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Questions tagged [root-systems]

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-7
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4answers
796 views

$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Edited 1/21/2018 to add the following: Here is a DropBox link https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0 to a PDF showing how my team used biomolecular first ...
1
vote
1answer
57 views

Duality isomorphism of representations of the maximale torus with respect to Steinberg's basis - is it an involution?

I am trying to apply Steinberg's basis of his paper "On a theorem of Pittie" for the case $G$ of type $A_2$ and the maximale torus $T$ itself as a maximal rank subgroup. Denote by $\alpha_1, \alpha_"$ ...
16
votes
0answers
731 views

Combinatorial identity involving the Coxeter numbers of root systems

The setup is: $R$ = irreducible (reduced) root system; $D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$; $\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$; $\...
7
votes
0answers
177 views

Reference for an “elementary” combinatorial fact

This is a question I've been meaning to ask for quite some time. Fact. For $n\in\mathbb N$ consider the set of segments $R=\{[i,j], 1\le i<j\le n\}$. Let a subset $E\subset R$ be nice iff $E$ is ...
6
votes
0answers
131 views

Macdonald's “Symmetric Functions and Hall Polynomials” Section 1.5 Example 9

I'm trying to follow Example 9 in Section 1.5 of the 2nd edition of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points. Before stating my ...
6
votes
0answers
199 views

Branching rules for type B/C/D Hall-Littlewood polynomials

For a root system $\Phi$ of rank $n$ with Weyl group $W$ and a dominant integral weight $\lambda$ consider the Hall-Littlewood polynomial $$P_\lambda(x_1,\ldots,x_n;t)=\frac1{W_\lambda(t)}\sum_{w\in W}...
6
votes
0answers
118 views

Root system inside the indefinite even unimodular lattice $II_{10,2}$

I apologize for asking questions that seem likely to be answered in Conway & Sloane's "Sphere Packings, Lattices, and Groups" if I knew where to look. Let $L$ be the unique* even unimodular ...
6
votes
0answers
255 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{...
5
votes
0answers
87 views

Gram matrix determinant in dimension 4 and $E_8$

Consider a determinant of a Gram matrix in dimension $4$. $$\begin{vmatrix} 1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\ -\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
5
votes
0answers
111 views

Can the Weyl orbits of fundamental weights tell us the Cartan matrix?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\Delta$ its root system contained in $\mathfrak{t}^{\vee}$ for a Cartan sub-algebra $\mathfrak{t}$ of $\mathfrak{g}$. Let $W$ be its Weyl group....
5
votes
0answers
139 views

Kostka polynomials in root systems other than A

The q, t - Kostka polynomials $K_{\lambda\mu}(q, t)$ are defined as follows (all notations I do not explain here come from the classical book by Macdonald: Symmetric Functions and Hall polynomials, ...
4
votes
0answers
71 views

Good range and fair range

Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
4
votes
0answers
145 views

Reference request for generalized root systems

Where can I find information on root systems where the inner product is other than the standard (all positive) signature?
4
votes
0answers
151 views

$N_{w \theta} \cap wN_{\theta}w^{-1}$ is normal in $N_{w \theta}$

Let $G$ be a connected reductive group over a field $k$, $A_0$ a maximal $k$-split torus, $\Phi = \Phi(A_0,G)$, and $\Delta$ a base of $\Phi$. Let $P_0$ be a minimal $k$-parabolic subgroup containing ...
4
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0answers
195 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
0answers
84 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 ...
4
votes
0answers
228 views

root system generalizations of Sekiguchi-Debiard (aka Laplace-Beltrami) operators

For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator: $$D_\alpha(X) =...
3
votes
0answers
113 views

Cohomology of root data

Let $(X,\Phi,X^\vee,\Phi^\vee)$ be a semisimple root datum (in the sense of SGAIII), and $W_0$ its (finite) Weyl group. What is known about the cohomology groups $H^n(W_0, X^\vee)$ ?
3
votes
0answers
51 views

Self-map of a set for which the sizes of fibers of iterates are given by polynomials

I am interested in functions $f\colon X\to X$ (where $X$ is some countable set) such that for every $x \in X$ there exists a polynomial $P_x$ such that $\#(f^k)^{-1}(x)=P_x(k)$ for all $k \geq 1$. ...
3
votes
0answers
120 views

How large is the intersection of the root system of a subalgebra of a compact Lie algebra with the original root system?

Let $\mathfrak{g}$ be a finite-dimensional real compact Lie algebra and $\mathfrak{t}\subset \mathfrak{g}$ a maximal abelian subalgebra. Let $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})\...
3
votes
0answers
126 views

How do I obtain Vandermonde identity from Weyl's denominator formula?

