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A partial permutation matrix $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. Some boxes get crossed out just horizontally, just vertically, neither, or both (such as the $1$s, which we think of as crossing themselves out both ways). Call these the SE hooks of $\pi$.

Similarly, we can define the NW hooks of $\rho$. Call $(\pi,\rho)$ complementary if when the SE hooks of $\pi$ are overlaid on the NW hooks of $\rho$, each box is crossed out, and no $1$ crosses out any other. It is easy to see that for any $\pi$, there exists a unique complementary $\rho$, and $\pi \mapsto w_0 \rho w_0$ is an involution.

Has this involution on partial permutations been studied before?

For me, it has come up in studying conormal varieties to Schubert varieties.

ADDED: Here's a familiar subcase. The diagram of $\pi$ is the set of boxes not crossed out in its SE hooks. If this is a partition in the NW corner, $\pi$ is called dominant, and every partition arises from a unique $\pi$. In this case, $w_0 \rho w_0$ is also dominant, and its diagram is (the rotation of) the complementary partition.

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I'm missing something. Consider the $2 \times 2$ case with $\pi$ just having a single 1 in the NE corner. The $\rho$ entries can only be in completely uncrossed boxes, i.e. in the SW corner alone, but this doesn't give a double crossing in the NW corner. –  Michael Albert May 19 '13 at 22:05
Sorry rather than "SW corner" that should be "W column" but the question still applies (if we put an entry in the NW corner, then the SW corner is still uncrossed, and if we put it in the SW corner then the NW corner is only crossed vertically). –  Michael Albert May 19 '13 at 22:16
Absolutely right, I will correct the question. –  Allen Knutson May 20 '13 at 1:28
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