &\node{\ydiagramb{5}}; &\node{\ydiagramb{4,1}}; &\node{\ydiagramb{3,2}}; &\node{\ydiagramb{3,1,1}}; &\node{\ydiagramb{2,2,1}}; &\node{\ydiagramb{2,1,1,1}}; &\node{\ydiagramb{1,1,1,1,1}};\\ \node{$\scriptstyle \varnothing$}; & \node{\sm{1}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.}; \\%&\node{.};&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{1}}; & \node{.};&\node{\sm{2}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.}; \\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{2}}; & \node{.};&\node{.};&\node{\sm{3}};&\node{\sm{1}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.}; \\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{1,1}}; & \node{.};&\node{.};&\node{\sm{1}};&\node{\sm{3}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.}; \\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{3}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{4}};&\node{\sm{2}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.}; \\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{2,1}}; 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\\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{2,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{4}};&\node{\sm{3}};&\node{\sm{9}};&\node{\sm{3}}; \\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ \node{\ydiagramb{1,1,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{1}};&\node{\sm{3}};&\node{\sm{5}}; \\%&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ % \node{\ydiagramb{5}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{6};&\node{ 4};&\node{ 2};&\node{.};&\node{ .};&\node{ .};&\node{.};\\ % \node{\ydiagramb{4,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{4};&\node{12};&\node{ 6};&\node{6};&\node{ 2};&\node{ .};&\node{.};\\ % \node{\ydiagramb{3,2}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{2};&\node{ 6};&\node{12};&\node{6};&\node{ 6};&\node{ 2};&\node{.};\\ % \node{\ydiagramb{3,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{ 6};&\node{ 6};&\node{4};&\node{ 6};&\node{ 6};&\node{.};\\ % \node{\ydiagramb{2,2,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{ 2};&\node{ 6};&\node{6};&\node{12};&\node{ 6};&\node{2};\\ % \node{\ydiagramb{2,1,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{ .};&\node{ 2};&\node{6};&\node{ 6};&\node{12};&\node{4};\\ % \node{\ydiagramb{1,1,1,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{ .};&\node{ .};&\node{.};&\node{ 2};&\node{ 4};&\node{6};\\ }; \end{tikzpicture} \] \subsection{The $k$-fold map on the monomial basis} We now turn to the study of the coefficients of the $k$-fold map in the monomial basis. Let $K_{\lambda,\mu} = [m_\mu]s_\lambda$ % \langle s_\lambda, h_\mu\rangle$ %MkW: was m_\mu which is wrong: it has to be h_\mu if we want to use the Hall inner product. But instead % we could take a coefficient and avoid using Hall inner product. be the \emph{Kostka coefficient} and let \smash{$K = (K_{\lambda,\mu})_{\lambda,\mu\in\Par}$} be the \emph{Kostka matrix}. We study the block-diagonal matrix \smash{$B(k) = K^{-1} \cdot D(k) \cdot K$}. \begin{Definition} For each $\lambda, \mu \in \Par$, define $b_\lambda^\mu(k)$ by $m_\lambda[k \mathbf{y}_\infty] = \sum_\mu b_\lambda^\mu(k) m_\mu(\mathbf{y}_\infty)$. \end{Definition} \begin{Definition} Given a multiset $A = \{\!