[Paper Review] A cellular approach to the Hecke-Clifford superalgebra

Masaki Mori|arXiv (Cornell University)|Jan 8, 2014

Algebraic structures and combinatorial models26 references4 citations

TL;DR

This paper introduces a generalized cellular algebra framework to classify simple modules of the Hecke–Clifford superalgebra by leveraging a right action of the Clifford superalgebra on its super-Specht module. Using a Morita context between the Hecke–Clifford superalgebra and the Clifford superalgebra, it establishes a one-to-one correspondence between simple modules of the former and certain simple modules of the latter, providing a new cellular approach that extends Graham and Lehrer’s theory to superalgebras.

ABSTRACT

The Hecke-Clifford superalgebra is a super-analogue of the Iwahori-Hecke algebra of type A. The classification of its simple modules is done by Brundan, Kleshchev and Tsuchioka using a method of categorification of affine Lie algebras. In this paper, we introduce another way to produce its simple modules with a generalized theory of cellular algebras which is originally developed by Graham and Lehrer. In our construction the key is that there is a right action of the Clifford superalgebra on the super-analogue of the Specht module. With the help of the notion of the Morita context, a simple module of the Hecke-Clifford superalgebra is made from that of the Clifford superalgebra.

Motivation & Objective

• To develop a generalized cellular algebra theory suitable for superalgebras, where matrix algebras over the Clifford superalgebra serve as new types of cells.
• To classify the simple modules of the Hecke–Clifford superalgebra using a Morita context between it and the Clifford superalgebra.
• To extend the classical cellular algebra framework—originally for algebras like the Iwahori–Hecke algebra—to superalgebras by incorporating supermodule structures and generalized bases.
• To provide an alternative to categorification methods used by Brundan, Kleshchev, and Tsuchioka for classifying simple modules of the Hecke–Clifford superalgebra.

Proposed method

• Introduce a generalized standardly based algebra structure on the Hecke–Clifford superalgebra using parametrized bases indexed by partitions.
• Define a right action of the Clifford superalgebra on the super-Specht module, enabling the construction of a Morita context between the Hecke–Clifford superalgebra and the Clifford superalgebra.
• Use the Morita context to relate simple modules of the Hecke–Clifford superalgebra to simple modules of the Clifford superalgebra via the image of the bimodule homomorphisms.
• Construct a bilinear form on the standard modules and define the simple quotient as the quotient by the radical of this form.
• Apply the generalized Morita correspondence theorem (Theorem 3.15) to establish a one-to-one correspondence between simple modules of the two algebras.
• Specialize the results to fields with various characteristics and quantum parameters, deriving classification theorems under different conditions on $ q $, $ a $, and the characteristic of the field.

Experimental results

Research questions

• RQ1 How can the classical cellular algebra framework be extended to include superalgebras with non-trivial superstructure, such as the Hecke–Clifford superalgebra?
• RQ2 What role does the Clifford superalgebra play in the representation theory of the Hecke–Clifford superalgebra, particularly through its action on the super-Specht module?
• RQ3 Can a Morita context be constructed between the Hecke–Clifford superalgebra and the Clifford superalgebra to relate their simple modules?
• RQ4 Under what conditions on the field and quantum parameter $ q $ does the classification of simple modules reduce to a combinatorial condition on partitions?
• RQ5 How does the new cellular approach compare with the categorification method used by Brundan, Kleshchev, and Tsuchioka for the same classification problem?

Key findings

• There is a one-to-one correspondence between the simple modules of the Hecke–Clifford superalgebra $ H_n^c $ and the simple modules of the Clifford superalgebra $ igoplus_{ ext{partition } u ext{ of } n} ext{Irr}^{ riangle_{ u} + heta_{ u}}_{ heta_{ u}}( ilde{ u}) $, as stated in Theorem 6.38.
• When $ q eq -1 $, $ 2aq[2] eq 0 $, and $ k $ is a field, the simple modules of $ H_n^c $ are in bijection with $ e $-restricted $ e_2 $-strict partitions of $ n $, as shown in Corollary 6.39.
• When $ q = -1 $, $ 2a eq 0 $, and $ k $ has characteristic $ p \neq 2 $, the simple modules of $ H_n^c $ are in bijection with $ 2p $-restricted $ p $-strict partitions of $ n $, as in Corollary 6.41.
• In the case $ q = -1 $, $ 2a = 0 $, the classification reduces to $ 2 $-restricted partitions of $ n $, as stated in Corollary 6.42.
• The results for $ 2a = 0 $ match those obtained via supercategorification using cyclotomic quiver Hecke superalgebras, confirming consistency with prior work.
• The generalized cellular framework successfully classifies simple modules by incorporating Clifford superalgebra modules as new types of cells, overcoming limitations of the original cellular algebra theory in the super setting.