Balanced semisimple filtrations for tilting modules

TL;DR

This paper demonstrates the existence of balanced semisimple filtrations in tilting modules for quantum groups at roots of unity, providing an efficient algorithm for character calculation consistent with existing formulas.

Contribution

It introduces the concept of balanced semisimple filtrations for tilting modules and proves their existence, linking them to character formulas and offering a new computational approach.

Findings

Balanced semisimple filtrations exist for tilting modules.

The proposed algorithm aligns with Soergel's character formula.

Filtrations are symmetric about a middle layer, simplifying character calculations.

Abstract

Let $U_l$ be a quantum group at an $l$th root of unity. Many tilting modules for $U_l$ have been shown to have what we call a balanced semisimple filtration, or a Loewy series whose semisimple layers are symmetric about some middle layer. The existence of such filtrations suggests a remarkably straightforward algorithm for calculating these characters if the irreducible characters are already known. We first show that the results of this algorithm agree with Soergel's character formula for the regular tilting modules. We then show that these balanced semisimple filtrations really do exist for these tilting modules.