# Tagged Questions

**2**

votes

**1**answer

158 views

### Which subgroups of a finite reflection group have distingushed coset representatives?

Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of ...

**4**

votes

**2**answers

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 ...

**5**

votes

**1**answer

135 views

### A subgroup of the Weyl group

Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2.
Let $Q=Q(D)$ denote the root lattice of $D$.
Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...

**4**

votes

**1**answer

132 views

### One-parameter subgroups of symplectic group associated to roots

I'm having trouble sorting out some basic definitions concerning Chevalley groups. The groups I'm interested in are the simply connected groups of type $C_n$, so the groups $\text{Sp}_{2n}$. The ...

**3**

votes

**1**answer

161 views

### Name for a class of parabolic subgroups

This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:
Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, ...

**9**

votes

**3**answers

703 views

### Kostant partition function: asymptotics and specifics

Let $\Phi$ denote a root system and let $\mathfrak P$ denote the associated Kostant partition function. Thus $\mathfrak P(\lambda)$ is the number of ways of writing $\lambda$ as a sum of elements of ...

**1**

vote

**0**answers

102 views

### Reference request: a verification of a nonstandard subgroup being a Tits subgroup.

I have a particular infinite-index subgroup $H$ of the genus 2 symplectic group $Sp(2, \mathbb{R})$. This subgroup is self-normalizing (ie. $gHg^{-1}=H$ only if $g\in H$). I am looking to determine ...

**2**

votes

**0**answers

163 views

### Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups

If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...

**2**

votes

**2**answers

562 views

### Possible Borel subgroups of GL_n?

I am trying to understand the interaction between Borel subgroups of $GL_n$ and its roots. Is it correct to say that for any choice of roots among each pair of reciprocal roots
there is a Borel ...

**3**

votes

**1**answer

320 views

### Applications of Chevalley groups theory for dummies

As an algebraist i frequently receive questions from my friends-mathematicians and non-mathematicians about applications of my topic "in real life". I study algebraic groups in the stream of ...

**4**

votes

**3**answers

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 ...