Chapter VI: Group-Theoretic Problems

Total Page:16

File Type:pdf, Size:1020Kb

Chapter VI: Group-Theoretic Problems CHAPTER VI GRouP - THEORETIC PROBI~I~S One of the major motivating factors in the investigation of graph isomorphism is the hope that a successful determination of its complexity status will provide new insights into P vs. NP problem. In the preceding chapters, we have developed a number of results which (partially) settle the complexity of testing isomorphism in certain subclasses of graphs. We have concentrated on the group-theoretic approach because it adds a wealth of algebraic properties which may be exploited when deter- mining the automorphism group of a graph. In this development, group-theoretic problems have assumed an auxiliary, although crucial role, and have served to reveal algebraic properties of the problem which were exploited in the design of the algo- rithms. In this chapter, we reverse this situation and discuss the complexity of group-theoretic problems in the abstract. Although we take graph isomorphism and other combinatorial problems as point of departure, these group-theoretic problems are of interest in their own right, not only as mathematically interesting questions, but also as algorithmic problems with considerable significance for Computer Sci- ence. In particular, there is a rich set of such problems which appear to be natural candidates for problems in NP that are neither in P nor are NP-eomplete~ I. Some Combinatorial Problems as Group-Theoretic Problems In this section, we translate a number of combinatorial problems, graph isomor- phism for one, into purely group-theoretic questions. The purpose of this translation is to expose a (sometimes hidden) algebraic nature of such problems, and to establish a continuity of the present subject with the preceding ones. In Chapter If, we interpreted the set of isomorphisms between two graphs as a coset of the automorphism group. Our first task is to translate the problem of graph isomorphism into a membership question in do~bL~ eos~ts of certain permutation groups, thereby entirely voiding the problem of its topological and combinatorial aspects. 232 Let A, B < S n be permutation groups of degree n. The double ~oset of A and B contain- ing an element ~r o~ S n is the set ATrB= IaTrfi t a EA,~ ~B t of permutations in S n. In particular, the set AB = f aft ! a E A, fl ~ B ~ is the double coset of A and B con- taining the identity permutation. Since right eosets of A are either disjoint or equal, the double coset A~B is the union of some right cosets of A. Similarly, it is the union of certain !eft eosets of B. Therefore, both the order of A and the order of B divide the eardinality of A~rB. Specifically, one knows The cardinality of the double coset A~rB is IA~B I = IAI' {B { IA"C~S I " The lemma follows from Lemma 20 of ChapLer III by putting the element ~Trfl in A~B into 1-1 correspondence with the element ~-la~rfl of A~B. Double cosets are either disjoint or equal. However, the double eoset ATrB need not be of cardina!ity equal to the cardinatity of the double coset ACD. Thus, double cosets partition the elements of Sn into classes of nonuniform size. We first show that the membership problem for certain double cosets is just graph isomorphism. Let X = (V,E) be a graph with n vertices, V = ~1 ..... nt. We consider X obtained by superimposing two structures, LX and Cx. Specifically, the structure Cx is the com- plete graph Kn = (~1 ..... hi, t(i,j) I 1 <- i < j -< n]), and the structure Lx is the pair (E, ~,), where E is the complement of E, i.e., E = t(i,j) I 1 -- i < j -< n, (i,j) ~t El. Now superim- posing Lx and Cx may be thought of as labelling the edges of Kn with labels of two kinds, "edge" and "not an edge". More abstractly, we consider superimposing Lx onto Cx as a permutation in Sym(ELjE), i.e., a 1-1 map from the set of all unordered pairs (i,j) in EL)E onto all unordered pairs (i,j) in the edge set of Kn. We first investigate when two permutations ~ and ~ in Sym(EuE) specify the graph X identically. Since we consider the labels in the sets E and E indistinguishable, it is clear that exactly the permutations in the right coset Sym(E)xSym(E)Tr specify identical graphs. We set A = Sym(E)xSym(E) and note that A is the automorphism