# Why are the cosets of the Bruhat decomposition independent of the representation of the Weyl group chosen?

Why are the cosets of the Bruhat decomposition independent of the representation of the Weyl group chosen?

For the sake of concreteness, let $$G=SL_n$$. Let $$B_+$$ the subgroup of upper triangular matrices and $$B_-$$ the subgroup of lower triangular matrices and let $$T = B_+ \cap B_-$$. Then the Weyl group $$W$$ of $$SL_n$$ be can canonically identified with $$\frac{N(T)}{T}$$, and furthermore we obtain two BRuhat decomposition's:

$$SL_n = \bigsqcup_{w \in W} B_+wB_+=\bigsqcup_{v \in W} B_-vB_-$$

The Weyl group here is being understood as the quotient group $$\frac{N(T)}{T}$$. Can somebody help me understand why the double coset $$B_+wB_+$$ is independent of the reprentative chosen for $$w$$ from the quotient group? Thanks

## 1 Answer

If you choose a different representative for $$w$$ in the quotient group, that amounts to multiplying $$w$$ by some element $$t \in T$$. Since $$T = B_+ \cap B_-$$, then $$t B_+ = B_+$$ and $$t B_- = B_-$$, i.e, the $$t$$ can be absorbed into the $$B_+$$ or $$B_-$$ of the double coset.