Linear preservers of Schur functions (Q1923149)

In the notation of the present paper, a Schur function is a function on \(n\times n\) matrices \(A=(a_{ij})\) over a field \(F\) of the form \[ d^H_\chi(A)= \sum_{\sigma\in H} \chi(\sigma)\Pi_ia_{i\sigma(i)}, \] where \(H\) is a subgroup of the symmetric group \(S_n\) and \(\chi\) a character of \(H\); when \(H=S_n\) and \(F=\mathbb{C}\), this is sometimes called an immanant. A linear transformation \(T\) on the space of \(n\times n\) matrices is called a preserver of \(d^H_\chi\) if \(d^H_\chi(T(A))= d^H_\chi(A)\) for all \(A\). The author surveys some recent results on Schur function preservers. These include the determination (under mild restrictions) of the preservers of the form \(T(A)=PAQ\) by \textit{M. A. Duffner} and the author [Linear Algebra Appl. 197/198, 177-188 (1994; Zbl 0798.15012)] and (again under mild restrictions) of all preservers in the immanant case by \textit{M. A. Duffner} [ibid. 197/198, 567-588 (1994; Zbl 0803.15010)] and \textit{M. P. Coelho} [ibid. 247, 265-271 (1996; Zbl 0860.20009)].