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Let $M$ be an arbitrary $n\times n$ complex matrix and $S_n$ be the symmetric group of order $n$. Given $\sigma\in S_n$, define $\sigma(M)$ to be the $n\times n$ matrix with elements $[\sigma(M)]_{ij} = M_{i\sigma(j)}$. With this, there are two well-known matrix functions (functions that take matrices as inputs) that interplay nicely:

\begin{align} \text{perm}(\sigma(M)) &= \text{perm}(M), \\ \det(\sigma(M)) &= \text{sgn}(\sigma)\det(M), \end{align} where perm and $\det$ are the permanent and determinant, respectively, and $\text{sgn}(\sigma)$ is the sign of the permutation $\sigma$. Note that these two equations have the following form: \begin{align*} f(\sigma(M)) = \rho(\sigma) f(M) \end{align*} where $f$ is some function of matrices and $\rho$ is a representation of the symmetric group. In the above examples, $\rho$ is the trivial representation (in the case of $f=\text{perm}$) or the sign representation (in the case of $f=\det$).

My question is this: given a representation $\rho$, can one find a matrix function $f$ such that $f(\sigma(M)) = \rho(\sigma)f(M)$? Subsequent questions regarding uniqueness follow naturally. Note that $f:\mathbb{C}^{n\times n}\to\mathbb{C}^d$ with $d$ the dimension of the representation.

As a concrete example, suppose $\rho$ is the two-dimensional representation of $S_3$. Is there a known function $f:\mathbb{C}^{3\times 3}\to \mathbb{C}^2$ (takes in matrices, outputs vectors) such that the aforementioned relationship holds?

Thanks in advance!

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    $\begingroup$ A somewhat similar concept is the immanant, though it isn't a vector-valued function. $\endgroup$
    – pregunton
    Commented Apr 11, 2021 at 19:47

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This is an interesting but slightly strange question. If you require $f$ to be linear (excluding at first glance your examples) then the answer is well-understood: you are asking for a morphism of representations between the vector space $M_n$ of $n\times n$ complex matrices with your action to some chosen representation of $S_n$. The representation $M_n$ is a direct sum of irreducible representations, and a nonzero such map exists if your chosen representation is a direct summand of $M_n$. The strange part is that your action on matrices is not very natural, in that it treats a matrix as just $n$ row vectors that happen to be drawn on top of each of other. This means that $M_n=(\mathbb{C}^n)^{\oplus{n}}$ as $S_n$-represenations, where $S_n$ acts by permutation on the entries. This representation decomposes into a dimension $n-1$ irreducible representation (vectors whose entries sum to zero) and the trivial representation. So, in the case you have asked about, there are in fact three such linear maps, but none are particularly interesting because they don't really treat $M$ as a matrix. In general there won't be any such linear maps for $n\geq 3$ because only two representations of $S_n$ ever appear.

For more interesting $f$ I hope there is an interpretation in terms of Springer theory, but I have nothing helpful to say there.

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