Representation Theory and Double Affine Hecke Algebras

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he PI intends to undertake a combinatorial study of structuresarising from affine algebras with applications to representationtheory, mathematical physics and q-series.The primary combinatorial objects are crystal graphs on the onehand and rigged configurations on the other hand. Crystal basesprovide a combinatorial description of the deep theory of crystalbases of modules over quantized universal enveloping algebrasdeveloped by Kashiwara and Lusztig: As the quantum parameterq tends to zero, these bases are described precisely by thecrystal graphs encoding nearly all the essential algebraic data.Rigged configurations on the other hand encode the particlestructure of the underlying physical model and lead to fermionicformulas. It is proposed to study the crystal structureon rigged configurations and to tackle the long-standing problemof a combinatorial expression for the fusion coefficients.These studies will have applications to q-series, in particularthe Bailey lemma, the X=M conjecture, and the theory of symmetricfunctions.There are several ways of solving certain models in statisticalmechanics, namely via the corner-transfer-matrix method andthe Bethe Ansatz. Even though the two methods lead to verydifferent looking answers, they describe the same solutions. From the mathematical perspective this suggest that there existsa one-to-one map between the two indexing sets that describe thesolution. The elements in the index set corresponding to thecorner-transfer-matrix method are called crystal bases. The elementsin the index set corresponding to the Bethe Ansatz are calledrigged configurations. The PI proposes to study the mapbetween the two indexing sets, its properties and generalizationsin detail. This will have applications in many diverse areasof mathematics and physics, such as representation theory,combinatorics, and statistical mechanics.