All Questions
10
questions
1
vote
0
answers
24
views
Bruhat Order of the Finite Symmetric Group
I am studying Theorem 2.1.5 in "Combinatorics of Coxeter Groups", but I am confused by a statement in the proof of the "if" direction (the part after $\textbf{However}$). Let me ...
- 63
1
vote
2
answers
119
views
Number of fixed points of generators of reflections (Coxeter) group
Say I have a group with presentation like
$$\langle s,t,u \mid s^2,t^2,u^2,(st)^2,(su)^3,(ut)^4\rangle,$$
faithful on set $S$ with exactly one orbit ($|S|$ is known). How could I determine $|\text{Fix}...
3
votes
1
answer
88
views
Orbits of pairs of multi-indices under the diagonal action of the symmetric group
This question concerns a statement made on page 168 of
Dipper, R. and Donkin, S., 1991. Quantum GLn. Proceedings of the London Mathematical Society, 3(1), pp.165-211.
I have tried to include all ...
- 2,251
1
vote
0
answers
100
views
Coxeter exchange condition in symmetric group
I would like to prove (for purposes of illustration mainly) that the symmetric group $S_n$ with the set $S$ of adjacent transpositions $(i, i+1)$ is a Coxeter group by proving that it satisfies the ...
- 103
4
votes
1
answer
121
views
Cardinality of a coxeter group
Let ${G}$ be a Coxeter group with the next presentation
\begin{equation}
G = \left\langle s_1,s_2,\cdots,s_{n-1} : (s_is_{i+1})^3=1 , \ (s_is_j)^2=1 \ ,\ |i-j| > 1 \right\rangle
\end{equation}
...
3
votes
1
answer
150
views
Reduced word of a transposition
Take the group $S_n$ of permutations of $n$ points and as generator set take $S = \{s_1 := (1,2), s_2 := (2,3),\ldots, s_{n-1} := (n-1,n)\}$ the set of Coxeter generators. Let $t = (i,j) \in S_n$ be a ...
- 6,409
3
votes
1
answer
1k
views
Finding the longest element $w_0$ of the reflection group $S_n$ with respect to the positive system $\{e_i - e_j, 1≤i < j≤1\}$
Preliminaries
Let $V = \mathbb R^n$, and $W = S_n$ the symmetric group. We consider the set $\Phi = \{±(e_i - e_j) \mid 1 ≤ i < j ≤ n\} $ (where the $e_i$ are the standard vectors), which is a ...
- 3,057
1
vote
2
answers
1k
views
What is the longest element of $S_n$ as a product of adjacent transpositions?
I can't seem to get this to work. According to wikipeda, the longest element of $S_n$ should be expressible as a product of $n(n-1)/2$ adjacent transpositions by
$$
(n, n-1)(n-1,n-2)\cdots(21)(n-1,n-2)...
- 1,950
3
votes
1
answer
225
views
reflection groups and hyperplane arrangement
We know that for the braid arrangement $A_\ell$ in $\mathbb{C}^\ell$: $$\Pi_{1 \leq i < j \leq \ell} (x_i - x_j)=0,$$
$\pi_1(\mathbb{C}^\ell - A_\ell) \cong PB_\ell$, where $PB_\ell$ is the pure ...
- 355
7
votes
2
answers
1k
views
Reflection groups and symmetric group
Define the action of $S_n$ on $\mathbb{R}^n$:
take any $x\in S_n$, consider the mapping $x: \mathbb{R}^n\to\mathbb{R}^n$, $e_1, e_2 ...e_n$ are the standard basis of $\mathbb{R}^n$, $x(e_k)=e_{x(k)}$...
- 93