Representations of wreath products on cohomology of De Concini-Procesi compactifications

Details Representations of wreath products on cohomology of De Concini-Procesi compactifications Representations of wreath products on cohomology of De Concini-Procesi compactifications

2003-07-30

arxiv.org/abs/math/0307383v1

Intern. Math. Res. Notices 2004:20 (2004), 983-1021

Mathematics

Representation Theory

38 pages

Scientific paper

The wreath product W(r,n) of the cyclic group of order r and the symmetric group S_n acts on the corresponding projective hyperplane complement, and on its wonderful compactification as defined by De Concini and Procesi. We give a formula for the characters of the representations of W(r,n) on the cohomology groups of this compactification, extending the result of Ginzburg and Kapranov in the r=1 case. As a corollary, we get a formula for the Betti numbers which generalizes the result of Yuzvinsky in the r=2 case. Our method involves applying to the nested-set stratification a generalization of Joyal's theory of tensor species, which includes a link between polynomial functors and plethysm for general r. We also give a new proof of Lehrer's formula for the representations of W(r,n) on the cohomology groups of the hyperplane complement.

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