Reflection Groups in Geometry - Essay Example

Reflection Groups in Geometry Abstract Reflection groups generated by reflection have widely been the subject of significant study. This paper presents a comprehensive exposition of the concept of reflection groups in Euclidean, spherical and hyperbolic geometry and the significance of these groups in several areas of algebraic geometry. A historical account of the development of reflection groups and various types of reflection groups is also presented. The concepts and equations governing reflection groups are discussed and proven including relevant illustrations. Table of Contents Abstract 1 Table of Contents 1 Introduction 2 Theory of Reflection Groups 3 Historical Account of Reflection Groups 4 Classification of Reflection Groups 8 Fundamental System 10 Coxeter Reflection Groups 10 Weyl Groups 13 Dynkin Diagrams 13 Pseudo-Reflection Groups 15 Acknowledgements 21 Appendices 21 Bibliography 22 Introduction Group is a fundamental concept in mathematics and group theory is widely applied in many areas of mathematics and other sciences. A reflection group is a distinct group produced by multiple reflections of a finite-dimensional (Euclidean) space. Weyl groups of simple lie algebras and symmetry groups of regular polytypes are examples of finite reflection groups while infinite groups comprise the Weyl groups of infinite-dimensional Kac–Moody algebras and the triangle groups similar to ordinary tessellations of the hyperbolic plane and Euclidean plane. With regard to symmetry, discrete isometry groups of broad Riemannian manifolds that are formed by reflections are grouped into classes leading to hyperbolic reflection groups (corresponding to hyperbolic space), affine (corresponding to Euclidean space) and finite reflection groups (the n-sphere). Coxeter groups are reflection groups that are finitely generated. Unlike reflection groups, Coxeter groups are abstract groups that have a certain structure generated by reflections. An investigation of the topology and geometry of reflection groups will help us comprehend the theoretic properties of the group. The concept of reflection in a Euclidean space and the hypothesis of discrete groups of motions resulting from reflections has it’s origin in the study of space polyhedral and plane regular polygons that goes back to early mathematics. In present day, reflection groups are common in many areas of mathematical research, and geometers encounter them as special convex polytopes or discrete groups of isometries of Riemannian spaces with even curvature. On the other hand, an algebraist encounters reflection groups in group theory, particularly in the representation theory, Coxeter groups and invariant theory. Other areas of mathematics where they may be encountered include the theory of arrangements of hyperplanes, theory of combinations and permutation, theory of modular forms and quadratic forms, low-dimensional topology, singularity theory, and the theory of hyperbolic real and complex manifolds (Yau 1986). Theory of Reflection Groups A reflection is a straight operator s on V that conveys a nonzero vector α to its negative and fixes the hyperplane Hα orthogonal to α. This statement can e represented mathematically as: s=sα where s=scα for any nonzero c Є R. This leads to the formula: (Coxeter 1989). A finite reflection group is a class of finite-dimensional Euclidean space that is produced by a number of reflections orthogonally across hyperplanes going through the origin. It is defined mathematically as: I. A pair (G, V) in which V is Euclidean space, G is a finite class of O (V) and G = {Sx : Sx Є G}produced by all reflections in G. For instance, in two-dimensional spaces, define Dn as the dihedral group of order 2n. Dn has n reflections and n rotations with an angle 2π/n and the dihedral group Dn is produced by reflections since a rotation through the angle can be gotten by multiplication of two reflections in relation to a pair of adjacent diagonals that meet at an angle of π/n. this can be represented mathematically by the equation below: Affine reflection groups are discrete classes belonging to the affine group of a Euclidean space of finite dimension produced by a number of affine reflections (Coxeter 1989). For a symmetric reflection group acting on {1,…,n} expand to action of Sn on Rn , ei Є basis of Rn, σ Є Sn, for all σ: Reflection because and since Sn is a symmetric group that is a reflection group. Historical Account of Reflection Groups According to existing accounts of the history of the theory of reflection groups, the modern theory has its roots in the works of geometers L. Schlafli and A. Mobins in the mid-nineteenth century. This concept was later broadened and used in the theory of Lie algebras in the research of W. Killing and E. Cartan towards the end of the nineteenth century, and ended in the works of M. Coxeter. The earliest instances of reflection groups in hyperbolic plane are dated back to H. Poincare and F. Klein at the end of the nineteenth century (American Mathematical Society, 1988). Finite and infinite