By Meinolf Geck
The modular illustration thought of Iwahori-Hecke algebras and this theory's connection to teams of Lie variety is a space of swiftly increasing curiosity; it's one who has additionally obvious a couple of breakthroughs lately. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of cohesion, this e-book is a very useful addition to present study during this box. utilizing the framework supplied by way of the Kazhdan-Lusztig conception of cells, the authors strengthen an analogue of James' (1970) "characteristic-free'' method of the illustration thought of Iwahori-Hecke algebras in general.
Presenting a scientific and unified therapy of representations of Hecke algebras at roots of cohesion, this booklet is exclusive in its technique and comprises new effects that experience no longer but been released in e-book shape. It additionally serves as heritage analyzing to additional lively components of present learn equivalent to the speculation of affine Hecke algebras and Cherednik algebras.
The major result of this booklet are acquired through an interplay of numerous branches of arithmetic, specifically the idea of Fock areas for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the idea of Kazhdan-Lusztig bases and cells, and computational methods.
This booklet may be of use to researchers and graduate scholars in illustration thought in addition to any researchers open air of the sphere with an curiosity in Hecke algebras.
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Extra resources for Representations of Hecke Algebras at Roots of Unity
Sample text
The following definition works for general weight functions. (In [132, Def. 11. Assume that L(s) > 0 for all s ∈ S. A W -graph for H (with respect to ) consists of the following data: (a) a set X together with a map I which assigns to each x ∈ X a subset I(x) ⊆ S; (b) a collection of elements {msx,y } in A, where x, y ∈ X and s ∈ S are such that s ∈ I(x) and s ∈ I(y). These data are subject to the following requirements. 6. Furthermore, let V be a free A-module with a basis {ey | y ∈ X}. For s ∈ S, define an A-linear map ρs : V → V by ρs (ey ) = ⎧ ⎨ vs ey + ⎩ ∑ x∈X: s∈I(x) −v−1 s ey msx,y ex if s ∈ I(y), if s ∈ I(y).
Let λ ∈ Λ and consider a matrix representation ρ λ : HK → Mdλ (K) afforded by Eελ . We say that ρ λ is a balanced representation if ε aλ ρ λ (Tw ) ∈ Mdλ (O0 ) for all w ∈ W . aλ λ In this case, we denote by cst w,λ ∈ K the constant term of ε ρst (Tw ), for any w ∈ W st and s, t ∈ M(λ ). The coefficients cw,λ will be called the leading matrix coefficients of H. 4. 2. If ρ λ is a balanced representation afforded by Eελ , then cw,λ = ∑ s∈M(λ ) css w,λ for all w ∈ W . 3 are refinements of Lusztig’s leading coefficients of character values.
M−1)/2 } {1W , sgn, σ1 , σ2 , . . , σ(m−2)/2 , sgn1 , sgn2 } if m is odd, if m is even, where 1W is the unit and sgn is the sign representation, all σ j are two-dimensional, and sgn1 , sgn2 are two further one-dimensional representations when m is even, in which case we fix the notation such that s1 acts as +1 in sgn1 and as −1 in sgn2 . 3. 4], one obtains the invariants aλ and f λ . 3, the columns are ordered such that the invariants aλ are increasing from left to right; the columns corresponding to sgn1 , sgn2 must be deleted if m is odd.