Root Cones and the Resonance Arrangement
TLDR
In this article, the authors studied the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts.Abstract:
We study the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance arrangement, the number of resonance chambers has only been computed up to the $n=8$ dimensional case. We focus on data structures for labeling chambers, such as sign vectors and sets of alternating trees, with an aim at better understanding the structure of the resonance arrangement, and, in particular, enumerating its chambers. Along the way, we make connections with similar (and similarly difficult) enumeration questions. With the root polytope viewpoint, we relate resonance chambers to the chambers of polynomiality of the Kostant partition function. With the hyperplane viewpoint, we clarify the connections between resonance chambers and threshold functions. In particular, we show that the base-2 logarithm of the number of resonance chambers is asymptotically $n^2$.read more
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The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations
TL;DR: In this paper, the dual action of Lie elements on faces of the adjoint braid arrangement, interpreted as the discrete differentiation of functions on faces across hyperplanes, was studied, and it was shown that functions which satisfy the Steinmann relations form a left comodule of the Lie cooperad.
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TL;DR: In this article, it was shown that any rational hyperplane arrangement is the minor of some large enough resonance arrangement, which is the adjoint of the Braid arrangement and is also called the all-subsets arrangement.
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Counting Integer Points of Flow Polytopes
TL;DR: The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes, and to reveal the geometry of these formulas.
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Counting integer points of flow polytopes
TL;DR: The Baldoni-Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways: on the one hand, these formulas are in terms of Kostant partition functions, connecting flow polytes to this classical vector partition function fundamental in representation theory; on the other hand, the Ehrart polynomials can be read off from the volume functions of flow polytoes as mentioned in this paper.
References
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Journal Article
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TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of 13,000 number sequences and is freely available on the Web (http://www.att.com/~njas/sequences/) and is widely used.
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Permutohedra, Associahedra, and Beyond
TL;DR: In this paper, the volume and number of lattice points of the permutohedron Pn are given by certain multivariate polynomials that have remarkable com- binatorial properties.
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Some recent applications of the theory of linear inequalities to extremal combinatorial analysis
TL;DR: Refractory shell molds are made by a method including the steps of forming a shell investment mold over a wax pattern, and immediately placing the mold and pattern in a heated environment whereby the pattern is removed by melting without cracking or otherwise damaging the mold.