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Invariant Theory and Superalgebras
Frank D. Grosshans, Gian-Carlo Rota, and Joel A. Stein
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1987; 80 pp; softcover
Number: 69
Reprint/Revision History:
reprinted 2000
ISBN-10: 0-8218-0719-6
ISBN-13: 978-0-8218-0719-4
List Price: US$26
Member Price: US$20.80
All Individuals: US$20.80
Order Code: CBMS/69
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This book brings the reader to the frontiers of research in some topics in superalgebras and symbolic method in invariant theory. Superalgebras are algebras containing positively-signed and negatively-signed variables. One of the book's major results is an extension of the standard basis theorem to superalgebras. This extension requires a rethinking of some basic concepts of linear algebra, such as matrices and coordinate systems, and may lead to an extension of the entire apparatus of linear algebra to "signed" modules. The authors also present the symbolic method for the invariant theory of symmetric and of skew-symmetric tensors. In both cases, the invariants are obtained from the symbolic representation by applying what the authors call the umbral operator. This operator can be used to systematically develop anticommutative analogs of concepts of algebraic geometry, and such results may ultimately turn out to be the main byproduct of this investigation.

While it will be of special interest to mathematicians and physicists doing research in superalgebras, invariant theory, straightening algorithms, Young bitableaux, and Grassmann's calculus of extension, the book starts from basic principles and should therefore be accessible to those who have completed the standard graduate level courses in algebra and/or combinatorics.

Readership

Table of Contents

  • The superalgebra super \([A]\)
  • Laplace pairings
  • The standard basis theorem
  • Invariant theory
  • Examples
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