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arXiv · 2012.06412

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

Eli Bagno·Riccardo Biagioli·Frédéric Jouhet·Yuval Roichman
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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

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Eli Bagno, Riccardo Biagioli, Frédéric Jouhet, Yuval Roichman. 2020-12-11. Block number, descents and Schur positivity of fully commutative elements in $B_n$. https://arxiv.org/abs/2012.06412

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