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Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux. (English) Zbl 0848.20007
St. Petersbg. Math. J. 7, No. 1, 77-127 (1996); translation from Algebra Anal. 7, No. 1, 92-152 (1995).
Summary: We construct certain families of piecewise linear representations (\(cpl\)-representations) of the symmetric group \(S_n\) and of the affine Weyl group \(\widetilde {S}_n\) of type \(A^{(1)}_{n -1}\) acting on the space of triangles \(X_n\). We find a nontrivial family of local \(cpl\)-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.

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory