Semilinear preservers of the immanants in the set of the doubly stochastic matrices

Abstract

Let $S_n$ denote the symmetric group of degree $n$ and $M_n$ denote the set of all $n$-by-$n$ matrices over the complex field, $\IC$. Let $\chi: S_n\rightarrow \IC$ be an irreducible character of degree greater than $1$ of $S_n$. The immanant $\dc: M_n \rightarrow \IC$ associated with $\chi$ is defined by $$ \dc(X) = \sum_{\sigma \in S_n} \chi(\sigma) \prod_{j=1}^n X_{j\sigma(j)} , \quad X = [X_{jk}] \in M_n. $$ Let $\Omega_n$ be the set of all $n$-by-$n$ doubly stochastic matrices, that is, matrices with nonnegative real entries and each row and column sum is one. We say that a map $T$ from $\Omega_n$ into $\Omega_n$ \begin{itemize} \item is semilinear if $T(\lambda S_1+(1-\lambda )S_2)=\lambda T(S_1)+(1-\lambda )T(S_2)$ for all $S_1,\ S_2\in \Omega_n$ and for all real number $\lambda$ such that $0\leq \lambda\leq 1$; \item preserves $d_{\chi }$ if $d_{\chi }(T(S))=d_{\chi }(S)$ for all $S\in\Omega_n$. \end{itemize} We characterize the semilinear surjective maps $T$ from $\Omega_n $ into $\Omega_n$ that preserve $\dc$, when the degree of $\chi$ is greater than one.