Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\ …

Consider the group $G = GL(\overline{\mathbb{F}_p}^n)$, and the Torus $T$ of diagonal matrices. Then the Weyl-Group $W = N_G(T)/T$ is isomorphic to $S_n$. Now there is a correspond …

Let $\Gamma$ be a Coxeter group on some generating set $S$, with reflection representation $V$. Then $\Gamma$ has two standard partial orders, the weak and strong Bruhat orders. M …

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 take …

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan suba …

Let $G$ be a semisimple algebraic group over an algebraically closed field, and let $T$ be a torus in $G$. If $T$ is a maximal torus, then $N_G(T)/Z_G(T)=N_G(T)/T$ is the Weyl gro …

I am looking for an automated way to draw diagrams of intervals in Weyl groups and in their various subsets such as minimal representatives of cosets $W/W_p$ for a parabolic Weyl g …

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$ …

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 $W$? Equivale …

Does anyone know a reference for the construction of cell modules for the Weyl group of type $D$, without any reference to the Weyl group of type $B$? What would be even better is …

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 expl …

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally c …

The Weyl Character formula tells us how to write the character of a representation as a linear combination of integral weights. Since characters are invariant under the action of t …

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 …

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 reduc …