reflection groups occur in the works of S. Kantor in 1885-1895 regarding the taxonomy of subgroups of the Cremona group of conversions of the complex projective plane. Kantor realized that the reflection action in the cohomology space of rational surfaces in algebra is blow-ups of the plane. Thus, all Weyl groups of type A2 x A1.A4.D5.E6.E7.E8 occur naturally when the 3-8 points are blown up. The final three groups occurred prior in algebraic geometry as the group of twenty seven lines on a cubic surface (type E6), the group of tritangent planes of a space sextic of genus 4 (type E8) and the group of bitangents on a plane quartic (type E8). In the early twentieth century, P.Schoute found a convex polytope in 6-dimensional space that has vertices in a bijective connection with twenty seven lines on a cubic surface and the group of symmetries is isomorphic to the group of twenty seven lines. Coxeter discovered a similar polytope in 7-dimensional space for the group of twenty eight bitangents of a plane quartic. In the 1930s, P. Du Val explained the connection between the cohomology space of the blow-up of the plane at six points and this six-dimensional space. He demonstrated that all Kantor groups are reflection groups in affine, Euclidean of hyperbolic spaces. The diagram below shows the Coxeter diagram of the Weyl group of type E8 that occurs when the number of points blown up is equal to eight. Fig.1 When nine points are entailed, we obtain a reflection group in affine space of dimension eight which is the type E8 affine Weyl group. The Coxeter diagram is attained by addition of one point in the long arm of the illustration in figure1 above. Beginning from ten points, we obtain Coxeter groups in hyperbolic space with Coxeter diagram of type En expanding the lengthy arm of the diagram in the figure above. In 1917, Arthur Coble generalized Kantor’s work and brought about the idea of a regular Cremona conversion of a higher-dimensional projective space and regarded more ordinary Coxeter groups of type W. This cortex diagram is developed from thee diagram by stretching the two upper arms. Reflection groups also occur in the class of surfaces of type K3; a class of algebraic surfaces in hyperbolic spaces. A multiple quartic surface in comprehensive three-dimensional projective space is a case of such a surface. In 1972, I. R. Shafarevich and I.I. Pyatetsky-Shapiro showed proof that the complex formation of a polarized algebraic K3-surface is established distinctively by the linear functional on the surface’s next cohomology space derived by performing an integration on an infinite holomorphic 2-form. The result of this investigation came to be known as the Global Torelli Theorem. As an outcome of this result, the researchers demonstrated the isomorphic nature of the set of automorphisms of a K3 surface. Therefore they decreased the subject of finiteness of the automorphism group to a matter of finiteness of the volume of the elementary polyhedron in a real hyperbolic space (Jain et al. 2008). E. Vinberg and V.Nikulin established the types of isomorphism of integral quadratic nature generated and that have the characteristic that the elementary polyhedron has finite volume. In principle, this was a solution of the issue of classification of fields of algebraic measurement two over C with an infinite group of automorphisms over C. During the 1930s, Patrick Du Val established the occurrence of Coxeter diagrams in the resolution of particular kinds of singularities on algebraic surfaces (simple singularities). However, Du Val failed to find any reflection groups related to these singularities. A. Grothendieck proposed a conjectural association to reflection groups and simple Lie algebras in the 1960s, which was validated by a construction of E. Brieskorn. In 1954, A. Todd and G.C. Shephard created the concept of finite complex reflection groups as a report on to the classical research on groups of projective transformations produced by homologies. Some instances of the collections of reflection hyperplanes and the hypersurfaces identified by polynomial invariants of the groups have been present in classical geometry since the 1890s. V.Popov classified infinite reflection groups having finite covolume that appear in complex affine spaces in 1982. These reflection groups appear in the concept of compactification of versal twist of plain elliptic singularities and symmetric surface singularities (Lehrer & Taylor 2009). Most notably, reflection groups appear in complex hyperbolic spaces with dimensions greater than one. Expanding from the research of H.Terada, G. Mostow and P.Deligne classified all hypergeometric functions with monodromy groups that are discrete reflection groups of finite covolume as complex hyperbolic crystallographic groups Br (c.h.c. groups). The compactification of the orbit spaces Br /T were isomorphic to some geometric invariant quotients P1(C) r+3//PGL(2,C) given that r≤9. With the exception of one missed c.h.c group in Deligne-Mostow’s file, no other c.h.c. groups of Br type had been ascertained until a couple of