\documentclass[ALCO]{cedram} \usepackage{amssymb, dsfont, fancyhdr, graphicx, color, tabularx, mathtools, hyperref,tikz,comment} \usepackage[shortlabels]{enumitem} \usepackage[all]{xy} \usepackage{ytableau} \usepackage{cite} \def\Imm{\operatorname{Imm}} \def\sgn{\operatorname{sgn}} \newcommand{\CC}{{\mathbb C}} \newcommand{\QQ}{{\mathbb Q}} \newcommand{\RR}{{\mathbb R}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\FF}{{\mathbb F}} \newcommand{\PP}{\mathbb P} \newcommand{\TT}{\mathbb T} \newcommand{\A}{{\mathcal A}} \newcommand{\X}{\mathcal X} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mf}[1]{\mathfrak{#1}} \newcommand{\Comment}[1]{{\color{blue} \sf ($\clubsuit$ #1 $\clubsuit$)}} \newcommand{\SComment}[1]{{\color{red} \sf ($\clubsuit$ #1 $\clubsuit$)}} \newcommand{\e}{{\varepsilon}} \newcommand{\Mat}{\text{Mat}} \newcommand{\res}[1]{|_{#1}} \newcommand{\gr}[1]{\Gamma[#1]} \newcommand{\gro}[1]{\Gamma(#1)} \newcommand{\wire}[2]{(#1,#2) } \newcommand{\lrangle}[2]{\langle #1 \rangle_{#2}} \newcommand{\spn}[1]{\sigma(#1)} \newcommand{\cl}[1]{\overline{#1}} \newcommand{\bbox}[1]{\mathbf{B}_{#1}} \newcommand{\multichoose}[2]{ \left.\mathchoice {\left(\kern-0.48em\binom{#1}{#2}\kern-0.48em\right)} {\big(\kern-0.30em\binom{\smash{#1}}{\smash{#2}}\kern-0.30em\big)} {\left(\kern-0.30em\binom{\smash{#1}}{\smash{#2}}\kern-0.30em\right)} {\left(\kern-0.30em\binom{\smash{#1}}{\smash{#2}}\kern-0.30em\right)} \right.} \makeatletter \newcommand{\specialcell}[1]{\ifmeasuring@#1\else\omit$\displaystyle#1$\ignorespaces\fi} \makeatother \title[$k$-positivity of 1324- and 2143-avoiding dual canonical basis elements]{$k$-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan--Lusztig immanants} \author[\initial{S.} Chepuri]{\firstname{Sunita} \lastname{Chepuri}} \address{Lafayette College\\ Pardee Hall\\ Easton\\ PA 18042 (USA)} \email{chepuris@lafayette.edu} \author[\initial{M.} Sherman-Bennett]{\firstname{Melissa} \lastname{Sherman-Bennett}} \address{University of Michigan\\ 2074 East Hall\\ Ann Arbor\\ MI 48109 (USA)} \email{msherben@umich.edu} \keywords{immanants, total positivity, dual canonical basis} \subjclass{15A15, 05E10, 20C30} \datereceived{2022-01-12} \daterevised{2022-07-07} \dateaccepted{2022-07-10} \equalenv{cor}{coro} \equalenv{lem}{lemm} \equalenv{defn}{defi} \equalenv{rmk}{rema} \equalenv{ex}{exam} \begin{abstract} In this note, we show that certain dual canonical basis elements of $\mathbb{C}[SL_m]$ are positive when evaluated on \emph{$k$-positive matrices}, matrices whose minors of size $k \times k$ and smaller are positive. Skandera showed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can be written in terms of \emph{Kazhdan--Lusztig immanants}, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan--Lusztig immanants indexed by 1324- and 2143-avoiding permutations. This extends previous work of the authors on Kazhdan--Lusztig immanants and uses similar tools, namely Lewis Carroll's identity (also known as the Desnanot-Jacobi identity). \end{abstract} \begin{document} \maketitle \section{Introduction}\label{sec:intro} For a function $f:S_n \to \CC$, the \emph{immanant} associated to $f$, $\Imm_f X:\Mat_{n \times n}(\CC) \to \CC$, is the function \begin{equation} \Imm_f X:= \sum_{w \in S_n} f(w) ~x_{1, w_1} \cdots x_{n, w_n}, \end{equation} where the $x_{i,j}$ are indeterminates. We evaluate $\Imm_f X$ on a matrix $M=(m_{i,j})$ by specializing $x_{i,j}$ to $m_{i,j}$ for all $i,j$. Immanants are a generalization of the determinant, where $f(w)=(-1)^{\ell(w)}$, and the permanent, where $f(w)=1$. Positivity properties of immanants have been studied since