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Register a new userPermutation representations on Schubert varieties
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Julianna S. Tymoczko
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This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.
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We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fultons essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix.
In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the representation varieties of Riemann surfaces with semi-simple conjugacy classes at the punctures. We proved several results which support this conjecture. Here we announce new results which are consequences of those of arXiv:0810.2076.
In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincare series). In case (B) we give a conjectural formula for the full mixed Poincare series.
We give a short proof based on Lusztigs generalized Springer correspondence of some results of [BrCh,BaCr,P].
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $mathcal C(mathfrak p) = {(X,Y) in mathfrak p times mathfrak p mid [X,Y] = 0}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $mathcal C(mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $mathfrak p$. As a consequence, for the case $P = B$ a Borel subgroup of $G$, we give a classification of when $mathcal C(mathfrak b)$ is irreducible; this builds on a partial classification given by Keeton. Further, in cases where $mathcal C(mathfrak p)$ is irreducible, we consider whether $mathcal C(mathfrak p)$ is a normal variety. In particular, this leads to a classification of when $mathcal C(mathfrak b)$ is normal.
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