The stabilizer of immanants (Q551254)

For any partition \(\pi\) of the number \(n\), \textit{D. E. Littlewood} [The theory of group characters and matrix representations of groups. Reprint of the 2nd ed. 1950. Providence, RI: AMS Chelsea Publishing (2006; Zbl 1090.20001)] introduced the immanant \(P_\pi(X)\) of the \(n\)-square matrix \(X\). This is a common generalization of the determinant and the permanent, more specifically, \(P_{(n)}(X)\) is the permanent of \(X\) and \(P_{(1,\dots, 1)}(X)\) is the determinant of \(X\). The paper under review is devoted to the study of the group \(G\) of those linear automorphisms of the space of complex \(n\)-square matrices that leave \(P_\pi\) invariant. Building on results obtained by \textit{M.\ A. Duffner} [Linear Algebra Appl. 197--198, 567--588 (1994; Zbl 0803.15010)] and by \textit{M.\ Purificação Coelho} [Linear Algebra Appl. 247, 265--271 (1996; Zbl 0860.20009)] it is proved that for \(n\geq 6\) and \(\pi\neq (n), (1,\dots, 1), (4,1,1,1)\), the identity component of \(G\) is the \(2n-2\) dimensional algebraic torus consisting of elementwise multiplications with rank 1 matrices whose diagonal elements multiply to 1. Based on results obtained by \textit{M.\ Purificação Coelho} and \textit{M.\ A. Duffner} [Linear Multilinear Algebra 44, No.~2, 111--130 (1998; Zbl 0931.15003)], it is also proved that for \(n\geq 5\) and \(\pi\neq (n), (1,\dots, 1)\) non-symmetric (i.e., different from its transpose), \(G\) is generated by the above-mentioned torus together with the group of conjugations by permutation matrices and with the operation of transposing. Unfortunately, the first-mentioned result is erroneously stated -- the identity component is claimed to be something else. Also, the semidirect product sign is sometimes used the wrong way around.