$W$-graph versions of tensoring with the $\S_n$ defining representation

TL;DR

This paper develops $W$-graph versions of tensoring with the defining representation of the symmetric group, providing combinatorial approximations and conjectures related to Hecke algebra structures and canonical bases.

Contribution

It introduces two new $W$-graph tensoring constructions for $ ext{S}_n$, analyzes their properties, and explores their relation to canonical bases and algebraic approximations.

Findings

Approximate canonical basis preservation as $rrow 0$

Conjecture on weak multiplicity-free restriction of $ ext{H}$ to $ ext{H}_J$

Partial results on the map $ ext{H} sr_{ ext{H}_J} ext{H} o ext{H}$

Abstract

We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin in \cite{HY1}, \cite{HY2}, focusing on the case $W = \S_n$. Our main application is to give two $W$-graph versions of tensoring with the $\S_n$ defining representation $V$, one being $\H \tsr_{\H_J} -$ for $\H, \H_J$ the Hecke algebras of $\S_n, \S_{n-1}$ and the other $(\pH \tsr_{\H} -)_1$, where $\pH$ is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding $W$-graph versions of the projection $V \tsr V \tsr - \to S^2 V \tsr -$. This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter $\u \to 0$. We make this approximation combinatorially explicit by determining it on cells. Also of interest is a combinatorial conjecture stating the restriction of $\H$ to $\H_J$ is "weakly multiplicity-free" for $|J| = n-1$, and a partial determination of the map $\H \tsr_{\H_J} \H \xrightarrow{\counit} \H$ on canonical basis elements, where $\counit$ is the counit of adjunction.