POLYNOMIAL FUNCTORS AND TWO-PARAMETER QUANTUM SYMMETRIC PAIRS

Abstract

AbstractWe develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups GLn, the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair ($$ {U}_{Q,q}^B $$ U Q , q B ($$ \mathfrak{gl} $$ gl n), Uq($$ \mathfrak{gl} $$ gl n)) which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra $$ {U}_{Q,q}^B $$ U Q , q B ($$ \mathfrak{gl} $$ gl n) appears in a Schur–Weyl duality with the type B Hecke algebra $$ {\mathcal{H}}_{Q,q}^B $$ H Q , q B (d). We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs.