Geometric methods in representation theory of finite p-groups.Mitya Boyarchenko, University of Chicago |
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April 9, 2009 AbstractLet F_q denote a finite field with q elements, and let G be a subgroup of the general linear group GL_n(F_q) that is defined as the zero set of a certain collection of polynomial equations with coefficients in F_q (the variables are the entries of an n-by-n matrix). In particular, G is a finite group. Some typical examples of such groups are: GL_n(F_q), SL_n(F_q) (the special linear group), Sp_{2n}(F_q) (the symplectic group of matrices of size 2n), UL_n(F_q) (the group of upper-triangular n-by-n matrices with 1's along the main diagonal). The same collection of equations defines an algebraic group G' over the algebraic closure of F_q. One can ask how the geometry of G' (as an algebraic variety) is related to (complex) representations of the finite group G. This question was classically studied by many mathematicians, notably George Lusztig, in the case where G' is a reductive group. My talk is devoted to the "opposite" case, where G' is a unipotent group, so that G is a finite p-group (where p=char(F_q)). I will present several nontrivial results relating the geometry of G' to complex irreducible representations of G. Among these results is a vast generalization of a 1966 conjecture of J. Thompson stating that the dimension of every irreducible representation of UL_n(F_q) is a power of q. (The conjecture is now a theorem of I.M. Isaacs.) The talk is based on joint work with Vladimir Drinfeld. |