The umbral symbolic method for supersymmetric tensors
The problem of finding a unified and effective symbolic method for both symmetric and skew-symmetric tensors remained unsolved until the publication of \textit{F. D. Grosshans}, \textit{G.-C. Rota} and \textit{J. A. Stein's} book [Invariant theory and superalgebras (1987; Zbl 0648.15020)]. The key idea of this work is, a posteriori, quite simple. Specifically, the algebra generated by the symbolic variables must be a supersymmetric algebra; that is, the symbolic letters have to be endowed with a \(\mathbb{Z}_ 2\)-gradation (or signature) which depends on the fact that the tensor they represent is either a symmetric or a skew-symmetric tensor. The commutation rules in the (supersymmetric) symbolic algebra depend on the signature of the variables. The algebra of polynomial functions on a finite set of ``generic'' symmetric and skew-symmetric tensors on a vector space \(V\) is viewed as the epimorphic image of the symbolic algebra under a \(\mathbf{GL}(V)\)-equivalent map \({\mathbf U}\), which is called the symbolic operator; thus the problem of finding the joint invariants of these tensors is reduced to the problem of finding the invariants of the supersymmetric symbolic algebra. By using the Straightening Formula for the supersymmetric letterplace algebra, the symbolic invariants are shown to be polynomials in ``supersymmetric brackets''; therefore, the First Fundamental Theorem remains valid for the joint invariants of a set of both symmetric and skew-symmetric tensors after the traditional brackets (determinants) are just replaced by their supersymmetric counterparts. Thus, the ideas and the philosophy of ``supersymmetric algebra'' turn out to be effective when applied to ordinary invariant theory, and the question arises of whether the symbolic method can be further extended to the superalgebraic setting of Lie superalgebra actions on varieties of supersymmetric tensors. In the paper under review the authors carry out this program, by deriving the portion of the theory that applies to the actions of general linear Lie superalgebras.
scientific article; zbMATH DE number 221234
scientific article