207 research outputs found
Vertices of Gelfand-Tsetlin Polytopes
This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns
arising in the representation theory \mathfrak{gl}_n \C and algebraic
combinatorics. We present a combinatorial characterization of the vertices and
a method to calculate the dimension of the lowest-dimensional face containing a
given Gelfand-Tsetlin pattern.
As an application, we disprove a conjecture of Berenstein and Kirillov about
the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can
construct for each a counterexample, with arbitrarily increasing
denominators as grows, of a non-integral vertex. This is the first infinite
family of non-integral polyhedra for which the Ehrhart counting function is
still a polynomial. We also derive a bound on the denominators for the
non-integral vertices when is fixed.Comment: 14 pages, 3 figures, fixed attribution
Minuscule Schubert varieties: poset polytopes, PBW-degenerated demazure modules, and Kogan faces
We study a family of posets and the associated chain and order polytopes. We
identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple
of a fundamental weight. We show that the character of such a Kogan face equals to
the character of a Demazure module which occurs in the irreducible representation of sln+1
having highest weight multiple of fundamental weight and for any such Demazure module
there exists a corresponding poset and associated maximal Kogan face. We prove that
the chain polytope parametrizes a monomial basis of the associated PBW-graded Demazure
module and further, that the Demazure module is a favourable module, e.g. interesting geometric
properties are governed by combinatorics of convex polytopes. Thus, we obtain for
any minuscule Schubert variety a flat degeneration into a toric projective variety which is
projectively normal and arithmetically Cohen-Macaulay. We provide a necessary and sufficient
condition on the Weyl group element such that the toric variety associated to the chain
polytope and the toric variety associated to the order polytope are isomorphic
A vector partition function for the multiplicities of sl_k(C)
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity
function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition
function. This allows us to apply known results about partition functions to
derive interesting properties of the weight diagrams. We relate this
description to that of the Duistermaat-Heckman measure from symplectic
geometry, which gives a large-scale limit way to look at multiplicity diagrams.
We also provide an explanation for why the weight polynomials in the boundary
regions of the weight diagrams exhibit a number of linear factors. Using
symplectic geometry, we prove that the partition of the permutahedron into
domains of polynomiality of the Duistermaat-Heckman function is the same as
that for the weight multiplicity function, and give an elementary proof of this
for sl_4(C) (A_3).Comment: 34 pages, 11 figures and diagrams; submitted to Journal of Algebr
Marked chain-order polytopes
We introduce in this paper the marked chain-order polytopes associated to a
marked poset, generalizing the marked chain polytopes and marked order
polytopes by putting them as extremal cases in an Ehrhart equivalent family.
Some combinatorial properties of these polytopes are studied. This work is
motivated by the framework of PBW degenerations in representation theory of Lie
algebras.Comment: 18 pages, title changed, the relation to string polytopes is remove
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