The root-systems tag has no usage guidance.

**5**

votes

**1**answer

488 views

### Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...

**4**

votes

**0**answers

138 views

### Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...

**4**

votes

**2**answers

162 views

### References about Hasse diagrams of root systems

This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical ...

**7**

votes

**1**answer

287 views

### A technical question about root systems

I'm studying root systems and coming up with an observation:
Let $\Phi$ be an irreducible root system and $\Phi^+$ be a positive root system. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ ...

**4**

votes

**1**answer

113 views

### Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...

**2**

votes

**2**answers

258 views

### Involution of $E_{8}$ lattice

Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are ...

**2**

votes

**1**answer

150 views

### Length of Weyl group element mapping highest root to a simple root

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the ...

**0**

votes

**2**answers

154 views

### Is the restricted root system of a simple real Lie group irreducible?

As the title asks, is the restricted root system of a simple real Lie group irreducible?
I believe this is true but I need a reference to cite.

**0**

votes

**0**answers

153 views

### How to find solutions for four polynomial equations with four unknown variables using Resultant Theory

Can I use resultant theory (or polynomial resultant method) to find solutions for four simultaneous polynomial equations with four unknown variables?
So far, I could only find examples which uses two ...

**3**

votes

**0**answers

104 views

### Synonyms for “labeling” of a graph

In Preprint 1 we write numerical labels 0 or 1 at each vertex of a Dynkin diagram $D$. We call it a labeling of the graph (Dynkin diagram) $D$.
In Preprint 2 we consider an extended (affine) Dynkin ...

**1**

vote

**1**answer

59 views

### roots in a root system which have nonzero coefficients with respect to each simple root

If we consider crystallographic root systems, then for each $k$ such that $n \leq k \leq d-1$ where $d$ is the Coxeter number, it seems to be the case that there is exactly one root of height $k$ with ...

**0**

votes

**0**answers

83 views

### Equivalent definitions of positive root system

$\bullet$ I begin with a definition of positive root systems of a root system over Euclidean space.
A subset $\Delta$ of root system $\Phi$ is called a simple root system (or base) in $\Phi$ if
(1) ...

**5**

votes

**4**answers

211 views

### From a (not positive definite) Gram matrix to a (Kac-Moody) Cartan matrix

Suppose I am given a symmetric matrix $G_{ij}$ with $G_{ii} = 2$. Can I always find an invertible integer matrix $S$ such that $(S^T G S)_{ii}=2$ and $(S^T G S)_{ij} \leq 0$ for $i \neq j$? Is there a ...

**5**

votes

**2**answers

241 views

### Is this characterization of (-1)-eigenspaces of the Weyl group of $E_6$ known?

I recently needed to know which circles $S$ in a maximal torus $T^6$ of the compact exceptional group $E_6$ yield one-dimensional subspaces $\mathfrak s$ of the Lie algebra $\mathfrak t^6$ that are ...

**2**

votes

**3**answers

507 views

### Root in positive Weyl chamber

Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g} $. Let $R\subset \mathfrak{t}^*$ the set of roots. We fix $\Pi\subset R$ the set of ...

**4**

votes

**2**answers

240 views

### The action of the center on the extended Dynkin diagram

Let $R$ be an irreducible root system with a basis $\Pi$.
We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$.
Let $Q^\vee\subset P^\vee$ ...

**0**

votes

**0**answers

113 views

### Reference about a formula of coroot in an affine root system

Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that
$$
(\alpha + p\delta)^{\vee} = \alpha^{\vee} + ...

**4**

votes

**0**answers

73 views

### Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...

**5**

votes

**2**answers

308 views

### Convention about “long” roots for simple Lie algebras of types ADE?

The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in ...

**2**

votes

**1**answer

98 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

**1**answer

256 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 ...

**7**

votes

**3**answers

428 views

### Can one easily pick out a basis of a simple Lie algebra after picking a convex order?

One of the basic results of Lie theory is that if one picks a Cartan subalgebra of a simple Lie algebra, there there is a canonical decomposition of $\mathfrak{g}$ into the Cartan and a bunch of ...

**2**

votes

**2**answers

230 views

### Which linear combinations of simple roots are roots

Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram.
Then for any two simple roots ...

**0**

votes

**1**answer

81 views

### A property of compact involutions of semi-simple Lie algebras?

I need to prove the statement below. Since my background on Lie theory is rather weak, I post it here.
Let $\frak{g}$ be a complex semi-simple Lie algebra. Fix a Cartan subalgebra $\frak{h}$ with ...

