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Friday, March 1, 2013 |
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Counting factorizations of Coxeter elements into products of reflections |
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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
S_n by any finite subgroup of GL_n 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). |