\hskip-1pt\{a_1, a_2,\ldots,a_n\}\!\hskip-1pt\}$, a \emph{multiset partition} of $A$ is a collection of multisets (\emph{parts}) $A^{(1)}, A^{(2)}, \ldots, A^{(\ell)}$ such that the union (with multiplicity) of the parts is $A$. The \emph{weight} of a multiset partition $A^{(1)}, A^{(2)}, \ldots, A^{(\ell)}$ is the tuple $w=(w_1, w_2, \ldots, w_\ell)$, where $w_i$ is the sum of elements of $A^{(i)}$ (with multiplicity). When $k \mid n$, let $\mathcal{P}_{k}(A,w)$ be the set of multiset partitions of $A$ of weight $w$, and in which each part has $k$ elements (with multiplicity). \end{Definition} \begin{Proposition} Let $|\lambda|=|\mu|$ and $A_\lambda = \{\!\hskip-1pt\{\lambda_i \mid i = 1, \ldots, k\ell(\mu)\}\!\hskip-1pt\}$. Then \vspace*{-6pt} \[ b_{\lambda}^\mu(k) = \sum_{\mathcal{P}_{k}(A,\mu)}~ \prod_{i=1}^{\ell(\mu)} \frac{k!}{m_0(A_\lambda^{(i)})!~m_1(A_\lambda^{(i)})!\cdots}, \] \vspace*{-1pt} \noindent where $m_j(A_\lambda^{(i)})$ is the multiplicity of $j$ in $A_\lambda^{(i)}$. \end{Proposition} If $b_\lambda^\mu(k)\neq 0$ then $\mathcal{P}_k(A_\lambda,\mu)\neq\varnothing$, so $\mu$ coarsens $\lambda$. In particular, $\mu\geq_{\mathrm{lex}}\lambda$, so $B(k)$ is lower-triangular. The diagonal is $(k^{\ell(\lambda)})_{\lambda\in\Par}$ since $B(k)$ is conjugate to the matrix of the $k$-fold map in the power sum basis, and $p_\lambda[k\y_\infty] = k^{\ell(\lambda)}p_\lambda(\y_\infty)$. %In particular, $B(k)$ is lower-triangular. The diagonal is $(k^{\ell(\lambda)})_{\lambda\in\Par}$ since $B(k)$ is conjugate to the matrix of the $k$-fold map in the power sum basis, and $p_\lambda[k\y_\infty] = k^{\ell(\lambda)}p_\lambda(\y_\infty)$. %\begin{Corollary} % If $b_\lambda^\mu(k) \ne 0$ then $\mu \ge \lambda$ in the lexicographic order. %\end{Corollary} %\begin{proof} % If there is a partition of $\{\!\hskip-1pt\{\lambda_i : i = 1, \ldots, k\ell(\mu)\}\!\hskip-1pt\}$ of weight $\mu$, then $\mu \ge \lambda$. %\end{proof} \begin{Example}\label{eg:B(2)} The following is the top-left corner of the infinite matrix $B(2)$. \end{Example} \vspace*{-24pt} \[ \ytableausetup{boxsize=.2em,centertableaux} \begin{tikzpicture} \node[matrix, row sep={1.2em,between origins}, column sep={1.5em,between origins}] (*) at (0,0) { & \node{$\varnothing$}; & \node{\ydiagramb{1}}; &\node{\ydiagramb{2}}; &\node{\ydiagramb{1,1}}; &\node{\ydiagramb{3}}; &\node{\ydiagramb{2,1}}; &\node{\ydiagramb{1,1,1}}; &\node{\ydiagramb{4}}; &\node{\ydiagramb{3,1}}; &\node{\ydiagramb{2,2}}; &\node{\ydiagramb{2,1,1}}; &\node{\ydiagramb{1,1,1,1}}; \\ \node{$\varnothing$}; & \node{\sm{1}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{1}}; & \node{.};&\node{\sm{2}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{2}}; & \node{.};&\node{.};&\node{\sm{2}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{1,1}}; & \node{.};&\node{.