DOCSLIB.ORG

GROUP ACTIONS on SETS 1. Group Actions Let X Be a Set and Let G Be

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
GROUP ACTIONS on SETS 1. Group Actions Let X Be a Set and Let G Be CHAPTER II GROUP ACTIONS ON SETS 1. Group Actions Let X be a set and let G be a group. As before, we say that G acts on X if we have a representation ρ : G S(X) or equivalently there is a binary operation G X X satisfying the rules ! × ! 1x = x for all x X; 2 (gh)x = g(hx) for allg; h G; x X: 2 2 Given such an action, define a relation on X by x y if and only if g G such that y = gx. It is not hard to see that this defines an equivalence relation∼ on X. (See the9 Exercises.)2 It follows that X is partitioned into disjoint equivalence classes which are also called orbits. The equivalence class containing x is Gx = gx g G , and is called the orbit of x. If there is only one orbit, the action is said to be f 2 g transitive. This is equivalent to the assertion: for each x; y G, g G such that y = gx. Since any set breaks up into a disjoint union of equivalence2 classes9 2 under an equivalence relation, it follows that if G acts on X, then X is a union of disjoint orbits. Examples. 1. Left multiplication. Let H G. Then G acts on X = G=H by g(xH) = (gx)H where g; x G. Note ≤ 1 1 1 1 2 that if xH = yH are the same coset, then (gx)− (gy) = x− g− gy = x− y H, so (gx)H = (gy)H. In other words, this action is well defined. It is also clear that 1(xH) = xH and2 (gh)(xH) = g(h(xH)), so it does define an action. Let 1 denote the trivial coset H. Then clearly X = G1 so the action is transitive. g 1 2. Conjugation. Let X = G and let G act on itself by x = gxg− where we use pre-exponential notation as indicated previously for the binary operation. The orbits of this action are called the conjugacy classes of G. 3. (a) More generally, let X be the set of all subsets of G (denoted in set theory by 2G). Let G act on X g 1 by S = gSg− for S X, i.e., for S G. (b) Alternately, let X be the set of all subgroups of G and let G act the same way. 2 ⊆ 4. Let G be the group of 3 3 real, orthogonal matrices, i.e., 3 3 real matrices A such that AAt = AtA = I. It is not hard to see that the× set of all such matrices forms a group,× called the orthogonal group and denoted O(3) or O(3; R), Such matrices determine linear transformations of R3 with preserve lengths. Hence, we get an action of the orthogonal group G = O(3) on Euclidean 3-space R3. The orbits are spheres centered at the origin. Suppose G acts on the set X. For x X, we set 2 Gx = g G gx = x : f 2 g Gx is a subgroup of G. For, certainly 1 Gx, and if gx = x and hx = x, it is clear that ghx = gx = x. 1 2 Moreover, gx = x implies x = g− x. Gx is called the isotropy subgroup or stabilizer of x. Note that Gx is not generally a normal subgroup of G. In fact, we have 1 gGxg− = Ggx: 1 1 1 For, h Ggx hgx = gx g− hgx = x g− hg Gx h gGxg− as claimed. 2 , , , 2 , 2 Typeset by -TEX AMS 15 16 II. GROUP ACTIONS ON SETS Proposition. Let G be a group acting on a set X and let x X. The mapping gGx gx provides 2 7! a one-to-one correspondence between G=Gx and Gx. Moreover, this correspondence is consistent with the actions of G on both sets. In particular, if G is finite, then Gx = (G : Gx) = G = Gx . j j j j j j Proof. First, note that the map is well defined. For, 1 1 hGx = gGx g− h Gx g− hx = x hx = gx: , 2 , , In fact, this argument read backwards, shows also that the map is one-to-one. The map is clearly onto. Finally, h(gGx) = (hg)Gx (hg)x = h(gx) 7! so the mapping is consistent with the two actions. Corollary. If G is a finite group acting on a set X, then every orbit is a finite set and its cardinality divides the order G of the group. j j Let G be a group, finite or infinite. Among the sets on which G acts, we may distinguish the coset spaces G=H for H a subgroup G. G acts transitively on such a set, and the proposition tells us that up to one-to-one action preserving correspondences, every set X on which G acts transitively may be assumed to be of this form. So up to such correspondence, any set on which G acts may be thought of as a disjoint union of coset spaces. Let G act on X. As mentioned before, we may associate to this action a representation ρ : G S(X). The kernel of this representation is the set of all g G which act as the identity on all of X. That! is, 2 Ker ρ = G \ x x X 2 is the intersection of all stabilizers of points in X. Exercises. 