Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories

TL;DR

This paper classifies indecomposable objects in Khovanov-Sazdanovic's categories, generalizing Deligne's categories, and describes their Grothendieck rings using representation theory over polynomial rings.

Contribution

It introduces a classification of indecomposables and describes Grothendieck rings for these generalized categories, extending the understanding of Deligne's interpolation categories.

Findings

Classified indecomposable objects in Khovanov-Sazdanovic categories.

Identified the associated graded Grothendieck rings as sums of representation categories.

Developed a categorification approach for Krull-Schmidt categories with filtrations.

Abstract

Khovanov and Sazdanovic recently introduced symmetric monoidal categories parameterized by rational functions and given by quotients of categories of two-dimensional cobordisms. These categories generalize Deligne's interpolation categories of representations of symmetric groups. In this paper, we classify indecomposable objects and identify the associated graded Grothendieck rings of Khovanov-Sazdanovic's categories through sums of representation categories over crossed products of polynomial rings over a general field. To obtain these results, we introduce associated graded categories for Krull-Schmidt categories with filtrations as a categorification of the associated graded Grothendieck ring, and study field extensions and Galois descent for Krull-Schmidt categories.