Cellular structures using ${\text{U}}_q$-tilting modules

Abstract

We use the theory of ${\text{U}}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group ${\text{U}}_q$ 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.

31 pages, lots of figures, substantially rewritten (following the suggestions of some referees), changed numbering, comments welcome

References in corpus (6)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Diagram categories for ${\text{U}}_q$-tilting modules at roots of unity
• Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality
• Semisimplicity of Hecke and (walled) Brauer algebras
• The degenerate affine walled Brauer algebra
• ${\mathfrak{sl}}_3$-web bases, intermediate crystal bases and categorification

Cited by in corpus (21)

• Semi-infinite highest weight categories
• The Modular Temperley-Lieb Algebra
• Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras
• SL2 tilting modules in the mixed case
• Quivers for SL(2) tilting modules
• Cellularity for weighted KLRW algebras of types $B$, $A^{(2)}$, $D^{(2)}$
• Relative cellular algebras
• Growth rates of the number of indecomposable summands in tensor powers
• $(\ell ,p)$-Jones-Wenzl Idempotents
• Monoidal categories, representation gap and cryptography
• Sandwich cellularity and a version of cell theory
• The center of SL2 tilting modules
• Cellularity of endomorphism algebras of tilting objects
• Fractal behavior of tensor powers of the two dimensional space in prime characteristic
• Algebraic properties of zigzag algebras
• The $q$-Schur category and polynomial tilting modules for quantum $G{L}_n$
• Diagrammatic Construction of Representations of Small Quantum ${\mathfrak{sl}}_2$
• On a symplectic quantum Howe duality
• On rank one 2-representations of web categories
• Graded triangular bases
• Growth problems in diagram categories