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The cone of cyclic sieving phenomena
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-2305-9764
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2019 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 342, no 6, p. 1581-1601Article in journal (Refereed) Published
Abstract [en]

We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.

Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod q^n-1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.

Place, publisher, year, edition, pages
Elsevier, 2019. Vol. 342, no 6, p. 1581-1601
Keywords [en]
cyclic sieving, stretched schur polynomial, convex polytope
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-250764DOI: 10.1016/j.disc.2019.01.037ISI: 000466833400006Scopus ID: 2-s2.0-85062678575OAI: oai:DiVA.org:kth-250764DiVA, id: diva2:1313631
Note

QC 20190510

Available from: 2019-05-05 Created: 2019-05-05 Last updated: 2019-05-29Bibliographically approved
In thesis
1. Combinatorics and zeros of multivariate polynomials
Open this publication in new window or tab >>Combinatorics and zeros of multivariate polynomials
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate $d$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. p. 42
Series
TRITA-SCI-FOU ; 2019:33
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-251303 (URN)978-91-7873-210-4 (ISBN)
Public defence
2019-05-24, D3, Lindstedsvägen 5, Stockholm, 14:00 (English)
Opponent
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Note

QC 20190510

Available from: 2019-05-10 Created: 2019-05-09 Last updated: 2019-05-10Bibliographically approved

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Amini, NimaAlexandersson, Per

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