2
votes
1answer
76 views

Double quotients of Coxeter groups have the chain property?

Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$. Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there ...
4
votes
1answer
213 views

Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...
4
votes
2answers
226 views

Root system automorphisms as inner automorphisms of extended Chevalley group

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
2
votes
0answers
72 views

Weyl group invariants of the representation ring of a split torus

Let $G$ be a semisimple split algebraic group, $T$ its split maximal torus and $W$ corresponding Weyl group. Let $T^*$ denote the character lattice of $T$ and $\Lambda$ denote the weight lattice, so ...
4
votes
3answers
576 views

Does -I belong to Weyl group?

Let $\Phi$ be an irreducible root system, with positive roots $\Phi^+$ relative to the base $\Delta$. If $W$ is the Weyl group, how can I determine if $-I$ belongs to $W$? Equivalently how can I see ...
4
votes
3answers
980 views

Longest element of a Weyl group

Let $G$ an algebraic (reductive) group. $T$ a maximal torus, $B$ a Borel subgroup containing $T$, and $w_0$ the longest element of the Weyl group. I'm looking for a reference explaining why when you ...