Showing papers on "Symmetric group published in 1991"

The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions

Abstract: Group Representations.- Representations of the Symmetric Group.- Combinatorial Algorithms.- Symmetric Functions.- Applications and Generalizations.

$q$-tensor space and $q$-Weyl modules

Abstract: We obtain the irreducible representations of the q-Schur algebra, motivated by the fact that these representations give all the irreducible representations of GLn(q) in the nondescribing characteristic The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction The theory of polynomial representations of general linear groups is equivalent to the representation theory of Schur algebras (see Green's book [5] and the bibliography therein) In [4], we defined q-analogues of Schur algebras When q = 1, these are the usual Schur algebras, and when q is a prime power, representations of q-Schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case, including all irreducible representations of these groups and important information about decomposition numbers It is natural to ask what features of the classical Schur algebras have qanalogues In this paper, we define q-analogues of tensor space, of Weyl modules, and of weight spaces, thereby generalizing the main reslllts which appear in Green's book [5] For example, we classify the irreducible modules for qSchur algebras, we determine bases for q-Weyl modules compatible with weight spaces, and we give results on composition multiplicities of the irreducible modules in q-Weyl modules The proofs are largely self-contained, so by specializing q to 1, we recover the corresponding results in [5] This paper, therefore, is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 THE q-SCHUR ALGEBRA Let r be a natural number, let R be an integral domain, and let q be a unit in R We denote the symmetric group on r letters by er The Hecke algebra Z is the R-free R-algebra with basis {Twlw E er} where the multiplication is determined by the following rule If a = (i, i + 1) is a basic transposition in Received by the editors August 15, 1989 1980 Mathematics Subject Classification ( 198 5 Revision) Primary 1 6A64, 1 6A6 5 ; Secondary 20C30 This research was supported in part by NSF Grant No DMS-8802290 The authors gratefully acknowledge support received from NATO Grant No 0222/87 (r) 1991 American Mathematical Society 0002-9947/91 $100 + $25 per page

A Frobenius formula for the characters of the Hecke algebras

Abstract: This paper uses the theory of quantum groups and the quantum Yang-Baxter equation as a guide in order to produce a method of computing the irreducible characters of the Hecke algebra. This approach is motivated by an observation of M. Jimbo giving a representation of the Hecke algebra on tensor space which generates the full centralizer of a tensor power of the “standard” representation of the quantum group\(U_q (\mathfrak{s}l(n))\). By rewriting the solutions of the quantum Yang-Baxter equation for\(U_q (\mathfrak{s}l(n))\) in a different form one can avoid the quantum group completely and produce a “Frobenius” formula for the characters of the Hecke algebra by elementary methods. Using this formula we derive a combinatorial rule for computing the irreducible characters of the Hecke algebra. This combinatorial rule is aq-extension of the Murnaghan-Nakayama for computing the irreducible characters of the symmetric group. Along the way one finds connections, apparently unexplored, between the irreducible characters of the Hecke algebra and Hall-Littlewood symmetric functions and Kronecker products of symmetric groups.

Permutation polynomials of the formx r f(x q?1)/d) and their group structure

Abstract: The object of this paper is to give a systematic treatment of permutation polynomials (over a finite fieldF q ) of the formx r f(x q−1)/d). In particular, a criterion is obtained for such a polynomial to be a permutation polynomial and it is proved that all such permutation polynomials form a group isomorphic to a generalized wreath product of certain abelian groups.

Quasi-exactly solvable Lie algebras of differential operators in two complex variables

Abstract: The authors completely classify all 'quasi-exactly solvable' Lie algebras of first-order differential operators in two complex variables. Applications to quasi-exactly solvable quantum problems are indicated.

