Open AccessMirković–Vilonen basis in type 𝐴₁
Author(s) -
Pierre Baumann,
Arnaud Demarais
Publication year - 2021
Publication title -
representation theory
Language(s) - English
Resource type - Journals
ISSN - 1088-4165
DOI - 10.1090/ert/582
Subject(s) - basis (linear algebra) , tensor product , mathematics , pure mathematics , type (biology) , equivalence (formal languages) , algebraic group , standard basis , algebra over a field , algebraic number , dual (grammatical number) , product (mathematics) , geometry , mathematical analysis , linguistics , ecology , philosophy , biology
Let G G be a connected reductive algebraic group over C \mathbb C . Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product V ( λ 1 ) ⊗ ⋯ ⊗ V ( λ r ) V(\lambda _1)\otimes \cdots \otimes V(\lambda _r) of irreducible representations of G G . We compute this basis in the case G = S L 2 ( C ) G=\mathrm {SL}_2(\mathbb C) and conclude that in this case it coincides with the dual canonical basis at q = 1 q=1 .