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

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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