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Events for June – October 2007

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Aug. 30, 2007 2pm (MATH 350)	Lie Theory Nat Thiem (CU) An introduction to combinatorial representation theory
Sep. 06, 2007 2pm (MATH 350)	Lie Theory Richard Green (CU) X Kazhdan--Lusztig polynomials are subtle objects that arise naturally in the context of Hecke algebras associated to Coxeter groups. I will discuss how methods from the theory of diagram algebras may be used to help compute these polynomials. Diagram algebras and Kazhdan-Lusztig polynomials
Sep. 20, 2007 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X Hecke algebras are fundamental tools that appear throughout mathematics including representation theory, number theory, geometry, and probability. In representation theory, they play an important role interpolating between groups and their combinatorics. Unipotent Hecke algebras are a family of Hecke algebras that "see" the representation theory of finite groups of Lie type but act much more like Weyl groups (Coxeter groups). Their structure varies widely, ranging from generalizations the usual Iwahori-Hecke algebra to a commutative algebra often studied in number theory. This talk will begin with a general construction of Hecke algebras and then specialize to the combinatorics of unipotent Hecke algebras. Unipotent Hecke algebras
Oct. 04, 2007 2pm (MATH 350)	Lie Theory Richard Green (CU) X This talk will introduce the notion of a $P$ -kernel, due to R.P. Stanley. I will explain how $P$ -kernels can regarded as an abstraction of Kazhdan--Lusztig theory in which the collection of all Kazhdan--Lusztig polynomials is treated as a single combinatorial object in terms of an incidence algebra. The relationship between $P$ -kernels and IC ("Intersection Cohomology") bases will also be discussed. P-kernels and IC bases
Oct. 11, 2007 2pm (MATH 350)	Lie Theory Hugh Denoncourt (CU) X Freely braided elements of simply laced Coxeter groups were introduced by Green and Losonczy as a generalization of fully commutative elements. We extend this definition to arbitrary Coxeter groups and obtain a characterization in terms of the number of commutation classes of the element. A characterization of freely braided elements in arbitrary Coxeter groups
Oct. 18, 2007 2pm (MATH 350)	Lie Theory Marty Walter (CU) X There are many examples of non-isomorphic finite groups of the same order that have the ``same representation theory (over the complex numbers).'' By this we mean that the groups in question have the same number of irreducible representations of a given dimension, and the tensor products of any two such representations decompose in like fashion. We look for properties of the ``fine structure'' of the representations which allow us to distinguish said groups. The search leads us to consider cohomology, notions of positivity and partial orderings, infinitely divisible laws in probability theory and other interesting topics. (The search is not over.) An analytic approach to the representation theory of finite groups

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