Generalized Polynomial Superfunctors

Abstract

We introduce a generalized notion of homogeneous strict polynomial functors defined over a superalgebra, $A$. In particular, we define two closely related families of categories $\mathsf{P}^d_A$ and $\mathsf{P}^d_{(A,\mathfrak{a})}$ which generalize the categories $\mathsf{P}_d$ of classical homogeneous strict polynomial functors studied by Friedlander and Suslin and the categories $\mathsf{Pol}_{d,\Bbbk}^{(\text{I})}$ and $\mathsf{Pol}_{d,\Bbbk}^{(\text{II})}$ of homogeneous strict polynomial superfunctors defined by Axtell. In particular, we exhibit equivalences between the categories $\mathsf{P}^d_A$, $\mathsf{P}^d_{(A,\mathfrak{a})}$ and the categories of left supermodules for generalized Schur algebras $S^A(m|n,d)$, $T^A_{\mathfrak{a}}(n,d)$, respectively (the latter of which were introduced by Kleshchev and Muth). Moreover, we establish a relationship between webs for $\mathfrak{gl}_n(A)$ and these generalized strict polynomial functors in the form of a faithful (and full under certain assumptions on $\Bbbk$) functor from the category of $\mathfrak{gl}_n(A)$-webs to $\mathsf{P}_{(A,\mathfrak{a})}$.