Cosets Of Groups Research Articles

On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups

On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups

Fundamental heap for framed links and ribbon cocycle invariants

A heap is a set with a certain ternary operation that is self-distributive and exemplified by a group with the operation [Formula: see text]. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap [Formula: see text]-cocycles as weights. This invariant corresponds to a rack cocycle invariant for the rack constructed by doubling of a heap, while colorings can be regarded as heap morphisms from the fundamental heap. For the construction of the invariant, first computational methods for the heap cohomology are developed. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one-dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. Computations of the cocycle invariants are presented using the cocycles constructed, and conversely, it is proved that the invariant values can be used to derive algebraic properties of the cohomology.

An efficient S-box design scheme for image encryption based on the combination of a coset graph and a matrix transformer

<abstract> <p>Modern block ciphers deal with the development of security mechanisms to meet the security needs in several fields of application. The substitution box, which is an important constituent in block ciphers, necessarily has sufficient cryptographic robustness to counter different attacks. The basic problem with S-box design is that there is no evident pattern in its cryptographic properties. This study introduces a new mathematical algorithm for developing S-box based on the modular group coset graphs and a newly invented mathematical notion "matrix transformer". The proficiency of the proposed S-box is assessed through modern performance evaluation tools, and it has been observed that the constructed S-box has almost optimal features, indicating the effectiveness of the invented technique.</p> </abstract>

Affine Deligne–Lusztig varieties and quantum Bruhat graph

In this paper, we consider affine Deligne–Lusztig varieties $$X_w(b)$$ and their certain union $$X(\mu ,b)$$ inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne–Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in the associated Iwahori–Weyl group and a dimension formula for $$X(\mu ,b)$$ . We show that one can considerably weaken the superregularity hypothesis and sometimes eliminate it, thus strengthening these existing results.

Counting Parabolic Double Cosets in Symmetric Groups

Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, and used it to compute $p_n$, the number of parabolic double cosets in $S_n$, for $n\leq13$. In this paper, we derive a new formula for $p_n$ and an efficient polynomial time algorithm for evaluating this formula. We use these results to compute $p_n$ for $n\leq5000$ and to prove an asymptotic formula for $p_n$ that was conjectured by Billey et al.

Novel integrable interpolations

A novel class of integrable σ-models interpolating between exact coset conformal field theories in the IR and hyperbolic spaces in the UV is constructed. We demonstrate the relation to the asymptotic limit of λ-deformed models for cosets of non-compact groups. An integrable model interpolating between two spacetimes with cosmological and black hole interpretations and exact conformal field theory descriptions is also provided. In the process of our work, a new zoom-in limit, distinct from the well known non-Abelian T-duality limit, is found.

Primary cosets in groups

A finite group G is called a generalized Frobenius group with kernel F if F is a proper nontrivial normal subgroup of G, and for every element Fx of prime order p in the quotient group G/F, the coset Fx of G consists of p-elements. We study generalized Frobenius groups with an insoluble kernel F. It is proved that F has a unique non- Abelian composition factor, and that this factor is isomorphic to $$ {\mathrm{L}}_2\left({3}^{2^{\mathrm{l}}}\right) $$ for some natural number l. Moreover, we look at a (not necessarily finite) group generated by a coset of some subgroup consisting solely of elements of order three. It is shown that such a group contains a nilpotent normal subgroup of index three.

Primary Cosets in Groups

Primary Cosets in Groups

On growth of double cosets in hyperbolic groups

Let [Formula: see text] be a hyperbolic group, [Formula: see text] and [Formula: see text] be subgroups of [Formula: see text], and [Formula: see text] be the growth function of the double cosets [Formula: see text]. We prove that the behavior of [Formula: see text] splits into two different cases. If [Formula: see text] and [Formula: see text] are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as [Formula: see text]. We can even take [Formula: see text]. In contrast, for quasiconvex subgroups [Formula: see text] and [Formula: see text] of infinite index, [Formula: see text] is exponential. Moreover, there exists a constant [Formula: see text], such that [Formula: see text] for all big enough [Formula: see text], where [Formula: see text] is the growth function of the group [Formula: see text]. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

Matrix Representations as a Gateway to Group Theory

ABSTRACT Almost all finite groups encountered by undergraduates can be represented as multiplicative groups of concise block-diagonal binary matrices. Such representations provide simple examples for beginning a group theory course. More importantly, these representations provide concrete models for “abstract” concepts. We describe Maple lab assignments which explore group actions, subgroups, normality, cosets, quotient groups, homomorphisms, isomorphisms, and kernels.

