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Events for December 2010 – April 2011

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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
Mar. 01, 2011 2pm (MATH 350)	Lie Theory Zajj Daugherty (St. Olaf) X The degenerate two-boundary Hecke algebra arises as a centralizer algebra, much like the group algebra of the symmetric group, in duality with Lie algebras of type gl_n and sl_n. We will use combinatorics, symmetric functions, and braid groups to study the representation theory of this algebra and reveal a connection to the type C graded Hecke algebra. Two-boundary Hecke algebras and the graded Hecke algebra of type C
Mar. 08, 2011 2pm (MATH 350)	Lie Theory Ryan Vinroot (William & Mary) X Let I(n) denote the number of elements whose square is the identity in SO(n,q), a special orthogonal group over a field with q elements. The topic of this talk is the fact that the sequence I(n) is semi-recursive in n, in that I(2m+3) depends on I(2m+1) and I(2m-2) linearly. The combinatorial proof of this fact brings to light mysterious equalities between the number of involutions in various finite classical groups. I will also give an application of this semi-recursion for I(n) to give a universal bound for the character degree sum for finite classical groups over fields of odd characteristic. This work is joint work with Feiqi Jiang, an undergraduate from the University of Michigan. Involutions in finite special orthogonal groups
Mar. 15, 2011 2pm (MATH 350)	Lie Theory Dana Ernst (Plymouth State) X Kazhdan--Lusztig polynomials are subtle objects that arise naturally in the context of Hecke algebras associated to Coxeter groups. Unfortunately, computing these polynomials efficiently quickly becomes difficult, even in finite groups of moderate size. Computing the Kazhdan--Lusztig polynomials would be simplified if one could quickly compute the leading coefficients. In this talk, I will discuss how methods from the theory of diagram algebras can be used to non-recursively compute the leading coefficients of certain Kazhdan--Lusztig polynomials. In particular, we will focus our attention on Hecke algebras of types $A$ , $B$ , and affine $C$ . Moreover, we will relay the current state of affairs of diagram algebras as tools for exploring Kazhdan--Lusztig theory. Diagram algebras and applications to Kazhdan--Lusztig theory
Mar. 29, 2011 2pm (MATH 350)	Lie Theory Nantel Bergeron (York University) X One of the best frameworks to investigate the combinatorial structure of many algebraic objects is the theory of combinatorial Hopf . Combinatorial objects can often be combined, or decomposed into simpler components. Enumeration and classification of these structures often give rise to an associated graded Hopf algebra where multiplication and comultiplication encode association and decomposition of the combinatorial objects. I will define combinatorial Hopf algebras, present related problems and results. Combinatorial Hopf algebras
Apr. 05, 2011 2pm (MATH 350)	Lie Theory Jonathan Comes (Tech. Univ. Munich) X A tensor category is a category equipped with a nicely behaved tensor product, for example take Rep(G) where G is a finite group. In this talk I will explain how to do calculations in tensor categories by manipulating certain pictures. I will also explain how various diagram algebras (such as Temperley-Lieb or Brauer algebras) arise as endomorphism algebras in some "universal" tensor categories. Diagram algebras from the viewpoint of tensor categories
Apr. 18, 2011 2pm (MATH 350)	Lie Theory Rahbar Virk (UC Davis) X The category O of (modules of) a complex semisimple Lie algebra was introduced by J. Bernstein, I. Gelfand and S. Gelfand in 1976. Since then it has found applications and motivated much progress in areas as diverse as knot theory, D-modules, the geometric Langlands program and combinatorics. I will explain how much of the structure of category O can be explained by viewing it as a `reasonably faithful module' for the Weyl group (or rather the associated Hecke algebra). I will situate myself on the algebraic side of this story, make this statement precise and present some of my own contributions to the subject. To keep things as elementary and concrete as possible I will mainly focus on sl2 and sl3. Time permitting, I will say something about the algebraic geometry underlying this story. Category O: from Lie theory to reflection groups via derived categories
Apr. 26, 2011 2pm (MATH 350)	Lie Theory Justin Keller (CU) X A supercharacter theory is an approximation to the usual character theory of a group. Even very coarse approximations still contain useful information about the group and have been exploited in situations when the full character table is unknown. We describe some of the ways that supercharacter theories arise in the context of the general linear group over a finite field. Supercharacter Theories of GL_n(F_q)

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