Counting factorization of Coxeter elements into products of reflections
|Room:||Mathematics and Computer Building (MC) 5158|
A classical formula asserts that the number of factorizations of the full cycle (1,2,...,n) into (n-1) transpositions is n(n-2). I will talk about two generalizations of this result. The first one deals with factorizations of "higher genus", i.e. into (n-1+2g) transpositions for g>0. It is due to Shapiro, Shapiro and Vainshtein in the context of Hurwitz numbers. The second one, where one replaces the symmetric group Sn by any finite subgroup of GLn generated by reflections, and the long cycle by a "Coxeter element" is due to Deligne, and Bessis. I will then present our new result, that generalizes both results simultaneously: we treat the case of "higher genus" factorizations in arbitrary well-generated complex reflection groups (in particular, in finite Coxeter groups).
Joint work with Christian Stump, Hanover.
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