Maximal Abelian Subgroups Research Articles

Embedding space approach to Lorentzian CFT amplitudes and causal spherical functions

Conformal field theory in a Minkowski setting is discussed in an embedding space approach, paying special attention to causality constraints for four-point amplitudes. The physics of dilatation and Lorentz boost is emphasized in specifying the noncompact maximal Abelian subgroup of SO(d,2). Reduction of a conformal field theory four-point amplitudes as functions of cross ratios is shown to be equivalent to enforcing H bi-invariance, i.e., F(hgh′)=F(g), with g∈SO(d,2) and H an appropriate subgroup. Causality is imposed by introducing appropriate semigroups. Causal zonal spherical functions are constructed, making contact with Minkowski conformal blocks introduced previously. Published by the American Physical Society 2024

Clifford orbits and stabilizer states

Abstract Stabilizer states serve as “classical objects” in the stabilizer formalism of quantum theory, and play an important role in quantum error correction, fault-tolerant quantum computation, and quantum communication. They provide an efficient description of many basic features of quantum theory and exhibit a rich structure. For prime dimensional systems, they may be defined by two quite different yet equivalent ways: The first is via stabilizer groups (maximal Abelian subgroups of the discrete Heisenberg-Weyl group). The second is via the orbits of the Clifford group acting on any computational basis state. However, in a general dimensional system, this equivalence breaks down, and consequently, it is desirable to classify the difference and relation between the above two approaches to stabilizer states. In this work, we show that these two approaches are equivalent if and only if the system dimension is square-free (i.e., has no square factor). Furthermore, we completely clarify the relation between the Clifford orbits and stabilizer states in an arbitrary dimensional system, derive the explicit expressions of the Clifford orbits and determine their cardinalities.

On p-groups with a maximal elementary abelian normal subgroup of rank k

There are several results in the literature concerning p-groups G with a maximal elementary abelian normal subgroup of rank k due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators etc. of a 2-group G in terms of k, which were previously known only for p>2. We also prove a theorem that is new even for odd primes. Namely, we show that if G has a maximal elementary abelian normal subgroup of rank k, then for any abelian subgroup A the Frattini subgroup Φ(A) can be generated by 2k elements (3k when p=2). The proof of this rests upon the following result of independent interest: If V is an n-dimensional vector space, then any commutative subalgebra of End(V) contains a zero algebra of codimension at most n.

The Margolis homology of the cohomology restriction from an extra-special group to its maximal elementary abelian subgroups

The Margolis homology of the cohomology restriction from an extra-special group to its maximal elementary abelian subgroups

Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory

We demonstrate the emergence of an effective “time” axis in the ground state of a gauged rotational symmetry theory in four-dimensional Euclidean space. In so doing, we remove the necessity of Wick rotation to Lorentz spacetime, an arbitrary and sometimes ill-defined procedure, especially for gravity-related theories. We begin by adapting the Cho-Duan-Ge decomposition to the gauge theory of the four-dimensional rotational symmetry group SO(4), where it identifies the maximal Abelian subgroup SO(2)⊗SO(2) in a gauge covariant manner. We then find the one-loop effective theory to have a stable condensate of monopoles corresponding to the reduction of SO(4) symmetry to SO(2)⊗SO(2). The construction of the condensate ensures that the four-dimensional spatial direction of its field strength must coincide with that of this embedding, and that a magnetic potential must be worked against to divert a trajectory away from this direction. Indeed, movement along this direction represents minimal potential energy. We take it to be the time direction. The gauge-dependent nature of the condensate is such that different gauge choices may lead to different time axes and we show on very general grounds that these different coordinate systems must be relatable by transformations of Lorentz form.

Weights for ℓ-local compact groups

In this note, we initiate the study of F-weights for an ℓ-local compact group F over a discrete ℓ-toral group S with discrete torus T. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when T is the unique maximal abelian subgroup of S up to F-conjugacy and every element of S is F-fused into T, the number of weights of F is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of F with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when F is simple and |S:T|=ℓ. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.