the early 1990's \cite{GJ,Greene,StemConj,HaiConj}. One of the main results in this area is that when $f$ is an irreducible character of $S_n$, then $\Imm_f(X)$ is nonnegative on \emph{totally nonnegative matrices}, that is, matrices with all nonnegative minors~\cite{StemTNN}. In this note, we will investigate positivity properties of functions closely related to \emph{Kazhdan--Lusztig immanants}, introduced by Rhoades and Skandera~\cite{RS}. \begin{defn} The \emph{Kazhdan--Lusztig immanant} $\Imm_v X: \Mat_{n \times n}(\CC) \to \CC$, for a permutation $v \in S_n$, is given by \begin{equation} \label{eq:immFormula1} \Imm_v X:= \sum_{w \in S_n} (-1)^{\ell(w)-\ell(v)} P_{w_0w, w_0v}(1) ~x_{1, w_1} \cdots x_{n, w_n} \end{equation} where $P_{x, y}(q)$ is the Kazhdan--Lusztig polynomial associated to $x,y \in S_n$, $w_0 \in S_n$ is the longest permutation, and we write permutations $w=w_1w_2\dots w_n$ in one-line notation. (For the definition of $P_{x, y}(q)$ and their basic properties, see e.g. \cite{BB}.) \end{defn} Our interest in Kazhdan--Lusztig immanants stems from their connection to the dual canonical basis of $\CC[SL_m]$. Using work of Du~\cite{Du}, Skandera~\cite{Skan} showed that the dual canonical basis elements of $\CC[SL_m]$ are exactly Kazhdan--Lusztig immanants evaluated on matrices of indeterminates with repeated rows and columns. Let $X=(x_{ij})$ be the $m \times m$ matrix of variables $x_{ij}$ and let $\multichoose{[m]}{n}$ denote the set of $n$-element multisets of $[m]:=\{1, \dots, m\}$. For $R, C\in \multichoose{[m]}{n}$ with $R=\{r_1 \leq \cdots \leq r_n\}$ and $C=\{c_1\leq \cdots \leq c_n\}$, we write $X(R,C)$ to denote the matrix $(x_{r_i, c_j})_{i, j=1}^n$ (see Definition~\ref{defn:RC}). \begin{prop}[\protect{\cite[Theorem 2.1]{Skan}}] \label{prop:canonicalImm} The dual canonical basis of $\CC[SL_m]$ consists of the nonzero elements of the following set: \[\left \{\Imm_v X(R, C): v \in S_n \text{ for some } n \in \mathbb{N} \text{ and } R, C \in \multichoose{[m]}{n}\right \}. \] \end{prop} The positivity properties of dual canonical basis elements have been of interest essentially since their definition, and are closely related to the study of total positivity. In 1994, Lusztig~\cite{LusTPGr} defined the totally positive part $G_{>0}$ of any reductive group $G$. He also showed that all elements of the dual canonical basis of $\mathcal{O}(G)$ are positive on $G_{>0}$. Fomin and Zelevinsky~\cite{FZTNNSemisimple} later proved that for semisimple groups, $G_{>0}$ is precisely the subset of $G$ where all \emph{generalized minors} are positive. Generalized minors are dual canonical basis elements corresponding to the fundamental weights of $G$ and their images under Weyl group action. Here, we study signs of dual canonical basis elements on a natural generalization of $G_{>0}$. Let $S$ be some subset of generalized minors and $G_{>0}^S$ the subset of $G$ where all elements of $S$ are positive. Which dual canonical basis elements are positive on all elements of $G_{>0}^S$? In this note, we consider the case where $G=SL_m$ and $S$ consists of the generalized minors corresponding to the first $k$ fundamental weights and their images under the Weyl group action. In this situation, $G_{>0}^S$ is the set of \emph{$k$-positive matrices}, matrices where all minors of size $k$ and smaller are positive. Cluster algebra structures, topology, and variation diminishing properties of these matrices have been previously studied in~\cite{BCM,CKST,Choud1,Choud2}. We call a matrix functional \emph{$k$-positive} if it is positive when evaluated on all $k$-positive matrices. Our