he PI intends to undertake a combinatorial study of structures arising from affine algebras with applications to representation theory, mathematical physics and q-series. The primary combinatorial objects are crystal graphs on the one hand and rigged configurations on the other hand. Crystal bases provide a combinatorial description of the deep theory of crystal bases of modules over quantized universal enveloping algebras developed by Kashiwara and Lusztig: As the quantum parameter q tends to zero, these bases are described precisely by the crystal graphs encoding nearly all the essential algebraic data. Rigged configurations on the other hand encode the particle structure of the underlying physical model and lead to fermionic formulas. It is proposed to study the crystal structure on rigged configurations and to tackle the long-standing problem of a combinatorial expression for the fusion coefficients. These studies will have applications to q-series, in particular the Bailey lemma, the X=M conjecture, and the theory of symmetric functions.
There are several ways of solving certain models in statistical mechanics, namely via the corner-transfer-matrix method and the Bethe Ansatz. Even though the two methods lead to very different looking answers, they describe the same solutions. From the mathematical perspective this suggest that there exists a one-to-one map between the two indexing sets that describe the solution. The elements in the index set corresponding to the corner-transfer-matrix method are called crystal bases. The elements in the index set corresponding to the Bethe Ansatz are called rigged configurations. The PI proposes to study the map between the two indexing sets, its properties and generalizations in detail. This will have applications in many diverse areas of mathematics and physics, such as representation theory, combinatorics, and statistical mechanics.