1. Let the group G act on the set X. (a) Show that the relation x y defined in the section is an equivalence relation. (b) Suppose G were only a monoid∼ but we defined the notion of G acting on X the same way. Would still be an equivalence relation? ∼ 2. Let the group Gl(n; R) of all n n invertible real matrices act on Rn the usual way where Rn is exhibited as the set of all n 1 column vectors.× What are the orbits? For each orbit, pick a point in that orbit and describe the isotropy× group. (If you pick the point properly, the description should be relatively simple.) 3. Let O(n) denote the group of all n n real orthogonal matrices, and let O(n) act on Rn the usual way. (a) Show that the orbits of O(n) are×n 1 spheres of different radii in Rn. − (b) What is the isotropy group of the unit vector e1 with first coordinate one and other coordinates zero. 2. Centralizers, Normalizers, and the Class Equation We consider some important examples of groups acting on sets. (1) Let G act on the set X of subsets of G by conjugation g 1 S = gSg− : The orbits of the action are families of conjugates subsets. The most interesting case is that in which the set is a subgroup H and the orbit is the set of all subgroups conjugate to H. The isotropy subgroup GS of the subset S is also denoted 1 NG(S) = g G gSg− = S f 2 g 3. CENTRALIZERS, NORMALIZERS, AND THE CLASS EQUATION 17 and is called the normalizer of S in G. Note that if H is a subgroup of G, then H is a subgroup of NG(H) and in fact NG(H) is the largest subgroup of G in which H is normal. (See the Exercises.) It follows from our basic proposition on the isotropy subgroup that in the case of a finite group, the number of distinct conjugates of a given subset S is (G : NG(S)), so that number divides G . g 1 j j (2) Let G act on X = G by conjugation of elements: x = gxg− . The orbit of the element x G is the set of all elements of G conjugate to x. The isotropy subgroup is denoted 2 CG(x) = g G gx = xg f 2 g and is called the centralizer of x in G. As above, the number of distinct conjugates of an element x is (G : CG(x)) so it divides G . The intersection of allj thej centralizers of elements of G is denoted Z(G) = g G gx = xg for all x G f 2 2 g and is called the center of the group G. Note that Z(G) is the kernel of the representation ρ : G Aut(G) defined by ! g 1 ρ(g)(x) = x = gxg− : Theorem. (The class equation) Let G be a finite group. Then G = Z(G) + (G : CG(x)): j j j j X x T 2 where T is a set of distinct representatives of the conjugacy classes of G of size larger than one. Proof. Let G act on itself by conjugation. Then G decomposes as a disjoint union of orbits, i.e., conjugacy classes. The number of elements in the class containing x is as above (G : CG(x)). However, this index is one if and only if G = CG(x), i.e., every element of G commutes with x, i.e., x Z(G). The above 2 equation then just reflects the decomposition of G into this disjoint union. Corollary. Let G = pn where p is a prime. Then p divides Z(G) . In particular, Z(G) is not the trivial subgroup. j j j j Proof. Use the class equation. The left hand side is a power of p. The terms in the sum on the right are all non-trivial divisors of pn, so they are all divisible by p. It follows that Z(G) is divisible by p. j j Corollary. Let p be a prime.
