A COMBINATORIAL STUDY OF AFFINE SCHUBERT VARIETIES IN THE AFFINE GRASSMANNIAN
| Autor: | Marc Besson, Jiuzu Hong |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Algebra and Number Theory
010102 general mathematics Lattice (group) Duality (order theory) Iwahori subgroup Affine Grassmannian (manifold) Type (model theory) 01 natural sciences Combinatorics Algebraic group Grassmannian 0103 physical sciences Lie algebra 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematics::Representation Theory Mathematics |
| Zdroj: | Transformation Groups. 27:1189-1221 |
| ISSN: | 1531-586X 1083-4362 |
| DOI: | 10.1007/s00031-020-09634-9 |
| Popis: | Let $$ {\overline{\mathrm{X}}}_{\uplambda} $$ be the closure of the I-orbit $$ {\overline{\mathrm{X}}}_{\uplambda} $$ in the affine Grassmanian Gr of a simple algebraic group G of adjoint type, where I is the Iwahori subgroup and λ is a coweight of G. We find a simple algorithm which describes the set Ψ(λ) of all I-orbits in $$ {\overline{\mathrm{X}}}_{\uplambda} $$ in terms of coweights. We introduce R-operators (associated to positive roots) on the coweight lattice of G, which exactly describe the closure relation of I-orbits. These operators satisfy Braid relations generically on the coweight lattice. We also establish a duality between the set Ψ(λ) and the weight system of the level one affine Demazure module of $$ {}^L\tilde{\mathfrak{g}} $$ indexed by λ, where $$ {}^L\tilde{\mathfrak{g}} $$ is the affine Kac–Moody algebra dual to the affine Kac–Moody Lie algebra $$ \tilde{\mathfrak{g}} $$ associated to the Lie algebra $$ \mathfrak{g} $$ of G. |
| Databáze: | OpenAIRE |
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