Multiplicity in root components via geometric Satake. Issue 6 (3rd June 2022)

• Record Type: Journal Article
• Authors: Besson, Marc; Jeralds, Sam; Kiers, Joshua
• Is Part Of: Communications in algebra. Volume 50:Issue 6(2022)
• Page Start: 2621
• Page End: 2636
• Publication Date: 2022-06-03
• Subjects: Affine Grassmannian -- geometric Satake -- representation theory; 20G05 -- 14M15 -- 22E57; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.2015361
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: In this note we explicitly construct top-dimensional components of certain cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands V ( λ + μ − N β ) ⊂ V ( λ ) ⊗ V ( μ ) for S L n + 1 ( C ), where N ≥ 1 and β is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when β is not a simple root. Finally, we demonstrate that not all such top-dimensional components can be realized as closures of orbits.