ИСТИНА

Let K be an algebraically closed field of characteristic zero. If we are given by an irreducible character f: S_n-->K of the symmetric group of degree n, we can consider the following function on the space M_n of square n-matrices $$Imm_f(a_{ij})=\sum_{\sigma\in \mathrm{S}_n}\chi(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}\ldots a_{n\sigma(n)}.$$ This function is called immanant. In particular cases of trivial and sign characters immanant gives permanent and determinant respectively. There are some works investigating linear automorphisms h:M_n-->M_n preserving an immanant. In case of determinant this is a classical problem is solved by Frobenius (1897). For permanent the answer is obtained in the paper due to M. Marcus and F.C. May (1962). In case of general immanants the problem is solved in the paper due to Duffner (1994). In the talk we discuss a new simple proof of all these results using language of algebraic geometry and generalizations of these results to the case of polynomial (non-linear) automorphisms.