The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ℂ. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.

Key Words:

• Lie superalgebras
• Symmetric and exterior powers
• Young tableux

Mathematics Subject Classification:

• 05E10
• 17B66

ACKNOWLEDGMENTS

This article has been submitted as part of a PhD thesis. I would like to thank my PhD advisor, Professor Amitai Regev, for his many helpful comments and suggestions.

Notes

^# Communicated by M. Cohen.