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Specht Filtrations for Hecke Algebras of Type A
Author(s) -
Hemmer David J.,
Nakano Daniel K.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005186
Subject(s) - mathematics , hecke algebra , functor , pure mathematics , filtration (mathematics) , cohomology , schur algebra , type (biology) , corollary , symmetric group , group (periodic table) , decomposition , algebra over a field , physics , classical orthogonal polynomials , ecology , gegenbauer polynomials , quantum mechanics , orthogonal polynomials , biology
Let H q ( d ) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive 1th root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l ⩾ 4 the multiplicities in a Specht or dual Specht module filtration of an H q ( d )‐module are well defined. A cohomological criterion is given for when an H q ( d )‐module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p ⩾ 5. These results are false in general for p = 2 or 3.