A Euclidean interpretation of Dynkin diagrams and its relation to root systems

1987 ◽  
Vol 22 (1) ◽  
Author(s):  
Daniel Drucker ◽  
Daniel Frohardt
Keyword(s):  
Root Systems ◽  
Dynkin Diagrams

Triple root systems, rational quivers and examples of linear free divisors

2018 ◽  
Vol 29 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Kazunori Nakamoto ◽  
Ayşe Sharland ◽  
Meral Tosun
Keyword(s):  
Comparison Theorem ◽  
Triple Point ◽  
Root Systems ◽  
Rational Singularities ◽  
Dynkin Diagrams ◽  
Free Divisors ◽  
Resolution Graphs

The dual resolution graphs of rational triple point (RTP) singularities can be seen as a generalization of Dynkin diagrams. In this work, we study the triple root systems corresponding to those diagrams. We determine the number of roots for each RTP singularity, and show that for each root we obtain a linear free divisor. Furthermore, we deduce that linear free divisors defined by rational triple quivers with roots in the corresponding triple root systems satisfy the global logarithmic comparison theorem. We also discuss a generalization of these results to the class of rational singularities with almost reduced Artin cycle.

scholarly journals Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 61
Author(s):  
Adam Brus ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Case Analysis ◽  
Root Systems ◽  
Trigonometric Functions ◽  
Root Lattice ◽  
Direct Evaluation ◽  
Trigonometric Transforms ◽  
Weyl Transforms ◽  
Dynkin Diagrams ◽  
Discrete Transforms ◽  
Unitary Transform

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems A1 and Cn are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms.

scholarly journals Twisted quadratic foldings of root systems

2021 ◽  
Vol 33 (1) ◽  
pp. 65-84
Author(s):  
M. Lanini ◽  
K. Zainoulline
Keyword(s):  
Root System ◽  
Equivariant Cohomology ◽  
Quaternion Algebra ◽  
Root Systems ◽  
Group Cohomology ◽  
Characteristic Classes ◽  
Reflection Groups ◽  
Content Type ◽  
Dynkin Diagrams ◽  
Projective Homogeneous Varieties

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of type E 8 E_8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H 4 H_4 . By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.

The Mutation Game, Coxeter–Dynkin Graphs, and Generalized Root Systems

2020 ◽  
Vol 27 (01) ◽  
pp. 55-78
Author(s):  
N.J. Wildberger
Keyword(s):  
Root Systems ◽  
Combinatorial Structure ◽  
Dynkin Diagrams ◽  
General Graphs ◽  
Numbers Game

We introduce the mutation game on a directed multigraph, which is dual to Mozes’ numbers game. This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more general graphs. We interpret Coxeter–Dynkin diagrams in this multigraph context and exhibit new geometric forms for the associated root systems.

Preliminary notions

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Root System ◽  
Maximal Torus ◽  
Reductive Group ◽  
Root Systems ◽  
Positive System ◽  
Reductive Groups ◽  
Dynkin Diagrams ◽  
Standard Construction ◽  
Minimal Type ◽  
Split Maximal Torus

This chapter considers some preliminary notions, starting with standard pseudo-reductive groups, Levi subgroups, and root systems. It reviews the “standard construction” of pseudo-reductive k-groups and shows that any connected reductive group equipped with a chosen split maximal torus is generated by that maximal torus and its root groups for the simple positive and negative roots relative to a choice of positive system of roots in the root system. It also discusses the basic exotic construction, noting that the only nontrivial multiplicities that occur for the edges of Dynkin diagrams of reduced irreducible root systems are 2 and 3. Finally, it explains the minimal type pseudo-reductive k-group G, along with quotient homomorphism between pseudo-reductive groups.

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