Spin polynomial functors and representations of Schur superalgebras

Abstract

We introduce categories of homogeneous strict polynomial functors, P o l d , k I \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk } and P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } , defined on vector superspaces over a field k \Bbbk of characteristic not equal 2. These categories are related to polynomial representations of the supergroups G L ( m | n ) GL(m|n) and Q ( n ) Q(n) . In particular, we prove an equivalence between P o l d , k I \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk } , P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } and the category of finite dimensional supermodules over the Schur superalgebra S ( m | n , d ) \mathcal {S}(m|n,d) , Q ( n , d ) \mathcal {Q}(n,d) respectively provided m , n ≥ d m,n \ge d . We also discuss some aspects of Sergeev duality from the viewpoint of the category P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } .