The spin-Brauer diagram algebra

Abstract

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