Presentations of Categories of Modules using the Cautis-Kamnitzer-Morrison Principle

Abstract

We use duality theorems to obtain presentations of some categories of modules. To derive these presentations we generalize a result of Cautis-Kamnitzer-Morrison [arXiv:1210.6437v4]: Let $\mathfrak{g}$ be a reductive Lie algebra, and $A$ an algebra, both over $\mathbb{C}$. Consider a $(\mathfrak{g} , A)$-bimodule $P$ in which (a) $P$ has a multiplicity free decomposition into irreducible $(\mathfrak{g} , A)$-bimodules. (b) $P$ is "saturated" i.e. for any irreducible $\mathfrak{g}$-module $V$, if every weight of $V$ is a weight of $P$, then $V$ is a submodule of $P$. We show that statements (a) and (b) are necessary and sufficient conditions for the existence of an isomorphism of categories between the full subcategory of $\mathcal{R}ep A$ whose objects are $\mathfrak{g}$-weight spaces of $P$, and a quotient of the category version of Lusztig's idempotented form, $\dot{\mathcal{U}} \mathfrak{g}$, formed by setting to zero all morphisms factoring through a collection of objects in $\dot{\mathcal{U}} \mathfrak{g}$ depending on $P$. This is essentially a categorical version of the identification of generalized Schur algebras with quotients of Lusztig's idempotented forms given by Doty in [arXiv:math/0305208]. Applied to Schur-Weyl Duality we obtain a diagrammatic presentation of the full subcategory of $\mathcal{R}ep S_d$ whose objects are direct sums of permutation modules, as well as an explicit description of the $\otimes$-product of morphisms between permutation modules. Applied to Brauer-Schur-Weyl Duality we obtain diagrammatic presentations of subcategories of $\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ and $\mathcal{R}ep \mathcal{B}_{r,s}^{(n)}$ whose Karoubi completion is the whole of $\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ and $\mathcal{R}ep \mathcal{B}_{r,s}^{(n)}$ respectively.