Double Cyclotomic Schur Algebras

Throughout this paper we study a class of toroidal symmetric groups by considering their finite dimensional quotients. The aim of the first three sections is to classify irreducible modules over a field. In the last two sections, we introduce the double cyclotomic Schur algebras and show that they are cellular in the sense of Graham-Lehrer [11]. Under certain assumption, the double cyclotomic Schur algebras can be quasi-hereditary in the sense of Cline-Parshall-Scott [2].In the first section we give the definitions of a toroidal symmetric group and its quotient Hr. Theorem 1.11 and 1.13 indicate respectively that Hr is a free module and a symmetric algebra. A new kind of multi-composition (resp. multi-partitions) is defined in this section, so called (n, m)-composition (resp. partition), which is constructed by n pairs of m-compositions (resp. partitions). Using this new kind of compositions (partitions), we construct a cellular basis of Hr and draw the conclusion that Hr is a cellular algebra. The action of the generators li over base elements (cf. Lemma 2.14) is of key importance in the second section.Lemma 2.14. Suppose Following the general results on cellular algebras, we classify simple Hr-modules over a field in the third section. The main theorem of this section is Theorem 3.4.Theorem 3.4 Let R be a field. D 0 if and only if the following conditions hold.(a) (ij) are e-restricted for all 1 < i < n, 1 < j < m.Motivated by [7], in the forth section, we construct a semi-standard basis or Murphy basis of the permutation module. The definition of semi-standard tableaux is generalized (cf. Definition 4.3). Using permutation module, we introduce a certain endomorphism algebra over a poset, called double cyclotomic Schur algebra in the last section. Let T C An>m, the double cyclotomic Schur algebra associated to T is defined by setting S(T) = Endnr (0Aer X>&r). Such algebra is a cellular algebra (cf.Theorem 5.6). By choosing the poset appropriately, we find that it can also be a quasi-hereditary algebra.