Finite Group Theory

Generalized "Coherent" States are the eigenstates of the lowering and raising operators of non-compact groups. In particular the discrete series of representations of SO (2, 1) are studied in detail: the resolution of the identity and the connection with the Hilbert spaces of entire functions of growth (1, 1). Also discussed are the application to the evaluation of matrix elements of finite group elements and the contraction to the usual coherent states.

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ r , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S n , the Weyl group of ?=?? n . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ?? n , to the algebra of S n -invariant differential operators with rational coefficients on the space ℂ n of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ n , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S n . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S n . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

This paper proposes an efficient two-pass protocol for authenticated key agreement in the asymmetric (public-key) setting. The protocol is based on Diffie-Hellman key agreement and can be modified to work in an arbitrary finite group and, in particular, elliptic curve groups. Two modifications of this protocol are also presented: a one-pass authenticated key agreement protocol suitable for environments where only one entity is on-line, and a three-pass protocol in which key confirmation is additionally provided. Variants of these protocols have been standardized in IEEE P1363 [17], ANSI X9.42 [2], ANSI X9.63 [4] and ISO 15496-3 [18], and are currently under consideration for standardization and by the U.S. government's National Institute for Standards and Technology [30].

The cherry on top of this stacky paper is the following: for any g>1 we give a finite group G such that the moduli space of connected admissible G-covers of genus g is a smooth, fine moduli space, which is a Galois cover of the moduli space of stable curves. The proof relies on methods introduced by Looijenga and Pikaart-De

We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them premodular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any t-th order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

It has been conjectured that every (2+1)-TQFT is a Chern-Simons-Witten (CSW) theory labeled by a pair (G, λ), where G is a compact Lie group, and λ ∈ H 4 (BG; Z) a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the E 6 subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mathematically constructed for every pair (G, λ). The cases that are constructed mathematically include:

Most of the colloidal clusters have been produced from oil-in-water emulsions with identical microspheres dispersed in oil droplets. Here, we present new types of binary colloidal clusters from phaseinverted water-in-oil emulsions using various combinations of two different colloids with several size ratios: monodisperse silica or polystyrene microspheres for larger particles and silica or titania nanoparticles for smaller particles. Obviously, a better understanding of how finite groups of different colloids self-organize in a confined geometry may help us control the structure of matter at multiple length scales. In addition, since aqueous dispersions have much better phase stability, we could produce much more diverse colloidal materials from water-in-oil emulsions rather than from oil-in-water emulsions. Interestingly, the configurations of the large microspheres were not changed by the presence of the small particles. However, the arrangement of the smaller particles was strongly dependent on the nature of the interparticle interactions. The experimentally observed structural evolutions were consistent with the numerical simulations calculated using Surface Evolver. These clusters with nonisotropic structures can be used as building blocks for novel colloidal structures with unusual properties or by themselves as light scatterers, diffusers, and complex adaptive matter exhibiting emergent behavior.

In this paper we study a special class of finite p-groups, which we call powerful p-groups. In the second part of this paper, we apply our results to the study of p-adic analytic groups. This application is possible, because a finitely generated prop group is p-adic analytic if and only if it is "virtually pro-powerful." These applications are described in the introduction to the second part, while now we describe the present part in more detail. In the first section we define a powerful p-group, as one whose subgroup of pth powers contains the commutator subgroup. We give several results

Mfound an obstruction to the solvability on S 2 coming from the conformal group. This same obstruction applied also to the problem on the standard S". More obstructions were found recently by Bourguignon and Ezin [7]. In an earlier paper J. Moser [2] showed that the problem has a solution for a function K on S 2 which is somewhere

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that nonisomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.

The purpose of this article is to give a proof of the Orbifold Theorem announced by Thurston in late 1981: If $O$ is a compact, connected, orientable, irreducible and topologically atoroidal 3-orbifold with non-empty ramification locus, then $O$ is geometric. As a corollary, any smooth orientation preserving non-free finite group action on $S^3$ is conjugate to an orthogonal action.

Zero-sum Ramsey theory is a newly established area in combinatorics. It brings to ramsey theory algebric tools and algebric flavour. The paradigm of zero-sum problems can be formulated as follows: Suppose the elements of a combinatorial structure are mapped into a finite group K. Does there exists a prescribed substructure the sum of the weights of its elements is 0 in K?

The Non-Abelian finite group PSL2 is the only simple subgroup of SU (3) with a complex three-dimensional irreducible representation. It has two maximal subgroups, S4 which, along with its own A4 subgroup, has been successfully applied in numerous models of flavor, as well as the 21 element Frobenius group Z7 ⋊Z3, which has gained much less attention. We show that it can also be used to generate tri-bimaximal mixing in the neutrino sector, while allowing for quark and charged lepton hierarchies. *

We study phase transitions in models of opinion formation which are based on the social impact theory. Two different models are discussed: (i) a cellular-automata based model of a finite group with a strong leader where persons can change their opinions but not their spatial positions, and (ii) a model with persons treated as active Brownian particles interacting via a communication field. In the first model, two stable phases are possible: a cluster around the leader, and a state of social unification. The transition into the second state occurs for a large leader strength and/or for a high level of social noise. In the second model, we find three stable phases, which correspond either to a "paramagnetic" phase (for high noise and strong diffusion), a "ferromagnetic" phase (for small nose and weak diffusion), or a phase with spatially separated "domains" (for intermediate conditions).

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z D (C). We use this result to obtain a criterion for C to be grouptheoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z n p , whenever p is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group G is reducible to instances of Translating Coset in G/N and N , for appropriate normal subgroups N of G. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group. * A preliminary version of this paper appeared in .

Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true.

