UC Berkeley

LLT polynomials were first introduced by Lascoux, Leclerc, and Thibon using the action of an affine Hecke algebra for S$_n$, and can be viewed as a $q$-generating function for both ribbon tableaux and tuples of semistandard Young tableaux. This definition has since been expanded to arbitrary Lie type, although with no combinatorial definition. We establish a combinatorial model for LLT polynomials in particular cases for Sp$_{2n}$ and further conjecture a similar model for the orthogonal Lie types. Our definition uses a new object we call an out-in tableau as well as a correspondence between oscillating tableaux and symplectic tableaux that we use to give a proof of a Cauchy identity for Sp$_{2n}$.