Symmetric group

We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold. We apply this to obtain a classification of polar actions with a fixed point on symmetric spaces.

We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.

We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.

In this paper we study the "holomorphic K -theory" of a projective variety. This K -theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in . This theory is built out of studying algebraic bundles over a variety up to "algebraic equivalence". In this paper we will give calculations of this theory for "flag like varieties" which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K -theory and the appropriate "morphic cohomology" groups, defined in in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the "symmetrized loop group" ΩU (n)/Σn where the symmetric group Σn acts on U (n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU (k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K -theory by inverting the Bott class, then rationally this is isomorphic to topological K -theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K -theory, and show that these obstructions vanish for generalized flag manifolds.

The theory of supercharacters, recently developed by Diaconis-Isaacs and André, is used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.

Throughout this paper, a semigroup S will denote a torsion free grading monoid, and it is a non-zero semigroup with 0. The operation is written additively. The aim of this paper is to study semigroup version of an integral domain ([1],[3],[4] and [티).

Let G be a finite group, and write cd(G) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees a, b ∈ cd(G), the total number of (not necessarily different) primes of the greatest common divisor gcd(a, b) is at most 2. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL 2 (q) for q ≥ 7.

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a ∆-product or * -product from the values of its corresponding supercharacters.

Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius-Wielandt group is a triple (G, H, L) where H/L is almost a Frobenius complement for G; A Camina pair is a pair (G, N ) where N is almost a Frobenius kernel for G; A Camina triple is a triple (G, N, M ) where (G, N ) and (G, M ) are almost Camina pairs. In this paper we study triples (G, N, M ) where (G, N ) and (G, M ) are almost Frobenius groups.

A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D n . Each of the 192 expressions is labeled by an element of D 4 . Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S 4 . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

The framework used to prove the multiplicative deformation of the algebra of Feynman-Bender diagrams is a twisted shifted dual law (in fact, twisted twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameter is a perturbation of the shuffle coproduct which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.

We show that the following conditions on a C*-algebra are equivalent: (i) it has the fixed point property for nonexpansive mappings, (ii) the spectrum of every self adjoint element is finite, (iii) it is finite dimensional. We prove that (i) implies (ii) using constructions given by Goebel, that (ii) implies (iii) using projection operator properties derived from the spectral and Gelfand-Naimark-Segal theorems, and observe that (iii) implies (i) by Brouwer's fixed point theorem.

This study investigates the number of homomorphisms from the quaternion group into various finite groups. Quaternion groups, denoted as Q8, possess unique algebraic properties that make them intriguing subjects for group theory inquiries. The research explores the enumeration of homomorphisms from Q8into specific finite groups, providing insights into the structural relationships between these groups. Here, we derive general formulae for counting the number of homomorphisms from quaternion group into each of quaternion group, dihedral group, quasi-dihedral group and modular group by using only elementary group theory

Full ranking a 1 ≻ • • • ≻ a n ⇔ Permutation σ ∈ S n that maps an item to its rank: σ(a i ) = i The variability of full rankings is therefore modeled by a probability distribution p over the set of permutations S n . p is called a ranking model. introduction Example: probability distribution over S 5 (APA dataset)

The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the "Nim field" (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations.

Let R m be the (unique) universal homogeneous m-edge-coloured countable complete graph (m ≥ 2), and G m its group of colourpreserving automorphisms. The group G m was shown to be simple by John Truss. We examine the automorphism group of G m , and show that it is the group of permutations of R m which induce permutations on the colours, and hence an extension of G m by the symmetric group of degree m. We show further that the extension splits if and only if m is odd, and in the case where m is even and not divisible by 8 we find the smallest supplement for G m in its automorphism group.

The operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m > 2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case. This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive. In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random mcoloured complete graph) where the group of switching-automorphisms is highly transitive. Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds.

The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the "Nim field" (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations.

The operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m > 2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case. This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive. In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random mcoloured complete graph) where the group of switching-automorphisms is highly transitive. Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds.

Let πn be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, the authors of showed that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2, . . . , n} in πn is n 2 2 (1o(1)) as n → ∞. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.

It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multiparticle systems since they contain spinor and vector representations within the same formalism, and, moreover, allow for a complete study of all classical Lie groups. In this work, together with an accompanying work also presented at this conference, an analysis of q -symmetry -for generic q 's -based on the ordinary symmetric groups is given for the first time. We construct q -Young operators as Clifford idempotents and the Hecke algebra representations in ideals generated by these operators. Various relations as orthogonality of representations and completeness are given explicitly, and the symmetry types of representations is discussed. Appropriate q -Young diagrams and tableaux are given. The ordinary case of the symmetric group is obtained in the limit q → 1. All in all, a toolkit for Clifford algebraic treatment of multi-particle systems is provided. The distinguishing feature of this paper is that the Young operators of conjugated Young diagrams are related by Clifford reversion, connecting Clifford algebra and Hecke algebra features. This contrasts the purely Hecke algebraic approach of King and Wybourne, who do not embed Hecke algebras into Clifford algebras.

