On the category of finite-dimensional representations of $\OSPrn$: Part I

TL;DR

This paper investigates the structure of finite-dimensional modules over the orthosymplectic Lie supergroup, providing combinatorial formulas, constructing related algebras, and connecting to geometric and representation-theoretic frameworks.

Contribution

It introduces a positive counting formula for homomorphism spaces, constructs algebras linked to blocks, and connects the module category to geometric and algebraic structures, refining previous results.

Findings

Derived a counting formula for homomorphism dimensions

Constructed algebras with combinatorics matching blocks of modules

Linked the module category to geometric objects like Grassmannians and Springer fibers

Abstract

We study the combinatorics of the category F of finite-dimensional modules for the orthosymplectic Lie supergroup OSP(r|2n). In particular we present a positive counting formula for the dimension of the space of homomorphism between two projective modules. This refines earlier results of Gruson and Serganova. Moreover, for each block B of F we construct an algebra A(B) whose module category shares the combinatorics with B. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It will turn out that A(B) is isomorphic to the endomorphism algebra of a minimal projective generator of B. This provides a direct link from F to parabolic categories O of type B or D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fibres of types C/D. We also indicate why F is not highest weight in general.