Reflection group

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.

Definition

Let E be a finite-dimensional Euclidean space. A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin).

The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field.

Examples

Plane

In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of and correspond to the Coxeter diagram Conversely, the cyclic point groups in two dimensions are not generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group.

Infinite reflection groups include the frieze groups and and the wallpaper groups , ,, and . If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.

Space

Finite reflection groups are the point groups Cnv, Dnh, and the symmetry groups of the five Platonic solids. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of R3 is an instance of the ADE classification.

Kaleidoscopes

Reflection groups have deep relations with kaleidoscopes, as discussed in (Goodman 2004).

Relation with Coxeter groups

A reflection group W admits a presentation of a special kind discovered and studied by H. S. M. Coxeter. The reflections in the faces of a fixed fundamental "chamber" are generators ri of W of order 2. All relations between them formally follow from the relations

expressing the fact that the product of the reflections ri and rj in two hyperplanes Hi and Hj meeting at an angle is a rotation by the angle fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group.

Finite fields

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).

Generalizations

Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered. The most important class arises from Riemannian symmetric spaces of rank 1: the n-sphere Sn, corresponding to finite reflection groups, the Euclidean space Rn, corresponding to affine reflection groups, and the hyperbolic space Hn, where the corresponding groups are called hyperbolic reflection groups. In two dimensions, triangle groups include reflection groups of all three kinds.

See also

References

Standard references include (Humphreys 1992) and (Grove & Benson 1996).

  • Coxeter, H.S.M. (1934), "Discrete groups generated by reflections", Ann. of Math., 35: 588–621, doi:10.2307/1968753
  • Coxeter, H.S.M. (1935), "The complete enumeration of finite groups of the form ", J. London Math. Soc., 10: 21–25
  • Goodman, Roe (April 2004), "The Mathematics of Mirrors and Kaleidoscopes" (PDF), American Mathematical Monthly, doi:10.2307/4145238
  • Humphreys, James E. (1992), Reflection groups and Coxeter groups, Cambridge University Press, ISBN 978-0-521-43613-7
  • Zalesskiĭ, Aleksandr E.; Serežkin, V N (1981), "Finite Linear Groups Generated by Reflections", Math. USSR Izv., 17 (3): 477–503, Bibcode:1981IzMat..17..477Z, doi:10.1070/IM1981v017n03ABEH001369
  • Kane, Richard, Reflection groups and invariant theory (review) (PDF)
  • Hartmann, Julia; Shepler, Anne V., Jacobians of reflection groups, arXiv:math/0405135, Bibcode:2004math......5135H
  • Dolgachev, Igor V., Reflection groups in algebraic geometry, arXiv:math.AG/0610938

External links

1991 São Toméan legislative election

Parliamentary elections were held in São Tomé and Príncipe on 20 January 1991. They were the first multi-party elections for the National Assembly, following a referendum the previous year. The result was a victory for the Democratic Convergence Party-Reflection Group, which won 33 of the 55 seats, defeating the former sole legal party, the Movement for the Liberation of São Tomé and Príncipe - Social Democratic Party. Voter turnout was 77.1%.

Alda Bandeira

Alda Bandeira Tavares Vaz da Conceição (born September 22, 1949) is a politician in São Tomé and Príncipe.Bandeira was born in Santana, São Tomé in 1949. Her father was a nurse. She studied modern languages at Eduardo Mondlane University in Maputo, Mozambique, received an MA in modern languages and literature from Lisbon University, and studied international relations at the Lisbon Instituto Superior de Ciências Sociais e Políticas. Bandeira taught in secondary schools in Maputo and São Tomé between 1975 and 1982.Bandeira began her political career as the director of multilateral cooperation in the Ministry of Foreign Affairs, a position she held from 1987 to 1990.She also served as national coordinator for US African Development Foundation programs in São Tomé from 1988-1990. Bandeira was one of the founding members of the Democratic Convergence Party-Reflection Group (PCD-GR), the first public opposition group in the country. After an overwhelming victory for the party in the 1991 elections, Bandeira held the office of foreign minister from 1991 until 1993. After her husband Norberto Costa Alegre was appointed prime minister, she stepped down from her position in the government to occupy her seat as a Member of Parliament and avoid conflicts of interest.Bandeira was elected president of the PCD-GR in 1995, an office she held until 2001. In 1996 she ran for President of the country, coming in third place with 15% of the vote. Bandeira taught at the Instituto Superior Politécnico in São Tomé from 2000-2002. In April 2002 she was appointed foreign minister again, but resigned later that year. Bandeira is currently serving as the Director General of the newly formed Instituto Marítimo e de Administracão Portuaria (Maritime and Port Administration Institute) in São Tomé.

