Cellular structures using $\textbf{U}_q$-tilting modules
Henning Haahr Andersen, Catharina Stroppel, Daniel Tubbenhauer
TL;DR
The paper proves that the endomorphism algebra of any U_q(g)-tilting module T is cellular, providing a general, representation-theoretic route to cellularity that applies across semisimple and nonsemisimple settings, including infinite-dimensional categories. Central to the approach is constructing cellular bases from Δ_q- and ∇_q-filtrations via lifts through indecomposable tilting summands T_q(λ), yielding cell modules, simple modules, and a semisimplicity criterion aligned with the tilting structure. The authors then instantiate the framework in a wide array of examples—symmetric groups, Hecke algebras, TL and spider algebras, Ariki–Koike algebras, Brauer-type algebras, and parabolic/affine O—often obtaining new (graded) cellular bases and insights into dimensions and simples. They also relate the theory to infinite-dimensional settings (highest weight categories) and compare graded structures with existing bases, notably in TL algebras. Overall, the work provides a unifying, flexible method to derive cellularity and detailed module information for a large family of centralizer algebras arising from quantum group tilting theory.
Abstract
We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras, and recover the cellularity of several known algebras (with partially new cellular bases) which all fit into our general setup.