Questions tagged [root-systems]
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256
questions
25
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889
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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 ...
22
votes
3
answers
2k
views
Number of triples of roots (of a simply-laced root system) which sum to zero
In a paper 1105.5073, the authors took a simply-laced root system $\Delta$ of type $G=A,D,E$, and then counted the number of unordered triples $(\alpha,\beta,\gamma)$ of roots which sum to zero: $\...
20
votes
1
answer
876
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 ...
17
votes
0
answers
902
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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 $0,1,2,...,r$;
$\...
16
votes
2
answers
541
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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 ...
15
votes
1
answer
1k
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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 ...
13
votes
3
answers
898
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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-...
13
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2
answers
1k
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Significance of half-sum of positive roots belonging to root lattice?
Let $\Phi$ be a (crystallographic) root system and $\Phi^{+}$ a choice of positive roots, with $\Delta$ the corresponding choice of simple roots. So the root lattice of $\Phi$ is just $\mathbb{Z}\...
13
votes
1
answer
527
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,\...
12
votes
2
answers
469
views
Lattice structure in the root poset
Let $W$ be a Coxeter group with simple generators $s_1$, $s_2$, ..., $s_r$. Let $\Phi^+$ be the corresponding positive root system, with $\alpha_i$ the positive root corresponding to $s_i$. Bjorner ...
11
votes
1
answer
736
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Motivation behind Kac's notation for affine root systems
I'm reading Kac's Infinite Dimensional Lie Algebras. In Chapter 4, he classifies the affine root systems. Bourbaki classified the affine Coxeter groups, but multiple root systems can give the same ...
11
votes
1
answer
308
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Generalized root systems and reflection groups
Consider the following alternative definition of finite reflection group:
Definition: A finite reflection group $\Gamma\subset\mathrm O(\Bbb R^d)$ is a finite group generated by orthogonal ...
10
votes
1
answer
448
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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(\...
10
votes
1
answer
374
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 ...
10
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1
answer
224
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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 ...
10
votes
0
answers
397
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Gram matrix determinant in dimension 4 and $E_8$
Consider a determinant of a Gram matrix in dimension $4$.
$$\begin{vmatrix}
1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\
-\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
9
votes
4
answers
1k
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Root systems and sums of squares
It is easy to see that the quadratic form for the root system $A_n$ is a sum of $n+1$ squares of integral linear forms:
$$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 + (...
9
votes
2
answers
572
views
Number of reduced decompositions of the longest element of the Weyl group
Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w_0 \in W$ the longest element of $W$. In general $w_0$ admits more than one reduced decomposition into a product of reflections, ...
9
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2
answers
1k
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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{...
9
votes
1
answer
679
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 (...
9
votes
2
answers
611
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 ...
9
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3
answers
1k
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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 $\...
9
votes
1
answer
507
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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\...
9
votes
1
answer
925
views
Technical lemma on root systems, reduced to linear algebra
Update: I have posted the case of $G = SL(n)$ as a different question here.
This is a technical lemma I am currently stuck at. Any suggestions about how to proceed are welcome.
Let $G$ be a split ...
9
votes
1
answer
294
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 ...
9
votes
0
answers
91
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A characterization of root systems via their intersections with halfspaces
In a recent preprint I obtained a nice characterization of root systems as a side product.
I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
8
votes
3
answers
622
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-...
8
votes
1
answer
745
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Non-reduced, non-crystallographic root systems
Let $V$ be a $n$-dimensional real vector space with standard inner product $(\cdot,\cdot)$. For any $\alpha \neq 0 \in V$, set $\alpha^\vee := \frac{2}{(\alpha,\alpha)}\alpha$. For $\alpha \neq 0,\...
8
votes
2
answers
566
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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,...
8
votes
1
answer
372
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Motivation behind Panyushev's "constant-averages-along-orbits" conjecture
In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594, Dmitri Panyushev discusses an interesting self-map on the set of antichains of a finite ...
8
votes
1
answer
401
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 ...
8
votes
0
answers
257
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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 ...
7
votes
3
answers
3k
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Longest element of a Weyl group
Let $G$ an algebraic (reductive) group. $T$ a maximal torus, $B$ a Borel subgroup containing $T$, and $w_0$ the longest element of the Weyl group.
I'm looking for a reference explaining why when you ...
7
votes
2
answers
1k
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Definition of $\textrm{GSpin}_{2n}$ and its root datum
I'm trying to get my hands on the general spin group $G = \textrm{GSpin}_{2n}$. It have seen it mentioned as a connected, reductive group whose derived group is $\textrm{Spin}_{2n}$, which is the ...
7
votes
1
answer
396
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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 ...
7
votes
1
answer
331
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Relation between different $E_8$ matrices
There are several rank-8 square matrices known to be related to $E_8$:
Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix
$$M_1=\left [\begin{array}{rr}
2 & -1 &...
7
votes
1
answer
215
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 ...
7
votes
1
answer
2k
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Does the classification of reductive groups follow from that of semisimple groups?
I had a question for anyone familiar with the proofs of the classification of reductive groups. I skipped most of the details of classification when I originally learned linear algebraic groups, and ...
7
votes
2
answers
489
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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 ...
7
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1
answer
466
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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 $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ ...
7
votes
1
answer
267
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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 $\{...
7
votes
1
answer
418
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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
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0
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675
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Is this construction related to the geometric Langlands program perhaps?
Given a complex Lie algebra $\mathfrak{g}$, a choice of Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a dominant integral weight $\lambda$ of $\mathfrak{g}$, there is a natural construction ...
7
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0
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331
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Why are fundamental weights denoted by omega?
In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...
7
votes
0
answers
173
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Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?
For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
7
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0
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188
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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
3
answers
1k
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Does -I belong to Weyl group?
Let $\Phi$ be an irreducible root system, with positive roots $\Phi^+$ relative to the base $\Delta$.
If $W$ is the Weyl group, how can I determine if $-I$ belongs to $W$? Equivalently how can I see ...
6
votes
3
answers
735
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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 $\mathfrak h$, ...
6
votes
2
answers
1k
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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.
...
6
votes
4
answers
1k
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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 ...