Load more

Recommended publications
  • Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19Th, 2020
    Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19th, 2020 Group Isomorphism Theorems + Group Actions The Isomorphism Theorems for Groups Group Actions Polynomial Invariants and An This material represents x3.2.3-3.3.2 from the course notes. Quotients and Homomorphisms, I Like with rings, we also have various natural connections between normal subgroups and group homomorphisms. To begin, observe that if ' : G ! H is a group homomorphism, then ker ' is a normal subgroup of G. In fact, I proved this fact earlier when I introduced the kernel, but let me remark again: if g 2 ker ', then for any a 2 G, then '(aga−1) = '(a)'(g)'(a−1) = '(a)'(a−1) = e. Thus, aga−1 2 ker ' as well, and so by our equivalent properties of normality, this means ker ' is a normal subgroup. Thus, we can use homomorphisms to construct new normal subgroups. Quotients and Homomorphisms, II Equally importantly, we can also do the reverse: we can use normal subgroups to construct homomorphisms. The key observation in this direction is that the map ' : G ! G=N associating a group element to its residue class / left coset (i.e., with '(a) = a) is a ring homomorphism. Indeed, the homomorphism property is precisely what we arranged for the left cosets of N to satisfy: '(a · b) = a · b = a · b = '(a) · '(b). Furthermore, the kernel of this map ' is, by definition, the set of elements in G with '(g) = e, which is to say, the set of elements g 2 N. Thus, kernels of homomorphisms and normal subgroups are precisely the same things.
    [Show full text]
  • The Fundamental Groupoid of the Quotient of a Hausdorff
    The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Ronald Brown,∗ Philip J. Higgins,† Mathematics Division, Department of Mathematical Sciences, School of Informatics, Science Laboratories, University of Wales, Bangor South Rd., Gwynedd LL57 1UT, U.K. Durham, DH1 3LE, U.K. October 23, 2018 University of Wales, Bangor, Maths Preprint 02.25 Abstract The main result is that the fundamental groupoidof the orbit space of a discontinuousaction of a discrete groupon a Hausdorffspace which admits a universal coveris the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author’s book on Topology [3]. This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader.
    [Show full text]
  • GROUP ACTIONS 1. Introduction the Groups Sn, An, and (For N ≥ 3)
    GROUP ACTIONS KEITH CONRAD 1. Introduction The groups Sn, An, and (for n ≥ 3) Dn behave, by their definitions, as permutations on certain sets. The groups Sn and An both permute the set f1; 2; : : : ; ng and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If we label the vertices of the n-gon in a definite manner by the numbers from 1 to n then we can view Dn as a subgroup of Sn. For instance, the labeling of the square below lets us regard the 90 degree counterclockwise rotation r in D4 as (1234) and the reflection s across the horizontal line bisecting the square as (24). The rest of the elements of D4, as permutations of the vertices, are in the table below the square. 2 3 1 4 1 r r2 r3 s rs r2s r3s (1) (1234) (13)(24) (1432) (24) (12)(34) (13) (14)(23) If we label the vertices in a different way (e.g., swap the labels 1 and 2), we turn the elements of D4 into a different subgroup of S4. More abstractly, if we are given a set X (not necessarily the set of vertices of a square), then the set Sym(X) of all permutations of X is a group under composition, and the subgroup Alt(X) of even permutations of X is a group under composition. If we list the elements of X in a definite order, say as X = fx1; : : : ; xng, then we can think about Sym(X) as Sn and Alt(X) as An, but a listing in a different order leads to different identifications 1 of Sym(X) with Sn and Alt(X) with An.
    [Show full text]
  • Mathematics of the Rubik's Cube
    Mathematics of the Rubik's cube Associate Professor W. D. Joyner Spring Semester, 1996{7 2 \By and large it is uniformly true that in mathematics that there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." John von Neumann COLLECTED WORKS, VI, p. 489 For more mathematical quotes, see the first page of each chapter below, [M], [S] or the www page at http://math.furman.edu/~mwoodard/mquot. html 3 \There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man's life to know them the little new that each man gets from life is very costly and the only heritage he has to leave." Ernest Hemingway (From A. E. Hotchner, PAPA HEMMINGWAY, Random House, NY, 1966) 4 Contents 0 Introduction 13 1 Logic and sets 15 1.1 Logic................................ 15 1.1.1 Expressing an everyday sentence symbolically..... 18 1.2 Sets................................ 19 2 Functions, matrices, relations and counting 23 2.1 Functions............................. 23 2.2 Functions on vectors....................... 28 2.2.1 History........................... 28 2.2.2 3 × 3 matrices....................... 29 2.2.3 Matrix multiplication, inverses.............. 30 2.2.4 Muliplication and inverses...............
