Andrea Brini

On the Exterior Calculus of Invariant Theory

The full extent of the tragedy of Hermann Grassmann, which has been unfolding since his death, by a succession of misadventures and misunderstandings of his work unique in the history of modern mathematics, is yet to be fully realized. It began with a deliberate ambiguity on the part of the author of the “Ausdehnungslehre,” whereby one and the same notation was chosen to designate two different algebraic operations, originally known as the progressive and the regressive product. This choice was not an oversight on Grassmann’s part, as we shall see, but was rather dictated by the author’s deep understanding of the philosophical reach of his discovery, an understanding which lay well beyond the powers of the

Recursive matrices and umbral calculus

Two major objectives can be seen to guide much recent work in enumeration: (1) to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide-enough type; (2) to recover from empirical data an underlying set-theoretic structure which would reveal the source of the given recursion. We are here concerned with the first of these objectives, though the eventual understanding of the second is tacitly present, if only as a goal. We noticed the coincidence of several computations which, similar as they are in retrospect, had failed to realize their kinship. On leafing through the unique assembly of recursively solvable combinatorial problems in Comtet’s and Sloane’s invaluable collections, one is struck by the repeated occurrence of one and the same kind of double recursion. More strikingly, the same recursion is seen to occur in the polynomial sequences of the Umbra1 Calculus of Roman and Rota (see [25]). Everywhere, the Lagrange inversion formula for power series plays a pivotal role. Much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula, both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs (Andrews, Foata, Garsia, Gessel, Joni, Raney, Reiner, Schtitzenberger, to name but a few). Whatever their origins, the identities abutting Lagrange inversion are expressed by integers alone. This suggests not only a hidden set-theoretic layer, but a characteristic-free generalization as well: this generalization is the central theme of our work. We define a monoid of infinite matrices-“recursive matrices” for short. The entries of these matrices give the sought-out recursion, for example, that for coefficients of binomial and Sheffer polynomials and factor sequences, as well as that of the special sequences recently introduced by Roman in [26]. 546 002 l-8693/82/040546-28%02.00/‘0

Some properties of characteristic polynomials and applications to T-lattices

Some intersection results for the characteristic polynomial of locally finite inf-semilattices are given; furthermore, we use these results to exhibit some characterizations and some enumeration formulas for lattices of triangular type.

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.

Polynomial sequences of integral type

Such sequences of polynomials have been intensively studied in recent years (see, e.g., [2, 4-6, S-101). Our point of view in this work will be to assume these sequences to be integer-valued. This assumption, far from being a restriction, leads to several unexpected simplifications in the theory and brings it closer to its combinatorial sources. We have therefore decided to develop the theory anew along these lines, that is, by working exclusively over the ring of integers. While some of the results relating to the operator calculus associated therewith may be gleaned from analogous results in the umbra1 calculus, there is nontheless a body of results which make sequences of integral type (as we shall call them) strikingly different in their behavior. To wit, a sequence of polynomials of integral type is uniquely associated with a shift-invariant operator Q-the delta operator of the umbra1 calculus-which in the present case can be uniquely expanded into a formal power series of the difference operator A:

Higher dimensional recursive matrices and diagonal delta sets of series

The idea to study infinite matrices whose entries are the coefficients of the powers of a given formal series is rather old and dates back at least to Schur’s posthumous papers on Faber polynomials [39-41]. In 1953, Jabotinsky reconsidered Schur and Shiffer’s [38] work on the subject and developed a systematic study of these matrices [20]. Since then, several applications confirmed that Jabotinsky matrices are a useful tool in computational analysis and in the theory of formal series (see, e.g., [ 16, 17, 21, 251). However, there exists a totally different motivation in order to undertake this kind of study. Not long ago, moving from considerations on the umbra1 calculus-the operator calculus associated with polynomial sequences of Sheffer type [ 1, 13, 28, 32, 36, 37]-the authors of [l] were led to introduce a class of matrices which turns out to be an extension of that studied by Jabotinsky. These matrices, called “recursive matrices,” can be cogently interpreted as matrices of integer evaluations of polynomial sequences of Sheffer type as well as matrices representing the groups of umbra1 and shift- invariant operators, respectively. A self-contained theory of recursive matrices has been worked out in [2] and provides generalized versions of Jabotinsky’s results; on the other hand, it can be used (see, e.g., [3]) to extend and simplify the umbra1 calculus and some related topics as, for example, the theory of binornial functions [29-321. The purpose of the present paper is to contribute to the higher-dimensional reformulation of [2]. This program is to be at least twofold since it has to reflect the fact that, in the multivariate case, two different definitions are available for the algebra of Laurent series. 315

A general umbral calculus in infinitely many variables

On etudie des suites de fonctions factorielles satisfaisant des analogues des recurrences binomiales et de Sheffer et on montre comment un calcul operatoriel ressemblant au calcul «umbral» peut etre construit dessus

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.

Remarks on invariant geometric calculus. Cayley-Grassmann algebras and geometric Clifford algebras

The invariant geometric calculus was founded by the German mathematician H.G. Grassmann in 1844 (Ausdehnungslehre [15, 16]). In this treatise, he introduced the modern notion of a vector in an abstract n-dimensional space and, in general, the notion of an extensor (decomposable antisymmetric tensor). Grassmann’s plan was radically innovative; his aim was to found an intrinsic algebraic calculus for (projective, affine, euclidean) geometry, that was alternative to the cartesian idea of linking algebra and geometry by a system of coordinates. To this aim, in the Ausdehnungslehre, several algebraic operations on extensors are introduced, for example, the progressive product, the regressive product and the Erganzung operation (in geometric language: projection, section and orthogonal duality).