Symmetry and combinatorial enumeration in chemistry

Abstract: 1 Introduction.- 2 Symmetry and Point Groups.- 2.1 Symmetry Operations and Elements.- 2.2 Conjugacy Glasses in Point Groups.- 2.3 Subgroups of Point Groups.- 2.4 Conjugate and Normal Subgroups of Point Groups.- 2.5 Non-Redundant Set of Subgroups for a Point Group.- 3 Permutation Groups.- 3.1 Permutations and Cycles.- 3.2 Permutation Groups.- 3.3 Transitivity and Orbits..- 3.4 Symmetric Groups.- 3.5 Parity.- 3.6 Alternating Groups.- 4 Axioms and Theorems of Group Theory.- 4.1 Axioms and Multiplication Tables.- 4.2 Subgroups.- 4.3 Cosets.- 4.4 Equivalence Relations.- 4.5 Conjugacy Classes.- 4.6 Conjugate and Normal Subgroups.- 4.7 Subgroup Lattices.- 4.8 Cyclic Groups.- 5 Coset Representations and Orbits.- 5.1 Coset Representations.- 5.2 Transitive Permutation Representations.- 5.3 Mark Tables.- 5.4 Permutation Representations and Orbits.- 6 Systematic Classification of Molecular Symmetries.- 6.1 Assignment of Coset Representations to Orbits.- 6.2 SCR Notation.- 7 Local Symmetries and Forbidden Coset Representations.- 7.1 Blocks and Local Symmetries.- 7.2 Forbidden Coset Representations.- 8 Chirality Fittingness of an Orbit.- 8.1 Ligands.- 8.2 Behavior of Cosets on the Action of a CR.- 8.3 Chirality Fittingness of an Orbit.- 9 Subduction of Coset Representations.- 9.1 Subduction of Coset Representations.- 9.2 Subduced Mark Table.- 9.3 Chemical Meaning of Subduction.- 9.4 Unit Subduced Cycle Indices.- 9.5 Unit Subduced Cycle Indices with Chirality Fittingness.- 9.6 Desymmetrization Lattice.- 10 Prochirality.- 10.1 Desymmetrization of Enantiospheric Orbits.- 10.2 Prochirality.- 10.3 Further Desymmetrization of Enantiospheric Orbits.- 10.4 Chiral syntheses.- 11 Desymmetrization of Para-Achiral Compounds.- 11.1 Chiral Subduction of Homospheric Orbits.- 11.2 Desymmetrization of Homospheric Orbits.- 11.3 Chemoselective and Stereoselective Processes.- 12 Topicity and Stereogenicity.- 12.1 Topicity Based On Chirality Fittingness of an Orbit.- 12.2 Stereogenicity.- 13 Counting Orbits.- 13.1 The Cauchy-Frobenius Lemma.- 13.2 Configurations.- 13.3 The Polya-Redfield Theorem.- 14 Obligatory Minimum Valencies.- 14.1 Isomer Enumeration under the OMV Restriction.- 14.2 Unit Cycle Indices.- 15 Compounds with Achiral Ligands Only.- 15.1 Compounds with Given Symmetries.- 15.2 Compounds with Given Symmetries and Weight.- 16 New Cycle Index.- 16.1 New Cycle Indices Based On USCIs.- 16.2 Correlation of New Cycle Indices to Polya's Theorem.- 16.3 Partial Cycle Indices.- 17 Cage-Shaped Molecules with High Symmetries.- 17.1 Edge Strategy.- 17.2 Tricyclodecanes with Td and Its Subsymmetries.- 17.3 Use of Another Ligand-Inventory.- 17.4 New Type of Cycle Index.- 18 Elementary Superposition.- 18.1 The USCI Approach.- 18.2 Elementary Superposition.- 18.3 Superposition for Other Indices.- 19 Compounds with Achiral and Chiral Ligands.- 19.1 Compounds with Given Symmetries.- 19.2 Compounds with Given Symmetries and Weights.- 19.3 Compounds with Given Weights.- 19.4 Special Cases.- 19.5 Other Indices.- 20 Compounds with Rotatable Ligands.- 20.1 Rigid Skeleton and Rotatable Ligands.- 20.2 Enumeration of Rotatable Ligands.- 20.3 Enumeration of Non-Rigid Isomers.- 20.4 Total Numbers.- 20.5 Typical Procedure for Enumeration.- 21 Promolecules.- 21.1 Molecular Models.- 21.2 Proligands and Promolecules.- 21.3 Enumeration of Promolecules.- 21.4 Molecules Based on Promolecules.- 21.5 Prochiralities of Promolecules and Molecules.- 21.6 Concluding Remarks.- 22 Appendix A. Mark Tables.- A.1 Td Point Group and Its Subgroups.- A. 2 D3h Point Group and Its Subgroups.- 23 Appendix B. Inverses of Mark Tables.- B. 1 Td Point Group and Its Subgroups.- B. 2 D3h Point Group and Its Subgroups.- 24 Appendix C. Subduction Tables.- C. 1 Td Point Group and Its Subgroups.- C. 2 D3h Point Group and Its Subgroups.- 25 Appendix D. Tables of USCIs.- D. 1 Td Point Group and Its Subgroups.- D. 2 D3h Point Group and Its Subgroups.- 26 Appendix E. Tables of USCI-CFs.- E. 1 Td Point Group and Its Subgroups.- E.2 D3h Point Group and Its Subgroups.- 27 Index.

