Cosets Of Groups Research Articles

Incidence properties of cosets in loops

AbstractWe study incidence properties among cosets of infinite loops, with emphasis on well‐structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a infinite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a infinite Bol loop S, can in some cases determine whether |S| divides |Q| for all infinite Bol loops Q with S⩽Q, and even whether there is a selection of left cosets of S that partitions Q. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop Q (in particular, in a commutative Moufang loop Q), two left cosets of S⩽Qare either disjoint or they intersect in a set whose cardinality equals that of some subloop of S.

On cosets in Coxeter groups

In this paper the notion of Coxeter length for a subset of a Coxeter group, as introduced in [9], is investigated for various subsets of a Coxeter group. Mostly cosets of various subgroups are examined as well as the associated idea of X-posets, which is a vast generalization of the Bruhat order.

A monomial matrix formalism to describe quantum many-body states

We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called here ‘M-spaces’; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M-spaces encompass various important state families, such as all Pauli stabilizer states and codes, the Affleck–Kennedy–Lieb–Tasaki (AKLT) model, Kitaev's (Abelian and non-Abelian) anyon models, group coset states, W states and the locally maximally entanglable states. We furthermore show how basic properties of M-spaces can be understood transparently by manipulating their monomial stabilizer groups. In particular, we derive a unified procedure to construct an eigenbasis of any M-space, yielding an explicit formula for each of the eigenstates. We also discuss the computational complexity of M-spaces and show that basic problems, such as estimating local expectation values, are NP-hard. Finally, we prove that a large subclass of M-spaces—containing, in particular, most of the aforementioned examples—can be simulated efficiently classically with a unified method.

Bundles over nearly-Kahler homogeneous spaces in heterotic string theory

We construct heterotic vacua based on six-dimensional nearly-Kahler homogeneous manifolds and non-trivial vector bundles thereon. Our examples are based on three specific group coset spaces. It is shown how to construct line bundles over these spaces, compute their properties and build up vector bundles consistent with supersymmetry and anomaly cancelation. It turns out that the most interesting coset is SU(3)/U(1)2. This space supports a large number of vector bundles which lead to consistent heterotic vacua, some of them with three chiral families.

The large sieve and random walks on left cosets of arithmetic groups

Building on Kowalski’s work on random walks on the groups \mathrm{SL}(n,ℤ) and \mathrm{Sp}(2g,ℤ) , we consider similar problems (we try to estimate the probability with which, after k steps, the matrix obtained has a characteristic polynomial with maximal Galois group or has no nonzero squares among its entries) for more general classes of sets: in \mathrm{GL}(n,A) , where A is a subring of ℚ containing ℤ that we specify, we perform a random walk on the set of matrices with fixed determinant D \in A^× . We also investigate the case where the set involved is any of the two left cosets of the special orthogonal group \mathrm{SO}(n,m)(ℤ) with respect to the spinorial kernel Ω(n,m)(ℤ) .

Superconformal indices for [formula omitted] theories with multiple duals

Following a recent work of Dolan and Osborn, we consider superconformal indices of four-dimensional N = 1 supersymmetric field theories related by an electric–magnetic duality with the SP ( 2 N ) gauge group and fixed rank flavour groups. For the SP ( 2 ) (or SU ( 2 ) ) case with 8 flavours, the electric theory has index described by an elliptic analogue of the Gauss hypergeometric function constructed earlier by the first author. Using the E 7 -root system Weyl group transformations for this function, we build a number of dual magnetic theories. One of them was originally discovered by Seiberg, the second model was built by Intriligator and Pouliot, the third one was found by Csáki et al. We argue that there should be in total 72 theories dual to each other through the action of the coset group W ( E 7 ) / S 8 . For the general SP ( 2 N ) , N > 1 , gauge group, a similar multiple duality takes place for slightly more complicated flavour symmetry groups. Superconformal indices of the corresponding theories coincide due to the Rains identity for a multidimensional elliptic hypergeometric integral associated with the B C N -root system.

Permutational isomers on a molecular skeleton with neighbor-excluding ligands

Permutational isomers of ligands substituted on a molecular skeleton are studied under a condition on the ligand placement which excludes a second ligand of the same type at a neighboring skeletal site. General theory to treat such a circumstance is developed to identify a correspondence between isomers and suitable double cosets of groups involving permutations of ligands amongst skeletal sites. Then this theory is illustratively applied for a selection of skeletons, including an experimentally realized hexamalono-buckminsterfullerene skeleton, with 12 ligation locations.

A recursion of the Pólya polynomial for the symmetric group

A recursion for the Pólya polynomial of the symmetric group is proved. This recursion will be used to compute the Pólya polynomial of the Young group. In turn, a theorem by George Pólya, generalized by deBruijn, will be used to compute the double cosets of Young groups embedded in an overall symmetric group. Connections to mathematical physics and combinatorics will be noted but not proved.

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570–600].

Non-Abelian coset string backgrounds from asymptotic and initial data

We describe hierarchies of exact string backgrounds obtained as non-Abelian cosets of orthogonal groups and having a space--time realization in terms of gauged WZW models. For each member in these hierarchies, the target-space backgrounds are generated by the ``boundary'' backgrounds of the next member. We explicitly demonstrate that this property holds to all orders in $\alpha'$. It is a consequence of the existence of an integrable marginal operator build on, generically, non-Abelian parafermion bilinears. These are dressed with the dilaton supported by the extra radial dimension, whose asymptotic value defines the boundary. Depending on the hierarchy, this boundary can be time-like or space-like with, in the latter case, potential cosmological applications.

