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Vertex algebras and combinatorial identities
by
Mirko Primc
Department of Mathematics, University of Zagreb
Rogers-Ramanujan identities, first discovered by Rogers in 1894, are two analytic identities expressing certain infinite products as infinite sums; the combinatorial Rogers-Ramanujan identities are obtained by equating the coefficients in the formal power series. These identities were a starting point of an analytic theory developed by Ramanujan, Watson, Bailey and Andrews, to mention just a few, with many connections with number theory and combinatorics. In 1980's Rogers-Ramanujan identities unexpectedly appeared in two seemingly unrelated contexts: in statistical mechanics through the work of Baxter and in representation theory of affine Lie algebras through the work of Lepowsky and Wilson. Further developments in mid 1980's showed a close relation of these results with quantum groups and vertex algebras associated with affine Lie algebras.
In this talk I'll describe some ideas and results related to Lepowsky-Wilson's approach to combinatorial identities of Rogers-Ramanujan type.
The starting point in 1978 was the observation of Lepowsky and Milne that, up to some factor, the principally specialized characters of standard A(1)1-modules of level 3 coincide with the product sides of Rogers-Ramanujan identities. Lepowsky and Wilson constructed bases of these modules, "up to a factor" parameterized by partitions satisfying difference 2 condition, and proved the Rogers-Ramanujan identities by applying the Weyl-Kac character formula. The main ingredient of the proof is the first vertex operator construction of representations of affine Lie algebras, which eventually led to other vertex operator constructions and to the theory of vertex operator algebras.
The ideas used in the Lepowsky-Wilson's proof of Rogers-Ramanujan identities were successfully applied to different standard modules for different affine Lie algebras, sometimes yielding new constructions of standard modules with underlying "classical" combinatorial identities (Lepowsky-Wilson, Misra, Mandia), sometimes giving new identities related to statistical physics (Lepowsky-Primc, Capparelli, Georgiev). If we forget some technical difficulties, the problem of constructing a basis of standard module is solved in two steps: (1) one should construct "enough" relations among elements of standard module to be able to reduce a PBW-spanning set to a smaller spanning set parameterized by partitions satisfying certain difference conditions and (2) one should prove linear independence of the obtained spanning set.
The proper setting for a description of relations among elements of standard module is the structure of (generalized) vertex operator algebras: "all possible" relations are the annihilating fields of the given representation and their structure is precisely described in terms of representation theory.
On the other hand, there is no universal recipe for a proof of linear independence; several methods were used so far, depending on a type of basis that is constructed. In a joint work with A. Meurman we developed an approach based on Verma modules and vertex algebras, in several aspects resembling the Groebner base theory. By using this approach we constructed bases of all standard A(1)1-modules, and a basis of the basic A(1)2-module, and as a result we obtained several infinite series of new Rogers-Ramanujan type identities. It seems that one part of this approach - the construction of relations among the annihilating fields of standard modules - will have a transparent description for all affine Lie algebras in terms of representation theory of vertex operator algebras. The combinatorial part of this approach gets to be very complicated even for the low rank affine Lie algebras A(1)1 and A(1)2, but some examples suggest that the combinatorial difficulties for affine Lie algebras of higher ranks might be circumvented by using the crystal base theory for quantum groups - a theory designed to generalize Baxter's results.
Date received: April 4, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-57.