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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

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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.

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