It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multiparticle systems since they contain spinor and vector representations within the same formalism, and, moreover, allow for a complete study of all classical Lie groups. In this work, together with an accompanying work also presented at this conference, an analysis of q -symmetry -for generic q 's -based on the ordinary symmetric groups is given for the first time. We construct q -Young operators as Clifford idempotents and the Hecke algebra representations in ideals generated by these operators. Various relations as orthogonality of representations and completeness are given explicitly, and the symmetry types of representations is discussed. Appropriate q -Young diagrams and tableaux are given. The ordinary case of the symmetric group is obtained in the limit q → 1. All in all, a toolkit for Clifford algebraic treatment of multi-particle systems is provided. The distinguishing feature of this paper is that the Young operators of conjugated Young diagrams are related by Clifford reversion, connecting Clifford algebra and Hecke algebra features. This contrasts the purely Hecke algebraic approach of King and Wybourne, who do not embed Hecke algebras into Clifford algebras.

We construct the Fock space representation of quantum affine algebras using combinatorics of Young walls. We also show that the crystal basis of the Fock space representation can be realized as the abstract crystal consisting of proper Young walls. We then generalize Lascoux-Leclerc-Thibon algorithm to obtain an effective algorithm for constructing the global bases of basic representations.

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.

In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which has been obtained by using a Kostant type cohomology formula for gl m|n . In general, we can obtain in a combinatorial way a Weyl type formula for various highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair.

We consider a category of gl ∞ -crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight gl ∞ -modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients.

We present two constructions in this paper : (a) A 10-vertex triangulation CP 2 10 of the complex projective plane CP 2 as a subcomplex of the join of the standard sphere (S 2 4 ) and the standard real projective plane (RP 2 6 , the decahedron), its automorphism group is A 4 ; (b) a 12-vertex triangulation (S 2 ×S 2 ) 12 of S 2 ×S 2 with automorphism group 2S 5 , the Schur double cover of the symmetric group S 5 . It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S 2 ×S 2 . Both constructions have surprising and intimate relationships with the icosahedron. It is well known that CP 2 has S 2 × S 2 as a two-fold branched cover; we construct the triangulation CP 2 10 of CP 2 by presenting a simplicial realization of this covering map The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S 2 × S 2 , different from the triangulation alluded to in (b). This gives a new proof that Kühnel's CP 2 9 triangulates CP 2 . It is also shown that CP 2 10 and (S 2 × S 2 ) 12 induce the standard piecewise linear structure on CP 2 and S 2 × S 2 respectively.

Let M be an n-vertex combinatorial triangulation of a Z 2 -homology d-sphere. In this paper we prove that if n ≤ d + 8 then M must be a combinatorial sphere. Further, if n = d + 9 and M is not a combinatorial sphere then M can not admit any proper bistellar move. Existence of a 12-vertex triangulation of the lens space L(3, 1) shows that the first result is sharp in dimension three. In the course of the proof we also show that any Z 2 -acyclic simplicial complex on ≤ 7 vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible.

A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.

We give a new construction of polyhedra with specified links and use this to investigate SQ universality of groups with certain presentations. In particular, we answer negatively a question of J. Howie from on SQ universality of groups with so-called special presentations.

Vertices of the 4-dimensional semi-regular polytope, snub 24-cell and its symmetry group W (D 4 ) : C 3 of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of E 8 root system. A simple method is employed to construct the E 8 root system in terms of icosians which decomposes into two copies of the quaternionic root system of the Coxeter group W (H 4 ), while one set is the elements of the binary icosahedral group the other set is a scaled copy of the first. The quaternionic root system of H 4 splits as the vertices of 24-cell and the snub 24-cell under the symmetry group of the snub 24-cell which is one of the maximal subgroups of the group W (H 4 ) as well as W (F 4 ). It is noted that the group is isomorphic to the semi-direct product of the Weyl group of D 4 with the cyclic group of order 3 denoted by W (D 4 ) : C 3 , the Coxeter notation for which is [3, 4, 3 + ]. We analyze the vertex structure of the snub 24-cell and decompose the orbits of W (H 4 ) under the orbits of W (D 4 ) : C 3 . The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group W (D 4 ) : C 3 . In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96 and 600=24+96+192+288 respectively under the group W (D 4 ) : C 3 . The dual polytope of the snub 24-cell is explicitly constructed. Decompositions of the Archimedean W (H 4 ) polytopes under the symmetry of the group W (D 4 ) : C 3 are given in the appendix.

