Cyclotomic 𝑞-Schur algebras associated to the Ariki-Koike algebra

Abstract

Let H n , r \mathcal {H}_{n,r} be the Ariki-Koike algebra associated to the complex reflection group S n ⋉ ( Z / r Z ) n \mathfrak {S}_n\ltimes (\mathbb {Z}/r\mathbb {Z})^n , and let S ( Λ ) \mathcal {S}(\varLambda ) be the cyclotomic q q -Schur algebra associated to H n , r \mathcal {H}_{n,r} , introduced by Dipper, James and Mathas. For each p = ( r 1 , … , r g ) ∈ Z > 0 g \mathbf {p} = (r_1, \dots , r_g) \in \mathbb {Z}_{>0}^g such that r 1 + ⋯ + r g = r r_1 +\cdots + r_g = r , we define a subalgebra S p \mathcal {S}^{\mathbf {p}} of S ( Λ ) \mathcal {S}(\varLambda ) and its quotient algebra S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} . It is shown that S p \mathcal {S}^{\mathbf {p}} is a standardly based algebra and S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of S ( Λ ) \mathcal {S}(\varLambda ) , which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic q q -Schur algebras associated to Ariki-Koike algebras H n i , r i \mathcal {H}_{n_i,r_i} of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra H ¯ p \overline {\mathcal {H}}^{\mathbf {p}} of type p \mathbf {p} , and prove the Schur-Weyl duality between H ¯ p \overline {\mathcal {H}}^{\mathbf {p}} and S ¯ p \overline {\mathcal {S}}^{\mathbf {p}} .