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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.

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