The Expected order of a Random Permutation

Abstract: Let fin be the expected order of a random permutation, that is, the arithmetic mean of the orders of the elements in the symmetric group Sn. We prove that log/in ~ c\/(n/\ogn) as n -> oo, where

The geometry of the chamber system of a semimodular lattice

Abstract: In this paper geometric properties of the following metric space C are studied. Its elements are called chambers and are the maximal chains of a semimodular lattice X of finite height and its metric d is the gallery distance. We show that X has many properties in common with buildings. More specifically, Tits [17] has recently described buildings in terms of “Weyl-group valued distance functions”. We consider the Jordan-Holder permutation π(C, D) corresponding to a pair C, D of chambers and show that it has most properties of such a distance with values in the symmetric group.

Mapping Class Groups of Low Genus and Their Cohomology

Abstract: Concerned with the calculation of the cohomology of the mapping class group of a closed oriented surface of genus two, this book uses methods involving braid groups, modular representations of symmetric groups and configuration spaces.

Polynomial representations of finite general linear groups in non-describing characteristic

Abstract: Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.

Covariants of the symmetric group and its analogs in Weyl algebras

Abstract: The principal object of attention in this theory is the structure of the graded G-module S(V) for a given G-module V. The odd analog of the symmetric algebra S(V) is the exterior algebra A(V), and its superanalog is the algebra E(V) = S(V) ® A(V). The important role of the latter in many algebraic and homological constructions was clarified by A. Weyl. We therefore call E(V) the Weyl algebra, It is bigraded: EP,q(v) = SP(V) ® Aq(v), and the structure of its isotypical components relative to the group action is conveniently described by the Poincare series:

Integral group rings: Groups of units and classical K-theory

Abstract: This survey contains results obtained in this area from the second half of the sixties to the present time. In the group of units of the group ring, normal periodic subgroups, elements of finite order, free subgroups, congruence subgroups, questions of conjugacy of finite subgroups, and matrix representations are considered. Moreover, questions connected with the calculation of the groups K0, K1 for group rings are discussed with a description of the structure of projective modules, groups of invertible matrices, etc.

Symmetry classes: functions of three variables

Abstract: The attempt to generalize this fact to functions of n variables led Alfred Young to develop his theory of symmetrizers, now subsumed in the theory of representations of the symmetric group. According to this theory, a function of n variables is uniquely expressible as the sum of Pn functions, each one belonging to a different symmetry class, where Pn is the number of partitions of the integer n. Unfortunately, a simple intuitive description of such symmetry classes has never been given except for n = 2. It is known that for functions of three variables, there is only one other symmetry class besides the two obvious symmetry classes of symmetric functions and of skew-symmetric functions. We give this third symmetry class a very simple characterization, one that seems to have been overlooked. We show that it consists of all cyclic-symmetric functions. We prove that every function of three variables is uniquely expressible as the sum of a symmetric function, a skew-symmetric function and a cyclic-symmetric function. To make this note self-contained, we have added a short derivation of some known formulas. We believe that the underlying idea of this note will extend to functions of n variables. We hope the present note will at least entice the reader to further study of the vast theory of symmetry classes.

Some useful results concerning the representation theory of the symmetric group

Abstract: A theorem concerning the explicit form of the eigenvalues of the class sums of the symmetric group (Sn) is derived and used to obtain the following results: (1) the centre of the Sn-algebra is generated by means of polynomials in the set of elements consisting of the generators of the centre of the Sn-k algebra augmented by the single cycle class sums ((2))n, ((3))n, . . ., ((k+1))n; (2) the irreps of Sn with up to k rows are fully specified by the class sums ((2))n, ((3))n, . . ., ((k))n. Furthermore, it is found that the k class sums ((2))n, ((3))n, . . ., ((k+1))n suffice to specify the irreps of Sn for all n >k.