# Questions tagged [root-systems]

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### Lie algebras, root systems and qubits

This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group ...
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### Numerical method for mixed system of equations and nonlinear inequalities

I am currently encountering challenges in determining the solution method for the following system of equations and inequalities: \begin{aligned} &F(x) = 0\\ &G(x) < 0\\ \end{aligned} ...
1 vote
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### Ultra-operations numbers (polynomials) [closed]

After Bring's root article, I became interested in understanding the theory of ultra numbers and their operations. There are very few vague concepts about these numbers on the Internet. I would be ...
103 views

### Calculating relative root systems

Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. ...
1 vote
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### Linear independence of reciprocals of products of closed sets of roots in type $A$ inversion sets

Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a ...
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### Describing characters of a reductive group in terms of characters of a maximal torus

Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-...
1 vote
104 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 ...
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 $(\... 7 votes 1 answer 319 views ### Relation between different$E_8$matrices There are several rank-8 square matrices known to be related to$E_8$: Cartan$E_8matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix $$M_1=\left [\begin{array}{rr} 2 & -1 &... 7 votes 0 answers 649 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 ... 2 votes 0 answers 100 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 ... 2 votes 2 answers 261 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 141 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 ... 1 vote 1 answer 120 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 ... 1 vote 1 answer 90 views ### Problem in understanding a fact about Belavin-Drinfeld triple A Belavin-Drinfeld triple associated to a simple Lie algebra L is a triple (\Gamma_1, \Gamma_2, \tau) where \Gamma_1, \Gamma_2 \subseteq \Gamma (\Gamma is a set of simple roots or fundamental ... 0 votes 1 answer 107 views ### Sub-coroot lattices [This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ] Let T be a maximal torus of a compact Lie group K, and let \Lambda \subset {\mathfrak t} be the ... 3 votes 1 answer 140 views ### Sub-coroot systems Let T be a maximal torus of a compact Lie group K, and let \Psi \subset {\mathfrak t} be the (finite) set of coroots for (K,T), where \mathfrak t is the Lie algebra of T. Assume now that ... 2 votes 0 answers 44 views ### Multiplicative invariants of non-reduced root systems It is a well known fact (cf.  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 ... 2 votes 1 answer 172 views ### Find an analogue of Weyl chamber structure Let G be a split reductive group and let T be a split maximal torus whose rank is l. Is it possible to find a base \gamma_1,..., \gamma_l of the weight lattice X(T) such that the cone C in ... 2 votes 1 answer 77 views ### Real roots along root strings Let A be a Cartan matrix, i.e. a n\times n matrix with integer entries such that A_{ii}=2 and A_{ij}\leq0 for i\neq j. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra \... 2 votes 1 answer 244 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-... -1 votes 1 answer 92 views ### finding positive roots for a polynomial [closed] I have a polynomial, and I want to get the conditions for the number of positive roots What are the different methods out there to determine these conditions? this is the polynomial: f(g)=A1g^5 + A2g^... 4 votes 0 answers 178 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 254 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 ... 2 votes 0 answers 46 views ### A construction of Weyl-equivariant maps from the space of regular Cartan triples to the space of tuples of complex polynomials (up to scalar factors) Let G be a compact semisimple Lie group and let T be a maximal torus in G. On the Lie algebra level, we have a real Lie algebra \mathfrak{g} and a (particular) real slice, say \mathfrak{t}, ... 0 votes 0 answers 35 views ### Intersection of certain subsets in a split connected reductive group G related to affine open cover of G/B Let k be a field of characteristic zero and G a split connected reductive group over k. Moreover, let T be a split maximal torus of G and B\supset T a Borel subgroup. Additionally, we ... 7 votes 1 answer 211 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 ... 16 votes 2 answers 469 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 ... 