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I am having trouble to understand the notations used in Coxeter presentations of symmetric groups. The notation is defined as follows.

Suppose $G$ is the symmetric group on the set $\left\{1, 2, 3, \ldots, n+1\right\}$ . Then, $G$ is isomorphic to a Coxeter group, with the following Coxeter presentation:

$$ G \cong \langle s_i | s^2_i = 1\forall1 \le i \le n, \left(s_i s_{i+1}\right)^3 = 1\forall1 \le i \le n - 1, \left(s_i s_j\right)^2 = 1\forall1 \le i < j \le n, |i - j| > 1 \rangle $$.

I am not sure how to read this expression, especially, $1\forall1$. It doesn't look like $\forall$ is the universal quantifier here.

  • 2
    The notation is a bit strange: I'd rewrite this perhaps as $G \cong \langle s_i \vert s_i^2 = 1 \, \forall i : 1 \leq i \leq n ; \cdots \rangle$. – 2017-02-18
  • 1
    @Travis, did you mean $G \cong \langle s_i \vert s_i^2 = 1, \forall i : 1 \leq i \leq n ; \cdots \rangle$? – 2017-02-18
  • 0
    @Travis, you are probably missing a comma. – 2017-02-18
  • 1
    think of it as $s_i^2 = 1$ for all $i \in [1..n]$ – 2017-02-18

1 Answers 1

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Presumably:

$$ G \cong \langle \ s_i \mid \forall i \ (1 \le i \le n \to s^2_i = 1) \text { and }$$

$$\forall i \ (1 \le i \le (n - 1) \to (s_i s_{i+1})^3 = 1) \text { and }$$

$$\forall i \ \forall j \ ((1 \le i < j \le n \text { and } |i - j| > 1) \to (s_i s_j)^2 = 1) \ \rangle$$