Graded Frobenius cluster categories

TL;DR

This paper extends the concept of gradings from 2-Calabi-Yau triangulated categories to Frobenius categories that categorify cluster algebras with coefficients, providing a K-theoretic interpretation and concrete examples.

Contribution

It introduces graded Frobenius cluster categories, generalizing previous triangulated cases, and demonstrates their application to categorify structures like partial flag varieties and Grassmannians.

Findings

Degrees are additive on exact sequences in graded Frobenius categories

Examples include categories categorifying cells in partial flag varieties

Categories of Jensen, King, and Su are examples of graded Frobenius cluster categories

Abstract

Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences. We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiss, Leclerc and Schroer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.