Crystal bases and simple modules for Hecke algebra of type Dn

Abstract

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