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Events for October 2010 – February 2011

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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
Nov. 02, 2010 2pm (MATH 350)	Lie Theory Matt Macauley (Clemson) X In this talk, I will describe how a scientist studying discrete dynamical systems in a bioinformatics institute stumbled into the field of Coxeter groups. Central to these diverse fields are involutions, and what has a group-theoretic interpretation on one hand may lead to insight to the dynamics on the other, and vice-versa. I will highlight some of the central themes and common structures, as well as discuss some novel approaches to some open and open-ended problems. From dynamical systems to Coxeter groups
Nov. 09, 2010 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Gaussian poset is a type of finite poset satisfying a certain technical algebraic condition that defies concise summary. I will explain how minuscule representations of Lie algebras give rise to interesting examples of Gaussian posets. It is conjectured that any Gaussian poset is a direct sum of Gaussian posets coming from minuscule representations, and this has been an open question since the 80s. Although a direct combinatorial characterization of Gaussian posets is not known, the set of irreducible Gaussian posets is the solution to another problem that is purely combinatorial in nature. These (expository) talks will also discuss applications of Gaussian posets to the combinatorics of plane partitions. Gaussian posets I
Nov. 16, 2010 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Gaussian poset is a type of finite poset satisfying a certain technical algebraic condition that defies concise summary. I will explain how minuscule representations of Lie algebras give rise to interesting examples of Gaussian posets. It is conjectured that any Gaussian poset is a direct sum of Gaussian posets coming from minuscule representations, and this has been an open question since the 80s. Although a direct combinatorial characterization of Gaussian posets is not known, the set of irreducible Gaussian posets is the solution to another problem that is purely combinatorial in nature. These (expository) talks will also discuss applications of Gaussian posets to the combinatorics of plane partitions. Gaussian posets II
Nov. 30, 2010 2pm (MATH 350)	Lie Theory Z. Strider McGregor-Dorsey (CU) X The notion of a full heap was developed by Richard Green as an affine analogue to the minuscule posets of simple Lie algebras. As such, one might expect that various properties of minuscule posets are shared by full heaps. (For example, convex subposets of both minuscule posets and full heaps correspond to their respective minuscule elements.) In this talk, we look to a few properties of minuscule posets and discuss their application to full heaps, if any. In particular, we will build new full heaps from others using the ideas from the Fundamental Theorem for Finite Distributive Lattices , as well as explore a somewhat unsatisfying relation to Gaussian posets. And more!, time permitting. Making Analogies with Full Heaps I
Dec. 07, 2010 2pm (MATH 350)	Lie Theory Z. Strider McGregor-Dorsey (CU) X The notion of a full heap was developed by Richard Green as an affine analogue to the minuscule posets of simple Lie algebras. As such, one might expect that various properties of minuscule posets are shared by full heaps. (For example, convex subposets of both minuscule posets and full heaps correspond to their respective minuscule elements.) In this talk, we look to a few properties of minuscule posets and discuss their application to full heaps, if any. In particular, we will build new full heaps from others using the ideas from the Fundamental Theorem for Finite Distributive Lattices , as well as explore a somewhat unsatisfying relation to Gaussian posets. And more!, time permitting. Making Analogies with Full Heaps II
Jan. 18, 2011 2pm (MATH 350)	Lie Theory Richard Green (CU) X Let $W$ be a finite Weyl group, and let $α_{p}$ be a simple root of $W$ corresponding to a minuscule weight of the associated Lie algebra. The set of positive roots, $R_{p}$ , of $W$ in which $α_{p}$ appears with nonzero coefficient is equipped with a natural partial order. Remarkably, this partial order makes $R_{p}$ into a distributive lattice. Also remarkably, the lattice $J(Rp)$ of ideals of $R_{p}$ is isomorphic to the lattice of weights of the minuscule representation. There are signs that this result is folklore, but I have never seen it stated. I will discuss the strangeness of this result, speculate on possible extensions of it, and lament the lack (to my knowledge) of a conceptual proof. Roots, weights, and distributive lattices
Jan. 25, 2011 2pm (MATH 350)	Lie Theory Z. Strider McGregor-Dorsey (CU) X The notion of a full heap was developed by Richard Green as an affine analogue to the minuscule posets of simple Lie algebras. As such, one might expect that various properties of minuscule posets are shared by full heaps. We continue the previous two talks by showing some strengths and limitations of these analogies. In this quest, we will use ideas from the Fundamental Theorem for Finite Distributive Lattices and Gaussian posets. We'll also pack a spaceship in a less hurried way than we did last semester, hopefully to the delight of the quartermaster. Making Analogies with Full Heaps III
Feb. 01, 2011 2pm (MATH 350)	Lie Theory Matthew Moore (CU) X Priestley (and Stone) dualities are stated in order to motivate the definition of a natural duality, which gives a natural and categorical way of associating algebraic structures to structured topological spaces. Useful duality theorems will be stated, and the sample application of reconstructing an algebra from its endomorphism monoid will be discussed. Natural Dualities, Distributive Lattices
Feb. 15, 2011 2pm (MATH 350)	Lie Theory Tom Halverson (Macalester) X Tensor powers of the defining matrix representation of the special linear group SL(2,C) lead naturally to the study of the combinatorics of Catalan numbers and the Temperley-Lieb algebra. Tensor space decomposes into irreducible modules whose multiplicities are Catalan numbers, and the Temperley-Lieb algebra is the centralizer algebra of transformations which commute with SL(2,C). We will tell this story and see the Temperley-Lieb algebra appear as a diagram algebra which models these commuting transformations. Then we will generalize this story in two ways: 1. We replace the defining matrix of SL(2,C) with a natural 3-dimensional representation. Then the tensor powers lead to the study of Motzkin numbers, which are less-famous cousins of the Catalan numbers, and the "Motzkin Algebra," a diagram centralizer algebra that generalizes Temperley-Lieb. 2. We look at finite subgroups G of SL(2,C), and we show that the centralizer algebra, in this case, reflects the beautiful structure of the McKay correspondence. In particular, it has dimension and a basis labeled by walks on the Dynkin diagram of G. SL(2) centralizer algebras
Feb. 22, 2011 2pm (MATH 350)	Lie Theory Matt Douglass (U. North Texas) X Suppose W is a finite Coxeter group. In this talk I'll explain a conjectural relation between (1) the representation of W on the cohomology of the complement of the arrangement of W and (2) the regular representation of W via the descent algebra of W. It is expected that both the group algebra of W and the cohomology of the complement can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of W. The conjecture is known to be true for symmetric groups, dihedral groups, and is being checked for all low rank Coxeter groups using GAP and CHEVIE. This is joint work with Gerhard Riehrle and Goetz Pfeiffer. The cohomology of hyperplane complements and Solomon's descent algebra

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