All Questions
Tagged with rt.representation-theory root-systems
80 questions
2
votes
0
answers
126
views
A property of matrices formed by pairing of roots and coroots
Let $A$ be an $n\times n$ integral matrix, define its level $l(A)$ as
$$l(A) := \begin{cases}0 &\det A = 0 \\ \text{smallest integer } N \text{ such that } NA^{-1} \text{ is integral} &\det A \...
- 528
3
votes
1
answer
193
views
Opposite convex order on the set of positive roots of a semisimple Lie algebra
Let $\mathfrak{g}$ be a semisimple Lie algebra of rank l and let $\Delta^+$ be its set of positive roots. Denote by $s_1,...,s_l$ the simple generators of its Weyl group and let $w_0$ be the longest ...
3
votes
1
answer
108
views
Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 528
3
votes
1
answer
162
views
Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
3
votes
1
answer
142
views
Relationships between the positive cone inside a root system and the dominant Weyl chamber
Let $G$ be a reductive group and fix a choice of positive roots inside the associated root system.
My question is about the relationship between the cone spanned by $\mathbb{Z}_{\geq 0}$-linear ...
- 33
3
votes
0
answers
194
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 673
6
votes
1
answer
284
views
Multiplication factors in folding root systems and Lie algebras by automorphisms
When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$.
$$\sum_{i \in J} \alpha_i .$$
Whereas ...
- 83
2
votes
1
answer
268
views
Stabilizer of a Levi subgroup in the Weyl group and its quotient
(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive ...
- 709
1
vote
0
answers
105
views
Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
3
votes
0
answers
139
views
Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
2
votes
0
answers
159
views
The Cartan is a complex vector space but the root system is real?
Let $\frak{g}$ be a complex semisimple Lie algebra with some choice of Cartan subalgebra $\frak{h}$. The dual space $\frak{h}^* = \mathrm{Hom}_{\mathbb{C}}(\frak{h},\mathbb{C})$ is a complex vector ...
- 1,144
3
votes
2
answers
492
views
Pairing a root with the half-sum of positive roots
Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
3
votes
0
answers
152
views
Disconnected reductive algebraic groups in Sage
All simply connected split simple groups have been implement on Sage and it is possible to find their highest roots, fundamental weights, Dynkin diagrams or compute the tensor of two of their ...
- 370
1
vote
1
answer
138
views
About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$
$\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$_2$.
We consider its root system as follows (though it is probably not necessary to ...
- 2,446
2
votes
0
answers
48
views
Multiplicative invariants of non-reduced root systems
It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
- 553
2
votes
1
answer
357
views
Tensor product of fundamental representations
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...
- 391
4
votes
0
answers
210
views
Schur polynomials are polynomials but also sequences on a lattice?
Monomial symmetric polynomials in $n$ variables $x_1, \ldots x_n$ form a natural basis for the space $\mathcal{S}_n$ of symmetric polynomials in $n$ variables and are defined by additive ...
5
votes
1
answer
310
views
Non-standard partial orders on root systems
Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a ...
7
votes
1
answer
228
views
Why is the fundamental group of $\mathsf E_n$ cyclic of order $9 - n$?
Several years ago, I mentioned offhandedly to a colleague that I had noticed that, if you extend the $\mathsf E_n$ series downwards in the natural way, by removing nodes from the long arm of $\mathsf ...
- 12.9k
17
votes
2
answers
656
views
Typos in Bourbaki's root-system tables
A while ago, a colleague told me that he thought he remembered that there were typos in Bourbaki's tables in the English translation of "Groupes et algèbres de Lie", but that he could no ...
- 12.9k
5
votes
1
answer
143
views
PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$
I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.
Alternatively I ...
8
votes
0
answers
267
views
A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
- 1,856
6
votes
1
answer
255
views
A weight generalization of root systems?
For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...
- 123
2
votes
1
answer
229
views
Action of the negative Cartan-Weyl generators on a highest weight element
Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha_1, \cdots, \alpha_r$ be the corresponding choice of simple roots, $X_{\alpha_i}, ...
- 123
1
vote
0
answers
69
views
Automorphism groups of which lattices act irreducibly on the ambient Euclidean space
(I asked this question on MSE a few days ago but it hasn't drawn any response yet.)
Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider ...
- 538
9
votes
2
answers
1k
views
Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?
My questions may turn out to be related to Schur functors.
If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{...
- 5,215
4
votes
0
answers
264
views
Singular del Pezzo surfaces and degeneration of root systems
Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes
$$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...
- 1,803
3
votes
1
answer
630
views
A formula for the dual Coxeter number
Let $\Phi$ be the root system of a finite dimensional simple Lie algebra $\mathfrak g$, with dual Coxeter number $h^\vee$.
Let $\alpha_0\in \Phi$ be a long root (if all the roots have the same length, ...
