Here is some basic information about this web page. (This web page is under construction.)
Kenneth Driessel created this web page in the August of 2008 for a seminar at Iowa State University. This seminar began in August, 2008. Kenneth Driessel, Irvin Hentzel and Jim Murdock organized this seminar. Many of the lectures are related to topics in the book "Computational Invariant Theory" by H. Derksen and G. Kemper (Springer, 2002).
This seminar meets at 2:10 on Tuesday in Carver 128.
Here is a list of the seminar talks.
December : The seminar will not meet during December.
November 25 : The seminar will not meet this week.
November 18: Kenneth Driessel, "The Reynolds Operator (continued)"
November 11: Calin Chindris, Department of Mathematics, University of Iowa, "Quiver Invariant Theory"
Abstract: A 'quiver' is a finite graph, and a 'quiver representation' assigns a finite dimensional vector space to each vertex and a linear map to each arrow. Quivers and their representations arise naturally in many areas of mathematics including the representation theory of finite dimensional algebras, invariant theory, and algebraic combinatorics.
A basic numerical invariant of a representation of a quiver Q is its 'dimension vector' which is defined to be the vector whose ith component is the dimension of the vector space at the ith vertex. By fixing a dimension vector \alpha, one can construct the representation space R(Q,\alpha) acted upon by a group G, which is just a product of general linear groups, in such a way that the G-orbits in R(Q,\alpha) are in one-to-one correspondence with the isomorphism classes of \alpha-dimensional representations. The goal of quiver invariant theory is to understand the various rings of invariants and the quotient varieties arising from this action.
In this talk, aftr presenting some of the main concepts and constructions from quiver invariant theory, I will explain how to realize the Littlewood-Richardson coefficients as dimensions of weight spaces of semi-invariants of star-shaped quivers. I will also discuss Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients. This work is based on joint work with H. Derksen and J. Weyman.
November 4: Kenneth Driessel, "The Reynolds Operator"
Abstract: An algebraic groups is "semi-simple" if its associated Lie algebra is semi-simple. We shall limit our attention to semi-simple groups. Let V be a finite dimensional vector space and let G be an semi-simple, algebraic group acting on V. Then we have an induced action of G on the space K[V] of polynomial functions on V. We write g.p for g in G and p in K[V]. An element p in K[V] is "invariant" if, for all g in G, g.p = p. Let K[V]^G denote the algebra of invariants. A "Reynolds operator" R is a linear projection map of K[V] onto K[V]^G which satisfies the following condition: for all g in G and p in K[V], R(g.p) = R(p). We shall see that we can use the Casimir operator as the basis of an algorithm for the Reynolds operator.
October 28: Kenneth Driessel, "Semi-simple and nilpotent matrices (continued) "
October 21: Kenneth Driessel, "Semi-simple and nilpotent matrices "
Abstract: Let V be a finite dimensional vector space over a field K and let S be a linear operator on V. Then S is 'semi-simple' if every S-invariant subspace of V has a complementary S-invariant subspace. We shall discuss the following result: If K has characteristic zero then, for every linear operator T on V, there is a semi-simple operator S on V and a nilpotent operator N on V such that T = S+N and SN = NS; furthermore, the operators S and N with these properties are unique and are polynomials in T. We shall also apply this result to solvable Lie algebras. (Only a basic knowledge of linear algebra will be needed to understand this discussion.)
October 14: Kenneth Driessel, "The Casimir Operator and Representations (continued)"
October 7: Kenneth Driessel, "The Casimir Operator and Representations"
Abstract: We shall review the definition of the Casimir operator of a Lie algebra. Then we shall show how it can be used to show that representations of the Lie algebra can be decomposed into irreducible representations.
September 30: Kenneth Driessel,"The Casimir Operator (continued)"
September 23: Kenneth Driessel, "The Casimir Operator (continued)"
September 16: Kenneth Driessel, "The Casimir Operator"
Abstract: Let L be a Lie algebra. Then the Casimir operator is a distinguished element of the center of the universal enveloping algebra of L. In any irreducible representation of L, by Schur's lemma, any member of the center of the universal enveloping algebra is proportional to the identity. This constant of proportionality can be used to classify the representations of L. (Physical mass and spin are examples of these constants, as are many other quantum numbers found in quantum mechanics.) We shall define the Casimir operator and prove its elementary properties. We shall explicitly compute the Casimir operator for the Lie algebra of SU(2). The Casimir operator is important to us because it plays a central role in Derksen and Kemper's algorithm for computing invariants.
September 9: Kenneth Driessel, "The Convolution Algebra of SU(2) (continued)"
September 2: Kenneth Driessel, "The Convolution Algebra of SU(2)"
Abstract: We shall review the definition of the convolution algebra of an matrix (algebraic) group and we shall discuss several examples. In particular, we shall discuss the convolution algebra of the special unitary group SU(2).
August 26: Ken Driessel, "The Universal Enveloping Algebra of a Lie Algebra"
Abstract: We shall discuss part of Chapter V 'Universal Enveloping Algebras' in the book 'Lie Algebras' by N. Jacobson.
Last update: November 16, 2008
Kenneth R. Driessel, driessel@iastate.edu