Classification Of Uniform Flag Triangulations Of The Boundary Of The Full Root Polytope Of Type A
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CLASSIFICATION OF UNIFORM FLAG TRIANGULATIONS OF THE BOUNDARY OF THE FULL ROOT POLYTOPE OF TYPE A R. EHRENBORG1,∗ , G. HETYEI2 and M. READDY1 1
2
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, U.S.A. e-mails: [email protected], [email protected]
Department of Mathematics and Statistics, UNC-Charlotte, Charlotte, NC 28223-0001, U.S.A. e-mail: [email protected] (Received March 27, 2020; accepted June 30, 2020)
Abstract. The full root polytope of type A is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and backward arrows. We prove the refined face counts only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.
1. Introduction Triangulations of root polytopes and of products of simplices have been a subject of intense study in recent years [2,5–7,13]. Motivated by an observation made in [9], we recently [11] established that the Simion type B associahedron [21] may be realized as a pulling triangulation of the full root polytope of type A, defined as the convex hull of all differences of pairs of standard basis vectors in Euclidean space. These vertices can be thought of ∗ Corresponding
author. The first and third authors thank the Institute for Advanced Study in Princeton, New Jersey for supporting a research visit in Summer 2018. This work was partially supported by grants from the Simons Foundation (#429370 to Richard Ehrenborg, #245153 and #514648 to G´ abor Hetyei, #422467 to Margaret Readdy). Key words and phrases: Bessel function, Catalan number, cyclohedron, Delannoy number, face vector, flag complex, matching ensemble, type B associahedron. Mathematics Subject Classification: primary 52B05, 52B12, secondary 05A15, 05E45.
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R. R. EHRENBORG, EHRENBORG, G. HETYEI and M. READDY
as arrows between numbered nodes. We also show that all pulling triangulations are flag. The full root polytope is the centrally symmetric variant of the type A positive root polytope whose lexicographic and revlex triangulations were studied by Gelfand, Graev and Postnikov [13]. A question naturally arises: Are there other reasonably uniform triangulations of the root polytope? In this paper we fully answer this question. We classify all flag triangulations that are uniform in the sense that the flag condition depends only on the relative order on the numbering of the basis vectors
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