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Mathematics, Representation Theory
Abstract:
We define the affine VW supercategory
$\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$, which arises from
studying the action of the periplectic Lie superalgebra $\mathfrak{p}(n)$ on
the tensor product $M\otimes V^{\otimes a}$ of an arbitrary representation $M$
with several copies of the vector representation $V$ of $\mathfrak{p}(n)$. It
plays a role analogous to that of the degenerate affine Hecke algebras in the
context of representations of the general linear group; the main obstacle was
the lack of a quadratic Casimir element in $\mathfrak{p}(n)\otimes
\mathfrak{p}(n)$. When $M$ is the trivial representation, the action factors
through the Brauer supercategory $\mathit{s}\mathcal{B}\mathit{r}$. Our main
result is an explicit basis theorem for the morphism spaces of
$\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$ and, as a consequence, of
$\mathit{s}\mathcal{B}\mathit{r}$. The proof utilises the close connection with
the representation theory of $\mathfrak{p}(n)$. As an application we explicitly
describe the centre of all endomorphism algebras, and show that it behaves well
under the passage to the associated graded and under deformation.
