On decomposition numbers of the cyclotomic q-Schur algebras

Abstract

Let S ( Λ ) be the cyclotomic q-Schur algebra associated to the Ariki–Koike algebra H . We construct a certain subalgebra S 0 ( Λ ) of S ( Λ ) , and show that it is a standardly based algebra in the sense of Du and Rui. S 0 ( Λ ) has a natural quotient S 0 ¯ ( Λ ) , which turns out to be a cellular algebra. In the case where the modified Ariki–Koike algebra H ♭ is defined, S 0 ¯ ( Λ ) coincides with the cyclotomic q-Schur algebra associated to H ♭ . In this paper, we discuss a relationship among the decomposition numbers of S ( Λ ) , S 0 ( Λ ) and S 0 ¯ ( Λ ) . In particular, we show that some important part of the decomposition matrix of S ( Λ ) coincides with a part of the decomposition matrix of S 0 ¯ ( Λ ) .