Grounded partitions of type $A_1^{(1)}$ at levels 1 and 2: bijections, affine crystal graphs, and partition identities

Abstract

Grounded partitions, introduced by Dousse and Konan, are coloured partitions satisfying difference conditions given by a matrix with nonnegative integer entries. For the matrices studied in this paper, the generating functions are known to be infinite products, corresponding to the principal specialisation of characters of highest weight modules of type $A_1^{(1)}$. We give the first bijective proof that the generating functions of grounded partitions at level $2$ are infinite products. We then give a new combinatorial model for affine crystal graphs of type $A_1^{(1)}$ at level $2$, where the vertices are grounded partitions and the arrows are given by explicit bracketing rules. The grounded partition model for affine crystal graphs of highest weights $\Lambda_0$, $\Lambda_1$ and $\Lambda_0 + \Lambda_1$ gives rise to new $q$-series identities obtained by decomposing the affine crystal graphs into the crystal graphs of finite type $A_1$ via the restricted representation.