NDLI: $h$-Polynomials of Reduction Trees

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mészáros, Karola
Copyright Year	2016
Abstract	We develop a method of proving nonnegativity of the coefficients of certain polynomials, also called reduced forms, defined by Kirillov in his quasi-classical Yang--Baxter algebra, its abelianization, and related algebras. It has been shown previously that the relations of the abelianization of the quasi-classical Yang--Baxter algebra, also called the subdivision algebra, encode ways of subdividing flow polytopes. In turn, these subdivisions can be represented as reduced forms, or as reduction trees. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We explain how a shelling of the canonical triangulation can be read off from the corresponding reduction tree in the subdivision algebra. We then introduce the notion of shellable reduction trees in the subdivision and related algebras and define $h$-polynomials of reduction trees. In the case of the subdivision algebra, the $h$-polynomials of the canonical triangulations of flow polytopes equal the $h$-polynomials of the corresponding reduction trees, which motivated our definition. We show that the reduced forms in various algebras, which can be read off from the leaves of the reduction trees, specialize to the shifted $h$-polynomials of the corresponding reduction trees. This yields a technique for proving nonnegativity properties of reduced forms. As a corollary we settle a conjecture of Kirillov.
Sponsorship	National Science Foundation
Starting Page	736
Ending Page	762
Page Count	27
File Format	PDF
ISSN	08954801
DOI	10.1137/140987882
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	30
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-04-14
Access Restriction	Subscribed
Subject Keyword	subdivision algebra quasi-classical Yang--Baxter reduction free Special polytopes Combinatorial aspects of groups and algebras $h$-polynomial flow polytope
Content Type	Text
Resource Type	Article
Subject	Mathematics

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