For n ∈ N let Xn = {1 < 2 < · · · < n} be a finite n-element chain and let Tn denote the full transformation semigroup, i.e. the semigroup of all mappings α : Xn → Xn under composition. We say that a transformation α ∈ Tn is increasing (decreasing) if x ≤ xα (x ≥ xα) for all x ∈ Xn. This paper investigates algebraic and rank properties of the semigroup Incn of all increasing mappings of Xn. Algebraic and rank properties of Tn have been studied over a long period and many interesting results have emerged (see, for example [2], [4], [7]). In [6], Umar considered algebraic and rank properties of the semigroup Dn of all decreasing transformations. Moreover, Umar ([5]) showed that the semigroups Incn and Dn are isomorphic. Finally, we characterize all maximal subsemigroups of Incn. We begin by recalling some notations and definitions that are used in the paper. For the standard terms and concepts in semigroup theory we refer to [3]. For every transformation α ∈ Tn, we denote by ...

We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.

We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.

We compute the Schur group of the cyclotomic fields Q(c,,,) and real quadratic fields Q(&/*) where d is a product of an even number of primes congruent to three modulo four. Some results are also given about the Schur group of certain subfields of cyclotomic fields.

The Team Orienteering Problem with Time Windows (TOPTW) is a well-known variant of the Vehicle Routing Problem (VRP) whose aim is to maximize the total amount of profit collected from the visited customers while taking into consideration some resource limitations. In this variant, customers must be served at most once in their time windows while respecting the predefined maximum travel time limit of each vehicle. In this paper we propose a column generation based algorithm to solve the linear relaxation of TOPTW. A dynamic programming algorithm is used to solve the subproblems in order to generate additional columns. Experiments conducted on the benchmark of TOPTW show the effectiveness of our algorithm and the strengthen of our formulation since we were able to prove the optimality of several instances by finding their integer solutions at the root node while solving the linear relaxation of the model.

Let K denote any field or the ring of the integers X, and V be a finite-dimensiwnal vector space over k', dim Z' = 2k, respectively; if K = 2, let V be a free Abelian group of rank 2k. Illorcover, let ( , ) be a bilinear nondegenerate antisymmetric form defined over V. Consider the homomorphism defined in the following way: where (vi , aj> stands for the scalar product of the vectors E, , 7:j and the symbol h means that the corresponding vector is omitted. The symplectic group G = Sp(V) acts on Z'"" with the action defined by g(0, @ .'. @ vn) = (gv, 0 ... @gqJ gEwV) and with respect to this action Ker vij is an invariant space.

Let S be a transformation semigroup of degree n. To each element s ∈ S we associate a permutation group G R (s) acting on the image of s, and we find a natural generating set for this group. It turns out that the R-class of s is a disjoint union of certain sets, each having size equal to the size of G R (s). As a consequence, we show that two R-classes containing elements with equal images have the same size, even if they do not belong to the same D-class. By a certain duality process we associate to s another permutation group G L (s) on the image of s, and prove analogous results for the L-class of S. Finally we prove that the Schützenberger group of the H-class of s is isomorphic to the intersection of G R (s) and G L (s). The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.

Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data. The first method is based on the known useful classification of circulant graphs in terms of $S$-rings and results in exhaustive listing (with the use of COCO and GAP) of all corresponding $S$-rings of the indicated orders. The latter method is conducted in the framework of a general approach developed earlier for counting circulant graphs of prime-power orders. It is a Redfield--P\'olya type of enumeration based on an isomorphism criterion for circulant graphs of such orders. In particular, five intermediate enumeration subproblems arise, which are refined further into eleven subproblems of this type (5 and 11 are, not accidentally, the 3d Catalan and 3d little Schr\"oder numbers, resp.). All of them are resolved for the fou...

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to projective representations of a finite group (Abelian), $G$ say, such that the corresponding twisted group ring has minimal center. The latter then translates to the fact that the corresponding 2-cocycle allows a minimal (none!) number of ray classes and this forces a decomposition of $G$ in cyclic components in a suitable way, cf. Zmud, M., Symplectic geometries and projective representations of finite Abelian groups, (Russian) Mat. Sb. (N.S.) 87(129) (1972), 3--17.. In this small paper, I will provide a way to represent an Abelian Group Clifford Algebra using a matrix, and then give a way to calculate whether or not the center is trivial.

We study the recurrence behaviour of random walks on partially oriented honeycomb lattices. The vertical edges are undirected while the orientation of the horizontal edges is random: depending on their distribution, we prove a.s. transience in some cases, and a.s. recurrence in other ones. The results extend those obtained for the partially oriented square grid lattices , ).

In this paper, a result of Albert, Atkinson, Handley, Holton, and Stromquist (Proposition 2.4 of ) which characterizes the optimal packing behavior of the pattern 1243 is generalized in two directions. The packing densities of layered patterns of type (1 α , α) and (1, 1, β) are computed.

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S 5 , S 6 , and S 7 for involutions.