Rouquier Complexes
نویسنده
چکیده
1.1. Setup. In this talk k is a field of characteristic 0. Let (W,S) be a Coxeter group with |S| = n < ∞. Also we let V = ∑ s∈S kes be the reflection representation of W , and R = k[V ]. R is a graded algebra with V ∗ in degree 2. We abbreviate M ⊗R N = MN for a right R-module M and a left R-module N . We let {αs}s∈S be the dual basis of {es}s∈S , which can be considered as elements in V ∗ ⊂ R. Thus R ' k[α1, · · · , αn] is a polynomial algebra of n variables. The natural action of W on V induces that on R, i.e. (w · f)(v) := f(w−1(v)), and thus on R−modgr (the category of graded R-module) and D(R−modgr) (its bounded derived category) in a way that for f ∈ R and m ∈ M we have f ·m = (w−1 · f)(m). Note that for M ∈ R−modgr, M ' RwM , where Rw is a R-bimodule such that Rw ' R as a left R-module and m · a = mw(a) for m ∈ Rw, a ∈ R,
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