$q$-Universal characters and an extension of the lattice $q$-universal characters

We consider two different subjects: the $q$-deformed universal characters $\widetilde S_{[\lambda,\mu]}(t,\hat t;x,\hat x)$ and the $q$-deformed universal character hierarchy. The former are an extension of $q$-deformed Schur polynomials, and the latter can be regarded as a generalization of the $q$-deformed KP hierarchy. We investigate solutions of the $q$-deformed universal character hierarchy and find that the solution can be expressed by the boson–fermion correspondence. We also study a two-component integrable system of $q$-difference equations satisfied by the two-component universal character.