We introduce a quantum double quasitriangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D φ (G) associated to a finite group G and group 3-cocycle φ. Keywords: quantum double-quasi-Hopf algebra-finite group-cocycle-category-reconstruction.

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].

We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the Yang-Baxter equation. The images of B n under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of Bn factoring over Temperley-Lieb algebras and the corresponding link invariants.

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra A Γ ⊂ A of local observables invariant under Γ. A set of positive energy A Γ modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of SL(2, Z). We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules.

We consider the standard semi-direct product Ó of finite groups . We show that with certain choices of generators for these three groups, the Cayley graph of Ó is (essentially) the zigzag product of the Cayley graphs of and . Thus, using the results of [RVW00], the new Cayley graph is ...

We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precisely those with property F.

Let X be a proper metric space and let νX be its Higson corona. We prove that the covering dimension of νX does not exceed the asymptotic dimension asdim X of X introduced by M.Gromov. In particular it implies that dim νR n = n for euclidean and hyperbolic metrics on R n . We prove that for finitely generated groups Γ ⊂ Γ with word metrics the inequality dim νΓ ≤ dim νΓ holds. Also we prove that a small action at infinity of a geometrically finite group Γ on some compactification X of the universal covering space X = EΓ enables one to map the Higson compactification onto X . In that case the rational acyclicity of X implies the conjecture by S. Weinberger for X which is a form of the Novikov Conjecture for Γ.

A standing conjecture in L 2 -cohomology is that every finite CWcomplex X is of L 2 -determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L 2 -acyclic, we also prove that the L 2 -determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L 2 -Betti numbers.

For a torsion unit $u$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$, and a prime $p$ which does not divide the order of $u$ (but the order of $G$), a relation between the partial augmentations of $u$ on the $p$-regular classes of $G$ and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable $G$, consequences concerning rational conjugacy of $u$ to a group element are discussed, considering as examples the symmetric group $S_{5}$ and the groups $\text{\rm PSL}(2,p^{f})$.

In this paper we obtain necessary and sufficient conditions for the crossed product R * G to be prime or semiprime under the assumption that R is prime. The main techniques used are the A-methods which reduce these questions to the finite normal subgroups of G and a study of the X-inner automorphisms of R which enables us to handle these finite groups. In particular we show that R * G is semiprime if R has characteristic 0. Furthermore, if R has characteristic p >0, then R *G is semiprime if and only if R *P is semiprime for all elementary abelian p-subgroups P of A+(G)O G,.,.

Using the isomorphism conjectures of Baum & Connes and Farrel & Jones, we compute the algebraic K-and L-theory and the topological K-theory of cocompact planar groups (= cocompact N.E.C-groups) and of groups G appearing in an extension 1 → Z n → G → π → 1 where π is a finite group and the conjugation π-action on Z n is free outside 0 ∈ Z n. These computations apply for instance to two-dimensional crystallographic groups and cocompact Fuchsian groups.

We present experimental results on hydrothermal traveling-waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels, i.e., within periodic or non-periodic boundary conditions. For periodic boundary conditions, by increasing the control parameter or changing the discrete mean-wavenumber of the waves, we produce modulated waves patterns. These patterns range from stable periodic phase-solutions, due to supercritical Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes and/or counter-propagating-waves competition, i.e., traveling sources and sinks. The transition from non-linearly saturated Eckhaus modulations to transient pattern-breaks by traveling holes and spatiotemporal defects is documented. Our observations are presented in the framework of coupled complex Ginzburg-Landau equations with additional fourth and fifth order terms which account for the reflection symmetry breaking at high wave-amplitude far from onset. The second part of this paper [1] extends this study to spatially non-periodic patterns observed in both annular and bounded channel.

CP T groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincaré group which describe orientable (extended) objects. A general method for construction of CP T groups of the fields of any spin is given. CP T groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. CP T groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of the same spin with different masses.

The partial group algebra of a group G over a field K, denoted by Kpar(G), is the algebra whose representations correspond to the partial representations of G over K-vector spaces. In this paper we study the structure of the partial group algebra Kpar(G), where G is a finite group. In particular, given two finite abelian groups G1 and G2, we prove that if the characteristic of K is zero, then Kpar(G1) is isomorphic to Kpar(G2) if and only if G1 is isomorphic to G2.

Let G be a finite group generated by reflections. It is shown that the elements of G can be arranged in a cycle (a ''Gray code'') such that each element is obtained from the previous one by applying one of the generators. The case G = ! 1 n yields a conventional binary Gray code. These generalized Gray codes provide an efficient way to run through the elements of any finite reflection group.

In the representation theory of finite groups, there is a well known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an Abelian defect group P, then A and its Brauer corresponding block B of the normalizer N G (P) of P in G are equivalent (Rickard equivalent). This conjecture is called Broué's Abelian defect group conjecture. We prove in this paper that Broué's Abelian defect group conjecture is true for a non-principal 3-block A with an elementary Abelian defect group P of order 9 of the Janko simple group J 4 . It then turns out that Broué's Abelian defect group conjecture holds for all primes p and for all p-blocks of the Janko simple group J 4 .

The diameter of a group G with respect to a set S of generators is the maximum over g E G of the length of the shortest word in S U S-' representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent, to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if IS1 is allowed to exceed (logIGI)'+C (c > 0), it plays a heavy role when 1. 51 = O(1). In particular, every nonabelian finite simple group has a set of 5 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index.-Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((nlnn)'/2(1 + o(1))) upper bound. In the average case, the upper bound improves to exp((lnn)'(l + o(1))). As a first step toward extending this result to simple groups other than A,, , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for A,. In the limited spa.ce of this article, we try to illuminate some of the basic underlying techniques.

A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that b n ≤ T (n) ≤ c n √ n n! for n ≥ 5, where b ≈ 1.719 and c ≈ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n × n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.