Bruhat decomposition in GAP

Question

Certain finite groups (of Lie type) have a normal form for their elements through Bruhat decomposition. Does GAP or any of its packages have any tools to calculate this?

Answer 1

For the special linear group $\text{SL}(n,q)$, the symplectic group $\text{Sp}(n,q)$, the special unitary group $\text{SU}(n,q)$ and special orthogonal group $\text{SO}(n,q)$ in the natural representation for $n \geq 4$ and $q = p^f$ where $p$ is a prime number and $f$ an positive integer, the package "BruhatDecomposition" can be used (except for $p = 2$ in the orthogonal case).

In each case there is a function "BruhatDecompositionSX(stdgens, g)" (X in [L,p,U,O]) which returns an MSLP S and the Bruhat decomposition of g. If the MSLP S is evaluated with the stdgens, then the output is g in its Bruhat decomposition. The stdgens can be obtained by the functions "LGOStandardGensSX(n,q)" and correspond to the Leedham-Green and O'Brien standard generators.

I hope this helps you at least in some cases.