};&\node{\sm{1}};&\node{\sm{4}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{3}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{2}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{2,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{2}};&\node{\sm{4}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{1,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{2}};&\node{\sm{8}};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{4}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{2}};&\node{.};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{3,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{2}};&\node{\sm{4}};&\node{.};&\node{.};&\node{.};\\ \node{\ydiagramb{2,2}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{1}};&\node{.};&\node{\sm{4}};&\node{.};&\node{.};\\ \node{\ydiagramb{2,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{4}};&\node{\sm{4}};&\node{\sm{8}};&\node{.};\\ \node{\ydiagramb{1,1,1,1}}; & \node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{.};&\node{\sm{1}};&\node{\sm{4}};&\node{\sm{16}};\\ }; \end{tikzpicture} \] Again, we can obtain a formula in terms of Kronecker products. \begin{Proposition}\label{p:b-expresion-Kronecker} For all $\lambda,\mu\in\Par$, we have $b_\lambda^\mu(k) = (m_\lambda*h_\mu)(1,\stackrel{k}{\ldots},1)$. \end{Proposition} \vspace*{-24pt} \section{Lifting the $q$-team-and-leader identity} %\vspace*{0pt} To establish Proposition~\ref{prop:t&l}, we translate both the left- and the right-hand side of the desired isomorphism through \eqref{eq: main iso} --- Proposition~\ref{prop:t&l} then becomes \vspace*{-6pt} \[ \Lambda_{\le n}[\z_{m}] \otimes \Lambda_{\le d}[\x_n, \y_m] \cong %MkW: no ? since we have proved this %\stackrel{?}{\cong} \Lambda_{\le d}[\x_n] \otimes \Lambda_{\le n+d}[\y_m]. \] This framework allows us to construct a simple map from the left- to the right-hand side of this expression as follows. We state the result for a larger class of maps, which we use in the next section. \begin{Proposition}\label{prop:injective} Suppose $\delta \le |\x|$, $\varepsilon\in\N_0$, and $|\y| = |\z|$. Then the following map is injective: \begin{align*} \pi_\delta: \Lambda_{\leq\delta}[\z]\otimes\Lambda_{\leq\varepsilon}[\x,\y] &\to \Lambda_{\leq\varepsilon}[\x] \otimes\Lambda_{\leq\varepsilon+\delta}[\y]\\ P(\x,\y,\z) &\mapsto P(\x,\y,\y). \end{align*} \end{Proposition} The proof of this proposition uses a novel application of \emph{Lagrange interpolation}, which almost miraculously recovers each variable in the alphabet $\z$ that disappeared in the substitution. The details of the proof can be found in~\cite[\S 4]{GMSW}. The map $\pi_\delta$ above is an extension of the $1$-fold map, so it is $\SL_2(\F)$-equivariant by Proposition~\ref{prop:k-fold-sl2}. When $\delta=|\x|=n$, $|\y|=|\z|=m$, and $\varepsilon=d$, dimension counting shows that the map is an isomorphism. This concludes our new proof of Proposition~\ref{prop:t&l}. %MkW trivial rephrasing % \begin{proof}[Sketch] % For each $1\leq j\leq |\y|$, let $t_j$ be the element of the group algebra $\F\SymG_{|\x|+|\y|}$ given by $\smash{\sum_{i=1}^{|\x|} (x_i, y_j)}$. The $j$th \emph{Lagrange interpolation operator} is defined as % \begin{align*}\label{eq:Lj} % \mathcal{L}_j: \F(\x,\y)[\z] & \longrightarrow \F(\x, \y)[\z]\\[-.5em] % P&\longmapsto (1+t_j)\Big(P\cdot\prod_{i=1}^{|\x|}\frac{z_{j}-x_i}{y_j-x_i}\Big). % \end{align*} % A computation shows that $f \mapsto \mathcal{L}_1\mathcal{L}_2\dots \mathcal{L}_{|\y|}(f)$ is a left inverse of $\pi_\delta$. % \end{proof} \section{Lifting Stanley's $q$-Hook-Content Formula for hooks} \label{hooks} A (non-obvious) application of Stanley's $q$-Hook-Content Formula for partitions of hook shape gives the following factorisation \[ s_{(m+1,1^{n-1})}(q^{-d},q^{2-d},\ldots,q^d) = \qbinom{m+n-1}{m} \qbinom{m+d+1}{m+n}. \] To lift this identity over $\F$, we construct an $\SL_2(\F)$-representation $\Delta^{(m+1, 1^{n-1})}\Sym^d\F^2$ whose character when $\F = \C$ is the left-hand side. Sparing the details for this abstract, we show in \cite{GMSW} that this representation is $\SL_2(\F)$-isomorphic to the intersection \begin{equation} \label{eq:Delta} V(\y_m)\Lambda_{\leq d-n+1}[\y_m]\otimes \Lambda_{\leq d}[\x_n,\y_m]~~\cap~~\ker\mu, \end{equation} where $V(\y_m)$ is the Vandermonde determinant and $\mu$ is a map on polynomials given by the action of $1-\sum_{i=1}^n (x_i, y_1)\in\F\SymG_{n+1}$ (where $(x_i, y_1)$ swaps $x_i$ with~$y_1$). \begin{Proposition} \label{prop:hooks} There is a characteristic-independent $\SL_2(\F)$-isomorphism \[ \Delta^{(m+1, 1^{n-1})}\Sym^d\hskip1pt\F^2 \cong \Sym_m\Sym^{n-1}\hskip1pt\F^2\otimes \Sym_{m+n}\Sym^{d-n+1}\hskip1pt\F^2. \] \end{Proposition} \begin{proof}[Sketch] Let $\pi_{n-1}$ be the injection from Proposition~\ref{prop:injective} (with $\varepsilon=d-n+1$). Let $\zeta_{n, d-n+1}$ be defined as in Proposition~\ref{prop:Wronskian}. Calculation shows that the image of the right-hand side under the map $(\zeta_{n, d-n+1}\otimes1)\circ\pi_{n-1}$ is inside the intersection \eqref{eq:Delta} and hence inside \smash{$\Delta^{(m+1, 1^{n-1})}\Sym^d\F^2$}. Conclude by counting dimensions. \end{proof} The case $m=0$ of this proposition is the Wronskian isomorphism~\eqref{eq:Wronskian iso}. \section{Towards a categorification of $U_q^0(\sl_2)$} The $\SL_2(\F)$-plethysms $\Sym_n\Sym^d\F^2$ appear to be intimately linked to the Cartan subalgebra of the quantum group $U_q(\sl_2)$. Recall the triangular decomposition \[U_q(\sl_2)\cong U^+_q(\sl_2)\otimes U_q^0(\sl_2)\otimes U_q^-(\sl_2).\] The Cartan subalgebra $U_q^0(\sl_2)$ has an integral form spanned over $\Z[q,q^{-1}]$ by Lusztig's elements \smash{$\kbinom{K\,;\,i}{t}\in \Z[q,q^{-1}](K)$}, which are a $q$-deformation of the elements \smash{$\binom{H+i}{t}$} in Kostant's $\Z$-form of the enveloping algebra $U(\sl_2)$. While categorification of \smash{$U^+_q(\sl_2)$} (or \smash{$U^-_q(\sl_2)$}) is well understood \cite{Lauda}, describing a monoidal categorification of the Cartan subalgebra $U^0_q(\sl_2)$ remains an open problem. In $U^0_q(\sl_2)$, we have the multiplication rule \begin{equation}\label{eq:Kmultiplication} \kbinom{K}{t}\kbinom{K}{s}=\sum_{i=0}^{\min(s,t)}\qbinom{t}{i}\qbinom{s}{i}\kbinom{K;i}{s+t}\end{equation} where we write $\kbinom{K}{t}:=\kbinom{K\,;\,0}{t}$. Evaluated at $K=q^d$, this identity is a special case of the \textit{$q$-Pfaff--Saalschütz identity}, which is one