# Bijective proof of Cauchy and Littlewood identities for q-Whittaker polynomials

Seminar website:

https://stanford.edu/~nap/lattice-seminar

To attend the talk, please, send me your email at naprienko@stanford.edu.

The Cauchy Identities are fundamental features of a number of families of symmetric functions. For the Schur polynomials Cauchy Identities can be proven bijectively using the RSK correspondence. For Macdonald polynomials producing an elementary proof of the same identities has remained an outstanding challenge.

In this talk I will show how we solve this problem in the case of the q-Whittaker polynomials, i.e. Macdonald polynomials with t parameter set to zero. It turns out that the RSK correspondence can be q-deformed in a bijective fashion by properly lifting its set of symmetries. For this we employ results coming from the theory of Kashiwara's crystals and the bijection will be a result of a novel affine bi-crystal structure respectively on the set of infinite matrices and semi-standard tableaux. Our arguments pivot around a combination of various theories that include Demazure crystals, the Box-Ball system or Sagan and Stanley's skew RSK correspondence.

This is a joint work with Takashi Imamura and Tomohiro Sasamoto and it is a continuation of last week's seminar by Tomohiro Sasamoto.