Groups generated by involutions, Gelfand–Tsetlin patterns, and combinatorics of Young tableaux

We construct ceratin families of piecewise linear representations ($spl$ representations) of the symmetric group $S_n$ and of the affme Weyl group $\widetilde S_n$ of type $A_{n-1}^{(1)}$ acting on the space of triangles $X_n$. We find a nontrivial family of local $spl$-invariants for the action of the symmetric group $S_n$ on the space $X_n$ and construct a global invariant with respect to the action of the affine Weyl group $\widetilde S_n$ (the so-called cocharge). We find continuous analogs for the Kostka–Foulkes polynomials and for the crystal graph. We give an algebraic version of some combinatorial transformations on the set of standard Young tableaux.