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Surname list
Coxeter–Dynkin diagram Dynkin system Dynkin's formula Doob–Dynkin lemma Dynkin index This page lists people with the surname Dynkin. If an internal link
Dynkin
Pictorial representation of symmetry
theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams
Dynkin_diagram
Russian mathematician (1924–2014)
algebras, and Markov processes. The Dynkin diagram, the Dynkin system, and Dynkin's lemma are named after him. Dynkin was born into a Jewish family, living
Eugene_Dynkin
Theorem in stochastic analysis
In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth function applied
Dynkin's_formula
Family closed under complements and countable disjoint unions
A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set Ω {\displaystyle \Omega } satisfying a set of axioms weaker
Dynkin_system
In mathematics, the Dynkin index I ( λ ) {\displaystyle I({\lambda })} of finite-dimensional highest-weight representations of a compact simple Lie algebra
Dynkin_index
Pictorial representation of symmetry
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter
Coxeter–Dynkin_diagram
Russian economist
Alexander A. Dynkin (Russian: Александр Александрович Дынкин; born 30 June 1948) is a Russian economist whose research interests and publications have
Aleksandr_Dynkin
Class of mathematical problems
In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order
Optimal_stopping
Card trick and probabilistic concept
Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling)
Kruskal_count
mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact
Vogan_diagram
Geometric arrangements of points, foundational to Lie theory
they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie
Root_system
Statement in probability theory
In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), characterizes the situation
Doob–Dynkin_lemma
Classifies quivers of finite type in terms of Dynkin diagrams
proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams. A quiver is of finite type if it has only finitely many isomorphism
Gabriel's_theorem
Type of Kac–Moody algebras
may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras. The attachment of an extra node to the Dynkin diagram of the corresponding
Affine_Lie_algebra
Uniform polytope
E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences. The
1_32_polytope
78-dimensional exceptional simple Lie group
cohomology (over a perfect field k) by the set H1(k, Aut(E6)) which, because the Dynkin diagram of E6 (see below) has automorphism group Z/2Z, maps to H1(k, Z/2Z)
E6_(mathematics)
Simple Lie group; the automorphism group of the octonions
−1). It has a non-algebraic double cover that is simply connected. The Dynkin diagram for G2 is given by . Its Cartan matrix is: [ 2 − 3 − 1 2 ] {\displaystyle
G2_(mathematics)
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
the complex numbers the semisimple Lie algebras are classified by their Dynkin diagrams, of types "ABCDEFG". If L is a real simple Lie algebra, its complexification
Simple_Lie_group
From an exceptional automorphism of a Dynkin diagram
Rimhak Ree, who constructed them from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing
Ree_group
Polytope in 8-dimensional geometry
semi-regular figure. Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, . The rectified
4_21_polytope
Uniform 6-polytope
polytope. Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He
2_21_polytope
Algebraic structure of set algebra
].} A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below). This theorem (or the
Σ-algebra
Positive-definite integral set of repeated points with Abelian group-rank 24
There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkin diagrams. The complete
Niemeier_lattice
248-dimensional exceptional simple Lie group
cohomology (over a perfect field k) by the set H1(k,Aut(E8)), which, because the Dynkin diagram of E8 (see below) has no automorphisms, coincides with H1(k,E8)
E8_(mathematics)
Uniform 7-dimensional polytope
semi-regular figure. Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences. The
3_21_polytope
Algebraic construct of interest in theoretical physics
generalized Dynkin diagrams. When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dynkin diagram)
