Block number, descents and Schur positivity of fully commutative elements in $B_n$

TL;DR

This paper investigates the distribution of Coxeter descents and block number in fully commutative elements of the hyperoctahedral group, proving Schur-positivity of related generating functions through combinatorial decompositions.

Contribution

It introduces a new decomposition of fully commutative elements in $B_n$ into fibers and extends descent-preserving involutions to type B, establishing Schur-positivity results.

Findings

Chow's Chow quasi-symmetric generating function is Schur-positive.

Decomposition of $ ext{FC}(B_n)$ into fibers provides new structural insights.

Comparison of two type B Schur-positivity notions enhances understanding of symmetric functions.

Abstract

The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group $B_n$, $\FC(B_n)$, is studied in this paper. We prove that the associated Chow quasi-symmetric generating function is equal to a non-negative sum of products of two Schur functions. The proof involves a decomposition of $\FC(B_n)$ into a disjoint union of two-sided Barbash-Vogan combinatorial cells, a type $B$ extension of Rubey's descent preserving involution on $321$-avoiding permutations and a detailed study of the intersection of $\FC(B_n)$ with $S_n$-cosets which yields a new decomposition of $\FC(B_n)$ into disjoint subsets called fibers. We also compare two different type $B$ Schur-positivity notions, arising from works of Chow and Poirier