By Meinolf Geck

The modular illustration idea of Iwahori-Hecke algebras and this theory's connection to teams of Lie kind is a space of quickly increasing curiosity; it really is person who has additionally noticeable a couple of breakthroughs lately. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of solidarity, this e-book is a very helpful addition to present examine during this box. utilizing the framework supplied through the Kazhdan-Lusztig conception of cells, the authors boost an analogue of James' (1970) "characteristic-free'' method of the illustration concept of Iwahori-Hecke algebras in general.

Presenting a scientific and unified remedy of representations of Hecke algebras at roots of team spirit, this booklet is exclusive in its procedure and comprises new effects that experience no longer but been released in booklet shape. It additionally serves as history interpreting to extra energetic parts of present learn equivalent to the speculation of affine Hecke algebras and Cherednik algebras.

The major result of this e-book are got by means of an interplay of a number of branches of arithmetic, particularly the idea of Fock areas for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the speculation of Kazhdan-Lusztig bases and cells, and computational methods.

This e-book might be of use to researchers and graduate scholars in illustration conception in addition to any researchers outdoors of the sector with an curiosity in Hecke algebras.

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**Example text**

Then ∑ ∑ λ ∈Λ s,t∈M(λ ) ts fλ−1 cst x,λ cy−1 ,λ = 1 0 if x = y, otherwise. (c) Given λ ∈ Λ and s, t ∈ M(λ ), there exists some w ∈ W such that cst w,λ = 0. Con= 0. In particular, versely, given w ∈ W , there exist some λ , s, t such that cst w,λ aλ = min{g ∈ Γ 0 λ | ε g ρst (Tw ) ∈ O0 for all w ∈ W and s, t ∈ M(λ )}. Proof. 12 yield δsv δut δλ μ ε aλ +aμ cλ = ∑ μ λ ε aλ ρst (Tw ) ε aμ ρuv (Tw−1 ) . 3, the right-hand side is congruent to ∑w∈W cst w,λ cw−1 ,λ modulo m. If λ = μ , the left-hand side is zero.

Thus, the constant term of x is 0 if x ∈ m; the constant term equals rx if x ∈ O0× . Warning: Note that the algebra H is not defined over O0 ! For example, the coefficients in the quadratic relations for Ts (s ∈ S, L(s) > 0) do not lie in O0 . 2. Let Γ = Z so that A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Then K = K(v) is the field of rational functions in v. In this case (which is the most important one for many applications), O0 is just the localisation of A in the prime ideal generated by v: O0 = f /g f , g ∈ K[v] such that g(0) = 0 ⊆ K(v).

1. We will consider a certain valuation ring O0 in K. Let us denote by K[Γ>0 ] the set of all K-linear combinations of terms ε g , where g > 0; the notation K[Γ 0 ] has a similar meaning. Note that 1 + K[Γ>0 ] is multiplicatively closed. Furthermore, every element x ∈ K can be written in the form x = r x ε gx 1+ p , 1+q where rx ∈ K, gx ∈ Γ and p, q ∈ K[Γ>0 ]; note that, if x = 0, then rx and gx indeed are uniquely determined by x; if x = 0, we have r0 = 0 and we set g0 := +∞ by convention. We set O0 := {x ∈ K | gx 0} and m := {x ∈ K | gx > 0}.