Compact Schur-Weyl duality and the affine Type B/C Brauer algebra

2019/09/25 by Kieran Calvert, Calvert, Kieran
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Affine transformation #Algebra over a field #Algebraic structures and combinatorial models #Duality (order theory) #FOS: Mathematics #Functor #Mathematics #Pure mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Type (biology) #Weyl group #math.RA #math.RT

paper · pdf · doi:10.48550/arxiv.1909.11428

42 pages,added in details, reworked for VW or Nazarov-Wenzl algebra, references updated

openalex publication_date 2019/09/25 · arxiv created 2020/02/14 · arxiv updated 2020/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on END_K(X ⊗ V^{⊗ k}) for Orthogonal and Symplectic groups. Thus we obtain a compact analogue of Schur-Weyl duality. We study functors F_μ,k from the category of admissible O(p,q) or Sp_2n(ℝ) modules to representations of the type B/C affine Brauer algebra \mathfrakB_k^θ. Thus providing a Akawaka-Suzuki-esque link between O(p,q) (or Sp_2n(ℝ)) and \mathfrakB_k^θ. Furthermore these functors take non spherical principal series modules to principal series modules for the graded Hecke algebra of type D_k, C_n-k or B_n-k. With this we get a functorial correspondence between admissible simple O(p,q) (or Sp_2n(ℝ)) modules and graded Hecke algebra modules.

Citations

• On Some Types of Topological Groups
• The blocks of the Brauer algebra in characteristic zero

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