Combinatorics of hypergeometric functions associated with positive roots

Abstract

ABSTRACT. In this paper we study the hypergeometric system on unipotent matrices. This system gives a holonomic D-module. We find the number of independent solutions of this system at a generic point. This number is equal to the famous Catalan number. An explicit basis of r -series in solution space of this system is constructed in the paper. We also consider restriction of this system to certain strata. We introduce several combinatorial constructions with trees, polyhedra, and triangulations related to this subject. CONTENTS

Root polytopes, triangulations, and the subdivision algebra. I

“…Recall that a triangulation of the polytope P is regular if there exists a concave piecewise linear function f : P → R such that the regions of linearity of f are the maximal simplices in the triangulation. It has been shown in [GGP,Theorem 6.3] that the noncrossing triangulation of P(A + n ) is regular. This result can be naturally extended to the canonical triangulation of any of the root polytopes P(T ).…”

“…This result can be naturally extended to the canonical triangulation of any of the root polytopes P(T ). An attractive proof uses the following concave function constructed by Postnikov for an alternative proof of [GGP,Theorem 6.3].…”

Root Polytopes and Borel Subalgebras

“…Some authors (see in particular [16] and [17]), intend by root polytope the convex hull of the positive roots together with the origin, first introduced in [9] for the root system of type A n . We call this the positive root polytope and, if confusion may arise, we call P Φ the complete root polytope.…”

Delannoy Orthants of Legendre Polytopes

“…3 explores a few facts that always hold when we consider the intersection of a centrally symmetric polytope and a hyperplane that contains the origin but no vertex. The graphs we introduce are directed generalizations of the variants of some graphs that appear in the work of Gelfand, Graev, and Postnikov [7], the key Lemma 3.4 is a generalization of a result originally due to Kapranov, Postnikov, and Zelevinski (see the first half of Lemma 12.5 in [16]). …”

“…The root polytopes P A + n were first studied by Gelfand, Graev and Postnikov [7]. Results on these polytopes were generalized by Postnikov [16] and Wungkum Fong [6].…”