# Questions tagged [root-systems]

For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.

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### On axiomatic definition of affine root systems

In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite ...
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### Set of roots such that sum of them with given positive root is not a root

Let $\Phi$ be a indecomposable root system of rank greater than 1 and fix a positive root $\alpha$. How can one describe the set $B = \{ \beta \in \Phi \;| \; \alpha + \beta \notin \Phi \}$? It's ...
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### The effect of a Cayley transform on a Cartan subalgebra

I'm trying to understand equation 6.65b from Knapp's 'Lie groups,' 2ed. Setup: Let $\mathfrak{g}_0$ be a real semisimple Lie algebra with an involution $\theta$. Let $B$ be a bilinear, symmetric, non-...
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### Root system for Lie algebras: If $\alpha, \beta, \alpha - \beta \in \Phi$ are all roots, is $(\alpha, \beta) > 0$?

Exercise 8.11 of Humphreys' Introduction to Lie Algebras and Representation Theory asks to prove that if $\alpha, \beta \in \Phi$ and $(\alpha,\beta) > 0$, then $\alpha - \beta \in \Phi$ (...
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### Angles of the Fundamental Alcove (Chamber?)

I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
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### Reflection Group of Type $D_n$

Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups: ($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
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### Reflection Group of Type $C_n$

In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way: Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
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### Explicit matrices $h_\alpha$ that correspond to the long roots $\alpha$ in a classical compact simple Lie algebra over the reals

While it is easy to find many sources that give expressions for the (co)roots of an abstract root system, it is less easy to find a reference that gives explicit matrices that are the "coroots&...
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### How are weights and roots connected? (In the context of the semisimple Lie algebras)

I am studying representations of complex semisimple Lie algebras, their root system, weight spaces etc. I am a very beginner so this question is about relating key definitions to one another. Question ...
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### Reduced adapted expression for the longest element given an orientation for a Dynkin graph

In Kirillov's book "Quiver representation and quiver varieties" in page 45 there's a Theorem 3.33 that he say is due to Lusztig that says: Given an orientation $\Omega$ of a Dynkin graph $Q$,...
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### About imaginary root of a symmetrizable Kac Moody algebra

The question is from Infinite dimensional Lie algebras by Victor G Kac (ex 5.17, Page 74) Let $\mathfrak{g}=\mathfrak{g}(A)$ be a symmetrizable Kac Moody algebra and let $\Delta$ be its set of roots. ...
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### Difference of two simple roots is zero (?)

I am studying simple Lie algebras and, on page 129 of the book "Group Theory A physicist's Survey", by Pierre Ramond, it is claimed at the bottom of the page that << Hence the ...
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### Determining the Weyl group from a given root system

1. Definitions For $V$ a vector space over $\mathbb C$ we call a subset $R \subset V$ an abstract root system if: (1) The set $R$ is finite, spans $V$ and $0 \notin R$. (2) For every $\alpha$ in $R$ ...
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### Intuition behind calculating weight systems using SU(2) strings of roots?

I am in a weird sitution in my Groups course where I can 'blindly follow the rules' in, for instance, finding a weight system from its highest weight by representing the weights in their Dynkin basis, ...
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### How to derive the mentioned relations (in part a-b-c of example 9) from the Weyl denominator formula?

I want to understand Example 9 (a-b-c) of section 1.5 of the book "Symmetric functions and Hall polynomials (https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)" (pages 78-79): In ...
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### A simple question on root systems

I read the Wikipedia page about root systems. It is very easy to verify that the 2 following images are showing some root systems. https://en.wikipedia.org/wiki/Root_system These 2 pictures are from ...
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### System of representatives in reflection groups and subgroups

I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
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### An Exercise on Abstract Weight Theory (Humphrey Exercise 13.8)

The exercise goes like this: Let $\Phi$ be an irreducible root system. Prove that each $\lambda_i$ is of the form $\sum_j q_{ij}\alpha_j$, where all $q_{ij}$ are positive rational numbers. Here, as ...
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### What does it mean for a root to be an element of the Lie algebra?

My understanding of roots in Lie algebra theory is as follows: given a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ with basis $\{H_i\}_{i=1}^{r}$, and root vectors $\{X_\alpha\}_\alpha$, the ...
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### Condition for a sum of images of fundamental dominant weights to lie on a wall

Let $\Delta$ be a base of a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of $W$. I would like ...
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### Cartan integers are preserved by isomorphism

I have this proposition and the demonstration, but there is some parts I don't understand, $E$ is a euclidean space and $\Phi$ is a root system with base $\Delta$. Proposition: Let $\Phi' \subset E'$ ...
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### I don´t understand root systems

I don´t understand root systems. The Wikipedia (and my university lectures) say it is some configuration of vectors with certan properties. The root vectors should span the whole space, which I ...
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### A question about roots and root spaces of semisimple Lie algebras

In the book "Lie Groups: Beyond and Introduction" by Anthony Knapp, the author describes roots of a semisimple (A Lie algebra without any non-zero solvable ideal) Lie algebra as elements of ...
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### A description of the $A_n$-root system in terms of $\mathbb{R}^{n}$

The root system $(V,\Delta)$ of type $A$ is usually defined as a subset of $\mathbb{R}^{n+1}$: $$\{ e_i - e_j ;\, 1 \leq i,j \leq n+1,\, i \neq j\}.$$ However these vectors span an $n$-dimensional ...
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### Dominance of $w\mu$ for dominant cocharacter $\mu$

NOTE: The question has now been posted on MathOverflow: Dominance of $w\mu$ for dominant cocharacter Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ ...
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### Dominance order on cocharacter group $X_*(T)$

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. By $X_*(T)$ we denote ...
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### Relation between product of reflections and angle

I am currently reading Chapter 3 Root Systems of John Humphreys book on Lie Algebras. It's known that Weyl Group $W$ is generated by set of reflections. If I consider an arbitrary element of Weyl ...
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Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider the finite group $$\mathrm{Aut}(L) = \{ f \in \mathrm{O}(V) \mid f(L) = L \},$$ ...