of the many names given to $q$-binomial identities expressing the product of two $q$-binomials as a sum of triple products of $q$-binomials. We refer to our article \cite{GSMW-Pfaff} for the full statement and a literature review; see also~\cite{Koepf}. We have used~\eqref{eq:Kmultiplication} to predict several new infinite families of characteristic-free isomorphisms, including Propositions~\ref{prop:t&l} and \ref{prop:hooks}. As an example, take $\Delta^{(2,1)}\Sym^d\hskip1pt\F^2$. Letting $t=2$ and $s=1$, the multiplication rule reads \[\kbinom{K}{2}\kbinom{K}{1}= {\color{red}\kbinom{K}{3}}+ {\color{blue}[2]_q\kbinom{K\,;\,1}{3}}.\] By the ``translation'' described below, this lifts to a short exact sequence \[0\to {\color{blue}\F^2\otimes \bwedge{3}\Sym^{d+1}\F^2} \to \bwedge{2}\Sym^{d}\F^2\otimes \bwedge{1}\Sym^d\F^2 \to {\color{red}\bwedge{3}\Sym^{d}\F^2} \to 0,\] which, since $\Delta^{(2,1)}\Sym^d\F^2$ is the kernel of $\bwedge{2}\Sym^d\hskip1pt\F^2\otimes\bwedge{1}\Sym^d\F^2\to\bwedge{3}\Sym^d\F^2$, predicts the isomorphism $\Delta^{(2,1)}\Sym^d\F^2\cong \F^2\otimes \bwedge{3}\Sym^{d+1}\F^2$. This prediction is confirmed by Proposition~\ref{prop:hooks}. More generally,~\eqref{eq:Kmultiplication} predicts the isomorphisms \begin{align*} \Delta^{(1,1,1)}\Sym^d\hskip1pt\F^2&\cong \Sym_3\Sym^{d-2}\hskip1pt\F^2\\ \Delta^{(2,1)}\Sym^d\hskip1pt\F^2&\cong \F^2\otimes \Sym_3\Sym^{d-1}\F^2\\ \Delta^{(3)}\Sym^d\hskip1pt\F^2&= \Sym_3\Sym^d\hskip1pt\F^2 \end{align*} which are all instances of Proposition~\ref{prop:hooks}. %Simpler than referring to Wronskian (which % doesn't do (2,1). Observe that the tensor factors on each right-hand side are plethysms of the form $\Sym_n\Sym^m\hskip1pt \F^2$. The ``translation'' procedure to obtain these conjectures is made more precise in the second author's doctoral thesis~\cite{Martinez}, where a programme to categorify the Cartan subalgebra $U^0_q(\sl_2)$ by means of $\SL_2(\F)$-plethysms is outlined. Two major steps towards the programme are Propositions~\ref{prop:t&l} and \ref{prop:hooks}. A complete lifting of the general multiplication formula for Lusztig's elements remains open, involving two-row partition plethysms. The $q$-binomial identity to be lifted is \vspace*{-6pt} \[ s_{(a+b,~a)}\circ s_d(q^{-1},q) = \sum_{k=0}^{a-1} \qbinom{a+b+d+k}{2a+b} s_{(k+b,~k)}\circ s_{a-k}(q^{-1},q), \] \vspace*{3pt} \noindent which may be proved using $q$-Pfaff--Saalschütz and a Jacobi--Trudi identity \cite{GutKrat}. The lift of this identity is conjectured below. \begin{Conjecture}\label{con:filtration} Let $a,b\in\mathbb{N}$. The $\SL_2(\F)$-plethysm $\Delta^{(a+b,~ a)}\Sym^d \F^2$ has the filtration: \vspace*{-16pt} \begin{align*} \Delta^{(b)}\hskip1pt\Sym^{a}\hskip1pt\F^2&\otimes \Sym_{2a+b}\Sym^{d-a}\hskip1pt\F^2\\ \Delta^{(1+b,~1)}\hskip1pt\Sym^{a-1}\hskip1pt\F^2&\otimes \Sym_{2a+b}\Sym^{d-a+1}\hskip1pt\F^2\\[-3pt] &{\,}~\vdots\\ \Delta^{(a+b-2,~ a-2)}\hskip1pt\Sym^2\hskip1pt\F^2&\otimes \Sym_{2a+b}\Sym^{d-2}\hskip1pt\F^2\\ \Delta^{(a+b-1,~ a-1)}\hskip1pt\Sym^1\hskip1pt\F^2&\otimes \Sym_{2a+b}\Sym^{d-1}\hskip1pt\F^2. \end{align*} \end{Conjecture} Besides the