A q-deformed type B Cauchy identity and Chow's quasisymmetric functions

The Cauchy identity is a fundamental formula in algebraic combinatorics that captures all the nice properties of the RSK correspondence. In particular, expanding both sides of the identity with Gessel's quasisymmetric functions allows to recover the descent preserving property, an essential tool to prove the Schur positivity of sets of permutations. We look at the type B generalisation of these results that involves the domino insertion algorithm. We introduce a q-deformation of the modified domino functions of our previous works to extend a type B Cauchy identity by Lam and link it with Chow's quasisymmetric functions. We apply this result to a new framework of type B q-Schur positivity and to prove new equidistribution results for some sets of domino tableaux.