Let $\Phi$ be a root system with Weyl group $\operatorname{Weyl}(\Phi)$, let $\Phi^+$ be a set of positive roots for $\Phi$ and $\rho$ be the half sum of the elements of $\Phi^+$. Then the Weyl's ...
3
votes
0answers
271 views

Lusztig's definition of quantum groups

In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form $U^{\mathbb{Q}(q)}_q$ that is rather different from the usual approach (see [1,Ch.9.1] for ...
3
votes
0answers
111 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 ...
3
votes
0answers
240 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
0answers
113 views

action of Weyl group element on Weyl vector

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...
2
votes
0answers
184 views

Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...
2
votes
0answers
104 views

Absolute and Relative Coroots

$G$ is a connected reductive group over a field $k$. $T$ is a maximal torus and $S \subset T$ is a maximal $k$-split torus. We have an embedding $X_*(S) \hookrightarrow X_*(T)$. Is it true that if $\...
2
votes
0answers
82 views

Kernel of some expressions in real Lie algebras

Let $\mathfrak{g}$ be a real Lie algebra and let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition of $\mathfrak{g}$. Let $\mathfrak{a}$ be maximal abelian subalgebra of $\...
2
votes
0answers
238 views

the root lattice, reflections, and a coxeter element

Question: Is is possible to realise the positive root lattice $\Phi_{\Delta}^{>0}$ (viewed as an abstract poset) of a root system $\Phi_\Delta$ associated to a Dynkin or affine Dynkin diagram $\...
2
votes
0answers
200 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
0answers
440 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 ...
2
votes
0answers
780 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 ...
1
vote
0answers
28 views

Description of real roots of Kac—Moody algebra

Let $\Delta$ be a root system associated to a generalized Cartan matrix, $\alpha_1,\ldots,\alpha_n$ its simple roots. It is known that if $\Delta$ is of finite, affine or hyperbolic type, $\alpha=\...
1
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0answers
55 views

Associated subgroup of Weyl group

Let $\Phi$ be a root system. For a weight $\lambda\in\mathfrak{h}^*$, start by defining $ \Phi_{[\lambda]}:=\{\alpha\in \Phi \ | \ \langle \lambda,\alpha^{\lor}\rangle\in\mathbb{Z} \} $ and $ W_{[\...
1
vote
0answers
56 views

Is a root system determined by the choice of an invariant lattice?

Let $\Phi$ be a root system in a real vector space $V$, and let $W = W(\Phi)$ denote its Weyl group, and let $Q = Q(\Phi) \subseteq P = P(\Phi) \subseteq V$ denote the root and weight sublattices. ...
1
vote
0answers
106 views

Question about Jantzen-Zuckerman's translation princinple

In the paper Representation type of the blocks of category $\mathcal{O}_S$ in types $F_4$ and $G_2$: Section 2.3, I quote " Assume from now on that $\mu$ is an integral weight and $\mu+\rho$ is ...
1
vote
0answers
101 views

Pairing half the sum of the roots with a simple coroot

I was calculating something with the root system $A_n$ and I think there might be a more general principle at work. Here is the example: let $G = \operatorname{GL}_5$, with maximal torus $T$ and ...
1
vote
0answers
83 views

A property of the Weyl vector of an irreducible root system

Let $R$ be one of the $A,B,C,D,E,F$ root systems, let $\Lambda$ be the lattice generated by the roots, $R^+$ a system of poitive roots and $\rho$ and $\alpha$ be the Weyl vector and the highest root ...
1
vote
0answers
78 views

Some questions about $\rho^{\vee}$ in Lie theory

Let $\mathfrak{g}$ be a semisimple Lie algebra and $I$ its vertices of Dynkin diagram. The weight $\rho$ is defined by $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, ...
1
vote
0answers
184 views

Description of the center of a reductive group using absolute and relative roots

Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...
1
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0answers
136 views

Regarding Weyl's integration formula - from “equality up to scalar” to “exact equality”

Teaching a course on representations of compact groups, I want to prove Weyl's integration formula. It seems to me that in a few sources, they prove the formula up to scalar, and don't even ...
1
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0answers
153 views

Element of the Weyl group acting on co-roots

Let $W$ be the Weyl group of a semisimple algebraic group defined over say $\mathbb C$. We have the root space $\mathfrak a^*$ spanned by the roots $\{\beta \in \Delta\}$ and also by the weights $\{ \...
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0answers
129 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 $...
1
vote
0answers
106 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 ...
0
votes
0answers
76 views

How the roots and weights changed under a folding?

Let $e_1,e_2,e_3,f_1,f_2,f_3$ be the generators of the Chevalley basis of the Lie algebra $sl_4$. Let $e_1' = e_1+e_3$, $e_2'=e_2$, $f_1'=f_1+f_3$, $f_2'=f_2$. Then the subalgebra generated by $e_1', ...
0
votes
0answers
52 views

Uniqueness of Certain Root System Elements

Suppose $\Phi$ is an irreducible classical root system with simple roots $\Delta=\{\alpha_1,\ldots,\alpha_n\}$ and consider a subset $\Delta'=\{\alpha_{k_1},\ldots,\alpha_{k_m}\}\subseteq\Delta$. ...
0
votes
0answers
434 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 ...
0
votes
0answers
163 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,...