[Paper Review] Sandwich cellularity and a version of cell theory

Daniel Tubbenhauer|arXiv (Cornell University)|Jun 14, 2022

Algebraic structures and combinatorial models7 citations

TL;DR

This paper introduces sandwich cellularity as a unifying framework that generalizes cellular algebras, Kazhdan–Lusztig bases, and monoid representation theory. It establishes H-reduction as a key theorem to classify simple modules using cells and sandwiched algebras, applying the theory to Hecke algebras, diagram algebras, and monoid algebras with explicit dimension computations and module classifications.

ABSTRACT

We explain how the theory of sandwich cellular algebras can be seen as a version of cell theory for algebras. We apply this theory to many examples such as Hecke algebras, and various monoid and diagram algebras.

Motivation & Objective

• To unify cellular algebras, Kazhdan–Lusztig bases, and monoid representation theory under a single framework.
• To introduce sandwich cellular algebras as a generalization that captures Green’s H-groups and asymptotic Hecke algebras.
• To establish H-reduction as a classification tool for simple modules using cells and sandwiched algebras.
• To apply the theory to concrete examples: Hecke algebras, Brauer algebras, Temperley–Lieb algebras, and transformation monoids.
• To compute dimensions of simple modules and demonstrate that many diagram algebras are cellular via this framework.

Proposed method

• Reformulate the definition of sandwich cellular algebras to make it practically usable and conceptually aligned with cell theory.
• Use sandwich cell data to define cells that partition the algebra, analogous to Green’s relations and Kazhdan–Lusztig cells.
• Employ H-reduction to classify simple modules via cells and sandwiched algebras, generalizing the Clifford–Munn–Ponizovskiĭ theorem.
• Construct sandwiched algebras as analogues of H-groups and asymptotic Hecke algebras in the context of monoids and Hecke algebras.
• Apply the theory to diagram algebras by embedding smaller algebras into larger ones and analyzing sandwich matrices as submatrices.
• Use permutation actions and module isomorphisms (e.g., Δ(λ,K) ≅ K^⊕k) to deduce simplicity and compute dimensions of modules.

Experimental results

Research questions

• RQ1 How can sandwich cellular algebras serve as a common generalization of cellular algebras, monoid theory, and Kazhdan–Lusztig theory?
• RQ2 In what way does H-reduction classify simple modules of sandwich cellular algebras, and how does it generalize existing theorems?
• RQ3 Which diagram algebras and monoid algebras admit natural sandwich cellular structures, and how can their simple modules be computed?
• RQ4 What is the role of sandwiched algebras in relating cells to H-groups and asymptotic Hecke algebras?
• RQ5 How do quantum and p-Kazhdan–Lusztig bases fit into the sandwich cellularity framework?

Key findings

• All algebras are sandwich cellular, but the key challenge lies in finding useful sandwich cell structures.
• H-reduction successfully classifies simple modules of sandwich cellular algebras using cells and sandwiched algebras.
• For the rook monoid algebra Ro_n(δ), the simple module Δ(λ,K) is isomorphic to K^⊕k where k = binom(n,λ).
• The dimensions of simple modules for Temperley–Lieb algebras TL_n(δ) and partition algebras Pa_n^p(δ) are computed via isomorphisms and known formulas.
• Brauer algebras, rook monoids, and their quantum versions (e.g., BMW algebras) are shown to be sandwich cellular, with sandwich matrices arising as submatrices of larger ones.
• The theory recovers known results on module dimensions and cellularity of diagram algebras, but provides a unified and conceptual framework where these were previously disparate.