years later. The earliest c.h.c group in B4 occurred in a striking structure of a composite ball uniformization of the moduli space of cubic surfaces developed by J. Carlson, D. Allcock and D. Toledo. New examples of c.h.c groups were later discovered in dimensions 6 and 8 utilizing an identical uniformization structure for moduli spaces of other surfaces (Bourbaki & Meldrum 1999). Latest research by Toledo, Allcock and Carlson in the area of complex ball uniformization generates a novel dimension 10 complex reflection group. A general account of the Deligne-Mostow theory compiled by G. Heckaman, W.Conwenberg and E.Looijenga offers other fresh examples of complex reflection groups of crystallographic nature. D. Allcock constructed a c.h.c group in a record high dimension thirteen. However, no geometrical explanations have been established so far for the subsequent ball quotients (Kane 2001). Classification of Reflection Groups This section presents various important concepts related to Euclidean reflection groups that are useful in describing them. Consider E as an l-dimensional Euclidean space, 0 ≠ α Є E and the hyperplane In which is the normal inner product. The reflections is defined by if . This is a linearity expansion and Hα is known as the reflecting hyperplane of sα. Let be linear, for all Represent the orthogonal group of E. A Euclidean reflection group is the group produced by reflections. This explanation only considers finite reflection groups. If is the set of reflecting hyperplanes of W, H = {H reflecting hyperplane for a reflection. Considering the vectors of H, a set is obtained which satisfies the following conditions: (R1) if , if and only if : (R2) if , . Thus the following definitions can be obtained: I. A root system is a finite set of non-zero vectors that satisfy (R1) and (R2). A root is an element of Δ. A root system that satisfies the additional condition (R3) for all is crystallographic. The significance of these root systems is that (R3) guarantees that the reflecting group W produced by {} is a class of GLl(Z) and not merely GLl(R). Another important condition for root systems is: (R4) Δ spans E. a root system that satisfies this condition is known as an essential. A root system can be made essential, if Δ spans a space, E can be replaced by EΔ and the induced action of W on EΔ is considered. It is important to remember that the space EΔ is stable under such action, since W permutes the elements of Δ. In addition, for the orthogonal complement of EΔ is and W acting trivially on such space, no considerable information is mislaid. If we let W (E) be a reflection group and each be a root system of W, then and hence the subspace can be defined as of a reflection group W (E). As previously mentioned, this expression gives the decomposition and . We can therefore define the concept o isomorphism: 2 reflection groups W(E), W’(E’) are stably isomorphic given that a linear isomorphism exists such that for all and . (Benson & Grove 1985). Fundamental System For any root system represents a fundamental system of Δ if I. Σ is linearly independent; II. Each element of Δ is a linear permutation of elements of Σ, in which the coefficients are all nonpositive or nonnegative. The elements of Σ are known as simple roots. Coxeter Reflection Groups This section introduces the notion of Coxeter groups that bears great significance in the understanding of the construction of finite Euclidean reflection groups. A group W is a Coxeter group if there exists a subset SW that satisfies the condition W = , Where mss=1 and mss’{2,3,…}U{} for all s≠s’. The pair (W,S) is known as a Coxeter system. It is important to note that a reflection group can have more than one Coxeter system. For the dihedral group D6, the typical representation of this group is , which represents a coxeter system. However, because and are Coxeter systems. Multiplication of the above can give another Coxeter system for D6 If we let (W,S) be a Coxeter system, (W,S) is finite if W is finite and given and where (Wi,Si) and is a Coxeter system for I = 1,2, (W,S) is said to be reducible, and irreducible if the condition above is not satisfied. If (W, S) is a Coxeter system, a Coxeter graph X allocated to (W,S) is created as: I. if s,s’S, an edge labelled by mss’ exists between s and s’ given mss’≥3; II. the elements of S are the vertices of X; III. if s,s’S, no edge exists between s and s’ given mss’=2 Theorem 1: It is possible to prove that (W, S) is a Coxeter system i.e. W (Δ) = The proof of this theorem is as a result of Steinberg and utilizes the Matsumoto exchange property. Also, ψ is injective and two finite fundamental reflection groups are isomorphic only if they satisfy the condition that their related Coxeter systems are isomorphic. The remaining part of this section is dedicated to the surjective nature of ψ. We can relate a bilinear form to a Coxeter system (W,S), represents an R-basis of E. Given (W, S) is finite, it can be illustrated that B is positive definite, limiting the possibilities for finite Coxeter systems, and the fact that each of these chances