Consistency of chiral symmetry breaking in chiral Yang-Mills theory with adiabatic continuity

We study the pattern of chiral symmetry breaking in the $\ensuremath{\psi}\ensuremath{\chi}\ensuremath{\eta}$ model [with the chiral fermion sector containing ${\ensuremath{\psi}}^{{ij}}$, ${\ensuremath{\chi}}_{[ij]}$, and ${\ensuremath{\eta}}_{i}^{A}$; see Armoni and Shifman Phys. Rev. D 85, 105003 (2012)] on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}_{L}^{1}$ and derive implications forcing ${\mathbb{R}}^{4}$ physics. Center-symmetric vacua are stabilized by a double-trace deformation. With the center symmetry maintained at small $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$, i.e., at weak coupling, no phase transitions are expected in passing to large $L({S}^{1})\ensuremath{\gg}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ (here, $\mathrm{\ensuremath{\Lambda}}$ is the dynamical Yang-Mills scale). Starting with the small-$L$ limit, we find the leading-order nonperturbative corrections in the given theory. The instanton-monopole operators induce the adjoint chiral condensate $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ at weak coupling, i.e., at $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$. Then adiabatic continuity tells us that $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ exists on ${\mathbb{R}}^{4}$, in full accord with the prediction from Bolognesi et al. [Phys. Rev. D 97, 094007 (2018)]. Simultaneously with $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{3}{\ensuremath{\delta}}_{k}^{i}$, the $\mathrm{SU}({N}_{c})$ gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.

Character degrees of normally monomial 𝑝-groups of maximal class

AbstractA finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.

Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms \mathcal{A}_m of an m -ary tree \mathcal{T}_m by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has s\geq 1 orbits. We prove that such a group A embeds in a self-similar abelian group A^* which is also a maximal abelian subgroup of \mathcal{A}_m . The construction of A^{*} is based on the definition of a free monoid \Delta of rank s of partial diagonal monomorphisms of \mathcal{A}_m . Precisely, A^{*} = \overline{\Delta(B(A))} , where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of \mathcal{T}_m , and bar denotes the topological closure. Furthermore, we prove that if A is non-trivial, then A^{*} = C_{\mathcal{A}_m} (\Delta(A)) , the centralizer of \Delta(A) in \mathcal{A}_m . When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also \Delta -invariant for s=2 . In the final section, we introduce for m=ns \geq 2 , a generalized adding machine a , an automorphism of \mathcal{T}_{m} , and show that its centralizer in \mathcal{A}_{m} to be a split extension of \langle a \rangle^{*} by \mathcal{A}_s . We also describe important \mathbb{Z}_n [\mathcal{A}_s] submodules of \langle a\rangle^{*} .

Maximal order Abelian subgroups of Coxeter groups

AbstractIn this note, we give a classification of the maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analog of Cartan’s theorem that all maximal tori in a connected compact Lie group are conjugate.

Centralizers of elements of infinite order in plane Cremona groups

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of the projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out an overall study of centralizers of elements of infinite order in $\operatorname{Cr}_2(\mathbf{K})$ which leads to a classification of embeddings of $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well as a classification of maximal non-torsion abelian subgroups of $\operatorname{Cr}_2(\mathbf{K})$ when $\operatorname{char}(\mathbf{K})=0$.

Counting maximal abelian subgroups of p-groups

Counting maximal abelian subgroups of p-groups

Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball

We prove that to every connected Abelian subgroup H of the biholomorphisms of the unit ball Bn we can associate a set of bounded symbols whose corresponding Toeplitz operators generate a commutative C⁎-algebra on every weighted Bergman space. These symbols are of the form a(z)=f(μH(z)), where μH is the moment map for the action of H on Bn. We show that, for this construction, if H is a maximal Abelian subgroup, then the symbols introduced are precisely the H-invariant symbols. We provide the explicit computation of moment maps to obtain special sets of symbols described in terms of coordinates. In particular, it is proved that our symbol sets have as particular cases all symbol sets from the current literature that yield Toeplitz operators generating commutative C⁎-algebras on all weighted Bergman spaces on the unit ball Bn. Furthermore, we exhibit examples that show that some of the symbol sets introduced in this work have not been considered before. Finally, several explicit formulas for the corresponding spectra of the Toeplitz operators are presented. These include spectral integral expressions that simplify the known formulas for maximal Abelian subgroups for the unit ball.

