# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.00111