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The Bruhat decomposition is the disjoint unionassociated with a BN-pair.

For a finite group $G$ with a BN-pair, the Iwahori-Hecke algebra over $k$ is the endomorphism algebra of the permutation module $k[G/B]$, up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.

A BN-pair consists of subgroups $B,N\le G$ for which $G=⟨B,N⟩$, the subgroup $H=B\cap N$ is normal in $N$, the quotient $W=N/H$ is generated by a distinguished set $S$ of involutions, and the Bruhat multiplication and nondegeneracy axioms hold. The quotient is the associated Weyl group, and the axioms give the Bruhat decomposition of a BN-pair

The Iwahori-Hecke algebra of a BN-pair may be defined, up to the usual opposite-algebra convention, byIts standard basis $(T_w{)}_{w\in W}$ is indexed by the Bruhat double cosets. For a simple generator $x_i$ represented by ${\dot{x}}_i\in N$, setThe double-coset multiplication rule is the generic rule from part a with $a_i$ specialized to $q_i\cdot 1_k$. Thus $H_k(G,B)$ is a specialization of the generic algebra.

Solved by gpt-5.6-sol high.