Khovanov algebras of type B and tensor powers of the natural $\mathrm{OSp}$-representation

TL;DR

This paper develops the theory of projective endofunctors for Khovanov algebras of type B, relating them to Brauer algebras and orthosymplectic supergroup representations, providing insights into tensor power decompositions.

Contribution

It introduces a new framework for projective functors on Khovanov algebras of type B and connects them to translation functors for $ ext{OSp}$ representations, with applications to tensor powers.

Findings

Computed composition factors and graded layers of simple modules under projective functors.

Established a relation between Khovanov and Brauer algebras via graded lifts.

Described Loewy layers of indecomposable summands in tensor powers of the natural $ ext{OSp}$-representation.

Abstract

We develop the theory of projective endofunctors for modules of Khovanov algebras $K$ of type B. In particular we compute the composition factors and the graded layers of the image of a simple module under such a projective functor. We then study variants of such functors for a subquotient $e\tilde{K}e$. Via a comparison of two graded lifts of the Brauer algebra we relate the Khovanov algebra to the Brauer algebra and use this to show that projective functors describe translation functors on representations of the orthosymplectic supergroup $\mathrm{OSp}(r|2n)$. As an application we get a description of the Loewy layers of indecomposable summands in tensor powers of the natural representation of $\mathrm{OSp}(r|2n)$.