    [Show full text]
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
    [Show full text]
  • Definition 1.1. a Group Is a Quadruple (G, E, ⋆, Ι)
    GROUPS AND GROUP ACTIONS. 1. GROUPS We begin by giving a definition of a group: Definition 1.1. A group is a quadruple (G, e, ?, ι) consisting of a set G, an element e G, a binary operation ?: G G G and a map ι: G G such that ∈ × → → (1) The operation ? is associative: (g ? h) ? k = g ? (h ? k), (2) e ? g = g ? e = g, for all g G. ∈ (3) For every g G we have g ? ι(g) = ι(g) ? g = e. ∈ It is standard to suppress the operation ? and write gh or at most g.h for g ? h. The element e is known as the identity element. For clarity, we may also write eG instead of e to emphasize which group we are considering, but may also write 1 for e where this is more conventional (for the group such as C∗ for example). Finally, 1 ι(g) is usually written as g− . Remark 1.2. Let us note a couple of things about the above definition. Firstly clo- sure is not an axiom for a group whatever anyone has ever told you1. (The reason people get confused about this is related to the notion of subgroups – see Example 1.8 later in this section.) The axioms used here are not “minimal”: the exercises give a different set of axioms which assume only the existence of a map ι without specifying it. We leave it to those who like that kind of thing to check that associa- tivity of triple multiplications implies that for any k N, bracketing a k-tuple of ∈ group elements (g1, g2, .
    [Show full text]
  • An Algebraic Approach to the Weyl Groupoid
    An Algebraic Approach to the Weyl Groupoid Tristan Bice✩ Institute of Mathematics Czech Academy of Sciences Zitn´a25,ˇ 115 67 Prague, Czech Republic Abstract We unify the Kumjian-Renault Weyl groupoid construction with the Lawson- Lenz version of Exel’s tight groupoid construction. We do this by utilising only a weak algebraic fragment of the C*-algebra structure, namely its *-semigroup reduct. Fundamental properties like local compactness are also shown to remain valid in general classes of *-rings. Keywords: Weyl groupoid, tight groupoid, C*-algebra, inverse semigroup, *-semigroup, *-ring 2010 MSC: 06F05, 20M25, 20M30, 22A22, 46L05, 46L85, 47D03 1. Introduction 1.1. Background Renault’s groundbreaking thesis [Ren80] revealed the striking interplay be- tween ´etale groupoids and the C*-algebras they give rise to. Roughly speaking, the ´etale groupoid provides a more topological picture of the corresponding C*-algebra, with various key properties of the algebra, like nuclearity and sim- plicity (see [ADR00] and [CEP+19]), being determined in a straightforward way from the underlying ´etale groupoid. Naturally, this has led to the quest to find appropriate ´etale groupoid models for various C*-algebras. Two general methods have emerged for finding such models, namely arXiv:1911.08812v3 [math.OA] 20 Sep 2020 1. Exel’s tight groupoid construction from an inverse semigroup, and 2. Kumjian-Renault’s Weyl groupoid construction from a Cartan C*-subalgebra (see [Exe08], [Kum86] and [Ren08]). However, both of these have their limi- tations. For example, tight groupoids are always ample, which means the cor- responding C*-algebras always have lots of projections. On the other hand, the Weyl groupoid is always effective, which discounts many naturally arising groupoids.