group of the ~abelting structure Lx. 233 Next, we ask when two permutations n, ~ c Sym(EuE ) specify isomorphic graphs. Let X, be the graph specified by n, X~ the graph specified by ~. Then X# must be obtainable from X~ by a permutation in S n, So, let B be the permutation group induced by S n on the set of all edges of Kn. Then X~ and X~ are isomorphic iff the per- mutation ~ may be obtained as the product an#, where a c A and fl E B, i.e., iff E AnB. Note that B is the symmetry group of the labelled structure Cx. We there- fore obtain PI~POSlTION 1 (Hoffmann) Let X = (V,E) and X' = (V,E') be two graphs with n vertices. Let E and E' be the com- plement edge sets of X and X', respectively. Then X and X' are isomorphic iff each per- mutation ~ ~ Sym(EuE) which maps E onto E' and E onto E' is in the double coset AB, where A = Sym(E)×Sym(E) and B is the group of actions of Sym(V) on the set EUE. The proof is straightforward. Note that if one such permutation ~ is in AB, then the set of all such permutations is precisely one of the right cosets of A contained in AB. EXAMPLE 1 Consider the graph X of Figure 1 below. Its edge set E is la = (1,2), b = (2,3), c = (3,4), 1 2\/ \5/ 3 4 Figure I d= (4,5), e = (1,5)~. The complement set is E = If= (I,3), g = (2,4), h= (3,5), i = (1,4), k = (2,5)~. X is specified by every permutation in A = Sym(Ia,b,c,d, el)xSym(~f,g,h,i,kl). The graph X' of Figure 2 below 2 5 Figure 2 234 has the edge set E' = ia,e,h,i,kl with the complement set E' = ib,d,e,f,gl. This graph may be specified by q/= (b,c,h,e,k,g,d,i,f), which is a permutation mapping E onto E' and E onto E'. One verifies that q/= cxfl, where a = (a,e)(b,c)(h,k,i) is m A and fl = (a,k,e)(b,h,f)(d,i,g). Let B be the permutation group induced by S 5 on the set EuE, Then fle 13 since it is induced by the permutation ~ = (1,2,G), thus ~ e A13, i.e., the two graphs are isomorphic. Note that the factorization of ~ is not unique. A different faetorization is, for example, a'fl', where a' = (b,d,e)(f,g,h) and fl'= (b,k,g) (e,h,d) (e,i,f). More generally, let Lx be a combinatorial structure with the automorphism group A, C x a combinatorial structure with the automorphism group B, where A and B are permutation groups with the same permutation domain V. Let Tr E Sym(V) be a per- mutation specifying how to superimpose L x onto Cx, thereby obtaining the "labelled" structure X~. Then a permutation ~ ~ Sym(V) specifies a structure isomorphic to X~ iff !/~ e ArrB. Next, the automorphism group of the structure specified by ~ is clearly A~v~B: Let ~ c Sym(V) be an automorphism of Xn. Then ~ mush also be an automorphism of Cx, hence is in B. Furthermore, since the structure LX has been mapped into (Lx)~, ~k must be in A n. Conversely, it is clear that any permutation in A=f~B is an automor- phism of X~. Theorem 7 of Chapter II is a special case of this observation. Having interpreted double eosets as isomorphism classes, we may consider the number of double cosets into which Sym(V) is partitioned by the groups A and B as the number of ~o~%~orrLorIJh~c ways in which the structures L X and C X may he super- imposed (see also Section 6 below). We therefore know COROLLARY 1 The number of nonisomorphic graphs with n vertices and p edges is equal to the number of double cosets into which Sp+q is partitioned by the subgroups A and B, where A = SpxSq, q = (~)-p, and B is the group induced by S n on the set of all unor- dered pairs (i,j), i ~ i,j-< n, assuming these pairs have been enumerated in some order. Other combinatorial applications include the number of nonisomorphie graph vertex labellings, e.g., the number of distinct necklaces with n beads and a prescribed number of beads of equal colors, and edge labellings of graphs other than the com- plete graph Kn, 235 E~tmLg 2 We determine the number of nonisomorphic graphs with 4 vertices and 3 edges. Assuming a lexicographJc enumeration of the unordered pairs (i,j), I -< i < j -< 4, the group B is generated by the two permutations (2,4)(3,5) and (1,4,6,3)(2,5). We have p = 3 and q = 3, so that A is S3xS 3. By exhaustive enumeration one can prove that /0, (3,4), (3,5,4)~ is a complete set of representatives for double cosets of A and B, thus there are exactly 3 nonisomorphic graphs with 3 edges and 4 vertices.
