Séminaire Lotharingien de Combinatoire, 80B.91 (2018), 12 pp.
Louis J. Billera, Sara C. Billey, and Vasu Tewari
Boolean Product Polynomials and Schur-Positivity
Abstract.
We study a family of symmetric polynomials that we refer to as the Boolean product polynomials.
The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the nonzero vectors in Rn all of whose coordinates are either 0 or 1. To this end, one approach is to compute the zeros of the Boolean product polynomials over finite fields.
The zero loci of these polynomials cut out hyperplane arrangements known as resonance arrangements, which show up in the context of double Hurwitz polynomials.
By relating the Boolean product polynomials to certain total Chern classes
of vector bundles, we establish their Schur-positivity by appealing to
a result of Pragacz relying on earlier work on numerical positivity by
Fulton-Lazarsfeld. Subsequently, we study a two-alphabet version of these polynomials from the viewpoint of Schur-positivity. As a special case of these polynomials, we recover symmetric functions first studied by
Désarménien and Wachs in the context of descents in derangements.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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