Representations of Wreath Products on Cohomology of De Concini-procesi Compactifications

نویسنده

ANTHONY HENDERSON

چکیده

The wreath product W (r, n) of the cyclic group of order r and the symmetric group Sn 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. Introduction Let r and n be positive integers. Write W (r, n) for the wreath product μr ≀ Sn of the cyclic group μr of complex rth roots of 1 and the symmetric group Sn. Let V (r, n) be the standard vector space on which W (r, n) acts irreducibly as a complex reflection group, with hyperplane arrangement A(r, n) (in this context W (r, n) is usually called G(r, 1, n)). Explicitly, V (1, n) = C/C(1, 1, · · · , 1), W (1, n) = Sn acts on V (1, n) by permuting the coordinates, and A(1, n) = {{(z1, · · · , zn) ∈ C n | zi = zj}/C(1, 1, · · · , 1) | 1 ≤ i 6= j ≤ n} . If r ≥ 2, V (r, n) = C, W (r, n) acts by permutations of coordinates composed with multiplying coordinates by rth roots of 1, and A(r, n) = {{zi = 0} | 1 ≤ i ≤ n}∪{{zi = ζzj} | 1 ≤ i 6= j ≤ n, ζ ∈ μr} . Write M(r, n) for the projective hyperplane complement, i.e. the set of points in P(V (r, n)) which (viewed as lines in V (r, n)) lie in none of the hyperplanes in A(r, n); this is a nonsingular irreducible affine This work was supported by Australian Research Council grant DP0344185. 1

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