For each signed permutation $w\in W$, choose its symplectic signed-permutation matrix $\dot w\in N_G(T)$. The <Bruhat decomposition of a reductive algebraic group> is the explicit disjoint union
$$
\operatorname{Sp}_{2n}
=\bigsqcup_{w\in(\mathbb Z/2)^n\rtimes S_n}B\dot wB.
$$

To prove existence, compare the standard isotropic flag $F_\bullet$ with $gF_\bullet$. The ranks
$$
r_{ij}=\dim(F_i\cap gF_j)
$$
together with the symplectic orthogonality relations determine a unique signed permutation $w$. Symplectic row and column operations from $B$ then reduce $g$ to $\dot w$, so $g\in B\dot wB$. Conversely, the same intersection dimensions are constant on a double coset and recover $w$, proving disjointness. This is symplectic Gaussian elimination and establishes the claimed decomposition.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 4 / iv / Solution

{c}
{wiki=Bruhat_decomposition}

For a connected reductive algebraic group, a <Borel subgroup> $B$, and a maximal torus $T\subseteq B$, the <Weyl group> $W=N_G(T)/T$ indexes the double cosets:
$$
G=\bigsqcup_{w\in W}B\dot wB.
$$

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