Demazure Modules Research Articles

A weight multiplicity formula for Demazure modules

We establish a formula for the weight multiplicities of Demazure modules (in particular for highest-weight representations) of a complex connected algebraic group in terms of the geometry of its Langlands dual.

Complexity of Homogeneous Spaces and Growth of Multiplicities

The complexity of a homogeneous space G/H under a reductive group G is by definition the codimension of general orbits in G/H of a Borel subgroup B\subseteq G. We give a representation-theoretic interpretation of this number as the exponent of growth for multiplicities of simple G-modules in the spaces of sections of homogeneous line bundles on G/H. For this, we show that these multiplicities are bounded from above by the dimensions of certain Demazure modules. This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity.

A Twining Character Formula For Demazure Modules

We prove a formula for the twining characters of certain Demazure modules, over a Borel subalgebra b of a finite dimensional complex semisimple Lie algebra g. This formula describes the twining character of the Demazure module by the ω-Demazure operator associated to an element of the Weyl group that is fixed by the Dynkin diagram automorphism ω of g. Our result is a refinement of the twining character formula for the irreducible highest weight g-modules of symmetric dominant integral highest weights, and also of the ordinary Demazure character formula.

Crystal Bases, Path Models, and a Twining Character Formula for Demazure Modules

We give a combinatorial proof of a twining character formula for Demazure modules, by combining the isomorphism theorem between path models and crystal bases with our previous result about Lakshmibai-Seshadri paths fixed by a diagram automorphism.

Standard Paths and Standard Monomials Fixed by a Diagram Automorphism

We show that the standard paths and the standard monomials fixed by a diagram automorphism for a Kac–Moody algebra g can be identified with the standard paths and the standard monomials for the orbit Lie algebra of g, respectively. By using these results, we give a combinatorial proof of twining character formulas for an integrable highest weight module and for a Demazure module within the framework of path models.

Standard Monomial Theory for Bott–Samelson Varieties

Bott–Samelson varieties are an important tool in geometric representation theory [1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the properties of Schubert varieties. They have an action of a Borel subgroup, and the projective coordinate ring of a Bott–Samelson variety splits into certain generalized Demazure modules (which also appear in other contexts [22, 23]). Standard Monomial Theory, developed by Seshadri and the first author [15, 16], and recently completed by the second author [20], gives explicit bases for the Demazure modules associated to Schubert varieties. In this paper, we extend the techniques of [20] to give explicit bases for the generalized Demazure modules associated to Bott–Samelson varieties, thus proving a strengthened form of the results announced by the first and third authors in [12] (see also [13]). We also obtain more elementary proofs of the cohomology vanishing theorems of Kumar [10] and Mathieu [25]; of the projective normality of Bott–Samelson varieties; and of the Demazure character formula.

Polytopes for Crystallized Demazure Modules and Extremal Vectors

Abstract We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.

Polytopes for Crystallized Demazure Modules and Extremal Vectors

We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.

Lakshmibai–Seshadri Paths Fixed by a Diagram Automorphism

We prove that the Lakshmibai–Seshadri paths fixed by a diagram automorphism ω of a Kac–Moody algebra can be identified with the Lakshmibai–Seshadri paths for the orbit Lie algebra, by introducing ω-root operators. We also give certain formulas for the twining characters of integrable highest weight modules over a symmetrizable Kac–Moody algebra and of Demazure modules for a finite-dimensional semi-simple Lie algebra.

Semi-Infinite Realization of Unitary Representations of the N=2 Algebra and Related Constructions

In the examples of the N=2 super-Virasoro algebra and the affine \(\widehat{s\ell }\left( 2 \right)\) algebra, we investigate the construction of unitary representations of infinite-dimensional algebras in terms of “collective excitations” over a filled Dirac sea of fermionic or bosonic operators satisfying a generalized exclusion principle and represented by semi-infinite forms in the modes of one of the generators. We develop the methods for investigating properties of semi-infinite spaces (polynomial realization of the dual space) and for constructing the appropriate algebra action on these spaces (a filtration by subspaces similar to Demazure modules). We also consider relations of the semi-infinite realizations to the Rogers–Ramanujan-type identities, to the expression of coinvariants through meromorphic functions on products of Riemann surfaces with a prescribed behavior on multiple diagonals, and to some combinatorial facts; we also consider the relation between modular functors and fusion rules for the N=2 and \(\widehat{s\ell }\left( 2 \right)\) theories.

On the Connection Between Macdonald Polynomials and Demazure Characters

We show that the specialization of nonsymmetric Macdonald polynomials at t = 0 are, up to multiplication by a simple factor, characters of Demazure modules for \(\widehat{sl(n)}\). This connection furnishes Lie-theoretic proofs of the nonnegativity and monotonicity of Kostka polynomials.

Lattice Polytopes Associated to Certain Demazure Modules of sln + 1

Let w be an element of the Weyl group of sl_n + 1. We prove that for a certain class of elements w (which includes the longest element w_0 of the Weyl group), there exist a lattice polytope Δ ⊂ R^e(w), for each fundamental weight ω_i of sl_n + 1, such that for any dominant weight λ e ∑_i e 1^n a_i ω_i, the number of lattice points in the Minkowski sum Δ_λ^w e ∑_i e 1^n a_iΔ_i^w is equal to the dimension of the Demazure module E(λ). We also define a linear map A : R^e(w) → P ⊗_ZR where P denotes the weight lattice, such that char E(λ) e e^λ ∑e^−A(x) where the sum runs through the lattice points x of Δ_λ^w.

Crystals for Demazure Modules of Classical Affine Lie Algebras

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of typesA(1)n,B(1)n,C(1)n,D(1)n,A(2)2n−1,A(2)2n, andD(2)n+1. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show that if the highest weight islΛ0, the Demazure crystal has a remarkably simple structure.

Demazure modules and vertex models: The sl̂(2) case

We characterize, in the case of sl∧(2),the crystal base of the Demazure module Ew(Λ) in terms of extended Young diagrams or paths for any dominant integral weight Λ and Weyl group element w. Its character is evaluated via two expressions, “bosonic” and “fermionic.”

Characters of Demazure modules and solvable lattice models

We study the path realization of Demazure crystals related to solvable lattice models in statistical mechanics. Various characters are represented in a unified way as the sums over one-dimensional configurations which we call unrestricted, classically restricted and restricted paths. As an application, characters of Demazure modules are obtained in terms of q-multinomial coefficients for several level-1 modules of classical affine algebras.

Résultats combinatoires sur les modules de Demazure

In this paper, we introduce the notion of polytopes with integral structure, which generalizes the notion of usual polytopes whose vertices have integral coordinates. Then we consider a symmetrizable Kac-Moody algebra. For every dominant weight λ and every element of the Weyl group w, we construct a polytope with integral structure such that its Ehrhart polynomial describes the dimension of the Demazure module of highest weight nλ associated to w.

Real Characters for Demazure Modules of Rank Two Affine Lie Algebras

Using Littelmann's path model for highest weight representations of Kac–Moody algebras, we obtain explicit combinatorial expressions for certain specialized characters of all Demazure modules ofA1(1)andA2(2).