Bruhat involution of a partial-swap frame

Let $K$ be a field and $p$ an $m\times m$ idempotent with a lower–lower Bruhat factorization $p=F_1\pi F_2$: $F_1,F_2$ are lower triangular with nonzero diagonal, and $\pi$ is a $0/1$ matrix with at most one $1$ in each row and each column. Order the $2m$ coordinates as $H_0,\dots,H_{m-1},D_0,\dots,D_{m-1}$, and consider the partial-swap frame

Then $S_p=E_1\,w\,E_2$ with $E_1,E_2$ lower triangular with nonzero diagonal, where $w$ is the permutation matrix

In words: the Bruhat involution of $S_p$ swaps $H_i\leftrightarrow D_j$ exactly for the pivots $(i,j)$ of $p$, and fixes every other coordinate.

This is the "runs" principle in Jain's flag-basis note. Frames along the edges of the two-stage bit interchange are partial swaps. By Lemma 10 of the OpenAI preprint, an edge then costs one recursive field interchange per transposition, and a run of consecutive pivots $(i,j),(i+1,j+1),\dots$ is a single interchange of contiguous chunks. So the child widths entering the moment certificates $\sum_w n_w(w/m)^{1-a}$ are read off from the pivots of the idempotents. With IntMul.OpenAI.idempotent_corner_profile this gives the partial-swap batching lemma: $m-a$ corner transpositions plus one aligned run of width $2a-m$.

Formalization Note The $2m$ coordinates are Fin (m + m) via finSumFinEquiv (left summand = $H$). Matrices are blocks built with Matrix.fromBlocks.