Here is some basic information about this web page (This web page is under construction.)
I created this web page in the February of 2008 for a seminar at Iowa State University. This seminar began in January, 2008. Jim Murdock and I organized this seminar. (Murdock also set up a web page for this seminar. Here is a link to that web page. ) Most 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 Thursdays in Carver 298.
Here is a list of the seminar talks.
May 1: Ken Driessel, "Reynold's Operators" There are some typed notes for this meeting.
April 24: Ken Driessel, "Reynold's Operators, Preliminaries". There are some typed notes for this meeting.
Abstract: Reynold's operators are very important tools in invariant theory. We shall discuss their computation. In particular, we shall discuss the theory of Reynold's operators which appears in the book by Derksen and Kemper.
April 17: Ken Driessel, "Invariant Theory for Compact Matrix Groups"
Abstract: We shall discuss algorithms for computing the polynomial invariants of matrix groups. In particular, we shall discuss some of the algorithms which appear in Chapter 4 "Invariant theory of reductive groups" in the book by Derksen and Kemper. For the most part, we shall limit our attention to compact matrix groups. (Note, for example, that the unitary groups are compact.) Such groups have applications in quantum computing.
April 10: Irv Hentzel, "The Integral Representations of the Symmetric Groups"
Abstract: The group ring on the symmetric group is isomorphic to a direct sum of complete matrix rings over the rational numbers. The images of the permutations are integral matrices and the algorithm for computing the matrices is easily implemented on a computer.
These representations are used for substitutional analysis. Because the matrices are over the rationals, the calculations are exact. The elimination of any round-off error is necessary because we use them for the proof of identities. These proofs have to decide either true or not true.
April 3: L. Cattaneo, "Local invariants in quantum computing" (continued)
March 27: L. Cattaneo, "Local invariants in quantum computing"
Abstract: A description of invariants under local unitary transformations using Lie algebraic tools.
March 13: K. Driessel, "Hilbert's Theorem on the Finiteness of the Full Invariant System" (continued).
March 6: K. Driessel, "Hilbert's Theorem on the Finiteness of the Full Invariant System" (continued).
February 28: K. Driessel, "Hilbert's Theorem on the Finiteness of the Full Invariant System" (continued).
February 21: K. Driessel, "Hilbert's Theorem on the Finiteness of the Full Invariant System".
We shall discuss some of the history of invariant theory. In particular, we shall discuss Hilbert's finiteness theorem. Here is a statement of the theorem in the case of binary forms. Let K[x]:=K[x_1,x_2] denote the space of binary forms (that is, polynomial functions in two variables) with coefficients from a field K with characteristic zero. Let Gl(2) act on this space by substitution: (g.p) (y) := p(x) where y:=g.x. Then a function f:K[x]->K is an "invariant of a form p" if f(g.p)=f(p) for all g. Note that the invariants of a form p form a ring.
Theorem (Finiteness of the Full Invariant System): Every binary form possesses a finite full invariant system such that each invariant of the form is a polynomial function of the invariants in the full invariant system.
In the summer of 1897 Hilbert gave an introductory course in invariant theory at the University of Goettingen. The book "Theory of Algebraic Invariants" (Cambridge, 1993) is an English translation of the handwritten notes taken by Hilbert's student Sophus Marxsen. The statement of the theorem appears in that book.
We shall discuss the proof of this theorem. In particular, we shall discuss the structure of this proof and indicate how we can generalize it.
Here are a few more references:
Olver, P. (1999) Classical Invariant Theory, London Math Soc.Hilbert, D. (1978) Hilbert's Invariant Theory Papers - Translated by M.Ackermann, Math Sci Press.
February 14: K. Driessel, "Equivalent Generating Sets for Ideals in a Polynomial Ring".
There is an analogy between reduction of a matrix to row echelon form and the computation of a Groebner basis by the Buchberger algorithm. We discuss this analogy. Let R:=k[x_1,...,x_n] be an algebra of polynomials with coefficients in a field k. Let f:=(f_1,...,f_s) and g:=(g_1,...,g_t) be two lists of polynomials in R. We say that f and g are "ideal equivalent" if they generate the same ideal. We say that they are "Gaussian equivalent" if there is a sequence of elementary row operations which transforms f^T into g^T. (Here we view f^T as a column vector in R^s and g^T as a column vector in R^t.) We prove that f is ideal equivalent to g iff f is Gaussian equivalent to g. Here is a link to a paper which provides details.
February 7: J. Murdock, "Elimination Ideals (continued)".
January 31: J. Murdock, "Elimination Ideals".
We consider elimination ideals. In particular, we see how we can use them to determine algebraic dependence or independence of a finite set of polynomials. We also consider the ideal of relations determined by such a finite set.
January 24: J. Murdock, "A survey of Groebner Bases".
We define Groebner bases. We describe Buchberger's algorithm for computing such bases. We prove the Hilbert Basis Theorem (for ideals), i.e., polynomial rings are Noetherian. Here is a reference for this material: "An Introduction to Groebner Bases", Adams and Loustaunau (AMS, 1994).
January 17: J. Murdock, "What is invariant theory and why do Groebner bases matter?"
We consider two examples in invariant theory. We also discuss how Grobner bases play a role in the solutions.
Last update: May 4, 2008
Kenneth R. Driessel, driessel@iastate.edu