Yang-Baxter, Lascoux-Schutzenberger, Temperley-Lieb, Jacobi-Trudi, Fomin-Kirillov, Gessel-Viennot

(from a joint work with S.Fomin)

A. Lascoux and M. P. Schutzenberger introduced the symmetric functions called "stable Schubert polynomials" and indexed by a permutation. S. Fomin and A. Kirillov gave a geometric interpretation of these functions, in relation with braids and the Yang-Baxter equation. In the case of a (321)-permutation (no decreasing subsequences of length 3), the functions reduce to skew Schur functions. I. Gessel and X. G. Viennot gave a path geometric intertation of these functions based on Jacobi-Trudi identities. The relation between the two geometric constructions is given,in connection with the Temperley-Lieb algebra.