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

Orbit measures, random matrix theory and interlaced determinantal processes

A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction mulltiplicities. We show that a large class of them are determinantal

Ehrhart positivity and Demazure characters

Demazure characters, also known as key polynomials, generalize the classical Schur polynomials. In particular, when all variables are set equal to $1$, these polynomials count the number of integer points in a certain class of Gelfand--Tsetlin polytopes. This property highlights the interaction between the corresponding polyhedral and combinatorial structures via Ehrhart theory. In this paper, we give an overview of results concerning the interplay between the geometry of Gelfand-Tsetlin polytopes and their Ehrhart polynomials. Motivated by strong computer evidence, we propose several conjectures about the non-negativity of the coefficients of such polynomials.Comment: To appear in the conference proceedings of the Summer workshop on lattice polytopes, Osaka 201

Fano Varieties and Fano Polytopes

The foundation of this thesis is the problem whether a given (normal) Gorenstein Fano variety can be degenerated to a toric Gorenstein Fano variety. We will only consider those degenerations that are compatible with the choice of an ample line bundle on the original variety and an ample rational Cartier divisor on the toric variety. This compatibility will be defined thoroughly and is always granted in applications in representation theory or Newton-Okounkov Theory. The main matter of this thesis contains the proof that in the setting of these compatible toric degenerations the originally chosen line bundle will be isomorphic to the anti-canonical line bundle if and only if the divisor on the toric variety is anti-canonical. The if-part is already known but the only-if-part is not. Its proof requires different methods from various areas of mathematical research. We will need multiple vanishing theorems and further results from algebraic geometry, methods from polyhedral geometry (especially Ehrhart theory), results on the Hilbert polynomial and facts about toric varieties. As a by-product we establish a connection between the Ehrhart quasi-polynomial of a rational polytope and the cohomology of an associated rational Weil divisor on a toric variety. Up until know, this connection was only known for polytopes with integral vertices and integral divisors. It allows us to interpret Ehrhart-Macdonald Reciprocity as a special case of Serre Duality. In the final chapter of this thesis we will show that there actually exists such a compatible toric degeneration for every partial flag variety of a complex classical group. The construction is done via so called string polytopes that have been established by Littelmann and Berenstein–Zelevinsky. For this purpose we need to prove a classification of integral string polytopes. The proof is done via a newly developed diagrammatic description of so called Gelfand-Tsetlin patterns in spirit of Hasse diagrams of partially ordered sets