Stefano Capparelli

THE ROGERS–RAMANUJAN RECURSION AND INTERTWINING OPERATORS

We use vertex operator algebras and intertwining operators to study certain substructures of standard -modules, allowing us to conceptually obtain the classical Rogers–Ramanujan recursion. As a consequence we recover Feigin–Stoyanovskys character formulas for the principal subspaces of the level 1 standard -modules.

The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators

Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon’s generalization of the Rogers–Ramanujan identities and with Andrews’ related identities. The present work generalizes the authors’ previous work on intertwining operators and the Rogers–Ramanujan recursion.

Elements of the Annihilating Ideal of a Standard Module

The construction of affine Lie algebras by means of vertex operators has proved to be a remarkably fruitful line of research by motivating, being motivated by, and linking together a disparate range of sectors of mathematics and physics. Vertex operators entered mathematics through the work of J. Lepowsky and R. L. Wilson who gave an explicit construction of the affne algebra A’,l’ [LWl]. Their vertex operators were similar to ones used by physicists in the string model. Later on, I. Frenkel, V. Kac [FK], and G. Segal [S], found representations of afhne algebras using “untwisted” vertex operators. A large amount of very interesting research has followed stimulated by this observation and by the connection between classical combinatorial identities and the Weyl-Kac character formula [K, LM, LPl-2, LW2-4, Ma, Mil-21. A deeper understanding of vertex operators has been reached more recently as a result of work by Borcherds [B] who investigated general vertex operator algebras, and by Frenkel, Lepowsky, and A. Meurman in a series of papers, culminating in the recent book [FLM]. In particular, these three authors showed that the Monster finite simple group is the automorphism group of a certain vertex operator algebra, thus giving a natural realization of this largest among sporadic groups. This also established a surprising connection between the Monster and two-dimensional conformal quantum field theory, which is the physicists’ analogue of the theory of vertex operator algebras; cf. [BPZ, DL, FHL, FJ, TK, T, ZF, FLM].

A collection of results on Hamiltonian cycle systems with a nice automorphism group

Abstract We collect some old and new results on Hamiltonian cycle systems of the complete graph (or the complete graph minus a 1-factor) having an automorphism group that satisfies specific properties.

On the span of polynomials with integer coefficients

Following a paper of R. Robinson, we classify all hyperbolic polynomials in one variable with integer coefficients and span less than 4 up to degree 14, and with some additional hypotheses, up to degree 17. We conjecture that the classification is also complete for degrees 15, 16, and 17. Besides improving on the method used by Robinson, we develop new techniques that turn out to be of some interest. A close inspection of the polynomials thus obtained shows some properties deserving further investigations.

On Some Theorems of Hirschhorn

Abstract It is found that four theorems of Hirschhorn [Hirschhorn, M. D. (1979). Some partition theorems of the Rogers–Ramanujan type. J. Comb. Th. A 27:33–37] correspond to standard representations of an affine Lie algebra. Using this correspondence two more theorems of the same type are found. It turns out that a classical result of Euler also appears in this context.