Weighted words at degree two, I: Bressoud’s algorithm as an energy transfer

Abstract

In a recent paper, we generalized a partition identity stated by Siladić in his study of the level one standard module of type A_2^{(2)} . The proof used weighted words with an arbitrary number of primary colors and all the secondary colors obtained from these primary colors, and a brand new variant of the bijection of Bressoud for Schur’s partition identity. In this paper, the first of two, we analyze this variant of Bressoud’s algorithm in the framework of statistical mechanics, where an integer partition is viewed as an amount of energy shared, according to certain properties, between several states. This viewpoint allows us to generalize the previous result by considering a more general family of minimal difference conditions. For example, we generalize the Siladić identity to overpartitions. In the second paper, we connect this result to the Glaisher theorem and give some applications to level one perfect crystals.

Weighted Words at Degree Two, II: Flat Partitions, Regular Partitions, and Application to Level One Perfect Crystals

“…The second result, Theorem 27, concerns weighted words at degree two, and energies satisfying = up to some exceptions. This second theorem uses Theorem 22 and the result from the first paper of this series [14], which we summarize in Section 2.1. In the particular case of representations of affine Lie algebras we study here, Theorem 27 allows us to connect the difference conditions of the result on Siladić's identity, given in the first paper, and the energy function of the square, in terms of tensor product, of the vector representation of A (2) 2n .…”

“…The remainder of the paper is organized as follows. We first provide in Section 2 the tools and the main result of the first paper [14], as well as the main results connecting flat and regular partitions at degree one and two. Second, assuming Theorem 22, we recover in Section 3 the Frenkel-Kac character formulas.…”

“…We distinguish two cases. A major subsequent work would be to find a suitable energy to define regular partitions for degree k and which would allow us to state an analogue of Theorem 8 at degree k. This problem appears to be closely related to the problem of finding a generalization to weighted words at degree k of the result stated in [14].…”