The spin-Brauer diagram algebra

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