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Events for January – May 2012

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Feb. 07, 2012 2pm (MATH 350)	Lie Theory Richard Green (CU) X There are remarkable invariant cubic and quartic forms associated with the simple Lie algebras of types $E6$ and $E7$ , respectively. I will explain what this means, and how these invariant forms may be understood in terms of Coxeter groups and the combinatorics of minuscule representations. Lie algebras and invariant forms
Feb. 14, 2012 2pm (MATH 350)	Lie Theory Tyrrell McAllister (Wyoming) X A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $ℂ^{n}$ modulo a twisted action of the maximal torus in $SL(n,C)$ . An historically important example is the ring $R$ of invariants of $n$ -tuples of points in the projective plane, modulo automorphisms of the plane. We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst $O(n2)$ . In particular, we get a linear upper bound of $2n ? 8$ for the degree in which $R$ is generated. We also discuss a certain toric degeneration $R'$ of $R$ that has been used to study the invariants of $n$ -tuples on the projective line. We show that this toric degeneration $R'$ ceases to be useful for higher-dimensional projective spaces. In contrast to the linear upper bound on the degrees needed to generate $R$ , $R'$ has essential generators whose degrees are exponential in $n$ . This is joint work with Benjamin J. Howard. Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups
Feb. 21, 2012 2pm (MATH 350)	Lie Theory Connor Sempek (CU) X Given a finite set X, a cyclic group G, and a polynomial f with positive integer coefficients, the data (X, G, f) is said to exhibit the cyclic sieving phenomenon if the values of f evaluated at certain roots of unity reveal combinatorial information about actions of G on X. In the talk, this definition will be made precise, examples will be given, and more telling equivalent definitions will be considered. The Cyclic Sieving Phenomenon
Feb. 28, 2012 2pm (MATH 350)	Lie Theory Rahbar Virk (CU) X I will report on my current (and ongoing) work on understanding Hecke correspondences on the flag variety. Hecke correspondences are generalizations of incidence correspondences on P^1 to general flag varieties. Since the pioneering work of Beilinson-Bernstein and Kashiwara-Brylinski (on D-modules), Goresky-Macpherson and Beilinson-Bernstein-Deligne's work (on intersection cohomology), and Lusztig-Kazhdan and Lusztig-Vogan's work (in representation theory), it has been understood that the representation theory of reductive groups is controlled by the geometry and combinatorics of these Hecke correspondences. I will present my own contributions to this study. Time permitting I will explain how this ties into efforts by Nat Thiem and myself to understand and generalize Lusztig's theory of character sheaves. On Hecke Correspondences
Mar. 20, 2012 2pm (MATH 350)	Lie Theory Tyson Gern (CU) X The Robinson--Schensted correspondence between the symmetric group and pairs of standard young tableaux gives rise for a way to compute cells of Coxeter groups of type A. A lesser known correspondence exists for signed permutation and pairs of so-called domino tableaux, which allows us to compute cells in Coxeter groups of type D. This talk will describe the computation of these cells and will explore applications to Kazhdan--Lusztig theory. Domino Tableaux and Cells of Coxeter Groups
Apr. 10, 2012 2pm (MATH 350)	Lie Theory Erica Shannon (CU) X The partition algebra arises naturally out of the representation theory of the symmetric group and the $k$ -fold tensor of its permutation module. I'll show how to construct the partition algebra by defining a multiplication of diagrams, and then discuss some interesting subalgebras. In exploring representations of the partition algebra we'll discover several connections between the partition algebra and the symmetric group. Partition Algebras and the Symmetric Group
Apr. 17, 2012 2pm (MATH 350)	Lie Theory Scott Andrews (CU) X Determining the irreducible characters of the group of unipotent upper triangular matrices over a finite field is a provably difficult problem. Kirillov constructed class functions of this group which he conjectured to be the irreducible characters. Although he was incorrect, the Kirillov functions have important applications to the study of the character theory of this group, as well as generalizations in other types. I will construct the Kirillov functions in type A and investigate their application in other types. Kirillov Functions
May. 01, 2012 2pm (MATH 350)	Lie Theory Rob Maier (U. Arizona) X Coxeter groups occur as both automorphism groups and monodromy groups of certain Fuchsian differential equations. This includes the Picard-Fuchs equations associated to families of elliptic curves (i.e., associated to elliptic surfaces). Such equations are second-order differential equations that are extensions of the Gauss hypergeometric equation (e.g., they may be of the Heun type). The appearance of Coxeter groups as their automorphism groups will be explained, and it will also be briefly explained how Coxeter groups appear as the monodromy groups of certain generalized hypergeometric equations (those with algebraic solutions), as has been known since work of Frits Beukers. Coxeter Groups and Certain Differential Equations

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