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A Coxeter element is a product of all simple generators in some order. For a finite irreducible Coxeter group, all Coxeter elements are conjugate.

The Coxeter number is the order of a Coxeter element in a finite irreducible Coxeter group.

A Coxeter system $(W,I,M)$ records a Coxeter group $W$, its set $I$ of simple generators, and its Coxeter matrix $M=(m_{ij})$.

A finite Coxeter group is a Coxeter group with finitely many elements. Its Coxeter Gram matrix is positive definite.

A Hecke algebra is a deformation of the group algebra of a Coxeter group, obtained by deforming the quadratic relations for its simple generators while retaining the braid relations.

The longest element $w_0$ is the unique element of maximal Coxeter length in a finite Coxeter group. It is an involution and satisfies $\ell (w_{0}w)=\ell (w_0)-\ell (w)$.

If a finitely generated Coxeter group $W$ is finite, its integer-valued Coxeter length has a maximum. Conversely, if some $w_0$ has globally maximal length $N$, every group element has a word of length at most $N$. There are only finitely many words of bounded length in the finite set of simple generators, so $W$ is finite.

Realize the finite group as the reflection group of a root system with fundamental system $\Delta$ and positive system $\Pi$. Maximality and the fact that multiplication by a simple generator changes Coxeter length by one giveThe positive-root criterion for Coxeter length therefore gives $w_0\Delta \subseteq -\Pi$. Since $w_0\Delta$ is itself fundamental, it must be the simple system $-\Delta$ of the positive system $-\Pi$.

If $v_0$ is another maximal-length element, the same argument gives $v_0\Delta =-\Delta =w_0\Delta$. Hence $v_0^{-1}{w}_0$ stabilizes $\Delta$, and part c gives $v_0^{-1}{w}_0=1$. The Longest element of a finite Coxeter group is therefore unique.

Solved by gpt-5.6-sol high.

Every Hecke parameter of a BN-pair $q_i$ divides $|B|$ and, because $B$ is a $q$-group, is a power of $q$. If $q\equiv 1(\mathrm{mod}p)$, then $q_i=1$ in the field $k$ of characteristic $p$. The specialized quadratic relation becomeswhile the braid relations are unchanged. These are the defining relations of the Coxeter group $W$, so $x_i\mapsto T_i$ induces a surjective homomorphismBoth algebras have bases indexed by $W$, hence the homomorphism is an isomorphism of group algebras.

Solved by gpt-5.6-sol high.