All Questions
7 questions
10
votes
4
answers
998
views
Longest Element of an Affine Weyl Group
I know that the Weyl groups of affine Lie algebras don't have a longest element, but are there any good substitutes for w_0. In particular, is there any good substitute for a reduced decomposition of ...
- 5,548
7
votes
1
answer
259
views
A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 415
6
votes
3
answers
993
views
Occurrences of a simple reflection in the longest element of a Weyl group?
While looking at a preprint I've just bumped into a question about the longest element $w_0$ of a Weyl group $W$ (say irreducible of a Lie type $A$ - $G$ and of rank $n>1$, to simplify). ...
- 52.9k
5
votes
1
answer
2k
views
Weyl groups of $E_6$ and $E_7$
The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...
- 951
4
votes
0
answers
213
views
The Weyl group of Kac-Moody algebra and Coxeter group
Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...
- 653
3
votes
2
answers
748
views
For a Weyl group, what is the connection between its exponents and lengths of its elements?
The following seems to be true: if $|W_q| := \sum {q^{l(w)}}$, where the sum is taken over the elements $w$, then $|W_q| = \prod {(1 + q +...+ q^{e_i})}$, where the product is taken over the exponents ...
- 55
1
vote
0
answers
107
views
Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra
Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...
- 31