Questions tagged [root-systems]

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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 ...
მამუკა ჯიბლაძე's user avatar
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: $\...
Yuji Tachikawa's user avatar
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 ...
Joel Kamnitzer's user avatar
17 votes
0 answers
902 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 $0,1,2,...,r$; $\...
Jeffrey Adams's user avatar
16 votes
2 answers
541 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 ...
LSpice's user avatar
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15 votes
1 answer
1k 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 ...
ಠ_ಠ's user avatar
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13 votes
3 answers
898 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-...
Vincent's user avatar
  • 2,437
13 votes
2 answers
1k views

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}\...
Sam Hopkins's user avatar
  • 22.5k
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,\...
Sam Hopkins's user avatar
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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 ...
David E Speyer's user avatar
11 votes
1 answer
736 views

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 ...
David E Speyer's user avatar
11 votes
1 answer
308 views

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 ...
M. Winter's user avatar
  • 12.5k
10 votes
1 answer
448 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(\...
David E Speyer's user avatar
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 ...
Sam Hopkins's user avatar
  • 22.5k
10 votes
1 answer
224 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 ...
Nathan Reading's user avatar
10 votes
0 answers
397 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)& -\...
Daniil Rudenko's user avatar
9 votes
4 answers
1k 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 + (...
VA.'s user avatar
  • 12.9k
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, ...
Bas Winkelman's user avatar
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{...
Malkoun's user avatar
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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 (...
Daniil Rudenko's user avatar
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 ...
emiliocba's user avatar
  • 2,279
9 votes
3 answers
1k 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 $\...
David Stewart's user avatar
9 votes
1 answer
507 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\...
Gro-Tsen's user avatar
  • 29.8k
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 ...
Abhishek Parab's user avatar
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 ...
Sam Hopkins's user avatar
  • 22.5k
9 votes
0 answers
91 views

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 ...
M. Winter's user avatar
  • 12.5k
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-...
Ben Webster's user avatar
  • 43.9k
8 votes
1 answer
745 views

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,\...
Sam Hopkins's user avatar
  • 22.5k
8 votes
2 answers
566 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,...
Sam Hopkins's user avatar
  • 22.5k
8 votes
1 answer
372 views

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 ...
James Propp's user avatar
  • 19.4k
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 ...
Ravi Jagadeesan's user avatar
8 votes
0 answers
257 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 ...
Cheng-Chiang Tsai's user avatar
7 votes
3 answers
3k views

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 ...
th.ng's user avatar
  • 311
7 votes
2 answers
1k views

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 ...
D_S's user avatar
  • 6,100
7 votes
1 answer
396 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 ...
Cehiju's user avatar
  • 81
7 votes
1 answer
331 views

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 &...
Марина Marina S's user avatar
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 ...
LSpice's user avatar
  • 11.2k
7 votes
1 answer
2k views

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 ...
D_S's user avatar
  • 6,100
7 votes
2 answers
489 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 ...
jdc's user avatar
  • 2,984
7 votes
1 answer
466 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 $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ ...
user's user avatar
  • 287
7 votes
1 answer
267 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 $\{...
Tony Licata's user avatar
7 votes
1 answer
418 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 ...
Mike Pierce's user avatar
  • 1,149
7 votes
0 answers
675 views

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 ...
Malkoun's user avatar
  • 4,981
7 votes
0 answers
331 views

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 ...
Igor Makhlin's user avatar
  • 3,493
7 votes
0 answers
173 views

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 ...
Domagoj Hranjec's user avatar
7 votes
0 answers
188 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 ...
Igor Makhlin's user avatar
  • 3,493
6 votes
3 answers
1k views

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 ...
1df5e76's user avatar
  • 63
6 votes
3 answers
735 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 $\mathfrak h$, ...
emiliocba's user avatar
  • 2,279
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. ...
Nadia SUSY's user avatar
6 votes
4 answers
1k 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 ...
Alexander's user avatar
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