    [Show full text]
  • A Class of Totally Geodesic Foliations of Lie Groups
    Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 57-64. ~2 Printed in India. A class of totally geodesic foliations of Lie groups G SANTHANAM School of Mathematics. Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India MS received 4 October 1988; revised 25 May 1989 A~traet. This paper is devoted to classifying the foliations of G with leaves of the form .qKh - t where G is a compact, connected and simply connected L,c group and K is a connected closed subgroup of G such that G/K is a rank-I Riemannian symmetric space. In the case when G/K =S", the homotopy type of space of such foliations is also given. Keywords. Foliations; rank-I Riemannian symmetric space; cutlocus. I. Introduction The study of fibrations of spheres by great spheres is a very interesting problem in geometry and it is very important in the theory of Blaschke manifolds. In [l], Gluck and Warner have studied the great circle fibrations of the three spheres. In that paper they have proved very interesting results. When we look at the problem group theoretically, we see that all the results of [1] go through, for foliations of G with leaves of the form gKh- 1 where G is a compact, connected and simply connected Lie group and K is a connected closed subgroup of G such that G/K is a rank- l Riemannian symmetric space (see [-2]), except perhaps the theorem 3 for the pair (G, K) such that G/K=CP n, HP ~ etc.
    [Show full text]
  • Distortion of Dimension by Metric Space-Valued Sobolev Mappings Lecture II
    Distortion of dimension by metric space-valued Sobolev mappings Lecture II Jeremy Tyson University of Illinois at Urbana-Champaign Metric Spaces: Analysis, Embeddings into Banach Spaces, Applications Texas A&M University College Station, TX 8 July 2016 Outline Lecture I. Sobolev and quasiconformal mappings in Euclidean space Lecture II. Sobolev mappings between metric spaces Lecture III. Dimension distortion theorems for Sobolev and quasiconformal mappings defined from the sub-Riemannian Heisenberg group f homeo in Rn, n ≥ 2 Gehring metric QC ) analytic QC ) geometric QC modulus) estimates (local) QS f : X ! Y homeo between proper Q-regular mms satisfying Q-PI, Q > 1 0 metric QC HK=)98 QS f : X ! Y homeo between proper Q-regular mms 0 . geometric QC T(98 QS Motivation: quasiconformal mappings in metric spaces The theory of analysis in metric measure spaces originates in two papers of Juha Heinonen and Pekka Koskela: ‘Definitions of quasiconformality', Invent. Math., 1995 `QC maps in metric spaces of controlled geometry', Acta Math., 1998 The latter paper introduced the concept of p-Poincar´einequality on a metric measure space, which has become the standard axiom for first-order analysis. f : X ! Y homeo between proper Q-regular mms satisfying Q-PI, Q > 1 0 metric QC HK=)98 QS f : X ! Y homeo between proper Q-regular mms 0 . geometric QC T(98 QS Motivation: quasiconformal mappings in metric spaces The theory of analysis in metric measure spaces originates in two papers of Juha Heinonen and Pekka Koskela: ‘Definitions of quasiconformality', Invent. Math., 1995 `QC maps in metric spaces of controlled geometry', Acta Math., 1998 The latter paper introduced the concept of p-Poincar´einequality on a metric measure space, which has become the standard axiom for first-order analysis.
    [Show full text]
  • Ideal Classes and SL 2
    IDEAL CLASSES AND SL2 KEITH CONRAD 1. Introduction A standard group action in complex analysis is the action of GL2(C) on the Riemann sphere C [ f1g by linear fractional transformations (M¨obiustransformations): a b az + b (1.1) z = : c d cz + d We need to allow the value 1 since cz + d might be 0. (If that happens, az + b 6= 0 since a b ( c d ) is invertible.) When z = 1, the value of (1.1) is a=c 2 C [ f1g. It is easy to see this action of GL2(C) on the Riemann sphere is transitive (that is, there is one orbit): for every a 2 C, a a − 1 (1.2) 1 = a; 1 1 a a−1 so the orbit of 1 passes through all points. In fact, since ( 1 1 ) has determinant 1, the action of SL2(C) on C [ f1g is transitive. However, the action of SL2(R) on the Riemann sphere is not transitive. The reason is the formula for imaginary parts under a real linear fractional transformation: az + b (ad − bc) Im(z) Im = cz + d jcz + dj2 a b a b when ( c d ) 2 GL2(R). Thus the imaginary parts of z and ( c d )z have the same sign when a b ( c d ) has determinant 1. The action of SL2(R) on the Riemann sphere has three orbits: R [ f1g, the upper half-plane h = fx + iy : y > 0g, and the lower half-plane. To see that the action of SL2(R) on h is transitive, pick x + iy with y > 0.