σ-Extensions of discrete multivalued groups

The paper is devoted to the theory of singly generated multivalued groups. We construct new classes of such groups and find criteria that allow to check whether such a group is a coset group. For this purpose, we introduce the construction of a σ-extension of a singly generated multivalued group with Hermitian generator. This construction is based on the relation of the theory of such groups with the theory of symmetric graphs. The main result of the paper is as follows: the suggested construction of σ-extensions of singly generated bicoset multivalued groups with Hermitian generators is equivariant with respect to morphisms of graph-theoretic nature. Bibliography: 11 titles.

Finite level geometry of fractional branes

Abstract In some CFT models of simple current type, which are used to describe string theory on orbifolds and (adjoint) cosets of Lie groups, there arise fixed points of the simple current group. In these cases, the standard procedure to associate functions to Ishibashi states by averaging out the action of the simple current group, gives functions with unsatisfactory properties. In some cases the averaged Ishibashi function simply vanishes, which we see explicitly in SO ( 3 ) at level k = 4 l + 2 . In this Letter, an alternative function assignment is suggested, and it is shown that in some cases the resulting Ishibashi functions are orthogonal.

Affinity of permutations of [formula omitted

It was conjectured that if n is even, then every permutation of F 2 n is affine on some 2-dimensional affine subspace of F 2 n . We prove that the conjecture is true for n = 4 , for quadratic permutations of F 2 n and for permutation polynomials of F 2 n with coefficients in F 2 n / 2 . The conjecture is actually a claim about ( AGL ( n , 2 ) , AGL ( n , 2 ) ) -double cosets in permutation group S ( F 2 n ) of F 2 n . We give a formula for the number of ( AGL ( n , 2 ) , AGL ( n , 2 ) ) -double cosets in S ( F 2 n ) and classify the ( AGL ( 4 , 2 ) , AGL ( 4 , 2 ) ) -double cosets in S ( F 2 4 ) .

Heterotic strings on homogeneous spaces

AbstractWe construct heterotic string backgrounds corresponding to families of homogeneous spaces as exact conformal field theories. They contain left cosets of compact groups by their maximal tori supported by NS‐NS 2‐forms and gauge field fluxes. We give the general formalism and modular‐invariant partition functions, then we consider some examples such as SU (2)/U (1) ~ S2 (already described in a previous paper) and the SU (3)/U(1)2 flag space. As an application we construct new supersymmetric string vacua with magnetic fluxes and a linear dilaton.

E11 and Spheric Vacuum Solutions of Eleven- and Ten dimensional Supergravity Theories

In view of the newly conjectured Kac-Moody symmetries of supergravity theories placed in eleven and ten dimensions, the relation between these symmetry groups and possible compactifications are examined. In particular, we identify the relevant group cosets that parametrise the vacuum solutions of AdS x S type.

The group theory of oxidation II: cosets of non-split groups

The oxidation program given in the first article of this series (see preceding article in this issue) is extended to cover oxidation of 3d sigma model theories on a coset G/ H, with G non-compact (but not necessarily split), and H the maximal compact subgroup. We recover the matter content, the equations of motion and Bianchi identities from group lattice and Cartan involution. Satake diagrams provide an elegant tool for the computations, the maximal oxidation dimension, and group disintegration chains can be directly read off. We give a complete list of theories that can be recovered from oxidation of a 3-dimensional coset sigma model on G/ H, where G is a simple non-compact group.

Wigner rotations, Bargmann invariants and geometric phases

The concept of the 'Wigner rotation', familiar from the composition law of (pure) Lorentz transformations, is described in the general setting of Lie group coset spaces and the properties of coset representatives. Examples of Abelian and non-Abelian Wigner rotations are given. The Lorentz group Wigner rotation, occurring in the coset space SL(2, R)/SO(2)$\simeq$ SO(2, 1)/SO(2), is shown to be an analytic continuation of a Wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations, belonging to the coset space SU(2)/U(1)$\simeq$ SO(3)/SO(2). The possibility of interpreting these two Wigner rotations as geometric phases is shown in detail. Essential background material on geometric phases, Bargmann invariants and null phase curves, all of which are needed for this purpose, is provided.

A chip-firing game and Dirichlet eigenvalues

We consider a variation of the chip-firing game in an induced subgraph S of a graph G. Starting from a given chip configuration, if a vertex v has at least as many chips as its degree, we can fire v by sending one chip along each edge from v to its neighbors. Chips are removed at the boundary δ S. The game continues until no vertex can be fired. We will give an upper bound, in terms of Dirichlet eigenvalues, for the number of firings needed before a game terminates. We also examine the relations among three equinumerous families, the set of spanning forests on S with roots in the boundary of S, a set of “critical” configurations of chips, and a coset group, called the sandpile group associated with S.

Constructing Finite Unlabeled Structures Using Group Actions

The aim of this review is to point to a particular aspect of the constructive enumeration of finite unlabeled structures: The use of group actions and of double cosets in finite groups. In order to do this we start with a brief list of unlabeled structures, i.e., of mathematical structures defined as equivalence classes on finite sets. Then it will be shown that in each case the equivalence relation can be replaced by a finite group action, and so the whole bunch of tools of finite group actions' theory can be applied in each of these cases. The finite structures in question will be introduced as orbits of finite groups on finite sets, and so they can be obtained from transversals of orbit sets by simple deleting the labels. Therefore the construction of representatives of orbits, using double cosets, as well as the generation of representatives uniformly at random will be briefly described.

Decomposition of double cosets in p-adic groups

Decomposition of double cosets in p-adic groups