Crystal bases and simple modules for Hecke algebra of type Dn
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Let HC(Bn) (respectively HC(Dn)) be the Hecke algebra of type Bn (respectively of type Dn) over the complex numbers field C. Let ζ be a primitive 2
th root of unity in C. For any Kleshchev bipartition (with respect to (ζ,1,−1)) λ=(λ(1),λ(2)) of n, let λ be the corresponding irreducible HC(Bn)-module. In the present paper we explicitly determine which λ split and which λ remains irreducible when restricts to HC(Dn). This yields a complete classification of all the simple modules for Hecke algebra HC(Dn). Our proof makes use of the crystal bases theory for the Fock representation of the quantum affine algebra Uq(2) and deep result of Ariki's proof of LLT's conjecture [J. Math. Kyoto Univ. 36 (1996) 789–808].
th root of unity in C. For any Kleshchev bipartition (with respect to (ζ,1,−1)) λ=(λ(1),λ(2)) of n, let λ be the corresponding irreducible HC(Bn)-module. In the present paper we explicitly determine which λ split and which λ remains irreducible when restricts to HC(Dn). This yields a complete classification of all the simple modules for Hecke algebra HC(Dn). Our proof makes use of the crystal bases theory for the Fock representation of the quantum affine algebra Uq(2) and deep result of Ariki's proof of LLT's conjecture [J. Math. Kyoto Univ. 36 (1996) 789–808].