Let 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.
Geometric methods in representation theory of finite p-groups
Apr. 14, 2009 2pm (MATH 350)
Lie Theory
Rachel Krieger (CU)
X
A supercharacter theory of the finite group of unipotent uppertriangular matrices Un(Fq) has been developed based on the existence of an underlying nilpotent associative algebra. In this talk, we consider a family of maps between the nilpotent associative algebra and Un(Fq). We first determine a method by which the image of any element in the algebra can be computed. We then show that the particular map chosen does not affect the location of the two-sided orbit, so that the superclasses defined in this way are independent of the map between the underlying algebra and the group. Hence, the superclasses arising in this particular supercharacter theory may have another natural definition for Chevalley groups.
Modified Exponential Maps and the Finite Group of Unipotent Uppertriangular Matrices
Apr. 21, 2009 2pm (MATH 350)
Lie Theory
Hugh Denoncourt (CU)
X
Many combinatorial questions that arise in the context of Coxeter groups rely on properties of reduced expressions for a Coxeter group element. We present a discrete geometric model for arbitrary Coxeter group elements of arbitrary Coxeter groups and show that the model can be used to answer questions about reduced expressions. In particular, we can ascertain the effect of deleting a generator from a reduced expression upon the length of a Coxeter group element. Additionally, we construct correspondences between reduced expressions of an element and tournaments placed "on top" of the element's discrete geometric model.
Word combinatorics for Coxeter groups I
Apr. 28, 2009 2pm (MATH 350)
Lie Theory
Hugh Denoncourt (CU)
X
Many combinatorial questions that arise in the context of Coxeter groups rely on properties of reduced expressions for a Coxeter group element. We present a discrete geometric model for arbitrary Coxeter group elements of arbitrary Coxeter groups and show that the model can be used to answer questions about reduced expressions. In particular, we can ascertain the effect of deleting a generator from a reduced expression upon the length of a Coxeter group element. Additionally, we construct correspondences between reduced expressions of an element and tournaments placed "on top" of the element's discrete geometric model.
Geometric methods in representation theory of finite p-groups