All Questions
7 questions with no upvoted or accepted answers
12
votes
0
answers
479
views
Writing an element of a free product of $C_2$'s as a product of order-$2$ elements
My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
- 1,298
6
votes
0
answers
99
views
Indices of Coxeter groups in themselves
Every Euclidean Coxeter group ($P_n$, $Q_n$, $R_n$, $S_n$, $T_7$, $T_8$, $T_9$, $U_5$, $V_3$, $W_2$) contains infinitely many scaled copies of itself as subgroups. What are all the possible indices of ...
- 2,773
3
votes
0
answers
308
views
Is G(4,7) a Coxeter group
Let $G(4, 7)$ be an abstract group with the presentation
$$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$
Richard Schwartz considered ...
- 2,083
3
votes
0
answers
117
views
Why inherit the Tits systems structure by a $B$-adapted homomorphism?
Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...
- 479
2
votes
0
answers
211
views
A question of braid words
Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...
- 534
1
vote
0
answers
130
views
Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
- 1,623
0
votes
0
answers
185
views
Isomorphic Coxeter groups
After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
- 2,773