The spin-Brauer diagram algebra

Abstract

We investigate the spin-Brauer diagram algebra, denoted ${\mathbf{SB}}_n(\delta )$, that arises from studying an analogous form of Schur-Weyl duality for the action of the pin group on ${\mathbf{V}}^{\otimes n}\otimes \Delta$. Here $\mathbf{V}$ is the standard $N$-dimensional complex representation of $\mathbf{Pin}(N)$ and $\Delta$ is the spin representation. When $\delta =N$ is a positive integer, we define a surjective map ${\mathbf{SB}}_n(N)\twoheadrightarrow {\mathrm{End}}_{\mathbf{Pin}(N)}({\mathbf{V}}^{\otimes n}\otimes \Delta )$ and show it is an isomorphism for $N\ge 2n$. We show ${\mathbf{SB}}_n(\delta )$ is a cellular algebra and use cellularity to characterize its irreducible representations.

24 pages, 22 figures

References in corpus (1)

• Infinite rank spinor and oscillator representations

Cited by in corpus (2)

• The representation theory of Brauer categories I: triangular categories
• Representation stability for sequences of 0-Hecke modules