Parabolic double cosets in Coxeter groups

Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type G2. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar questions over more general fields seem to come up for the first time. In the Appendix, we link thedoublecosets with Cech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass’s finiteness conjecture in K-theory.

Non-abelian T-duality and Yang-Baxter deformations of Green-Schwarz strings

We perform non-abelian T-duality for a generic Green-Schwarz string with respect to an isometry (super)group G, and we derive the transformation rules for the supergravity background fields. Specializing to G bosonic, or G fermionic but abelian, our results reproduce those available in the literature. We discuss also continuous deformations of the T-dual models, obtained by adding a closed B-field before the dualization. This idea can also be used to generate deformations of the original (un-dualized) model, when the 2-cocycle identified from the closed B is invertible. The latter construction is the natural generalization of the so-called Yang-Baxter deformations, based on solutions of the classical Yang-Baxter equation on the Lie algebra of G and originally constructed for group manifolds and (super)coset sigma models. We find that the deformed metric and B-field are obtained through a generalization of the map between open and closed strings that was used also in the discussion by Seiberg and Witten of non-commutative field theories. When applied to integrable sigma models these deformations preserve the integrability.

Duality symmetries behind solutions of the classical simple pendulum

Describing the motion of the classical simple pendulum is one of the aims in every undergraduate classical mechanics course. Its analytical solutions are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions this symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ=Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. A discussion of the physical consequences that this symmetry has on the motions of the simple pendulum is presented in this contribution and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. Thus by studying deeper a very familiar mechanical system, it is possible to get an insight to more abstract physical and mathematical concepts. In particular a single solution of pure imaginary time for all allowed values of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

Conformal embeddings and higher-spin bulk duals

It is well-known that conformal embeddings can be used to construct non-diagonal modular invariants for affine lie algebras. This idea can be extended to construct infinite series of non-diagonal modular invariants for coset CFTs. In this paper, we systematically approach the problem of identifying higher-spin bulk duals for these kind of non-diagonal invariants. In particular, for a special value of the 't Hooft coupling, there exist a class of partition functions that have enhanced supersymmetry, which should be reflected in a bulk dual. As a illustration of this, we show that a partition function of a orthogonal group coset CFT has a $\mathcal N=1$ supersymmetric higher-spin bulk dual, in the 't Hooft limit. We also propose that two of the series of CFT partition functions, obtained from conformal embeddings, are equal, generalising the well-known dual interpretation of the 3-state Potts model as a $\frac{SU(2)_3 \otimes SU(2)_1}{SU(2)_4}$ and also as a $\frac{SU(3)_1 \otimes SU(3)_1}{SU(3)_2}$ coset model.

COSET OF A HYPERCOMPLEX NUMBER SYSTEM IN CLIFFORD ANALYSIS

We give certain properties of elements in a coset group with hypercomplex numbers and research a monogenic function and a Clifford regular function with values in a coset group by defining differential operators. We give properties of those functions and a power of elements in a coset group with hypercomplex numbers.

Quantizations from (P)OVM's

We explain the powerful role that operator-valued measures can play in quantizing any set equipped with a measure, for instance a group (resp. group coset) with its invariant (resp. quasi-invariant) measure. Coherent state quantization is a particular case. Such integral quantizations are illustrated with two examples based on the Weyl-Heisenberg group and on the affine group respectively. An interesting application of the affine quantization in quantum cosmology is mentioned, and we sketch a construction of new coherent states for the hydrogen atom.

DOUBLE COSETS IN FREE GROUPS

In this paper, we study double cosets of finite rank free groups. We focus our attention on cancellation types in double cosets and their formal language properties.

Nambu–Goldstone dynamics and generalized coherent-state functional integrals

This paper gives a new method of attack on the Nambu–Goldstone dynamics in spontaneously broken theories. Since the target space of the Nambu–Goldstone fields is a group coset space, their effective quantum dynamics can be naturally phrased in terms of generalized coherent-state functional integrals. As an explicit example of this line of reasoning, we construct a low-energy effective Lagrangian for the Heisenberg ferromagnet in a broken phase. The leading field configuration in the WKB approximation leads to the Landau–Lifshitz equation for quantum ferromagnet. The corresponding linearized equations allow one to identify the Nambu–Goldstone boson with a ferromagnetic magnon.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

On Classes of Local Unitary Transformations

We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed.