On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\). (English) Zbl 0888.20011

Ariki, Susumu

J. Math. Kyoto Univ. 36, No. 4, 789-808 (1996).

Let \(H_A\) be the Hecke algebra over the polynomial ring \(A=\mathbb Z[v_1,\dots,v_m,q,q^{-1}]\) defined by generators \(a_1,\dots,a_n\) and relations \((a_1-v_1)\cdots(a_1-v_m)=0\), \((a_i-q)(a_i+q^{-1})=0\) (\(2\leq i\leq n\)), \(a_1a_2a_1a_2=a_2a_1a_2a_1\), \(a_ia_j=a_ja_i\) \(\forall j\geq i+2\), \(a_ia_{i+1}a_i=a_{i+1}a_ia_{i+1}\) (\(2\leq i\leq n-1\)). This algebra is \(A\)-free and when specializing \(v_i\) & \(q\) with \(v_i\in\mathbb C\) & \(q\in\mathbb C^\times\) one gets \(H_\mathbb C\). Let \(u_n\) be the Grothendieck group of the category of \(H_\mathbb C\)-modules and let \(u=\bigoplus u_n\).
The main aim of this paper is to show that the graded dual of \(u\) is a highest weight module of \(g(A_\infty)\) (resp. \(g(A^{(1)}_{r-1})\)) if \(q^2\) is not a root of unity (resp. a primitive \(r\)-th root of unity), and the dual basis of irreducible modules coincides with the canonical basis. The proof depends on Lusztig’s theory of affine Hecke algebras and quantum groups, and Ginzburg’s theory of affine Hecke algebras.

Reviewer: A.Khammash (Makkah)

MSC:

20C08	Hecke algebras and their representations
17B37	Quantum groups (quantized enveloping algebras) and related deformations
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

Hecke algebras; generators; relations; Grothendieck groups; highest weight modules; irreducible modules; affine Hecke algebras; quantum groups