Quantum_group
Shape with six sides
orientations. The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have
Hexagon
Concept in mathematics
closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie
Reductive_group
Topics referred to by the same term
assumes its shape or conformation Folding (Dynkin diagram), in Lie theory, a way of obtaining one Dynkin diagram from another Fold change, a measure
Folding
bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are
E9_honeycomb
52-dimensional exceptional simple Lie group
construction of E8. In older books and papers, F4 is sometimes denoted by E4. The Dynkin diagram for F4 is: . Its Weyl/Coxeter group G = W(F4) is the symmetry group
F4_(mathematics)
Mathematical classification
where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications
ADE_classification
Soviet American mathematician
thesis On one problem from the diffusion process theory supervised by Eugene Dynkin. At MSU Molchanov graduated in 1967 with Russian Candidate degree (Ph.D
Stanislav_Molchanov
Formula in Lie theory
actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947). The history of the formula is described in detail in the article
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
Relationship between certain vector spaces
vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional
Triality
Uniform 6-polytope
vertices). Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are
1_22_polytope
Formula in Lie group theory
~i_{r}+j_{r}>0,~1\leq r\leq k.} This is Dynkin's formula. The striking similarity with (99) is not accidental: It reflects the Dynkin–Specht–Wever map, underpinning
Derivative of the exponential map
Derivative_of_the_exponential_map
Type of 7-polytope
7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of . Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)
Hexicated_7-simplexes
Uniform polytope in 8 dimensional geometry
E8 group. Its Coxeter symbol is 241, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences. The rectified
2_41_polytope
Term in mathematics
of Lie algebras and Lie groups, Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems that are relevant in
Satake_diagram
Uniform 6-dimensional polytope
operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group
Uniform_6-polytope
Kepler–Poinsot polyhedron with 20 faces
(nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having
Great_icosahedron
Graph with nodes connected linearly
Gibbons (1985), or Diestel (2005). In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and
Path_graph
Linear algebraic group
groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram. All split groups (those with a split maximal torus) are quasi-split
Quasi-split_group
Uniform Polytope
E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch. The rectified
2_31_polytope
Topics referred to by the same term
used before vowels Ân (digraph) An, in mathematics, a root system and its Dynkin diagram An, in mathematics, conventional notation for the alternating group
AN
Family of sets closed under intersection
of the fact that the π-𝜆 theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically prove the same results via monotone
Pi-system
finite-dimensional theory is greatly governed by a theory of root systems and Dynkin diagrams, strikingly similar to those of semisimple Lie algebras. A comprehensive
Nichols_algebra
Ukrainian American mathematician
career Fields Mathematics, Stochastic differential equations, Markovian processes Institutions Michigan State University Academic advisors Eugene Dynkin
Anatoliy_Skorokhod
Star polygon with 7 sides
Regular star polygon Edges and vertices 7 Schläfli symbol {7/2} Coxeter–Dynkin diagrams Symmetry group Dihedral (D7) Internal angle (degrees) ≈77.143°
Heptagram
Mathematical coincidence
associated Lie algebras instead.) There are some exceptional isomorphisms of Dynkin diagrams, yielding isomorphisms of the corresponding Coxeter groups and
Exceptional_isomorphism
Quadrilateral with sides of equal length
parallelogram, kite Edges and vertices 4 Schläfli symbol { } + { } {2α} Coxeter–Dynkin diagrams Symmetry group Dihedral (D2), [2], (*22), order 4 Area K = p ⋅
Rhombus
Type of geometrical object
