# Cyclotomic Specht filtrations and A-filtrations

Cramer, Wesley, Department of Mathematics, University of Virginia

Parshall, Brian, Department of Mathematics, University of Virginia

Scott, Leonard, Department of Mathematics, University of Virginia

Wang, Weiqiang, Department of Mathematics, University of Virginia

Chang, Theodore, Department of Statistics, University of Virginia

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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PHD (Doctor of Philosophy)

English

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2008/05/01