Ariki Koike Algebra Research Articles

Morita equivalences of Ariki-Koike algebras

We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle.

On certain bases for Ariki-Koike algebras arising from canonical bases for Uv(slm)

Frenkel, Khovanov and Kirillov showed that the parabolic Kazhdan-Lusztig basis of Iwahori-Hecke algebra associated to Sn can be obtained as the canonical basis of a weight subspace of V⊗n, where V is the vector representation of the quantum group Uv(slm). In this paper, a similar problem for the case of Ariki-Koike algebra ℋn,r is discussed. We construct a certain basis of ℋn,r, which is fixed by the involution and is closely related to the canonical basis of V⊗n, by making use of the representation of ℋn,r on V⊗n. In the case where r = 2, i.e., in the case of Iwahori-Hecke algebra of type Bn, this gives a basis different from the Kazhdan-Lusztig basis of ℋn,r.

SPECHT MODULES FOR ARIKI-KOIKE ALGEBRAS

Specht modules for an Ariki-Koike algebra H of type G(m, 1, r) have been investigated recently in the context of cellular algebras (see [GL] and [DJM]). Thus, these modules are defined as quotient modules of certain “permutation” modules, that is, defined as “cell modules” via cellular bases. So cellular bases play a decisive roˆle in these work. However, the classical theory [C] or the work [DJ1], [DJ2] in the case when m = 1,2 (i.e., the case of type A and B Hecke algebras) suggest that a construction as submodules without using cellular bases should exist. Following our previous work [DR], we shall introduce in this paper Specht modules for an Ariki-Koike algebra as submodules of those “permutation” modules, and prove that they are isomorphic to those (as quotient modules) defined in [DJM]. This results in some new versions of several known results such as standard basis and branching theorems. Under the hypothesis of the Morita equivalence theorem given in [DR], we shall also prove that Brundan's result which generalizes Kleshchev's branching rules [K] for symmetric groups to type A Hecke algebras can be lifted to Ariki-Koike algebras in this case. We point out that, using quantum group approach, I. Grojnowski [[G], 9.14] has established branching rules for Ariki-Koike algebras in general. However, it is not yet clear even in the type A case if one can use these rules to obtain Kleshchev's branching rules or vise versa.

Ariki-Koike algebras with semisimple bottoms

We investigated the representation thoery of an Ariki-Koike algebra whose Poincare polynomial associated with the bottom, i.e., the subgroup on which the symmetric group acts, is non-zero in the base field. We proved that the module category of such an Ariki-Koike algebra is Morita equivalent to the module category of a direct sum of tensor products of Hecke algebras associated with certain symmetric groups. We also generalized this Morita equivalence theorem to give a Morita equivalenve between a $q$-Schur$^m$ algebra and a direct sum of tensor products of certain $q$-Schur algebras.

The Jantzen sum formula for cyclotomic 𝑞–Schur algebras

The cyclotomic q q -Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen’s sum formula for the cyclotomic q q -Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.

A Frobenius Formula for the Characters of Ariki–Koike Algebras

Let Hn,r be the Ariki–Koike algebra associated to the complex reflection group Wn,r=G(r,1,n). In this paper, we give a new presentation of Hn,r by making use of the Schur–Weyl reciprocity for Hn,r established by M. Sakamoto and T. Shoji (1999, J. Algebra, 221, 293–314). This allows us to construct various non-parabolic subalgebras of Hn,r. We construct all the irreducible representations of Hn,r as induced modules from such subalgebras. We show the existence of a partition of unity in Hn,r, which is specialized to a partition of unity in the group algebra CWn,r. Then we prove a Frobenius formula for the characters of Hn,r, which is an analogy of the Frobenius formula proved by A. Ram (1991, Invent. Math.106, 461–488) for the Iwahori–Hecke algebra of type A.

Schur–Weyl Reciprocity for Ariki–Koike Algebras

Let g=glm1⊕···⊕glmr be a Levi subalgebra of glm, with m=∑ri=1mi, and V the natural representation of the quantum group Uq(g). We construct a representation of the Ariki–Koike algebra Hn,r on the n-fold tensor space of V, commuting with the action of Uq(g), and prove the Schur–Weyl reciprocity for the actions of Uq(g) and Hn,r on it.