- 43.2k
9
votes
2
answers
383
views
Action of Weyl group on regions of Shi arrangement
This is an elaboration of a question which was aked on MO several years ago, which was unanswered but deleted by the question-asker. I hope it is okay to elaborate on a deleted question like this; for ...
- 24.2k
3
votes
1
answer
239
views
when a set of roots extend to a system of simple roots
Given a set of roots in a root system, assume that the pairing of each two roots in this set is not positive. Then clearly the set gives a closed root subsystem. My question is, when this set extends ...
- 1,457
4
votes
1
answer
235
views
A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$
This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here.
$G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...
- 675
5
votes
1
answer
372
views
Table of products for Lie algebra inner product of roots and weights
For a simple Lie algebra $\frak{g}$, it is usual to scale the inner product so that the shortest simple root has length $2$. With this conventions, where can I find a table (online) of the following ...
- 993
5
votes
1
answer
505
views
How to determine a highest weight corresponding to a parabolic subgroup?
Let $G$ be a simply connected, semisimple algebraic group over $\mathbb C$ with maximal torus $T$ and Borel subgroup $B$ containing $T$. If $(V,\pi)$ is an irreducible representation of $G$, then $(V,...
- 6,180
8
votes
2
answers
619
views
Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity?
$\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, Lusztig's $q$-analog of weight multiplicty $K_{\lambda,...
- 24.2k
4
votes
0
answers
154
views
Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?
Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots.
We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\...
- 9,019
4
votes
1
answer
206
views
Can we have a nontrivial division of a irreducible root system as the union of two closed sub-root systems?
The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the ...
- 9,019
1
vote
1
answer
241
views
Can we have a nontrivial division of a irreducible root system as $\Phi=\Phi_{[\lambda]}\cup \Phi_{[\mu]}$?
Let $(\mathfrak{g},\mathfrak{h},\Phi)$ be a root system of a complex simple Lie algebra, where $\Phi$ is the set of all roots. For each $\alpha\in \Phi$, let $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$ be ...
- 9,019
2
votes
0
answers
68
views
Piecewise linear $\sigma_i$ - notation question
In cluster algebra framework, in order to get root clusters, a modified version of a simple reflection is used. Define $\sigma_i:\Phi_{\geq -1} \to \Phi_{\geq -1}$ by setting:
$ \sigma_i(\alpha) =
\...
- 225
4
votes
0
answers
99
views
About the geometry of the set of weights that is strongly linked to $\lambda$
Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\...
- 1,875
3
votes
2
answers
398
views
Partial ordering on $\mathfrak{h}^*$ and Bruhat ordering
In section 5.2 (p.95) of Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$.
Let $\mu\le \lambda$ if $\lambda-\mu\in \Gamma$, where $\Gamma$ is the set of $\mathbb{Z}^{\ge 0}$...
- 1,875
2
votes
0
answers
241
views
Intersection of Levi subgroups via intersection of their Weyl groups
Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T \subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
- 163
1
vote
2
answers
124
views
Confusion about $\lambda\in\mathfrak{h}^*$ such that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$
I am reading this paper: Representation type of the blocks of category $\mathcal{O}_S$
On p. 199, it said that
While on p. 183 (Section 9.2) of Representations of Semisimple Lie Algebras in the BGG ...
- 1,875
4
votes
0
answers
132
views
Panyushev's conjectured duality for root poset antichains
In his 2004 paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380), Panyushev conjectured (Conjecture 6.1) the ...
- 24.2k
6
votes
2
answers
487
views
For a fixed dominant weight $\lambda$, are almost all dominant weights in the same coset above it?
First some notation as in e.g. the book by Humphreys on Lie Algebras.
Let $E$ be an Euclidean space with inner product $(-,-)$, and denote $\langle v,w \rangle = \frac{2(v,w)}{(w,w)}$. Let $\Phi$ be ...
- 2,821
0
votes
1
answer
399
views
Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$
Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s_1\dots s_n$ ...
- 291
6
votes
3
answers
772
views
Existence of a weight of a representation in the fundamental Weyl chamber
Let $\mathfrak g$ be a complex simple Lie algebra.
Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system.
Pick a partial order on $\mathfrak h$, ...
- 2,446
6
votes
2
answers
1k
views
Non-faithful irreducible representations of simple Lie groups
For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
...
- 835
5
votes
0
answers
140
views
$q$-Kostant partition function and flow polytopes?
The Kostant partition function is known to be related to volumes and Ehrhart polynomials of flow polytopes of graphs (see e.g. https://link.springer.com/article/10.1007/s00031-008-9019-8 or https://...
- 24.2k
1
vote
1
answer
185
views
Duality isomorphism of representations of the maximal 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" (MSN) for the case $G$ of type $A_2$ and the maximal torus $T$ itself as a maximal rank subgroup. Denote by $\alpha_1, \...
- 21
2
votes
2
answers
1k
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}...
- 6,180