Decomposition of Matrix Algebras
Much of abstract algebra, including the study of groups, rings, and fields, developed from a study of concrete examples of these structures arising in (or, being used to model) physical systems. We generally need some sort of concrete realization, or "representation," of such a structure if we wish to study it and perform computations using it.
In time, the abstract structures became objects of study in their own right. However, the kinds of "representations" that exist for them have been studied too. Representation theory is an area of mathematics that concerns a study of these concreate realizations of abstract objects and their properties.
Mathematicians and physicists have been performing computations that involve these representations for decades. The associated computational problems have been studied more recently by theoretical computer scientists. Largely due to the work of Lajos Rónyai and his collaborators, the complexity of many of these computations is now reasonably well understood.
My work in this area has generally concerned the design and analysis of randomized algorithms that can be used to solve problems somewhat more efficiently than is possible deterministically.
C. W. Curtis and I. Reiner
Representation Theory of Finite Groups and Associative
Algebras
Wiley-Interscience, 1962
This is certainly not the only text on representation theory that you will find, nor is the most up-to-date. However, I consider it to be an extremely readable introduction to this area, and recommend it as a good place to start.
G. Ivanyos and L. Rónyai
Computations in Associative and Lie Algebras
Chapter 5 in Some Tapas of Computer Algebra,
A. M. Cohen, H. Cuypers, and H. Sterk (Eds.)
Springer, 1999; pp. 91–120
These are notes from a mincourse given at Eindhoven University of Technology. They introduce the fundamental algorithms for computations on associative algebras, along with a discussion of more recent work, and current implementations in computer algebra systems.
Randomized Computatations for Algebras and Representations of Groups
W. Eberly
Computations for Algebras and Group Representations
Technical Report 225/89, Department of Computer Science,
University of Toronto, 1989
Ph.D. Thesis:
[ Postscript ]
[ PDF ]
W. Eberly
Decomposition of Algebras over Finite Fields and
Number Fields
Computational Complexity 1 (1991),
pp. 183–210
Please send me email
to obtain a copy of this paper.
W. Eberly
Decomposition of Algebras over R and C
Computational Complexity 1 (1991),
pp. 211–234
Please send me email
to obtain a copy of this paper.
Randomized algorithms were presented for the decomposition of semi-simple algebras over number fields, and for the decomposition of matrix algebras over the real and complex numbers.
Virtually all of the original material in the PhD thesis appeared, in improved form, in the Computational Complexity papers. A small amount of material concerning computations of character tables, in Section 3.3 of the thesis, has not appeared elsewhere and may still be of limited interest.
More Efficient Decompositions of Associative Algebras
W. Eberly and M. Giesbrecht
Efficient Decomposition of Associative Algebras
Proceedings, 1996 International Symposium on Symbolic
and Algebraic Computation
Zurich, Switzerland, 1996; pp. 170–178
(Preliminary Report)
W. Eberly and M. Giesbrecht
Efficient Decomposition of Associative Algebras
over Finite Fields
Journal of Symbolic Computation 29 (2000),
pp. 441–458
Available online at
ScienceDirect.
W. Eberly and M. Giesbrecht
Efficient Decomposition of Separable Algebras
Journal of Symbolic Computation, in press
Available online at
ScienceDirect.
New algorithms were presented for the decomposition of matrix algebras over finite fields and for the decomposition of separable algebras over arbitrary fields.
This page was most recently changed on Saturday, September 10, 2005 by Wayne Eberly.
