Blocks of cyclotomic Hecke algebras

This paper classifies the blocks of the cyclotomic Hecke algebras of type G(r,1,n) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of `Jantzen equivalence'. We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999. Unfortunately, Cox has discovered that this previous proof is incomplete.Comment: Final version. To appear in Advances in Mathematic

Decomposition numbers for Hecke algebras of type G(r,p,n)G(r,p,n): the (ϵ,q)(\epsilon,q)-separated case

The paper studies the modular representation theory of the cyclotomic Hecke algebras of type $G(r,p,n)$ with (\eps,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 $1\le s\le r$ and $1\le m\le n$. 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 (\eps,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 \textit{$l$-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 $G(r,p,n)$.Comment: Final versio

Elementary divisors of Specht modules

Let H_q(S_n) be the Iwahori-Hecke algebra of the symmetric group. This algebra is semisimple over the rational function field Q(q), where q is an indeterminate, and its irreducible representations over this field are q-analogues S_q(lambda) of the Specht modules of the symmetric group. The q-Specht modules have an "integral form" which is defined over the Laurent polynomial ring Z_[q,q^{-1}] and they come equipped with a natural bilinear form with values in this ring. Now Z[q,q^{-1}] is not a principal ideal domain. Nonetheless, we try to compute the elementary divisors of the Gram matrix of the bilinear form on S_q(lambda). When they are defined, we give a precise relationship between the elementary divisors of the Specht modules S_q(lambda) and S_q(lambda'), where lambda' is the conjugate partition. We also compute the elementary divisors when lambda is a hook partition and give examples to show that in general elementary divisors do not exist.Comment: Sign mistake in 6.5 ff. corrected. European J. Combinatorics (to appear

Cyclotomic Carter-Payne homomorphisms

We construct a new family of homomorphisms between (graded) Specht modules of the quiver Hecke algebras of type A. These maps have many similarities with the homomorphisms constructed by Carter and Payne in the special case of the symmetric groups, although the maps that we obtain are both more and less general than these.Comment: This paper has been updated. The formula for the degree shift in Theorem 3.28 has been corrected and Examples 3.31 and 3.36 have been changed accordingl