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- Equivariant Hodge-Deligne polynomials of symmetric products of algebraic groupsPublication . Silva, JaimeLet X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products Sym(n) X when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials mu(Sn)(Xn)(t, u, v), codifying the permutation action of S-n as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products mu(Symn X) (t, u, v). These formulas are then applied to the case of linear algebraic groups.
- Hodge-deligne polynomials of character varieties of free abelian groupsPublication . Florentino; Silva, JaimeLet F be a finite group and X be a complex quasi-projective F-variety. For r is an element of N, we consider the mixed Hodge-Deligne polynomials of quotients X-r/F, where F acts diagonally, and compute them for certain classes of varieties X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X is the maximal torus of an affine reductive group G, and F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E-polynomials of (the distinguished component of) G-character varieties of free abelian groups. In the cases G = GL(n, C) and SL(n, C), we get even more concrete expressions for these polynomials, using the combinatorics of partitions.
- Mixed Hodge structures on character varieties of nilpotent groupsPublication . A. A. Florentino, Carlos; Lawton, Sean; Silva, JaimeLet Hom0(Ƭ, G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety Hom0(Ƭ, G) and on the character variety Hom0(Ƭ, G)//G. We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.