Carlos Westendorp

Carlos Westendorp y Cabeza (born 7 January 1937 in Madrid) is a Spanish diplomat and current Secretary General of the Club of Madrid.

He joined the Spanish Diplomatic Service in 1966.

Following several assignments abroad (1966 -1969: Deputy Consul General in São Paulo, Brazil; 1975-1979: Commercial and Economic Counsellor at the Spanish Embassy in the Hague, the Netherlands) and in Spain (1969–1975: Head of Economic Studies at the Diplomatic School; Director of Technological Agreements in the Ministry of Foreign Affairs; Chief of Cabinet of the Minister of Industry) he dedicated a great part of his professional career to the process of integration of Spain into the European Communities.

Between 1979 and 1985 at the Ministry of European Affairs, he successively served as Advisor to the Minister, as Head of the Minister’s Private Office and as Secretary General, presiding over the technical team in charge of the accession negotiations.

In 1986, when Spain joined the European Communities, he was appointed its first Ambassador Permanent Representative. He chaired the Committee of Permanent Representatives (COREPER) during the first Spanish Presidency of the EEC in 1989.

From 1991 to 1995 he was Spain’s Secretary of State for the European Union. He was centrally involved in the Spanish Presidency of the EU in 1995, which coincided with the adoption of the Euro, the launching of the Barcelona process and the signing of the transatlantic agenda.

In this last capacity, he chaired the Reflection group set up to prepare the negotiations on treaty change which led to the Treaties of Amsterdam and subsequently, Nice.

In December 1995, he was appointed Minister of Foreign Affairs and served in that capacity until the end of the last government presided by Felipe González.In July 1996 he was appointed Ambassador Permanent Representative of Spain to the United Nations in New York.

From 1997 to 1999 he served as the High Representative of the International Community in Bosnia-Herzegovina. At his request, the Bonn conference empowered him to take the necessary decisions to implement the peace agreements (the flag, the national anthem, the single currency, common license plates, the laws on citizenship, privatization and others, refugee return, removal of elected officials, etc.)

In 1999 he was elected Member of the European Parliament representing the PSOE. He served as Chairman of the Parliament’s Committee on Industry, Trade, Energy and Research until 2003.

In 2003 he was elected Member of the Madrid Regional Assembly and Speaker on Economy of the Socialist Group.

He was co-founder and Executive Vice-President of the Toledo Center for Peace and is now member of its board. After the elections of 2004 he was appointed Ambassador to the United States of America, a position he occupied until 2008.

He is currently principal advisor to Felipe González, Chairman of the Reflection Group established by the EU Heads of State and Government to assist the European Union to anticipate and meet the challenges facing in the period 2020 to 2030.

He is President of Westendorp International S.L., a private consulting company. He has addressed conferences and lectures and has written articles and books mostly on European Affairs, for which he was awarded the Salvador de Madariaga Prize of Journalism. He has been awarded various Spanish and foreign decorations, including the Great Cross of the Order of Charles III and Officier de la Légion d’Honneur.

Complex reflection group

In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.

Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra).

Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934) as abstractions of reflection groups, and finite Coxeter groups were classified in 1935 (Coxeter 1935).

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.

Standard references include (Humphreys 1992) and (Davis 2007).

Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge). An unlabeled branch implicitly represents order-3.

Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.

Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.

Daniel Daio

Daniel Lima dos Santos Daio (born 1947) is a former Prime Minister of São Tomé and Príncipe. The first person freely elected to the position, he held the post from 7 February 1991 to 16 May 1992. He is a member of the Democratic Convergence Party – Reflection Group

Democratic Convergence Party (São Tomé and Príncipe)

The Democratic Convergence Party-Reflection Group (Portuguese: Partido de Convergência Democrática-Grupo de Reflexão) is a political party in São Tomé and Príncipe. It was founded on 4 November 1990 by Movement for the Liberation of São Tomé and Príncipe (MLSTP) dissidents, independents, and young professionals. Leonel Mário d'Alva is the party leader.

The party won the 1991 parliamentary election and was the ruling party from 1991 to 1994. Since 1994 its power has declined, and it has maintained coalitions with other parties, first the MLSTP and then the Forces for Change Coalition of current President Fradique de Menezes.It was led for several years by Alda Bandeira.

