Every $g \in \operatorname{SL}(n,\mathbb{R})$ can be written as $g=g_1 s g_2$, where $g_i$ upper triangular and $s$ monomial.

Question

Every $g \in \operatorname{SL}(n,\mathbb{R})$ can be written as $g=g_1 s g_2$, where $g_1, g_2$ are upper triangular and $s$ is monomial (generalized permutation matrix where the entries can be $\neq 1$).

This would mean that there is a decomposition $NWAN=\operatorname{SL}(n, \mathbb{R})$ with

• $N$ the group of upper triangular matrices with diagonal elements $1$,
• $W$ the group of permutation matrices permutation matrices
• $A$ the diagonal matrices with determinant $1$.

I found this statement in Metric spaces of non-positive curvature by Bridson-Haefliger in the proof of Prop. 10.77 on page 339.

Is this decomposition well known? Bridson-Haefliger proceed to give a proof in Lemma 10.80 for $n=3$ only. I think this proof becomes more complicated for larger $n$. There they use the proof of the Jordan-Hoelder theorem for vectorspaces. Is there a different proof?

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Edit: The Borel subgroup is $B=AN=NA$, i.e. the diagonal entries can be different from $1$. Then the statement $BWB=\operatorname{SL}(n,\mathbb{R})$ holds. Otherwise it might not.

Answer 1

This should be the decomposition you want, after moving $A$ to the diagonal of $N$:

Bruhat Decomposition