The spin-Brauer diagram algebra

TL;DR

This paper introduces the spin-Brauer diagram algebra, establishing its structure, cellularity, and representation theory, and proves its isomorphism with certain endomorphism algebras related to the pin group for large enough N.

Contribution

It defines the spin-Brauer diagram algebra, proves its cellularity, and characterizes its irreducible representations, extending Schur-Weyl duality to the pin group context.

Findings

The algebra ${f SB}_n( ext{delta})$ is cellular.

A surjective map from ${f SB}_n(N)$ to endomorphisms is an isomorphism for $N \\geq 2n$.

Cellularity helps classify irreducible representations.

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.