three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: Selected regular and uniform 10-polytopes from each family include:
Uniform_10-polytope
24-dimensional repeating pattern of points
roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II25,1. By comparison, the Dynkin diagrams of
Leech_lattice
Polygons which have an accompanying imaginary dimension for each real dimension
= I, (R2R1)2R2 = (R1R2)2R1. Coxeter also generalised the use of Coxeter–Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is
Regular_complex_polygon
Uniform 8 dimensional polytope
E8 group. Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified
1_42_polytope
{\displaystyle r\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}} Coxeter-Dynkin diagram or 4-faces 27 6 r{3,3,3} 6 rr{3,3,3} 15 {}x{3,3} Cells 135 30 {3
Cantellated_5-simplexes
Seven-dimensional geometric object
four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: The A7 family has symmetry of order 40320 (8 factorial). There
Uniform_7-polytope
Truncated 5-simplex Type Uniform 5-polytope Schläfli symbol t{3,3,3,3} Coxeter-Dynkin diagram 4-faces 12 6 {3,3,3} 6 t{3,3,3} Cells 45 30 {3,3} 15 t{3,3} Faces
Truncated_5-simplexes
7-simplex Type uniform 7-polytope Schläfli symbol t0,4{3,3,3,3,3,3} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 2240 Vertices 280 Vertex
Stericated_7-simplexes
Complex simple Lie Algebra
mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of
Exceptional_Lie_algebra
Concept in Lie algebra mathematics
{\displaystyle {\mathfrak {g}}} , there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed
Simple_Lie_algebra
4-D object; direct sum of a cube and a segment
Type Polyhedral bipyramid Schläfli symbol {4,3} + { } dt{2,3,4} Coxeter-Dynkin Cells 12 {4}∨{ } (2×6) Faces 30 triangles (2×12+6) Edges 28 (2×8+12) Vertices
Cubical_bipyramid
Subgroup of a root system's isometry group
and β {\displaystyle \beta } are in Δ {\displaystyle \Delta } , then the Dynkin diagram for Φ {\displaystyle \Phi } relative to the base Δ {\displaystyle
Weyl_group
Topics referred to by the same term
dn (elliptic function), one of Jacobi's elliptic functions Dn, a Coxeter–Dynkin diagram Dn, a dihedral group Dn, a Dirichlet kernel Decinewton (symbol dN)
DN
Mathematical concept
an affine Cartan matrix, whose Dynkin diagram is given. The multiplicities of each fiber are indicated in the Dynkin diagram. This table can be found
Elliptic_surface
Geometric operation
that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a shortcut term which has a different
Omnitruncation
8-simplex Type uniform 8-polytope Schläfli symbol rr{3,3,3,3,3,3,3} Coxeter-Dynkin diagram 7-faces 6-faces 5-faces 4-faces Cells Faces Edges 1764 Vertices
Cantellated_8-simplexes
In mathematics, a type of algebra
Simple Lie algebra Loop algebra Affine Lie algebra Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification
Solvable_Lie_algebra
5-orthoplex Type uniform 5-polytope Schläfli symbol t{3,3,3,4} t{3,31,1} Coxeter-Dynkin diagrams 4-faces 42 10 32 Cells 240 160 80 Faces 400 320 80 Edges 280 240
Truncated_5-orthoplexes
mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k
En_(Lie_algebra)
6-simplex Type uniform 6-polytope Schläfli symbol t0,4{3,3,3,3,3} Coxeter-Dynkin diagrams 5-faces 105 4-faces 700 Cells 1470 Faces 1400 Edges 630 Vertices
Stericated_6-simplexes
Direct sum of simple Lie algebras
subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year-old Eugene Dynkin in 1947. Some minor modifications have been made
Semisimple_Lie_algebra
Symbolic representation of information using visualization techniques
diagram Dendrogram Dependency diagram Deployment diagram – from UML 9/9 Dynkin diagram Dot and cross diagram Double bubble map – used in education Drakon-chart
Diagram
Group of irregular uniform polytopes
and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams. The Coxeter symbol for these figures has the form ki,j, where
Gosset–Elte_figures
Canadian geometer (1907–2003)
Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Coxeter was born in Kensington
H.S.M._Coxeter
Maximal compact connected Abelian Lie subgroup
semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram. The unitary group U(n) has as a maximal torus the subgroup of all