Maximal abelian subgroups of the finite symmetric group

‎Let $G$ be a group‎. ‎For an element $ain G$‎, ‎denote by $cs(a)$ the second centralizer of~$a$ in~$G$‎, ‎which is the set of all elements $bin G$ such that $bx=xb$ for every $xin G$ that commutes with $a$‎. ‎Let $M$ be any maximal abelian subgroup of $G$‎. ‎Then $cs(a)subseteq M$ for every $ain M$‎. ‎The emph{abelian rank} (emph{$a$-rank}) of $M$ is the minimum cardinality of a set $Asubseteq M$ such that $bigcup_{ain A}cs(a)$ generates $M$‎. ‎Denote by $S_n$ the symmetric group of permutations on the set $X={1,ldots,n}$‎. ‎The aim of this paper is to determine the maximal abelian subgroups of $gx$‎ ‎of $cor$~$1$ and describe a class of maximal abelian subgroups of $gx$ of $cor$ at most~$2$‎.

The Steenrod algebra from the group theoretical viewpoint

In the paper “The Steenrod algebra and its dual” [2], J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme Gp represented by the dual Hopf algebra of the mod p Steenrod algebra. Then, Gp assigns a graded commutative algebra A⁎ over a prime field of finite characteristic p to a set of isomorphisms of the additive formal group law over A⁎, whose group structure is given by the composition of formal power series ([4] Appendix). The aim of this paper is to show some group theoretic properties of Gp by making use of this presentation of Gp(A⁎). We give a decreasing filtration of subgroup schemes of Gp which we use for estimating the length of the lower central series of finite subgroup schemes of Gp. We also give a successive quotient maps Gp→ρ0Gp〈1〉→ρ1Gp〈2〉→ρ2⋯→ρk−1Gp〈k〉→ρkGp〈k+1〉→ρk+1⋯ of affine group schemes over a prime field Fp such that the kernel of ρk is a maximal abelian subgroup.

On a type of maximal abelian torsion free subgroups of connected Lie groups

For an arbitrary real connected Lie group G we define $$\mathrm {p}(G)$$ as the maximal integer p such that $$\mathbb {Z}^p$$ is isomorphic to a discrete subgroup of G and $$\mathrm {q}(G)$$ is the maximal integer q such that $$\mathbb {R}^q$$ is isomorphic to a closed subgroup of G. The aim of this paper is to investigate properties of these two invariants. We will show that if G is a noncompact connected Lie group, then $$1\le \mathrm {q}(G)\le \mathrm {p}(G)\le \dim (G/K)$$ where K is a maximal compact subgroup of G. In the cases when G is an exponential Lie group or G is a connected nilpotent Lie group, we give explicit relationships between these two invariants and a well known Lie algebra invariant $$\mathcal M(\mathfrak {g})$$ , i.e. the maximum among the dimensions of abelian subalgebras of the Lie algebra $$\mathfrak {g}:=\mathrm {Lie}(G)$$ .

On Recognizability of PSU3(q) by the Orders of Maximal Abelian Subgroups

Li and Chen in 2012 proved that the simple group A1(pn) is uniquely determined by the set of orders of its maximal abelian subgroups. Later the authors proved that if L = A2(q), where q is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as L is isomorphic to L or an extension of L by a subgroup of the outer automorphism group of L. In this paper, we prove that if L = PSU3(q), where q is not a Fermat prime, then every finite group with the same set of orders of maximal abelian subgroups as L is an almost simple group with socle PSU3(q).

The restriction principle and commuting families of Toeplitz operators on the unit ball

On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of \widetilde{\SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of $\widetilde{\SU(n,1)}$, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly develop the restriction principle for each (conjugacy class of) maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform. In particular, we obtain a multiplicity one result for the restriction of the holomorphic discrete series to all maximal abelian subgroups. We also observe that the Segal-Bargman transform is (up to a unitary transformation) a convolution operator against a function that we write down explicitly for each case. This can be used to obtain the explicit simultaneous diagonalization of Toeplitz operators whose symbols are invariant by one of these maximal abelian subgroups.

Quasirecognition of E6(q) by the orders of maximal abelian subgroups

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

Large abelian normal subgroups

In this paper we study the family of finite groups with the property that every maximal abelian normal subgroup is self-centralizing. It is well known that this family contains all finite supersolvable groups, but it also contains many other groups. In fact, every finite group G is a subgroup of some member \(\Gamma \) of this family, and we show that if G is solvable, then \(\Gamma \) can be chosen so that every abelian normal subgroup of G is contained in some self-centralizing abelian normal subgroup of \(\Gamma \).