Antonio Teolis

The umbral symbolic method for supersymmetric tensors

0. Introduction. 1. The Basic Plethystic Algebras. 2. The Symbolic Method for S”(Sk( V)) and A”(Sk( V)). 3. Two Straightening Formulas for S”(S’( V)). 4. Two Gordan-Capelli Series for S”(S*( V)). 5. Applications: E. Pascal Theorems for Orthogonal and Symplectic Invariants. 6. Gordan-Capelli Series and Straightening Formulas for A”(S2( V)). I. Contragradient Actions and Umbra1 Calculus. 8. The First Fundamental Theorem. 9. The Second Fundamental Theorem. 10. Symbolic-Umbra1 Operators and Weitzenbdck’s Method of “Complex Symbols.” Appendix. Left and Right Superderivations on Supersymmetric Algebras.

The Method of Virtual Variables and Representations of Lie Superalgebras

We provide a brief account of Capelli’s method of virtual variables and of its relations with representations of general linear Lie superalgebras. More specifically, we study letterplace superalgebras regarded as bimodules under the action of superpolarization operators and exhibit complete decomposition theorems for these bimodules as well as for the operator algebras acting on them.

Capelli’s Method of Variabili Ausiliarie, Superalgebras and Geometric Calculus

Capelli’s technique of variabili ausiliarie [8] — or “virtual variables”, as we prefer to call them — has been proved to be a useful tool in order to unify concepts and simplify computations in a variety of problems of invariant theory and its applications. In our opinion, Capelli’s technique finds its natural setting and acquires a much greater effectiveness in the context of supersymmetric algebras and Lie superalgebra actions; specifically, the technique of virtual variables acquires a special suppleness when the virtual variables are allowed to have a different signature than the signature of the variables one starts with.

Grassmann geometric calculus, invariant theory and superalgebras

The idea of exploring and developing the deep connections between the theory of Cayley-Grassmann algebras and the invariant theory of skew-symmetric tensors was a recurrent theme of Rota’s mathematical work.

Remark on the Branching theorem and supersymmetric algebras

Supersymmetric algebras have already proved useful in giving trans- parent proofs of a number of basic results of representation theory [3, 7-9, 1 l-l 51. Specifically, the technique of introducing virtual variables, which may have a different signature than the signature of variables to be delt with, often cuts down the amount of computation. Furthermore, the extension of results of representation theory to the superalgebraic setting sheds new light, and permits us to establish natural correspondences that were formally missing. In this note we carry out this program by deriving a superalgebraic version of the Branching Rules for the representations of the general linear group. While the statement of supersymmetric branching rules is in all respects similar to the ordinary one (and differing from it in our allowing variables of two signature), the proof yields a useful dividend, namely, a simple combinatorial construction of a canonical basis for the decomposi- tion of a restriction of a representation. As an application we give a supersymmetric generalization of Pieri’s formula, as well as a proof of this formula which is perhaps as short as it can be whittled down to. This application has been inspired by some recent work of Bofli [S, 61. We have benefited from the pioneering work of Berele and Regev [3], who were first to state such a supersymmetric extension of branching rules, as well as from the insights of Balentekin and Bars [2]. 255

Grassmann Progressive and Regressive Products and CG-Algebras

Let v be a finite dimensional vector space, dim(v) = n, and let (G(v), ∨) be its exterior algebra. We will denote by ∨ the exterior product (equivalently, the wedge or Grassmann’s progressive product) in order to stress its close analogy with its geometric lattice companion; we call this operation the join. Given two extensors (i. e. decomposable antisymmetric tensors) A and B, they represent two subspaces and of v), respectively.

Combinatorics and representation theory of lie superalgebras over letterplace superalgebras

We state three combinatorial lemmas on Young tableaux, and show their role in the proof of the triangularity theorem about the action of Young-Capelli symmetrizers on symmetrized bitableaux. As an application, we describe in detail the way to specialize general results to the representation theory of the symmetric group and to classical invariant theory.