**4**

votes

**3**answers

1k views

### Modular Forms and Root Systems

In the study of semisimple Lie groups, lattices appear all over the place. In the theory of elliptic functions and modular forms, (equivalence classes of) lattices correspond to elliptic curves and to ...

**4**

votes

**2**answers

335 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 ...

**9**

votes

**4**answers

922 views

### Root systems and sums of squares

It is easy to see that the quadratic form for the root system $A_n$ is a sum of $n+1$ squares of integral linear forms:
$$q_{A_n} = 2 \sum_{i=1}^n x_i^2 - 2 \sum_{i=1}^{n-1} x_i x_{i+1} =
x_1^2 + ...

**4**

votes

**2**answers

245 views

### Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.
For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...

**4**

votes

**1**answer

269 views

### The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...

**3**

votes

**2**answers

551 views

### Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).
In this paper, in page 7, it ...

**3**

votes

**2**answers

254 views

### How to find faces of polytope defined by a Weyl orbit

A few days ago I asked the following question at MSE and received no answer. I thought I would try here.
Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let ...

**2**

votes

**1**answer

188 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 ...

**7**

votes

**1**answer

135 views

### Quantum Cartan matrices and Coxeter elements

Let $\Gamma$ be a bipartite graph, with the vertices partitioned into disjoint sets $\Pi_1$ and $\Pi_2$. Let $W$ be the associated Weyl group, with Coxeter generators $\{s_i\}_{i\in \Gamma}$. Let ...

**1**

vote

**0**answers

92 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 ...

**5**

votes

**1**answer

159 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 ...

**5**

votes

**1**answer

224 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 ...

**9**

votes

**2**answers

397 views

### Extension of the Weyl dimension formula

Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...

**6**

votes

**0**answers

165 views

### Non-crystallographic cluster algebras

Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed ...

**3**

votes

**1**answer

209 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, ...

**2**

votes

**0**answers

155 views

### simple roots of a reflection subgroup

Consider a Hermitian symmetric pair of complex Lie algebras $(\mathfrak{g},\mathfrak{k})$ and split the set of roots into compact roots (i.e. roots of $\mathfrak{k}$) and noncomapt roots $\Delta = ...

**2**

votes

**1**answer

101 views

### Compute roots of sum_i c_i/(a_i + b_i x)^p

How to compute the (real) roots of
$$\sum_{i=1}^n \frac{c_i}{(a_i + b_i \cdot x)^p}$$
for given reals $a_i, b_i, c_i$, and positive integers $n, p$? The cases $p=1, ..., 5$ and $n=6, ..., 20$ would ...

**4**

votes

**3**answers

287 views

### lines through A_n reflection arrangement and permutations

(updated; apologies for way too much room left for interpretation in the original post)
Let $\mathcal{A} =A_{n-1}$ be the $A_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes ...

**3**

votes

**1**answer

707 views

### A possible mistake in Kac's “Infinite Dimensional Lie Algebras”

I have a paperback 3rd edition and on page 65 you can find Proposition 5.8. My question is about part (c):
If $A$ is of indefinite type, then
$$ \overline{X} = \{ h \in \{ \frak h_{\mathbb{R}} ...

**9**

votes

**3**answers

830 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

**2**answers

120 views

### Non-Uniform Root of Polynomial in Open Cube

I'm looking to find a root $(x_1,\dots,x_n)$ of a polynomial $p \in {\mathbb R}[x_1,\dots,x_n]$ such that $0 \leq x_i < 1$ for all $i$. Further, I know in advance that setting $x_1 = \cdots = x_n$ ...

**1**

vote

**0**answers

105 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

673 views

### dual Coxeter number, affine algebra, invariants under twisting

Sometime ago we came across invariant quantities under twisting of all affine algebra. (See the appendix E of http://arxiv.org/abs/hep-th/0403076 .) Choose the convention so that the longest root has ...

**2**

votes

**0**answers

187 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

**0**answers

329 views

### analogues of power sum polynomials for symmetric Laurent polynomials

To deal with root systems of type B C D, one needs to understand symmetric Laurent polynomials $\Lambda$. I am wondering if the naive definition of power sum symmetric Laurent polynomials form a basis ...

**5**

votes

**1**answer

343 views

### schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups

I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...