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theses

In this dissertation, we propose a set of new partition identities, arising from a twisted vertex operator construction of the level 4 standard modules for the affine Kac-Moody algebra of type A(2)2 . These identities have an interesting new feature, absent from previously known examples of this type. This work is a continuation of a long line of research of constructing standard modules for affine Kac-Moody algebras via vertex operators, and the associated combinatorial identities. The interplay between representation theory and combinatorial identities was exemplified by the vertex-algebraic proof of the famous Rogers-Ramanujan-type identities using standard A(1)1-modules by J. Lepowsky and R. Wilson. In his Ph.D. thesis, S. Capparelli proposed new combinatorial identities using a twisted vertex operator construction of the standard A(2)2-modules of level 3, which were later proved independently by G. Andrews, S. Capparelli, and M. Tamba-C. Xie. We begin with an obvious spanning set for each of the level 4 standard modules for A(2)2 , and reduce this spanning set using various relations. Most of these relations come from certain product generating function identities which are valid for all the level 4 modules. There are also other ad-hoc relations specific to a particular module of level 4. In this way, we reduce our spanning sets to match with the graded dimensions of the said modules as closely as possible. We conjecture and present strong evidence for three partition identities based on the spanning sets for the three standard A(2)2-modules of level 4. One surprising result of our work is the discovery of relations of arbitrary length. Consequently, the partitions corresponding to these spanning sets cannot be described by difference conditions of finite length. The spanning set result proves one inequality of the proposed identities. There is strong evidence for the validity of the conjecture (i.e., the opposite inequality), since it has been verified to hold for partitions of n ≤ 170, and n = 180, 190 and 200.

ETD_5940

Ph.D.

Includes bibliographical references

by Debajyoti Nandi

doi:10.7282/T3154G81

ETD doctoral

The author owns the copyright to this work.