2 votes 0 answers 139 views ### Root systems and subroot systems Given the root system E_{6} with basis \alpha_{1},\dotsc,\alpha_{6}. How would I find all subroot systems of E_{6} (up to Weyl equivalence) where I can write the basis of each subroot system in ... 2 votes 0 answers 226 views ### The Weyl dimension formula for fundamental weights The Weyl dimension formula is an equation to calculate the dimension of a simple \frak{g}-module V_{\lambda}, of highest weight \lambda, for \frak{g} a complex semisimple Lie algebra. ... 4 votes 0 answers 88 views ### Interpretation of the coefficients in the sum of positive roots Take a finite Cartan datum with index set I, simple roots \{\alpha_i\mid i \in I\} and positive roots \Phi^+. Let 2\rho=\sum_{\alpha\in\Phi^+}\alpha be the sum of the positive roots and write ... 5 votes 1 answer 125 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 ... 6 votes 1 answer 183 views ### Order ideals of positive root systems and avoiding group elements in the Weyl group Let X be the poset of positive roots of a finite root system of Dynkin type Q. Question 1: In Dynkin type A_n, is it true that the poset of order ideals of X is isomorphic to the poset of [2,... 9 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 ... 1 vote 1 answer 294 views ### The principal congruence subgroup of the symplectic group over the integers Consider the symplectic group \text{Sp}_{2g}(\mathbb{Z}) over the integers. It has a classical root system C_g and associated root subgroups U_\varphi for \varphi\in C_g. These subgroups are ... 3 votes 0 answers 81 views ### How to determine sublattices S of a root lattice R Let R be a root lattice of a irreducible root system \Phi. Suppose W is a Weyl group of \Phi and S is a sublattice of R which is W-stable and satisfies |R/S|<\infty. For example, ... 1 vote 1 answer 200 views ### A nice/simple relationship between the Chevalley generators of \mathfrak{sp}_n and the Chevally generators of \mathfrak{sl}_n? The Lie algebra \mathfrak{sl}_n is defined to be the trace free matrices in M_n(\mathbb{C}). The Lie algebra \mathfrak{so}_n is defined to be the matrices A in M_n(\mathbb{R}) satisfying A +... 4 votes 1 answer 170 views ### Structure of the permutation groups acting on the root systems of Niemeier lattices of type A_{k}^n I have been doing research on the Niemeier lattices with root systems of type, A_{k}^n and I am particularly interested in the finite groups permuting the constituent root systems. These groups ... 1 vote 0 answers 221 views ### Condition for a sum of images of fundamental dominant weights to lie on a wall Let \Delta be a system of simple roots in a root system with Weyl group W. For \alpha\in\Delta, let \varpi_\alpha be the corresponding fundamental dominant weight. Let w\neq r be elements of ... 4 votes 0 answers 96 views ### Why is the A-series root system best written in a vector space of one dimension higher? In the classification of root systems, we have four families A_n,B_n,C_n, and D_n, and six exceptionals E_6,E_7,E_8, F_4, and G_2. For every non-exceptional case except A_n, the root system ... 3 votes 1 answer 344 views ### Conjugation of root subgroups by the Weyl group Fix a field k of characteristic zero, and let G be a connected reductive algebraic k-group of isotropic rank \ge 1. Fix a maximal k-split torus S, and let \Phi_k be the relative root ... 2 votes 0 answers 187 views ### Errata in N. A'Campo's "Tresses, monodromie et le groupe symplectique" There are many small mistakes in this article. A great amount of them are concentrated in Lemma 2. The setup for this lemma is the following. Let R be a commutative ring and n=2g+1 or n=2g a ... 4 votes 1 answer 189 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 ... 2 votes 1 answer 213 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}, ... 3 votes 0 answers 144 views ### The group of fixed points of an involution of a Weyl group Let R be a reduced root system in a vector space V over \mathbb Q. Let W=W(R) denote its Weyl group. Let S\subset R be a basis of R (a system of simple roots). Let D=D(R,S) denote the ... 1 vote 0 answers 59 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 ... 2 votes 0 answers 119 views ### Existence and uniqueness of solution of a nonlinear system I need a proof of the following result to calculate a Nash equilibrium in the Showcase Showdown game. For all n>1, the system of equations$$\left\{ \begin{aligned} (1+e^{x}(-1+x))^{n-2}&=\... 3 votes 1 answer 136 views ### Action of split torus on positive root spaces LetG$be a connected reductive group over a field$k$(not necessarily algebraically closed). Let$S$be a maximal split torus in$G$with relative root system$\Phi = \Phi_k(S,G)$. Let$\Phi^+$... 9 votes 2 answers 1k views ### Which representations of$\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{...
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\},\$...