53 research outputs found
Tensor product structure of affine Demazure modules and limit constructions
Let \Lg be a simple complex Lie algebra, we denote by \Lhg the
corresponding affine Kac--Moody algebra. Let be the additional
fundamental weight of \Lhg. For a dominant integral \Lg--coweight
\lam^\vee, the Demazure submodule V_{-\lam^\vee}(m\Lam_0) is a
\Lg--module. For any partition of \lam^\vee=\sum_j \lam_j^\vee as a sum of
dominant integral \Lg--coweights, the Demazure module is (as \Lg--module)
isomorphic to \bigotimes_j V_{-\lam^\vee_j}(m\Lam_0). For the ``smallest''
case, \lam^\vee=\om^\vee a fundamental coweight, we provide for \Lg of
classical type a decomposition of V_{-\om^\vee}(m\Lam_0) into irreducible
\Lg--modules, so this can be viewed as a natural generalization of the
decomposition formulas in \cite{KMOTU} and \cite{Magyar}. A comparison with the
U_q(\Lg)--characters of certain finite dimensional U_q'(\Lhg)--modules
(Kirillov--Reshetikhin--modules) suggests furthermore that all quantized
Demazure modules V_{-\lam^\vee,q}(m\Lam_0) can be naturally endowed with the
structure of a U_q'(\Lhg)--module. Such a structure suggests also a
combinatorially interesting connection between the LS--path model for the
Demazure module and the LS--path model for certain U_q'(\Lhg)--modules in
\cite{NaitoSagaki}. For an integral dominant \Lhg--weight let
V(\Lam) be the corresponding irreducible \Lhg--representation. Using the
tensor product decomposition for Demazure modules, we give a description of the
\Lg--module structure of V(\Lam) as a semi-infinite tensor product of
finite dimensional \Lg--modules. The case of twisted affine Kac-Moody
algebras can be treated in the same way, some details are worked out in the
last section.Comment: 24 pages, in the current version we added the case of twisted affine
Kac--Moody algebra
Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions
We study finite dimensional representations of current algebras, loop
algebras and their quantized versions. For the current algebra of a simple Lie
algebra of type {\tt ADE}, we show that Kirillov-Reshetikhin modules and Weyl
modules are in fact all Demazure modules. As a consequence one obtains an
elementary proof of the dimension formula for Weyl modules for the current and
the loop algebra. Further, we show that the crystals of the Weyl and the
Demazure module are the same up to some additional label zero arrows for the
Weyl module.
For the current algebra \Lgc of an arbitrary simple Lie algebra, the fusion
product of Demazure modules of the same level turns out to be again a Demazure
module. As an application we construct the \Lgc-module structure of the
Kac-Moody algebra \Lhg-module V(\ell\Lam_0) as a semi-infinite fusion
product of finite dimensional \Lgc--modules
One-skeleton galleries, the path model and a generalization of Macdonald's formula for Hall-Littlewood polynomials
We give a direct geometric interpretation of the path model using galleries
in the skeleton of the Bruhat-Tits building associated to a semi-simple
algebraic group. This interpretation allows us to compute the coefficients of
the expansion of the Hall-Littlewood polynomials in the monomial basis. The
formula we obtain is a "geometric compression" of the one proved by Schwer, its
specialization to the case turns out to be equivalent to
Macdonald's formula.Comment: 43 pages, 3 pictures, some improvements in the presentation,
semistandard tableaux for type B and C define
Richardson Varieties and Equivariant K-Theory
We generalize Standard Monomial Theory (SMT) to intersections of Schubert
varieties and opposite Schubert varieties; such varieties are called Richardson
varieties. The aim of this article is to get closer to a geometric
interpretation of the standard monomial theory. Our methods show that in order
to develop a SMT for a certain class of subvarieties in G/B (which includes
G/B), it suffices to have the following three ingredients, a basis for the
space of sections of an effective line bundle on G/B, compatibility of such a
basis with the varieties in the class, certain quadratic relations in the
monomials in the basis elements. An important tool will be the construction of
nice filtrations of the vanishing ideal of the boundary of the varieties above.
This provides a direct connection to the equivariant K-theory, where the
combinatorially defined notion of standardness gets a geometric interpretation.Comment: 38 page
Equations defining symmetric varieties and affine Grassmannians
Let be a simple involution of an algebraic semisimple group and
let be the subgroup of of points fixed by . If the restricted
root system is of type , or and is simply connected or if the
restricted root system is of type and is adjoint, then we describe a
standard monomial theory and the equations for the coordinate ring
using the standard monomial theory and the Pl\"ucker relations of an
appropriate (maybe infinite dimensional) Grassmann variety.Comment: 48 page
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