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Symbolic method in invariant theory.

G C Rota1, J A Stein

  • 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1986
PubMed
Summary
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A new symbolic method represents joint invariants for symmetric and skew-symmetric tensors. This approach is valid for skew-symmetric tensors across all field characteristics.

Area of Science:

  • Algebraic Geometry
  • Representation Theory
  • Symbolic Computation

Background:

  • Joint invariants are crucial in understanding symmetries of tensors.
  • Existing methods for representing these invariants can be complex and computationally intensive.
  • The need for efficient algorithms applicable to various algebraic structures is recognized.

Purpose of the Study:

  • To develop a novel symbolic method for representing joint invariants of symmetric and skew-symmetric tensors.
  • To extend the applicability of tensor invariant representation to infinite fields of arbitrary characteristic.
  • To provide a more accessible and potentially efficient computational tool for algebraic manipulation.

Main Methods:

  • An extension of the classical straightening algorithm is employed.

Related Experiment Videos

  • Symbolic computation techniques are utilized for tensor representation.
  • The method is specifically adapted for both symmetric and skew-symmetric tensor structures.

    • Main Results:

    • A symbolic representation for joint invariants of symmetric and skew-symmetric tensors has been successfully developed.
    • The method demonstrates robustness for skew-symmetric tensors over infinite fields, regardless of characteristic.
    • The proposed algorithm offers a new perspective on tensor invariant theory.

    Conclusions:

    • The developed symbolic method provides an effective means for representing joint invariants.
    • The method's validity over arbitrary characteristic fields broadens its theoretical and practical implications.
    • This work contributes to the advancement of symbolic computation in tensor algebra and related fields.