Question from me and a colleague:

Given a matrix \begin{equation} U = \begin{bmatrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix} \quad \text{with } U_{22} \neq 0, \end{equation} denote its determinant by $\Delta$ and let \begin{equation} \label{defineswap} \operatorname{sw}(2 \overline{2})(U) = \begin{bmatrix} \Delta/U_{22} & U_{12}/U_{22}\\ -U_{21}/U_{22} & 1/U_{22} \end{bmatrix}, \end{equation} a matrix with determinant $U_{11}/U_{22}$. (The notation is meant to suggest the word ``swap.") Suppose that the indeterminates $U_{11}$, $U_{12}$, $U_{21}$, $U_{22}$ are replaced by rational functions of four indeterminates $z(1)$, $z(2)$, $z(\overline{1})$, $z(\overline{2})$. Then these two equations are equivalent: \begin{equation} U \begin{bmatrix} z(1) \\ z(2) \end{bmatrix} = \begin{bmatrix} z(\overline{1}) \\ z(\overline{2}) \end{bmatrix} \quad \Leftrightarrow \quad \operatorname{sw}(2 \overline{2})(U) \begin{bmatrix} z(1) \\ z(\overline{2}) \end{bmatrix} = \begin{bmatrix} z(\overline{1}) \\ z(2) \end{bmatrix}. \end{equation}

This seems to be the tip of an iceberg: there is a representation $$ \operatorname{sw}: S_{4} \to \operatorname{Crem}(M_2) $$ of the symmetric group $S_{4}$ in the group of Cremona transformations of the space of $2$-by-$2$ matrices. We think of $S_{4 }$ as the group of permutations of the set $\{1,2,\overline{1},\overline{2}\}$, and for each such permutation $\pi$ we have \begin{equation} \operatorname{sw}(\pi)(U) \begin{bmatrix} z(\pi(1)) \\ z(\pi(2)) \end{bmatrix} = \begin{bmatrix} z(\pi(\overline{1})) \\ z(\pi(\overline{2})) \end{bmatrix}. \end{equation} If $\pi$ is an element of the subgroup $S_2 \times S_2$, then $\operatorname{sw}(\pi)$ is just the appropriate simultaneous permutation of rows and columns.

This leads us to ask whether there is a representation $$ \operatorname{sw}: S_{2n} \to \operatorname{Crem}(M_n) $$ with similar properties. Suppose that $U$ is an $n$-by-$n$ matrix of indeterminates, and that its entries have been replaced by rational functions in the indeterminates $z(1), z(2), \dots, z(n), z(\overline{1}), z(\overline{2}), \dots, z(\overline{n})$ so as to satisfy this equation: \begin{equation} U \begin{bmatrix} z(1) \\ z(1) \\ \vdots \\ z(n) \end{bmatrix} = \begin{bmatrix} z(\overline{1}) \\ z(\overline{2}) \\ \vdots \\ z(\overline{n}) \end{bmatrix}. \end{equation} Think of $S_{2n}$ as the group of permutations of the set $\{1,2,\dots,n,\overline{1},\overline{2},\dots,\overline{n}\}$. We demand that for each such permutation $\pi$ we have \begin{equation} \operatorname{sw}(\pi)(U) \begin{bmatrix} z(\pi(1)) \\ z(\pi(2)) \\ \vdots \\ z(\pi(n)) \end{bmatrix} = \begin{bmatrix} z(\pi(\overline{1})) \\ z(\pi(\overline{2})) \\ \vdots \\ z(\pi(\overline{n})) \end{bmatrix}, \end{equation} and we demand that if $\pi$ is an element of the subgroup $S_n \times S_n$, then $\operatorname{sw}(\pi)$ is the appropriate simultaneous permutation of rows and columns.