At the legislative elections, March 3, 2002, the party won together with the Force for Change Democratic Movement-Liberal Party (MDFM-PL) 39.4% of the popular vote and 23 out of 55 seats. The same alliance won at the legislative election, held on 26 March 2006, 36.79% and 23 out of 55 seats. The party supported incumbent Fradique de Menezes in the 30 July 2006 presidential election. He was re-elected with 60.58% of the vote.

Force for Change Democratic Movement – Liberal Party

The Force for Change Democratic Movement-Liberal Party (Portuguese: Movimento Democrático das Forças da Mudança-Partido Liberal) is a former political party in São Tomé and Príncipe. It was formed after the 29 July 2001 presidential elections by supporters of the elected president Fradique de Menezes. Tomé Vera Cruz became the party's first Secretary General.

At the legislative elections, 3 March 2002, the party won together with the Democratic Convergence Party-Reflection Group 39.4% of the popular vote and 23 out of 55 seats.

The same alliance won at the legislative election, held on 26 March 2006, 36.79% and 23 out of 55 seats.

Fradique de Menezes, who represented the party in the 30 July 2006 presidential election, was re-elected with 60.58% of the vote.

Francisco da Silva (politician)

Francisco da Silva (17 August 1957 – 14 April 2010) was the president of the National Assembly of São Tomé and Príncipe. He was provisionally replaced by Evaristo Carvalho. He is a member of the Democratic Convergence Party-Reflection Group (PCD-GR).

Hessian group

In mathematics, the Hessian group is a finite group of order 216, introduced by Jordan (1877) who named it for Otto Hesse. It may be represented as the group of affine transformations with determinant 1 of the affine plane over the field of 3 elements. It also acts on the Hesse pencil of elliptic curves, and forms the automorphism group of the Hesse configuration of the 9 inflection points of these curves and the 12 lines through triples of these points.

The triple cover of this group is a complex reflection group, 3[3]3[3]3 or of order 648, and the product of this with a group of order 2 is another complex reflection group, 3[3]3[4]2 or of order 1296. It has a normal subgroup that is an elementary abelian group of order 32, and the quotient by this subgroup is isomorphic to the group SL2(3) of order 24.

Hurwitz surface

In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms (Hurwitz 1893). They are also referred to as Hurwitz curves, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2).

The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the (ordinary) (2,3,7) triangle group. The finite quotient group is precisely the automorphism group.

Automorphisms of complex algebraic curves are orientation-preserving automorphisms of the underlying real surface; if one allows orientation-reversing isometries, this yields a group twice as large, of order 168(g − 1), which is sometimes of interest.

A note on terminology – in this and other contexts, the "(2,3,7) triangle group" most often refers, not to the full triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which is index 2. The group of complex automorphisms is a quotient of the ordinary (orientation-preserving) triangle group, while the group of (possibly orientation-reversing) isometries is a quotient of the full triangle group.

Leonel Mário d'Alva

Leonel Mário d'Alva (born 1935) is a São Toméan politician. He served as Prime Minister of São Tomé and Príncipe from 21 December 1974 until 12 July 1975, when the country gained independence from Portugal.

From late 1975 to 1980, D'Alva was President of the São Toméan National Assembly. After the country's first democratic elections in 1991, he was again elected National Assembly President. He also served as foreign minister from 1975 to 1978 and was acting president from 4 March to 3 April 1991.

D'Alva co-founded the Democratic Convergence Party – Reflection Group (PCD–RG) in 1991 and subsequently led it for many years.

Maria Tebús

Maria dos Santos Lima da Costa Tebús Torres (born 2 September 1958) is a São Toméan politician. She was born in the village of Santa Filomena in the island of São Tomé. Since 21 April 2006, she has been the country's Deputy Prime Minister and Minister of Planning and Finance in the government of Tomé Vera Cruz. She also held the position of Minister of Planning and Finance from 2002 to August 2003.

Tebus is a member of the Democratic Convergence Party-Reflection Group (PCD-GR).

National Assembly (São Tomé and Príncipe)

The unicameral National Assembly is São Tomé and Príncipe's legislative body.

The current National Assembly, formed following elections held on 12 October 2014, has a total of 55 members elected in 7 multi-member constituencies using the party-list proportional representation system. Members serve four-year terms.

Norberto Costa Alegre

Norberto José d'Alva Costa Alegre (born 1951) is a former prime minister of São Tomé and Príncipe. He held the post from 16 May 1992 to 2 July 1994. He is a member of the Democratic Convergence Party-Reflection Group (PCD-GR) and is married to former foreign minister Alda Bandeira.

Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.

Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center.

Restricted root system

In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group.

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