Abstract
We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. Our results demonstrate an interesting combination of the representation theory of finite Coxeter groups and their 0-Hecke algebras, including a two-sided duality between the induced and restricted representations.
Highlights
Let W := S :mst = 1, ∀s, t ∈ S be a Coxeter group generated by a finite set S with relationsmst = 1 for all s, t ∈ S, where mss = 1 for all s ∈ S and mst = mts ∈ {2, 3, . . .} ∪ {∞} for all distinct s, t ∈ S
In this paper we investigate the representation theory of the algebra HS(q) when (W, S) is a -laced Coxeter system
Our result shows an interesting combination of the representation theory of Coxeter groups and 0-Hecke algebras, but there are certain features of the representation theory of HS(q) that are unlike both symmetric groups and 0-Hecke algebras
Summary
The representation theory of Hecke algebras at roots of unity has been studied to some extent, with connections to other topics found (see Geck and Jacon [11]), but has not been completely determined yet even in type A (see Goodman and Wenzl [12]) Another interesting specialization of HS(q) is the 0-Hecke algebra HS(0), which is different from but closely related to the group algebra of W. For a finite Coxeter system (W, S), Norton [25] studied the representation theory of HS(0) over an arbitrary field F using the triangularity of the product in HS(0) Her main result is a decomposition of HS(0) into a direct sum of 2|S| many indecomposable submodules; this decomposition is similar to the decomposition of the group algebra of W (over the field of rational numbers) by Solomon [28].
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