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Events for September 2007 – January 2008

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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
Nov. 01, 2007 2pm (MATH 350)	Lie Theory Darren Funk-Neubauer (CSU-Pueblo) X In this talk I will define a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite dimensional vector space that act bidiagonally on each others eigenspaces. I will discuss how bidiagonal pairs are connected to the quantum group $Uq (sl2)$ . In particular, I will give a presentation of $Uq (sl2)$ by generators and relations which differs from the usual presentation. Using this new presentation I will discuss how to classify bidiagonal pairs in terms of finite dimensional represenatations of $Uq (sl2)$ . Bidiagonal pairs and the quantum group $Uq (sl2)$
Nov. 15, 2007 2pm (MATH 350)	Lie Theory Dana Ernst (CU) X Star operations, introduced by D. Kazhdan and G. Lusztig in 1979, are a type of raising and lowering operation performed on the elements of a Coxeter group $W$ . We say that $W$ is star reducible if every fully commutative element can be reduced to a product of commuting generators via lowering star operations. This concept generalizes the notion of cancellability which arises in the work of C.K. Fan. The star reducible Coxeter systems were classified by R.M. Green. In this talk, we will define a weaker generalization of Fan's cancellability, called weak star reducible, and we classify the weak star irreducible elements in types $B$ and affine $C$ . I will define all of the necessary terms in the talk. Weak star reduciblity in Coxeter groups
Nov. 29, 2007 2pm (MATH 350)	Lie Theory Shawn Baland (CU) X Many of the structures encountered in algebraic Lie theory have their origin in the study of finite-dimensional complex semisimple Lie algebras. In this talk, we will examine the existence and conjugacy of the Cartan subalgebras of a finite-dimensional complex Lie algebra. We will then study the module structure induced on a Lie algebra $g$ via the adjoint action of one of its Cartan subalgebras $h$ . This will allow us to decompose $g$ in to a direct sum of eigenspaces with with respect to $h$ . In the semisimple case, the linear functionals associated with these eigenspaces tell us a lot about a Lie algebra's structure. This information will be used in my second lecture to study the geometry and combinatorics of root systems. Semisimple Lie Algebras
Dec. 06, 2007 2pm (MATH 350)	Lie Theory Shawn Baland (CU) X The roots of a finite-dimensional complex semisimple Lie algebra span a Euclidean space with respect to the Lie algebra's Killing form. By assigning a total ordering to this space, we can partition the roots into what are called positive and negative root systems. In this talk, we will examine a subset of the positive system that forms a basis of the Euclidean space. The relative lengths and angles between the roots in this subset will be encoded into a matrix called the Cartan matrix. We will then relate the Cartan matrix to a graph called the Dynkin diagram, and it will be shown that there are only a limited number of possible connected Dynkin diagrams. Finally, we will discus the correspondence between these diagrams and the finite-dimensional complex simple Lie algebras. The combinatorics of root systems
Jan. 22, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X The universal enveloping algebra of a Lie algebra is an associative algebra with the same representation theory. This sequence of talks will be a (possibly intermittent) mini-course in Lie algebras and enveloping algebras, using only MATH 6130/6140 as a prerequisite. It is recommended that students take notes. Topics to be covered include Lie algebras, enveloping algebras, Hopf algebras, the Serre presentation, the triangular decomposition, Verma modules, simple modules and quantized enveloping algebras. The talks will focus on Lie theory and representation theory, not universal algebra as the title might imply. Introduction to universal enveloping algebras I
Jan. 29, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X The universal enveloping algebra of a Lie algebra is an associative algebra with the same representation theory. This sequence of talks will be a (possibly intermittent) mini-course in Lie algebras and enveloping algebras, using only MATH 6130/6140 as a prerequisite. It is recommended that students take notes. Topics to be covered include Lie algebras, enveloping algebras, Hopf algebras, the Serre presentation, the triangular decomposition, Verma modules, simple modules and quantized enveloping algebras. The talks will focus on Lie theory and representation theory, not universal algebra as the title might imply. Introduction to universal enveloping algebras II

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