main result is as follows: \begin{thm} \label{thm:main} Let $v \in S_n$ be $1324$-, $2143$-avoiding and suppose that for all $i\lambda_q$. Further, because $R$ is listed in weakly increasing order, $p>a$. We would like to argue that rows $p':=\delta_{a, n}(p)$ and $q':=\delta_{a, n}(q)$ of $A_{a, n}^{b, n}$ have distinct support. Since $p>a$, we have $(\lambda_{a, n}^{b, n})_{p'}=\lambda_p-1$ and $(\lambda_{a, n}^{b, n})_{q'}=\lambda_q-1$, so~$(\lambda_{a, n}^{b, n})_{p'}>(\lambda_{a, n}^{b, n})_{q'}$. An analogous argument shows that columns of $A_{a, n}^{b, n}$ with the same index have distinct support. Similarly, $\lambda_a^b$ is $(R\setminus\{r_{i_a}\},C\setminus\{c_{j_b}\})$-admissible, $\lambda_a^n$ is $(R\setminus\{r_{i_a}\},C\setminus\{c_{j_n}\})$-admissible, and $\lambda_n^b$ is $(R\setminus\{r_{i_n}\},C\setminus\{c_{j_b}\})$-admissible. On the other hand, $\lambda_n^n$ may not be $(R\setminus\{r_{i_n}\},C\setminus\{c_{j_n}\})$-admissible (consider $R=(1,1,2)$, $C=(1,2,3)$, $\lambda=(3,2,1)$, for example). Taking all of this together we find that the $(\det A_a^b~\det A_n^n)$ term in \eqref{eq:CarrolPartition} may be zero but that $\det A_{a, n}^{b, n}$ and $(\det A_a^n~\det A_n^b)$ are always nonzero. By induction, $\det A_{a, n}^{b, n}$ has sign $(-1)^{|\mu|+a+b+1}$ and $(\det A_a^n~\det A_n^b)$ has sign $(-1)^{a+b}$. If $(\det A_a^b~\det A_n^n)$ is nonzero it has sign $(-1)^{a+b+1}$. Thus, $(\det A_a^b~\det A_n^n-\det A_a^n~\det A_n^b)$ always has sign~$(-1)^{a+b+1}$ and $\det A$ is always nonzero with sign $(-1)^{|\mu|}$. \end{proof} \begin{cor}\label{cor:partitioncompdet} Let $\lambda \subseteq n^n$ be a Young diagram and let $\mu:= n^n /\lambda$. Suppose $\mu$ has Durfee square of size at most $k$, $M$ is a $k$-positive $m \times m$ matrix, and $R,C\in \multichoose{[m]}{n}$. Then \[ (-1)^{|\lambda|}\det M(R,C)\res\mu \geq 0 \] and equality holds if and only if $ (n^n/ (n-1, n-2, \dots, 1, 0) ) \nsubseteq \mu$ (or equivalently, $\lambda \nsubseteq (n-1, n-2, \dots, 1, 0)$) or if $\mu$ is not $(R,C)$-admissible. \end{cor} \begin{proof} Let $\dot{w_0}$ denote the matrix with ones on the antidiagonal and zeros elsewhere. For a multiset $J=\{j_1 \leq \cdots \leq j_n\}$, let $\overline{J}:=\{\overline{j}_1 \leq \cdots \leq \overline{j}_n\}$ where $\overline{j}_i:=n+1-j_{n+1-i}$. Let $M'$ be the antidiagonal transpose of $M$; in symbols, $M'=\dot{w_0} M^T \dot{w_0}$. Taking antidiagonal transpose does not effect the determinant, so $M'$ is also $k$-positive. If we transpose $M(R,C)\res{\mu}$ across the antidiagonal, we obtain the matrix $$N:=M'(\overline{C}, \overline{R})\res{\nu},$$ where $\nu$ is the Young diagram obtained from the skew-shape $\mu$ by reflecting across the antidiagonal. Applying Proposition~\ref{prop:partitiondet}, we have that $\det N$ has sign $(-1)^{|\lambda|}$ if $\nu$ is $(\overline{C}, \overline{R})$-admissible and is zero otherwise. It is not hard to check that $\nu$ is $(\overline{C}, \overline{R})$-admissible if and only if $\mu$ is $(R, C)$-admissible. \end{proof} We can use Proposition~\ref{prop:immDet} to rewrite Proposition~\ref{prop:partitiondet} and Corollary~\ref{cor:partitioncompdet} in terms of immanants. \begin{cor}\label{cor:partitionimm} Let $v \in S_n$ avoid $1324$ and $2143$. Suppose $\gr{v,w_0}$ is a Young diagram $\lambda$ with Durfee square of size at most $k$. If $M$ is a $k$-positive $m \times m$ matrix and~$R,C\in \multichoose{[m]}{n}$ such that $\lambda$ is $(R,C)$-admissible, then $\Imm_v M(R,C)>0$. \end{cor} \begin{proof} Note that $\gro{w_0} \subseteq \gr{v, w_0}$ implies $\lambda$ contains the partition $(n, n-1, \dots, 1)$. So, by Proposition~\ref{prop:partitiondet}, we know that $(-1)^{|\mu|}\det M(R,C)\res{\gr{v,w_0}}>0$ where $\mu=n^n/\lambda$. Notice that if a box of $\mu$ is in row $r$ and column $c$ then $v(r)0$. By Proposition~\ref{prop:immDet}, this means $\Imm_v M(R,C)>0$. \end{proof} \begin{cor}\label{cor:partitioncompimm} Let $v \in S_n$ avoid $1324$ and $2143$. Suppose $\gr{v,w_0}$ is $\lambda=n^n/\mu$ for some partition $\mu$ and the largest square in $\lambda$ is of size at most $k$. If $M$ is a $k$-positive $m \times m$ matrix and $R,C\in \multichoose{[m]}{n}$ such that $\lambda$ is $(R,C)$-admissible, then $\Imm_v M(R,C)>0$. \end{cor} \begin{proof} Note that $\gro{w_0} \subseteq \gr{v,w_0}$ implies $\lambda$ contains $(n^n/(n-1, n-2, \dots, 1,0))$. So, by Corollary~\ref{cor:partitioncompdet}, we know that $(-1)^{|\mu|}\det M(R,C)\res{\gr{v,w_0}}>0$. As in the proof of Corollary~\ref{cor:partitionimm}, there is a bijection between boxes of $\mu$ and inversions of $v$. So, we know $(-1)^{\ell(v)}\det M(R,C)\res{\gr{v,w_0}}=(-1)^{|\mu|}\det M(R,C)\res{\gr{v,w_0}}>0$. By Proposition~\ref{prop:immDet}, this means $\Imm_v M(R,C)>0$. \end{proof} \subsection{General Case}\label{sec:general} The following proposition will allow us to restrict to permutations that are not elements of a maximal parabolic subgroup of $S_n$. To state the lemma we temporarily denote the longest permutation in $S_j$ by $w_{(j)}$. \begin{prop}[\protect{\cite[Corollary 4.9]{CSB}}] \label{prop:ImmMultipleBlock} Suppose $v \in S_n$ is $1324$-, $2143$-avoiding and $\gr{v, w_0}$ is block-antidiagonal. Let $v_1 \in S_j$ and $v_2 \in S_{n-j}$ be permutations such that the upper-right antidiagonal block of $\gr{v, w_0}$ is equal to $\gr{v_1,w_{(j)}}$ and the other antidiagonal block is equal to $\gr{v_2,w_{(n-j)}}$. Then \[\Imm_v M=\Imm_{v_1} M([j], [n-j+1, n])\cdot \Imm_{v_2}M([j+1, n], [n-j]). \] \end{prop} \begin{figure} \centering \includegraphics[height=0.3\textheight]{figures/blockAntiDiagEx2.pdf} \caption{An example where $\gr{v,w_0}$ is block-antidiagonal. Here, $v=6475132$. In the notation of Proposition~\ref{prop:ImmMultipleBlock}, $j=4$, $v_1=3142$, and $v_2=132$.} \label{fig:blockAnti} \end{figure} See Figure~\ref{fig:blockAnti} for an example illustrating a block-antidiagonal $\gr{v, w_0}$ and the notation of Proposition~\ref{prop:ImmMultipleBlock}. To analyze the determinants appearing in Lewis Carroll's identity for computing $\det X(R, C)\res{\gr{v, w_0}}$, we will use the following two propositions. \begin{prop} \label{prop:deleteDotDetAnti} Let $v\in S_n$ be $2143$- and $1324$-avoiding, and $i\in [n]$. Let $x \in S_{n-1}$ be the permutation $x: \delta_i(j) \mapsto \delta_{v_i}(v_j)$ (that is, $x$ is obtained from $v$ by deleting~$v_i$ from $v$ in one-line notation and shifting the remaining numbers appropriately). Then \begin{enumerate} \item $\gr{x, w_0}=\left(\gr{v,w_0} \setminus \left\{(p, q): \parbox{1.75in}{\begin{center}(p, q) \text{ is sandwiched only by}\\ \text{ a non-inversion involving }i \end{center}} \right\}\right)_i^{v_i}$ \item If $(i, v_i)$ is not a spanning corner of $\gr{v,w_0}$, then $\gr{x,w_0}=\gr{v,w_0}_i^{v_i}$. \item For all $i$, $\det M\res{\gr{x,w_0}}=\det M\res{\gr{v,w_0}_{i}^{v_i}}.$ \end{enumerate} \end{prop} \begin{proof} Statement (1) follows from Lemma~\ref{lem:graphCharacterization}. Statements (2) and (3) are Proposition 4.17 from~\cite{CSB}. \end{proof} \begin{prop}\label{prop:sgn} Let $v\in S_n$ be $1324$- and $2143$-avoiding such that the last bounding box of $\gr{v,w_0}$ is $\bbox{n, v_n}$, and the second to last box is $\bbox{a, v_a}$ for some $a