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Events for June – October 2010

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Aug. 31, 2010 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X At a recent AIM workshop, the participants found a remarkable connection between two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are Hopf isomorphic. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras. This talk will be in two parts: (1) I will give an introduction to combinatorial Hopf algebras with our favorite examples (2) I will describe supercharacters and how they connect up to one of our favorite Hopf algebras. Supercharacters and combinatorial Hopf algebras I
Sep. 07, 2010 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X At a recent AIM workshop, the participants found a remarkable connection between two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are Hopf isomorphic. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras. This talk will be in two parts: (1) I will give an introduction to combinatorial Hopf algebras with our favorite examples (2) I will describe supercharacters and how they connect up to one of our favorite Hopf algebras. Supercharacters and combinatorial Hopf algebras II
Sep. 21, 2010 2pm (MATH 350)	Lie Theory Jacob Harper (CU) X This talk will be split into two parts. In the first part I will describe a hypersimplex and go over concepts of Coxeter groups and topology in order to get a better understanding of this polytope. In the second talk I will describe a matching on the face lattice of a hypersimplex and use discrete Morse theory to classify all subcomplexes whose reduced homology groups are concentrated in a single degree. These homology groups support a natural action of the symmetric group and I will finish my talk by describing the characters that this action produces. Homology representations arising from a hypersimplex I
Sep. 28, 2010 2pm (MATH 350)	Lie Theory Jacob Harper (CU) X This talk will be split into two parts. In the first part I will describe a hypersimplex and go over concepts of Coxeter groups and topology in order to get a better understanding of this polytope. In the second talk I will describe a matching on the face lattice of a hypersimplex and use discrete Morse theory to classify all subcomplexes whose reduced homology groups are concentrated in a single degree. These homology groups support a natural action of the symmetric group and I will finish my talk by describing the characters that this action produces. Homology representations arising from a hypersimplex II
Oct. 12, 2010 2pm (MATH 350)	Lie Theory Daniel Swenson (Black Hills State) X The Steinberg Complex is a generalization, to the case of an arbitrary finite group G, of the Steinberg Representation of a finite group of Lie type. It is defined in terms of the conjugation action of G on its poset of p-subgroups (analogous to the building in the classical case). I will describe the Steinberg Complex and what is known about it in the general case, and I will indicate some directions for further research. The Steinberg Complex of an Arbitrary Finite Group
Oct. 19, 2010 2pm (MATH 350)	Lie Theory Tom Denton (UC Davis) X The variety of J-trivial monoids does not enjoy the semi-simplicity of groups, but has other properties which make their representation theory approachable and inherently combinatorial. I will describe these combinatorics, and also present important examples of some J-trivial monoids, including the zero-Hecke monoid, the monoid of regressive order-preserving functions on a poset, and the upper-unitriangular matrices over the boolean semiring. Combinatorial Representation Theory of J-Trivial Monoids

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