Meurman, Arne; Primc, Mirko **(2001)**. A basis of the basic $germ sgerm l(3,Bbb C)sp sim$-module..* Communications in Contemporary Mathematics, 3, (No. 4)*, 593 - 614

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Authors:

Meurman, Arne
;
Primc, Mirko

Department:

Mathematics (Faculty of Sciences)

Algebra

Algebra

Research Group:

Algebra

Abstract:

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard $A_2^{(2)}$-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard $A_1^{(1)}$-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic $A_2^{(1)}$-module and, by using the principal specialization of the Weyl-Kac character formula, we obtain a Rogers-Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky-Wilson's approach for affine Lie algebras of higher ranks, say for $A_n^{(1)}, $nge 2$, in a way parallel to the next level of complexity seen when passing from the Rogers-Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli $ge 7$.

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard $A_2^{(2)}$-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard $A_1^{(1)}$-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic $A_2^{(1)}$-module and, by using the principal specialization of the Weyl-Kac character formula, we obtain a Rogers-Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky-Wilson's approach for affine Lie algebras of higher ranks, say for $A_n^{(1)}, $nge 2$, in a way parallel to the next level of complexity seen when passing from the Rogers-Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli $ge 7$.

Keywords:

partition ideals. ;
colored partitions ;
Rogers-Ramanujan identities ;
standard modules ;
vertex operator formula ;
vertex operator algebras ;
Affine Lie algebras

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2a5e8968-ea37-454c-892c-01131c40943b | Link: https://lup.lub.lu.se/record/2a5e8968-ea37-454c-892c-01131c40943b
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