{c}

The Coxeter Gram matrix is the <Gram matrix> $G=(-2\cos(\pi/m_{ij}))_{i,j\in I}$ of the distinguished basis in the <Geometric representation of a Coxeter group>.

Coxeter Gram matrix

Let $V$ be the real <vector space> with basis $(e_i)_{i\in I}$. The <Coxeter Gram matrix> defines the symmetric <bilinear form>
$$
\langle e_i,e_j\rangle=G(W)_{ij}=-2\cos\left(\frac{\pi}{m_{ij}}\right).
$$
Its diagonal entries are $2$. The <Geometric representation of a Coxeter group> is generated by the reflections
$$
\sigma(x_i)(v)=v-\langle v,e_i\rangle e_i.
$$
Each has square one, and on $\operatorname{span}\{e_i,e_j\}$ the product $\sigma(x_i)\sigma(x_j)$ has order $m_{ij}$. The reflections therefore satisfy the Coxeter relations and define a <group representation> $\sigma:W\to GL(V)$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 2 / a / Solution

Set
$$
a_{ij}=2\cos\left(\frac{\pi}{m_{ij}}\right),
\qquad c_i=x_1\cdots x_i,
\qquad c_0=1.
$$
In the <Geometric representation of a Coxeter group>,
$$
\sigma(x_i)e_j=e_j+a_{ij}e_i,
$$
including $i=j$, since $a_{ii}=2\cos\pi=-2$. As $\beta_i=\sigma(c_{i-1})e_i$, telescoping gives
$$
\sigma(c_i)e_j-\sigma(c_{i-1})e_j
=a_{ij}\beta_i.
$$
Summing from $i=1$ to $j-1$ yields
$$
\beta_j=e_j+\sum_{i<j}a_{ij}\beta_i,
$$
or equivalently
$$
e_j=\beta_j-\sum_{i<j}2\cos\left(\frac{\pi}{m_{ij}}\right)\beta_i.
$$
Summing the same telescoping identity all the way to $n$ and substituting this first formula gives
$$
\begin{aligned}
\sigma(c)e_j
&=e_j+\sum_{i=1}^na_{ij}\beta_i\\
&=\beta_j+\sum_{i\geq j}2\cos\left(\frac{\pi}{m_{ij}}\right)\beta_i,
\end{aligned}
$$
which is the second required identity.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 4 / b / Solution

{c}
{wiki=Coxeter_group#Geometric_representation}

For a finite-rank <Coxeter system>, let $V$ have basis $(e_i)_{i\in I}$ and symmetric <bilinear form>
$$
\langle e_i,e_j\rangle=-2\cos(\pi/m_{ij}).
$$
Its geometric representation sends the generator $x_i$ to the reflection
$$
\sigma(x_i)v=v-\langle v,e_i\rangle e_i.
$$

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 Incoming links: Geometric representation of a Coxeter group