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

Abstract: In a paper, we generalized a partition identity stated by Siladi\'c 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 Siladi\'c identity to overpartitions. In the second paper, we connect this result to the Glaisher theorem and give some applications to level one perfect crystals.