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Events for October 2009 – February 2010

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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
Nov. 02, 2009 2pm (MATH 350)	Lie Theory Aaron Lauve (Texas A&M) X In [Bialgebras of type one, 1978], Warren Nichols introduced a family of algebras that initially received little attention. Perhaps MathSciNet gives a clue as to why, calling the paper, "a fairly technical investigation into the structure of certain sorts of Hopf algebras." No surprise that it took more than a decade, and independent discovery by others, for Nichols' algebras to find purchase in the Hopf algebra community. Today, Nichols algebras play a critical role in the classification of finite dimensional pointed Hopf algebras. They also have a perfectly natural (and non-technical!) definition. I will start here, then give an overview of the ongoing classification program. The ubiquitous Cartan matrices make an appearance. Nichols algebras and pointed Hopf algebras
Nov. 12, 2009 2pm (MATH 350)	Lie Theory Tyrrell McAllister (Wyoming) X Polyhedral geometry has long provided insight into the representations of Lie algebras. An early example of the polyhedral approach was the introduction of the Gelfand--Tsetlin polytopes in the 1950's. More recently, Berenstein and Zelevinsky in 2001 encoded tensor product multiplicities for semi-simple Lie algebras as the number of integer lattice points in certain families of polyhedra. These results have theoretical implications as well as concrete computational applications. We discuss a particular combinatorial structure on Gelfand--Tsetlin polytopes, which we call their tiling poset. We use this combinatorial structure to compute the degrees of so-called "stretched Kostka coefficients", which express the dimension of a weight space as the parametrizing weights are scaled by an integer parameter. We also discuss applications of polyhedral algorithms to the computational complexity of tensor-product multiplicities. We mention some conjectures motivated by the polyhedral interpretation of these coefficients, including a conjecture that, if true, generalizes the Saturation Theorem of A. Knutson and T. Tao. Applications of polytopes to the representation theory of Lie algebras
Nov. 19, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Chevalley basis for a Lie algebra over the complex numbers is a basis for the algebra all of whose structure constants are integers. One of the standard constructions of Chevalley bases for simple Lie algebras over the complex numbers involves a certain function $ε$ taking values in the set {+1, -1}; this function is sometimes called Kac's asymmetry function. Although the definition of this function may appear mysterious at first, I will argue that it has some natural combinatorial interpretations. Along the way, I will present combinatorial constructions of some interesting nonassociative algebras, including the loop algebras associated to simple Lie algebras over the complex numbers. Chevalley bases for Lie algebras and the combinatorics of Kac's asymmetry function

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