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7
votes
1answer
287 views

Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
0
votes
1answer
123 views

Bounds on the smallest real positive root of a polynomial

I'm trying to find upper and lower bounds of the smallest positive root of a polynomial, stated in terms of its coefficients. As I appreciate it might be a very general problem, My specific interest ...
6
votes
1answer
97 views

Involutions and Little Adjoint Representations of Simple Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
5
votes
1answer
510 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
0answers
146 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
2answers
173 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
1answer
292 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
1answer
114 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
2answers
261 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 isomorphic)...
2
votes
1answer
163 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
2answers
169 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
0answers
171 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
0answers
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
1answer
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
0answers
90 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
4answers
217 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
2answers
244 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
3answers
582 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 ...
5
votes
2answers
253 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
0answers
120 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} + \frac{2p}{(\alpha,...
4
votes
0answers
75 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
2answers
325 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 Bourbaki'...
2
votes
1answer
99 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
260 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
3answers
431 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 1-...
2
votes
2answers
270 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
1answer
82 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
3answers
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
2answers
346 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
4answers
925 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
2answers
254 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
1answer
276 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 ...
4
votes
2answers
585 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
2answers
260 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
1answer
189 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
1answer
137 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
0answers
99 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
1answer
164 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
1answer
238 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
2answers
407 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
0answers
187 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 $(\mathbf{...
4
votes
1answer
219 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
0answers
158 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
1answer
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
3answers
288 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 $H_{ij}...
3
votes
1answer
710 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
3answers
843 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
2answers
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
0answers
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
0answers
690 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 ...