Cyclotomic Specht filtrations and A-filtrations

From the time the q-Schur algebra S(n, r) was defined by Dipper-James [DJ2], the close relationship between the representation theory of S(n, r) and Hecke algebras H has been apparent. In [CPS3], Cline-Parshall-Scott developed techniques that revealed the intricate relationship between S(n, r) and H. In [PS1], Parshall-Scott used these techniques to give a contravariant isomorphism between the full exact category of S(n, r)-modules with ∆-filtrations and the full exact category of H-modules with Specht filtrations. With the introduction of cyclotomic q-Schur algebras S ♮ , by Dipper-James-Mathas [DJM], and then the modified cyclotomic q-Schur algebras S by Shoji [Sh], one can apply the techniques of [CPS3] to these algebras. Central to the subject matter of [CPS3] are certain filtrations similar to Specht filtrations and ∆-filtrations. With the goal of exploring the relationship between Specht modules and ∆- modules, we begin by recalling the basics of quasi-hereditary and cellular algebras. Under certain assumptions, we give an explicit description of standard modules for a quasi-hereditary cellular algebra. Chapter 2 recalls the cellular structures described in [DJM] and [SS]. In Chapter 3, we take the definition of a Specht module S λ from Du-Rui [DR2] for the Ariki-Koike algebra H ♮ , and a similar definition for the modified Ariki-Koike algebra H. We then show (under certain conditions) that the permutation modules have Specht filtrations. In Chapter 5, we collect some basic results about tilting modules for S ♮ and S. Chapter 5 shows that under certain conditions ii the standard module ∆(λ) for S can be realized as Hom H (S λ , T ), where T is the direct sum of the permutation modules. Chapter 6 examines H and S through the theory of stratified categories. We show under certain restrictions on the base ring, the triple (H, T , S) satisfies the Integral Stratification Hypothesis, implying (under further restrictions on the base ring) the existence of a contravariant isomorphism between the full exact category of S-modules with ∆-filtrations and the full exact category of H-modules with Specht filtrations.

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