is a result of some finite reflection group. (Humphreys 1990) Theorem 2: Given (W, S) is a finite irreducible Coxeter system, its Coxeter graphs are: Theorem 3: Every Coxeter group system represented by the Coxeter graph Al, Bl,…G2(m) is due to a finite reflection group. Therefore the map ψ is surjective giving a bijection as necessary (Humphreys 1990). Weyl Groups In this section, a discussion of Weyl groups is presented. These are special finite reflection groups of Euclidean nature defined over Z. Definition: a lattice that has a rank l is a free Z-module L=Zl. A finite Euclidean reflection group WO (E) that admits a W-invariant lattice LE where is a Weyl group. It is significant to note that for all , the following identity exists: where is the angle between these vectors. Therefore we have . (Evens 2012) since Δ is a crystallographic root system. The only options for θ are π/2, 2π/2, 3π/4 and 5π/6. Dynkin Diagrams We change Coxeter graphs notation and introduce the concept of Dynkin Diagrams: Σ, a basic system of Δ and for Δ an essential root system of crystallographic nature. A graph X is assigned to Δ as: I. if α ≠ β Є Σ and θ is the angle between the two, 0,1,2 or 3 edge(s) connecting α and β by the rule Theorem 3: Given Δ is irreducible, then a maximum of two root lengths appear in Δ. If Δ has two root lengths, the roots are termed as short and long. This is denoted by an arrow which points in the direction of the shorter root in the Coxeter graph of Δ: given , we obtain or . This is the representation of a Dynkin diagram. Theorem 4: given Δ is an irreducible essential root system of crystallographic type, the Dynkin diagram can be any from the list below: Only one of the G2(m), G2 = G2 (6) is crystallographic. Theorem 5: a root system of crystallographic type exists having each of Al, Bl,…F4, G2 as in the Dynkin diagram above. Pseudo-Reflection Groups In this section, we look at a remarkable general account of Euclidean reflection groups known as pseudo-reflection groups. This category of reflection groups come up naturally and offer a superior working context. Definition: given that F is a field and V is an F-vector space of finite dimension, a pseudo-reflection on V gives a linear isomorphism s: VV that is not the identity but leaves a hyperplane H V pointwise invariant. If the group GGL (V) is generated by pseudo-reflections, it is known as a pseudo-reflection group. The theorem below proves that the regular nature of the ring of invariants S(V)G is a characteristic of pseudo-reflection groups. Theorem 6: the equivalent assertions include: I. Algebra S (V) G is a polynomial algebra. II. S (V) G [G]-module S (V) is independent of rank one. III. Group G is produced by pseudo-reflections. Given that K C R, is an actual reflection. It must not have order 2 when K = C. the group generated due to complex pseudo-reflections in then referred to as a complex reflection group. The theorem of rationality for representing finite Coxeter groups or Weyl groups also applies to pseudo-reflection groups and the group algebra of G over K is obtained by direct multiplication of matrix algebras over K. Shephard and Todd classified the irreducible complex reflection groups into two infinite series: the groups G(p,q,n), the groups and thirty-four exceptional groups. In describing the groups G(p,q,n), we establish that the groups are well presented; a general presentation of Coxeter groups with a few common properties. Particularly, these groups are presented by a set S that constitutes n or n + 1 pseudo-reflections and two types of reflections: finite order relations and homogeneous relations. The group presented similarly, but with no finite order relations is considered as an analog of the homogeneous defined in (8) for real reflection groups. Examples: I. G(p,1,n) This is the group of n by n monomial complex matrices with nonzero entries that are p-th roots of unity. This group is characterized by a semi-direct multiple decomposition G(p,1,n) = , where (Z/pZ)n represents the subgroup of diagonal matrices and represents the subgroup of combination matrices. Given that, where ξ is a primitive p-th root of unity, and retaining the notations of ξ2, G(p,1,n) is produced by the group of pseudo-reflections that satisfy he following relations: and In practice, this presents G(p,1,n) by relations and generators. The convenient means of encoding the relations is the use of a general account of the Coxeter diagrams: G(2,1,n) = Bn and the above construction represent the Coxeter presentation. For q/q, G(p,q,n) is defined as the subgroup of G(p,1,n) that consists of matrices with the product of the nonzero elements being a (p/q)-th root of unity (Ratcliffe 2006). II. G(p,p,n) This group is generated by the set of pseudo-reflections where and satisfy the relations below: and This presents G(p,p,n) as a set of relations and geneators which may be encoded in the diagram below: It is important to note that G (p, p, 2) = I2 (p) and the diagram above is a Coxeter presentation. In addition, G (2, 2, n) = Dn. (HöRmander 1971). III. G(p,q,n) For q/p, q ≠ p and q≠1, d= p/q. this great is produced by the set of pseudo-reflections where, that satisfy the relations: and This is a presentation of G(p,q,n) obtained by relations and generators that are encoded in the diagram below: (R. J. & Sher 2002). Reflection Groups from Polyhedral Cones Given that is a finite reflection group and taking the assumption that a fundamental domain, X exists for the natural permutation of G on the polyhedral cone Rn. Dim X = n Let the facets of X be denoted as and for . We represent the dual (Rn)* with Rn using a G-invariant, fixed symmetric, positive definite bilinear on Rn . XV Rn. the reflection σi is now defined as the orthogonal transformation of Rn that sends and fixing the hyperplane orthogonal to . hence is the space produced by as (Borovik & Borovik 2010). Theorem 7: if is a finite group in which a fundamental domain, X exists for the action of G on the polyhedral cone Rn. G is a reflection group generated by the reflections. (Campbell & Wehlau 2004). Proof: We start by showing that the boundary δX of X occurs in Y: =. This is a claim that holds for any fundamental element for the action of G. Suppose an exists, and X is closed, all sets g(X) are closed and Y is closed. A neighbourhood B of x exists that satisfies the condition . Conversely, since , exists. Hence . This is a contradiction because X is a fundamental domain for the action of G on Rn, therefore, which is a proof to the above claim. The notations presented above are then used to prove that all the reflections belong to G. Taking p as a point in the facet of X; . Considering the above claim, exists such that, hence and. Since X is a fundamental element, this statement only holds if. is arbitrarily chosen, proving that . for some therefore gi is a non-identity orthogonal conversion of Rn which sets point-wise. As a result, gi maps to itself. Since gi is of a finite order ≠ 1, the only options for is –vi. hence gi = σi and which proves our second claim. If we let G0 be the subgroup of G produced by, X is a fundamental element for the action of G on Rn, implying that . Therefore a point exists , making for all 1 ≠ g є G. particularly, for any and i = 1,…,m. thus, if p = g0(hi) for some hi Є Hi then which contradicts the previous condition since is not an identity. For the closed chamber Where go runs across all elements of Go, and is a notation of the closed half space inside containing the point p. at this point, exists since . since is a fundamental element for G-action on Rn , exists for any and such that . Therefore and , which forces hence . Since the above condition holds for arbitrary g Є G, G = G0 which proves the theorem (Armstront, 2006). Acknowledgements Appendices Bibliography American Mathematical Society. (1988). Journal of the American Mathematical Society. [Providence, R.I.], The Society. Armstront, D. D. (2006). Generalized noncrossing partitions and combinatorics of coxeter groups. Thesis (Ph.D.)--Cornell University, Aug., 2006. Benson, C. T., & Grove, L. C. (1985). Finite reflection groups. New York, Springer-Verlag. Borovik, A., & Borovik, A. (2010). Mirrors and reflections: the geometry of finite reflection groups. New York, Springer. Bourbaki, N., & Meldrum, J. (1999). Elements of the history of mathematics. Berlin [u.a.], Springer. Campbell, H. E. A. E., & Wehlau, D. L. (2004). Invariant theory in all characteristics. Providence (R.I.), American mathematical society. Coxeter, H. S. M. (1989). Introduction to geometry. New York, Wiley. Daverman, R. J., & Sher, R. B. (2002). Handbook of geometric topology. Amsterdam, Elsevier. http://www.sciencedirect.com/science/book/9780444824325. Evens, S. (2012). Mathematical aspects of quantization: Center for Mathematics at Notre Dame, Summer School and Conference, May 31-June 10, 2011, Notre Dame University, Notre Dame, Indiana. HöRmander, L. (1971). On the existence and the regularity of solutions of linear pseudo-differential equations. Genève, Enseignement mathématique. Université. Humphreys, J. E. (1990). Reflection groups and coxeter groups. Cambridge [England], Cambridge University Press. Jain, S. K., Parvathi, S., & Khurana, D. (2008). Noncommutative rings, group rings, diagram algebras, and their applications: international conference, December 18-22, 2006, University of Madras, Chennai, India. Providence, R.I., American Mathematical Society. J.J. Sylvester Symposium On Algebraic Geometry, & Igusa, J.-I. (1977). Algebraic geometry: the Johns Hopkins centennial lectures : supplement to the American journal of mathematics. Baltimore, Johns Hopkins University Press. Kane, R. M. (2001). Reflection groups and invariant theory. New York, NY [u.a.], Springer. Lehrer, G., & Taylor, D. E. (2009). Unitary reflection groups. Cambridge, UK, Cambridge University Press. Ratcliffe, J. G. (2006). Foundations of hyperbolic manifolds. New York, Springer. http://site.ebrary.com/id/10229376. Yau, S.-T. (1986). Nonlinear analysis in geometry / lectures given at the ETH-Zurich under the sponsorship of the International Mathematical Union. 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