Weighted words at degree two. II: Flat partitions, regular partitions, and application to level one perfect crystals (Q2121800)

Summary: In a recent work, \textit{X. Xiong} and \textit{W. J. Keith} [Ramanujan J. 49, No. 3, 555--565 (2019; Zbl 1470.11266)] gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series on Siladić's identity. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modules for the type \(A_{2n}^{(2)}, D_{n+1}^{(2)}\) and \(B_n^{(1)}\), and also to make the connection between the result stated in paper one and the representation theory. For Part I see [the author, ``Weighted words at degree two. I: Bressoud's algorithm as an energy transfer'', Ann. Inst. Henri Poincaré, Probab. Stat. (to appear); \url{arXiv:2001.10927}].