M. Kashiwara [Duke Math. J. 63, 465–516 (1991; Zbl 0739.17005)] has defined a so-called crystal basis to the quantized enveloping algebra associated to a symmetrizable Kac-Moody Lie algebra. This basis describes the combinatorial behaviour of the algebra when the deformation parameter is specialized to zero. (In the physical model, this parameter corresponds to temperature, and the simplifying behaviour of the algebra when the parameter is specialized to zero corresponds to the physical behaviour close to absolute zero).
In this paper, the authors discuss the relationship between such crystal bases and \(q\)-series identities (such as the Rogers Ramanujan identities). Such identities can be obtained from crystal bases by consideration of two different methods of evaluating generating functions of tensor products of crystals. In particular, the authors give some new identities associated to the affine Lie algebra \(C_n^{(1)}\), but other cases are also discussed. The links with the hard hexagon model, which is a two-dimensional lattice model of a gas with hard (or non-overlapping) particles, are also considered.
For the entire collection see [Zbl 0980.00024].