TY - JOUR

T1 - Decomposition numbers for Hecke algebras of type G(r, p, n)

T2 - The (ε, q)-separated case

AU - Hu, Jun

AU - Mathas, Andrew

PY - 2012/5

Y1 - 2012/5

N2 - The paper studies the modular representation theory of the cyclotomic Hecke algebras of type G(r, p, n) with (ε, q)-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type G(s, 1, m), where 1 ≤ s ≤ r and 1 ≤ m ≤ n. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero.In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type G(r, p, n) when the parameters are (ε, q)-separated.The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the l-splittable decomposition numbers and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type G(r, p, n).

AB - The paper studies the modular representation theory of the cyclotomic Hecke algebras of type G(r, p, n) with (ε, q)-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type G(s, 1, m), where 1 ≤ s ≤ r and 1 ≤ m ≤ n. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero.In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type G(r, p, n) when the parameters are (ε, q)-separated.The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the l-splittable decomposition numbers and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type G(r, p, n).

UR - http://www.scopus.com/inward/record.url?scp=84861079166&partnerID=8YFLogxK

U2 - 10.1112/plms/pdr047

DO - 10.1112/plms/pdr047

M3 - Article

AN - SCOPUS:84861079166

SN - 0024-6115

VL - 104

SP - 865

EP - 926

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 5

ER -