Maximal_torus
Directed graph which is also a multigraph
of the root system of the Dynkin diagram. Dlab & Ringel (1973) found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite dimensional
Quiver_(mathematics)
Shape with eleven sides
Type Regular polygon Edges and vertices 11 Schläfli symbol {11} Coxeter–Dynkin diagrams Symmetry group Dihedral (D11), order 2×11 Internal angle (degrees)
Hendecagon
Polygon with 257 sides
Type Regular polygon Edges and vertices 257 Schläfli symbol {257} Coxeter–Dynkin diagrams Symmetry group Dihedral (D257), order 2×257 Internal angle (degrees)
257-gon
Soviet and Russian mathematician (1933–2019)
groups and their geometrical applications. Onishchik was a student of Eugene Dynkin, under whose guidance he got his PhD at Moscow State University in 1960
Arkady_Onishchik
Topics referred to by the same term
Boron nitride, a chemical compound Bulimia nervosa, an eating disorder Dynkin diagram Bn, in mathematical analysis BN (biscuit), a Franco-British brand
BN
Group that admits a formal description in terms of reflections
(undirected) Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram
Coxeter_group
Mathematical group
preserving the structure of the associated Dynkin diagram. In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G)
Outer_automorphism_group
Uniform 6-polytope
Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram. This configuration matrix represents the expanded 6-simplex, with
Pentellated_6-simplexes
omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of . The full snub 5-simplex or omnisnub 5-simplex, defined as an
Stericated_5-simplexes
Russian mathematician
processes. Krylov studied at Lomonosov University, where he in 1966 under E. B. Dynkin attained a doctoral candidate title (similar to a PhD) and in 1973 a Russian
Nikolay Krylov (mathematician, born 1941)
Nikolay_Krylov_(mathematician,_born_1941)
Soviet mathematician (1903–1987)
Vladimir Arnold Sergei N. Artemov Grigory Barenblatt Roland Dobrushin Eugene Dynkin Israil Gelfand Boris Gnedenko Leonid Levin Valerii Kozlov Per Martin-Löf
Andrey_Kolmogorov
Group of unitary complex matrices with determinant of 1
\\(&0,0,0,\dots ,1,-1).\end{aligned}}} So, SU(n) is of rank n − 1 and its Dynkin diagram is given by An−1, a chain of n − 1 nodes: .... Its Cartan matrix
Special_unitary_group
Polytope contained by 7-polytope facets
four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: Selected regular and uniform 8-polytopes from each family include:
Uniform_8-polytope
Complicated polygon
1200 Vertices 120 Vertex figure {5,3} Schläfli symbol {5/2,5,3} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Icosahedral 120-cell Properties
Small_stellated_120-cell
disphenoid tetrahedral honeycomb Type convex uniform honeycomb dual Coxeter-Dynkin diagram Cell type Tetragonal disphenoid Face types isosceles triangle {3}
Tetragonal disphenoid honeycomb
Tetragonal_disphenoid_honeycomb
133-dimensional exceptional simple Lie group
cohomology (over a perfect field k) by the set H1(k, Aut(E7)) which, because the Dynkin diagram of E7 (see below) has no automorphisms, coincides with H1(k, E7
E7_(mathematics)
classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynkin diagram of
Real_form_(Lie_theory)
8-orthoplex Type uniform 8-polytope Schläfli symbol t1{3,3,3,3,3,3,4} Coxeter-Dynkin diagrams 7-faces 272 6-faces 3072 5-faces 8960 4-faces 12544 Cells 10080
Rectified_8-orthoplexes
Polygon shape with eight sides
Regular polygon Edges and vertices 8 Schläfli symbol {8}, t{4} Coxeter–Dynkin diagrams Symmetry group Dihedral (D8), order 2×8 Internal angle (degrees)
Octagon
6-orthoplex Type uniform 6-polytope Schläfli symbol 2r2r{3,3,3,3,4} Coxeter-Dynkin diagrams 5-faces 4-faces Cells Faces Edges 5760 Vertices 960 Vertex figure
Stericated_6-orthoplexes
Polygon with one edge and one vertex
arc edge. Type Regular polygon Edges and vertices 1 Schläfli symbol {1} or h{2} Coxeter–Dynkin diagrams or Symmetry group [ ], Cs Dual polygon Self-dual
Monogon
Regular star 4-polytope
720 Vertices 120 Vertex figure {3,5} Schläfli symbol {5/2,3,5} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Grand 120-cell Properties Regular
Great_stellated_120-cell
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