Elementary divisors of Specht modules
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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.
Sign mistake in 6.5 ff. corrected. European J. Combinatorics (to appear)
Sign mistake in 6.5 ff. corrected. European J. Combinatorics (to appear)