Recommended publications
  • Double Coset Formulas for Profinite Groups 1
    DOUBLE COSET FORMULAS FOR PROFINITE GROUPS PETER SYMONDS Abstract. We show that in certain circumstances there is a sort of double coset formula for induction followed by restriction for representations of profinite groups. 1. Introduction The double coset formula, which expresses the restriction of an induced module as a direct sum of induced modules, is a basic tool in the representation theory of groups, but it is not always valid as it stands for profinite groups. Based on our work on permutation modules in [8], we give some sufficient conditions for a strong form of the formula to hold. These conditions are not necessary, although they do often hold in interesting cases, and at the end we give some simple examples where the formula fails. We also formulate a weaker version of the formula that does hold in general. 2. Results We work with profinite groups G and their representations over a complete commutative noetherian local ring R with finite residue class field of characteristic p. There is no real loss ˆ of generality in taking R = Zp. Our representations will be in one of two categories: the discrete p-torsion modules DR(G) or the compact pro-p modules CR(G). These are dual by the Pontryagin duality functor Hom(−, Q/Z), which we denote by ∗, so we will usually work in DR(G). Our modules are normally left modules, so dual means contragredient. For more details see [7, 8]. For any profinite G-set X and M ∈ DR(G) we let F (X, M) ∈ DR(G) denote the module of continuous functions X → M, with G acting according to (gf)(x) = g(f(g−1x)), g ∈ G, f ∈ F (X, M), x ∈ X.
    [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]
  • Decomposition Theorems for Automorphism Groups of Trees
    DECOMPOSITION THEOREMS FOR AUTOMORPHISM GROUPS OF TREES MAX CARTER AND GEORGE WILLIS Abstract. Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, double cosets of the group of label preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups are enumerated. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range. 1. Introduction Totally disconnected locally compact groups (t.d.l.c. groups here on after) are a broad class which includes Lie groups over local fields. It is not known, however, just how broad this class is, and one approach to investigating this question is to attempt to extend ideas about Lie groups to more general t.d.l.c. groups. This approach has (at least partially) motivated works such as [4, 8, 9]. That is the approach taken here for Bruhat and Cartan decompositions of gen- eral linear groups over local fields, which are respectively double-coset enumerations of GLpn, Qpq over the subgroups, B, of upper triangular matrices, and GLpn, Zpq, of matrices with p-adic integer entries. Such decompositions of real Lie groups are given in [11, §§VII.3 and 4], and of Lie groups over local fields in [2, 3, 7, 10]. Automorphism groups of regular trees have features in common with the groups GLp2, Qpq (in particular, PGLp2, Qpq has a faithful and transitive action on a reg- ular tree, [12, Chapter II.1]). Here, we find double coset enumerations of automor- phism groups of trees that correspond to Bruhat and Cartan decompositions.
    [Show full text]
  • Separability Properties of Free Groups and Surface Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 78 (1992) 77-84 77 North-Holland Separability properties of free groups and surface groups G.A. Niblo School of Mathemuticul Sciences, Queen Mary and Westfield College, Mile End Road, London El 4NS, United Kingdom Communicated by K.W. Gruenberg Received II March 1991 Abstruct Niblo, G.A., Separability properties of free groups and surface groups, Journal of Pure and Applied Algebra 78 (1992) 77-84 A subset X of a group G is said to be separable if it is closed in the profinite topology. Separable subgroups are very useful in low-dimensional topology and there has been some interest in separable double cosets. A new method for showing that a double coset is separable is introduced and it is used to obtain a short proof of the result of Gitik and Rips, that in a free group every double cosct of finitely generated subgroups is separable. In addition it is shown that this property is shared by Fuchsian groups and the fundamental groups of Seifert fibred 3-manifolds. 1. Introduction Let G be a group regarded as acting on itself by left and right multiplication. In order to define a topology on G which is equivariant with respect to both of these actions, we only need to specify a base 93 for the neighbourhoods of the identity element. Definition. Let !93 be the base consisting of the finite-index normal subgroups of G.
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [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]
  • 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]
  • Affine Permutations and Rational Slope Parking Functions
    AFFINE PERMUTATIONS AND RATIONAL SLOPE PARKING FUNCTIONS EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI ABSTRACT. We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincar´epolynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials. 1. INTRODUCTION Parking functions are ubiquitous in the modern combinatorics. There is a natural action of the symmetric group on parking functions, and the orbits are labeled by the non-decreasing parking functions which correspond natu- rally to the Dyck paths. This provides a link between parking functions and various combinatorial objects counted by Catalan numbers. In a series of papers Garsia, Haglund, Haiman, et al. [18, 19], related the combinatorics of Catalan numbers and parking functions to the space of diagonal harmonics. There are also deep connections to the geometry of the Hilbert scheme. Since the works of Pak and Stanley [29], Athanasiadis and Linusson [5] , it became clear that parking functions are tightly related to the combinatorics of the affine symmetric group.
    [Show full text]