    [Show full text]
  • Graphs of Groups: Word Computations and Free Crossed Resolutions
    Graphs of Groups: Word Computations and Free Crossed Resolutions Emma Jane Moore November 2000 Summary i Acknowledgements I would like to thank my supervisors, Prof. Ronnie Brown and Dr. Chris Wensley, for their support and guidance over the past three years. I am also grateful to Prof. Tim Porter for helpful discussions on category theory. Many thanks to my family and friends for their support and encouragement and to all the staff and students at the School of Mathematics with whom I had the pleasure of working. Finally thanks to EPSRC who paid my fees and supported me financially. ii Contents Introduction 1 1 Groupoids 4 1.1 Graphs, Categories, and Groupoids .................... 4 1.1.1 Graphs ................................ 5 1.1.2 Categories and Groupoids ..................... 6 1.2 Groups to Groupoids ............................ 12 1.2.1 Examples and Properties of Groupoids .............. 12 1.2.2 Free Groupoid and Words ..................... 15 1.2.3 Normal Subgroupoids and Quotient Groupoids .......... 17 1.2.4 Groupoid Cosets and Transversals ................. 18 1.2.5 Universal Groupoids ........................ 20 1.2.6 Groupoid Pushouts and Presentations .............. 24 2 Graphs of Groups and Normal Forms 29 2.1 Fundamental Groupoid of a Graph of Groups .............. 29 2.1.1 Graph of Groups .......................... 30 2.1.2 Fundamental Groupoid ....................... 31 2.1.3 Fundamental Group ........................ 34 2.1.4 Normal Form ............................ 35 2.1.5 Examples .............................. 41 2.1.6 Graph of Groupoids ........................ 47 2.2 Implementation ............................... 49 2.2.1 Normal Form and Knuth Bendix Methods ............ 50 iii 2.2.2 Implementation and GAP4 Output ................ 54 3 Total Groupoids and Total Spaces 61 3.1 Cylinders .................................
    [Show full text]
  • Regular Representations and Coset Representations Combined with a Mirror-Permutation
    MATCH MATCH Commun. Math. Comput. Chem. 83 (2020) 65-84 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 Regular Representations and Coset Representations Combined with a Mirror-Permutation. Concordant Construction of the Mark Table and the USCI-CF Table of the Point Group Td Shinsaku Fujita Shonan Institute of Chemoinformatics and Mathematical Chemistry, Kaneko 479-7 Ooimachi, Ashigara-Kami-Gun, Kanagawa-Ken, 258-0019 Japan [email protected] (Received November 21, 2018) Abstract A combined-permutation representation (CPR) of degree 26 (= 24 + 2) for a regular representation (RR) of degree 24 is derived algebraically from a multiplica- tion table of the point group Td, where reflections are explicitly considered in the form of a mirror-permutation of degree 2. Thereby, the standard mark table and the standard USCI-CF table (unit-subduced-cycle-index-with-chirality-fittingness table) are concordantly generated by using the GAP functions MarkTableforUSCI and constructUSCITable, which have been developed by Fujita for the purpose of systematizing the concordant construction. A CPR for each coset representation (CR) (Gin)Td is obtained algebraically by means of the GAP function CosetRepCF developed by Fujita (Appendix A). On the other hand, CPRs for CRs are obtained geometrically as permutation groups by considering appropriate skeletons, where the point group Td acts on an orbit of jTdj=jGij positions to be equivalent in a given skeleton so as to generate the CPR of degree jTdj=jGij. An RR as a CPR is obtained by considering a regular body (RB), the jTdj positions of which are considered to be governed by RR (C1n)Td.
    [Show full text]