outstanding question of why this connection between $\SL_2(\F)$-plethysms and $U^0_q(\sl_2)$ exists, another challenge is computing homomorphism spaces between $\SL_2(\F)$-plethysms. The characteristic-free setup appears to be quite rigid, in the sense that isomorphisms are rare, and the ones that we find seem to be unique up to scalar multiplication, as expected for ``irreducible objects'' by Schur's Lemma. %MkW This motivates the problem of determining the (integral) endomorphism algebras of $\Sym_n\Sym^d\F^2$. \section{Towards the generalised Foulkes Conjecture for $\SL_2(\C)$} In this section we return to working in $\C$. The problem of understanding $\SL_N(\C)$-plethysm coefficients is difficult even in its simplest cases. The \emph{Foulkes Conjecture} \cite{Foulkes} has been widely open for decades~\cite{McKay, EvseevPagetWildon, CIM}. \begin{Conjecture}[Foulkes] Let $a, b\in\N$ with $a\le b$. Then $ \Sym^a \Sym^b \C^N$ is an $\SL_N(\C)$-sub\-repre\-sentation of $\Sym^b \Sym^a \C^N$. \end{Conjecture} The conjecture does not hold over arbitrary characteristic \cite[Ch.~3]{O'Donovan}. For $N=2$ the conjecture holds trivially, since both plethysms are isomorphic by Hermite reciprocity~\eqref{eq:hermite}. In contrast, several generalisations of the Foulkes Conjecture remain open for $\SL_2(\C)$. The following is a special case of conjectures found in \cite{Doran, Vessenes, AC,Bergeron} and \cite{Ardonne}, where the motivation is the particle entanglement spectrum of certain quantum states. \begin{Conjecture}%[Generalised Foulkes conjecture for $\SL_2$] \label{con:gen-Fou} Let $a, b, c, d \in \N$, with $a = \min\{a,b,c,d\}$ and $ab=cd$. Then $\Sym^a\Sym^b\C^2$ is an $\SL_2(\C)$-subrepresentation of $\Sym^c\Sym^d\C^2$. Equivalently, \vspace*{-6pt} \[ \qbinom{c+d}{c} \ge_q \qbinom{a+b}{a}\,. \] \end{Conjecture} If $a$ divides $c$, then the $(c/a)$-fold map is an $\SL_2(\C)$-homomorphism $\Sym^b\Sym^a\C^2\to\Sym^d\Sym^c\C^2$ by Proposition~\ref{prop:k-fold-sl2}. %(recall from \S\ref{sec:intro} that $\Sym^n V\cong\Sym_n V$ over $\C$). Applying Hermite reciprocity to both sides, and using our new model for plethysms, we obtain a conjecture which implies Conjecture~\ref{con:gen-Fou}. \begin{Conjecture} Let $a, b, c, d \in \N$, with $a = \min\{a,b,c,d\}$ and $c/a=b/d\in\N$. Then the $(c/a)$-fold map $\Lambda_{\le a}[\x_b]\to\Lambda_{\leq c}[\y_d]$ is injective. \end{Conjecture} Using the {\sc Magma} %\cite{Magma} code \texttt{kFoldMatrices.m} available from the fourth author's website \url{www.ma.rhul.ac.uk/~uvah099/}, the conjecture has been verified for all $a,b,c,d \le 10$. Setting $c = b$ and $d=a$ in the conjecture predicts a new \emph{explicit} isomorphism for Hermite reciprocity~\eqref{eq:hermite} when $a$ divides $b$. %\begin{Remark} % In \cite{Ardonne}, Ardonne, Estienne, and Garjani pose a similar conjecture arising %\end{Remark} \vspace*{-7pt} \section*{Acknowledgements} \vspace*{-6pt} The authors thank Abdelmalek Abdesselam, François Bergeron, Darij Grinberg, Nate Harman, Mikhail Khovanov, and Christian Krattenthaler for insightful conversations. \vspace*{-6pt} \printbibliography \end{document}