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Events for June – October 2009

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Sep. 03, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Coxeter element is an element of a Coxeter group formed by multiplying the Coxeter generators together in some order. A fully commutative element of a Coxeter group is an element all of whose reduced expressions are equivalent by short braid relations. These talks discuss recent and ongoing joint work with Dana Ernst (Plymouth State), Matt Macauley (Clemson) and others, which introduces and studies a class of elements generalizing Coxeter elements and properly contained in the fully commutative elements. Cyclically fully commutative elements of Coxeter groups I
Sep. 10, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Coxeter element is an element of a Coxeter group formed by multiplying the Coxeter generators together in some order. A fully commutative element of a Coxeter group is an element all of whose reduced expressions are equivalent by short braid relations. These talks discuss recent and ongoing joint work with Dana Ernst (Plymouth State), Matt Macauley (Clemson) and others, which introduces and studies a class of elements generalizing Coxeter elements and properly contained in the fully commutative elements. Cyclically fully commutative elements of Coxeter groups II
Sep. 24, 2009 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X In the first talk, we will review the concept of a supercharacter theory with an eye towards applying these ideas to algebraic structures that are not finite groups. In particular, there is much to be explored in the realm of finite dimensional algebras and infinite groups. Supercharacters and the combinatorics of set-partitions I
Oct. 01, 2009 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X In the second talk, we will explore the combinatorics of set-partitions via a supercharacter theory of the group of finite unipotent upper-triangular matrices. This classical example was the original motivation for defining supercharacters, and the resulting combinatorics is analogous to the integer partitions in the representation theory of the symmetric group. By studying the decomposition of restrictions and tensor products of supercharacters into supercharacters, this talk finds a surprising analogue to the combinatorics of the Littlewood--Richardson coefficients of the symmetric group. Joint with S. Lewis. Supercharacters and the combinatorics of set-partitions II
Oct. 15, 2009 2pm (MATH 350)	Lie Theory Shawn Baland (Aberdeen) X One of the most powerful tools used in the representation theory of finite groups is the idea of a group character. When the underlying field k is the complex numbers, we are given an ordinary character table whose rows are indexed by the irreducible characters of G, whose columns are indexed by the conjugacy classes of G, and whose entries are the respective irreducible character values. But when the characteristic of k is a prime dividing \|G\|, we are forced to take a different approach to character theory in order to obtain results analogous to those in the ordinary case. In this lecture, I will introduce the concepts of representation rings, Grothendieck groups, and Brauer characters, which will allow us to introduce the Brauer character table. Finally, I will use these structures to prove a theorem of Brauer that relates the number of irreducible kG-modules to the number of conjugacy classes in G whose elements have order coprime to the field characteristic. A survey of modular representation theory I: Brauer characters
Oct. 22, 2009 2pm (MATH 350)	Lie Theory Shawn Baland (Aberdeen) X The main idea behind group representation theory is to use well-understood results from linear algebra to study the structure of finite groups. Similarly, modular representation theory seeks to use well-understood results from ordinary representation theory to study representations where the field characteristic divides the group order. In this lecture, we will examine p-modular systems which, in a manner of speaking, allow us to move back and forth between ordinary representations and modular representations. We will discuss decomposition matrices, Cartan matrices and block idempotents, and then use these structures to calculate the modular representations of the dihedral group of order 12. Time permitting, we will also take a look at block theory and defect groups. A survey of modular representation theory II: p-modular systems and blocks

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