Abstract
In the 1980's, J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers–Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2,C)~. When applied to other representations and affine Lie algebras, Lepowsky and Wilson's approach yielded a series of other combinatorial identities of the Rogers–Ramanujan type. At about the same time, R. Baxter rediscovered Rogers–Ramanujan identities within the context of statistical mechanics. The work of R. Baxter initiated another line of research which yielded numerous combinatorial and analytic generalizations of Rogers–Ramanujan identities. In this note, we describe some ideas and results related to Lepowsky and Wilson's approach and indicate the connections with some results in combinatorics and statistical physics.
Similar content being viewed by others
References
Adams, W. W. and Loustaunau, P.: An Introduction to Gröbner Bases, Grad. Stud. Math. 3, Amer. Math. Soc., Providence, 1996.
Alladi, K., Andrews, G. E. and Gordon, B.: Refinements and generalizations of Capparelli's conjecture on partitions, J. Algebra 174 (1995), 636-658.
Andrews, G. E.: Some new partition theorems, J. Combin. Theory 2 (1967), 431-436.
Andrews, G. E.: A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970), 297-305.
Andrews, G. E.: The Theory of Partitions, Encyclop. Math. Appl., Addison-Wesley, Amsterdam, 1976.
Andrews, G. E., Baxter, R. J. and Forrester, P. J.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), 193-266.
Baxter, R. J.: Rogers-Ramanujan identities in the hard hexagon model, J. Statist. Phys. 26 (1981), 427-452.
Berkovich, A. and McCoy, B. M.: Rogers-Ramanujan identities: A century of progress from mathematics to physics, In: Proc. ICMBerlin 1998, Vol. III: Invited Lectures, Documenta Math. (1998), 163-172.
Berkovich, A., McCoy, B. M. and Schilling, A.: Rogers-Schur-Ramanujan type identities for the M(p,p') minimal models of conformal field theory, Comm. Math. Phys. 191 (1998), 325-395.
J. Borcea: Vertex operator algebras, annihilating fields, and a duality-like property for rank two affine Lie algebras, Preprint 1998:9, Dept. Math., Lund Univ.
Bressoud D.: A generalization of the Rogers-Ramanujan identities for all moduli, J. Combin. Theory 27 (1979), 64-68.
Burge, W. H.: A correspondence between partitions related to generalizations of the Rogers-Ramanujan identities, Discrete Math. 34 (1981), 9-15.
Capparelli, S.: On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra 154 (1993), 335-355.
Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Path space realization of the basic representation of A( 1 ) n, In: V. G. Kac (ed.), Infinite Dimensional Lie Algebras and Groups, Adv. Ser. Math. Phys. 7, World Scientific, Singapore, 1989, pp. 108-123.
Dong, C. and Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators, Progr. in Math. 112, Birkhäuser, Boston, 1993.
Feigin, B., Kedem, R., Loktev, S., Miwa, T. and Mukhin, E.: Combinatorics of the s^l2 spaces of coinvariants, Transformation Groups 6 (2001), 25-52.
Feigin, B. and Miwa, T.: Extended vertex operator algebras and monomial bases, math.QA/9901067.
Feigin, B. L. and Stoyanovsky, A. V.: Quasi-particles models for the representations of Lie algebras and geometry of flag manifold, hep-th/9308079.
Frenkel, I. B. and Kac, V. G.: Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980), 23-66.
Frenkel, I. B., Huang, Y.-Z. and Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104(494) (1993).
Frenkel, I. B. and Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
Georgiev, G.: Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace, J. Pure Appl. Algebra 112 (1996), 247-286; II. Parafermionic space, q-alg/9504024.
Gordon, B.: A combinatorial generalization of the Rogers-Ramanujan identities,Amer. J. Math. 83 (1961), 393-399.
Jimbo, M., Misra, K. C., Miwa, T. and Okado, M.: Combinatorics of representations of U q (s^l(n)) at q = 0, Comm. Math. Phys. 136 (1991), 543-566.
Jing, N., Misra, K. C. and Savage, C. D.: On multi-color partitions and the generalized Rogers-Ramanujan identities, CO/9907183.
Kac, V. G.: Infinite-Dimensional Lie Algebras, 3rd edn, Cambridge Univ. Press, Cambridge, 1990.
Kang, S.-J., Kashiwara, M., Misra, K. C., Miwa, T., Nakashima, T. and Nakayashiki, A.: Affine crystals and vertex models, Internat. J. Modern Phys. A 7, Suppl. 1A, Proc. RIMS Res. Project 1991, 'Infinite Analysis', World Scientific, Singapore, 1992, pp. 449-484.
Kedem, R., Klassen, T. R., McCoy, B. M. and Melzer, E.: Fermionic quasiparticle representations for characters of G (1) 1 x G (1) 1 /G (1) 2, Phys. Lett. B 304 (1993), 263-270.
Lepowsky, J. and Milne, S.: Lie algebraic approaches to classical partition identities, Adv.Math. 29 (1978), 15-59.
Lepowsky, J. and Primc, M.: Structure of the standard modules for the affine Lie algebra A (1) 1, Contemporary Math. 46 (1985).
Lepowsky, J. and Wilson, R. L.: The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), 199-290; II: The case A (1) 1, principal gradation, Invent. Math. 79 (1985), 417-442.
Li, H.-S.: Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), 143-195.
Mandia, M.: Structure of the level one standard modules for the affine Lie algebras B (1), F (1) 4 and G (1) 2, Mem. Amer. Math. Soc. 362 (1987).
Meurman, A. and Primc, M.: Annihilating ideals of standard modules of s^l(2,ℂ)∼ and combinatorial identities, Adv. Math. 64 (1987), 177-240.
Meurman, A. and Primc, M.: Annihilating fields of standard modules of s^l(2,ℂ)∼ and combinatorial identities, Mem. Amer. Math. Soc. 652 (1999).
Meurman, A. and Primc, M.: A basis of the basic s^l(3,ℂ)∼-module, QA/98120029.
Misra, K. C.: Structure of certain standard modules for A (1) n and the Rogers-Ramanujan identities, J. Algebra 88 (1984), 196-227.
Primc, M.: Vertex operator construction of standard modules for A (1) n, Pacific J. Math. 162 (1994), 143-187.
Primc, M.: Basic representations for classical affine Lie algebras, J. Algebra 228 (2000), 1-50.
Tsuchiya, A., Ueno, K. and Yamada, Y.: Conformal field theory on the universal family of stable curves with gauge symmetry, Adv. Stud. Pure Math. 19 (1989), 459-466.
Warnaar, S. O.: The Andrews-Gordon identities and q-multinomial coefficients, Comm. Math. Phys. 184 (1997), 203-232.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Primc, M. Vertex Algebras and Combinatorial Identities. Acta Applicandae Mathematicae 73, 221–238 (2002). https://doi.org/10.1023/A:1019747408807
Issue Date:
DOI: https://doi.org/10.1023/A:1019747408807