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Questions tagged [root-systems]

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5
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0answers
106 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....
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 $\alpha_1, \alpha_"$ ...
1
vote
0answers
54 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_{[\...
3
votes
1answer
48 views

Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
0
votes
2answers
177 views

Some confusion about weights and roots in parabolic root systems

I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...
2
votes
2answers
89 views

Definition of the weight lattice for a nonreduced root system

Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee}...
2
votes
0answers
110 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 ...
6
votes
0answers
127 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 ...
4
votes
1answer
90 views

Reference for parabolic root systems

Let $G$ be a connected reductive group with maximal split torus $A_0$, and $P = MN$ a parabolic subgroup with Levi $M$ containing $A_0$. Let $A_M$ be the split component of $\mathfrak a_M^{\ast} = X(...
7
votes
1answer
285 views

Subtori of groups of type E6

Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_{\bar{k}}$ is a semisimple algebraic group in the usual sense), and ...
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. ...
5
votes
1answer
145 views

How big can the index inside the root lattice of the lattice generated by a subset of roots be?

Let $\Phi$ be an irreducible crystallographic root system in a Euclidean vector space $V$. Let $S\subseteq \Phi$ be some subset of roots for which $\mathrm{Span}_{\mathbb{R}}(S)=V$. Question: How big ...
1
vote
0answers
105 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 ...
3
votes
2answers
181 views

Each $w\in W$ can be expressed as product of distinct reflections?

For a Weyl group $W$, I would like to know whether each $w\in W$ can be expressed as $w=s_{\alpha_1}s_{\alpha_2}\cdots s_{\alpha_k}$ for some distinct positive roots $\{\alpha_1, \alpha_2, \cdots, \...
7
votes
1answer
215 views

Del Pezzo surfaces and Picard-Lefschetz theory

Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (...
2
votes
0answers
182 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\...
5
votes
4answers
335 views

Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group

Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This ...
1
vote
0answers
97 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 ...
0
votes
0answers
71 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', ...
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 ...
9
votes
1answer
214 views

How many facets does the convex hull of all the roots of a root system have?

Let $V$ be an $n$-dimensional Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$ and $\Phi$ an irreducible crystallographic root system in $(V,\langle\cdot,\cdot\rangle)$. Question ...
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vote
0answers
80 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
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0answers
76 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 $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, ...
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 $H^n(W_0, X^\vee)$ ?
0
votes
1answer
146 views

Finding closed form expression for the roots of $f(x) = \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}$

Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows: \begin{align} f(x)\triangleq \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}, \end{align} ...
21
votes
1answer
516 views

Are (semi)simple Lie groups some sort of “homotopy quotient groups” of their maximal tori?

Warning: non-specialist writing, some rubbish possible. The formula $h^*(BG)\cong h^*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal ...
5
votes
2answers
185 views

Difference of adjacent dominant weights is a root?

The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 ...
13
votes
1answer
418 views

Minuscule weights of parabolic sub-root systems are not far from dominant

Let $\Phi$ be a crystallographic root system in an $n$-dimensional Euclidean vector space $(V,\langle\cdot,\cdot\rangle)$. For a root $\alpha\in \Phi$ we use $\alpha^\vee := \frac{2}{\langle \alpha,\...
5
votes
1answer
249 views

Realizing root-system roots as polynomial roots without Lie theory

The vectors of a root-system were originally called "roots" because they are the zeros of a characteristic polynomial that comes from the connection of (crystallographic) root-systems to classifying ...
7
votes
1answer
206 views

Shortest vectors in a root lattice

Let $R$ be a simply-laced root system in a Euclidean vector space $E$, with inner product normalized so that every root has length $\sqrt{2}$. Let $L \subseteq E$ be the lattice spanned by $R$. Is ...
1
vote
1answer
75 views

Reduced decomposition for Weyl group elements which support a Bessel function

Let $\Delta$ be a set of simple roots for a reduced root system, and let $(W,S)$ be the associated Coxeter system, where $W$ is the Weyl group and $S$ is the set of simple reflections corresponding to ...
10
votes
1answer
156 views

Dominance relation among Cartan matrices implies containment of root systems: Is this known?

Suppose $A$ and $A'$ are symmetrizable (generalized) Cartan matrices, in the sense of Kac's book Infinite-dimensional Lie algebras. Say $A$ dominates $A'$ if every entry of $A$ has weakly greater ...
2
votes
1answer
124 views

Dual Coxeter Number for Superalgberas

I am looking for a reference that gives the definition and has summarized the dual Coxeter number for superalgebras, especially for $\mathfrak{u}(m|n)$ (the Lie algebra of unitary supergroup $U(m|n)$)....
-7
votes
4answers
792 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 ...
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 $\Delta'=\{\alpha_{k_1},\ldots,\alpha_{k_m}\}\subseteq\Delta$. ...
-1
votes
1answer
131 views

Question about proof of positive roots under reflection

Since I did not receive a lot of responses on Math Stack Exchange I would like to repost this question here. Let $(W, S)$ be a finite Coxeter system. Furthermore, let $V$ be a real vector space with ...
8
votes
1answer
260 views

Why is the root poset is graded by height?

Let $\Phi$ be a finite crytallographic root system. Let $\Phi^+$ be the positive roots and $\alpha_1$, ..., $\alpha_n$ be the simple roots. For $\beta = \sum c_i \alpha_i$ in $\Phi^+$, we define $h(\...
3
votes
0answers
50 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
119 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})\...
1
vote
0answers
169 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 $Z(G)$, ...
4
votes
0answers
144 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?
2
votes
0answers
103 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 $\...
6
votes
1answer
260 views

What is the square of the Weyl denominator?

Let $\Phi$ be a (crystallographic) root system with Weyl group $\mathcal{W}$, and $\Phi^+$ a choice of positive roots, and $$ q := \prod_{\alpha\in\Phi^+} (\exp(\alpha/2) - \exp(-\alpha/2)) = \sum_{w\...
20
votes
1answer
626 views

Curious fact about number of roots of $\mathfrak{sl}_n$

The Lie algebra $\mathfrak{sl}_n $ has many special features which are not shared by other simple Lie algebras, for example all of its fundamental representations are minuscule. I recently discovered ...
4
votes
0answers
150 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 ...
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 ...
4
votes
1answer
151 views

The image of a base of absolute roots is a base of relative roots

$G$ is a connected, reductive group over a field $k$, $S$ is a maximal $k$-split torus of $G$, and $_k \Phi = \Phi(S,G)$ is the set of roots of $S$ in $G$. Equivalently, $_k \Phi$ is the restriction ...
13
votes
3answers
664 views

Where does the really nice '8-dimensional' description of the $E_7$ root system come from?

The Wikipedia page on $E_7$ tells me: Even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7-dimensional subspace of an 8-...
-1
votes
0answers
44 views

Solving Fractional Nonlinear Schroedinger Equation (System of Nonlinear Equations)

I also posted this on Math StackExchange last night. I'm solving the fractional nonlinear (time-independent) Schroedinger equation of the form $$\frac{1}{2}u-\frac{1}{2}\frac{\partial ^{\alpha}u}{\...
1
vote
1answer
201 views

Definition of Affine Root System: What is $\alpha_{+}$?

I was beginning to read Bruhat and Tits article Groupes Reductifs sur un Corps Locale and was confused on a point in the beginning of the section on affine root systems. $\mathbf{A}$ is a finite ...