Compact Group Research Articles

On the validity of the Cauchy–Schwarz inequality for the bracket map

Due to its properties, the bracket map associated with a dual integrable unitary representation of a locally compact group can be viewed as a certain operator-valued inner product; however, in the non-commutative setting, the Cauchy–Schwarz property for bracket is no longer present in its full strength. In this paper, we show that fulfillment of the property, even in weaker forms, has strong consequences on the underlying group \(G\) and the corresponding von Neumann algebra \(\mbox{VN}(G)\). In particular, we show that for unimodular group \(G\), positive elements of the \(L^1\) space over \(\mbox{VN}(G)\) which are affiliated with the commutant of \(\mbox{VN}(G)\) are precisely those for which the weaker variant of the inequality is fulfilled, and that the validity of the Cauchy–Schwarz property for the appropriate set of elements indicates the existence of a closed abelian subgroup or abelian von Neumann subalgebra of \(\mbox{VN}(G)\).

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A Gabor orthonormal basis, on a locally compact Abelian (LCA) group A, is an orthonormal basis of L2(A) that consists of time-frequency shifts of some template f∈L2(A). It is well known that, on Rd, the elements of such a basis cannot have a good time-frequency localization. The picture is drastically different on LCA groups containing a compact open subgroup, where one can easily construct examples of Gabor orthonormal bases with f maximally localized, in the sense that the ambiguity function of f (i.e., the correlation of f with its time-frequency shifts) has support of minimum measure, compatibly with the uncertainty principle. In this note we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in L2(A) that are maximally localized in the time-frequency space in the above sense – an issue that is open even for finite Abelian groups. As a byproduct, on every LCA group containing a compact open subgroup we exhibit the complete family of optimizers for Lieb's uncertainty inequality, and we also show previously unknown optimizers on a general LCA group.

A note on matrix-valued orthonormal bases over LCA groups

It is well known that orthonormal bases for a separable Hilbert space [Formula: see text] are precisely collections of the form [Formula: see text], where [Formula: see text] is a linear unitary operator acting on [Formula: see text] and [Formula: see text] is a given orthonormal basis for [Formula: see text]. We show that this is not true for the matrix-valued signal space [Formula: see text], [Formula: see text] is a locally compact abelian group which is [Formula: see text]-compact and metrizable, and [Formula: see text] and [Formula: see text] are positive integers. This problem is related to the adjointability of bounded linear operators on [Formula: see text]. We show that any orthonormal basis of the space [Formula: see text] is precisely of the form [Formula: see text], where [Formula: see text] is a linear unitary operator acting on [Formula: see text] which is adjointable with respect to the matrix-valued inner product and [Formula: see text] is a matrix-valued orthonormal basis for [Formula: see text].

Novel transcriptomic panel identifies histologically active eosinophilic oesophagitis

Background and aimsEosinophilic oesophagitis (EoE) is characterised by symptoms of esophageal dysfunction and oesinophil tissue infiltration. The EoE Diagnostic Panel (EDP) can distinguish between active and non-active EoE using a...

Supersymmetry and trace formulas. Part I. Compact Lie groups

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.

Actions of Compact and Discrete Quantum Groups on Operator Systems

Abstract We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.

TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES

AbstractFor closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$ . Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$ . Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

Heyde Theorem for Locally Compact Abelian Groups Containing No Subgroups Topologically Isomorphic to the 2-Dimensional Torus

Heyde Theorem for Locally Compact Abelian Groups Containing No Subgroups Topologically Isomorphic to the 2-Dimensional Torus

The five gradients inequality on differentiable manifolds

The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds.

Factoring determinants and applications to number theory

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of [Formula: see text]-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove that, in certain cases, these unitary averages factor as polynomials into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the “swap” terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from random matrix theory.

Open mappings of locally compact groups

Abstract The aim of this note is to insert in the literature some easy but apparently not widely known facts about morphisms of locally compact groups, all of which are concerned with the openness of the morphism.

Division rings for group algebras of virtually compact special groups and $3$-manifold groups

Let k be a division ring and let G be either a torsion-free virtually compact special group or a finitely generated torsion-free 3 -manifold group. We embed the group algebra kG in a division ring and prove that the embedding is Hughes-free whenever G is locally indicable. In particular, we prove that Kaplansky’s Zero Divisor Conjecture holds for all group algebras of torsion-free 3 -manifold groups. The embedding is also used to confirm a conjecture of Kielak and Linton. Thanks to the work of Jaikin-Zapirain and Linton, another consequence of the embedding is that kG is coherent whenever G is a virtually compact special one-relator group. If G is a torsion-free one-relator group, let \overline{kG} be the division ring containing kG constructed by Lewin and Lewin. We prove that \overline{kG} is Hughes-free whenever a Hughes-free kG -division ring exists. This is always the case when k is of characteristic zero; in positive characteristic, our previous result implies that this happens when G is virtually compact special.

An Analogue of the Klebanov Theorem for Locally Compact Abelian Groups

An Analogue of the Klebanov Theorem for Locally Compact Abelian Groups

Word Measures on Unitary Groups: Improved Bounds for Small Representations

Abstract Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}\in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $\mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $\mu _{w,U(n)}$.

“Reticulata irises”: a nomenclatural and taxonomic synopsis of the genera Alatavia and Iridodictyum (Iris subg. Hermodactyloides auct., Iridaceae)

The “Reticulata irises” are dwarf irises highly appreciated in horticulture, which are characterised by their tuberiform bulbs, with a single fleshy inner tunic clothed mostly with reticulate or reticulate-hairy outer tunics, and basal leaves bifacial, angulose or finely sulcate in section. The aggregate is often accepted as a taxonomically compact group to which the name Iris subg. Hermodactyloides (Iris sect. Reticulatae) is often applied. It includes between 10–22 taxa (species and subspecies) occurring disjunctly from central Türkiye and the Transcaucasus throughout the Middle East to western China. Molecular work shows that Iris subg. Hermodactyloides, is polyphyletic as frequently delineated. Alternatively, analytic treatments accept two genera, Alatavia and Iridodictyum, exhibiting clear differences in morphology, biogeography, and phylogenetic connections. Recently, new field prospection across scarcely prospected vast territories led to the description of many new taxa in the “Reticulata irises.” In this context, an updated synopsis of the bulbous genera Alatavia (four species) and Iridodictyum (18 + 2 species) is reported. For each accepted taxon, main synonyms, type indication, chromosome numbers, distribution areas, and taxonomic or nomenclatural remarks are reported. Further, five new specific combinations are introduced, and also four neotypes, two lectotypes, two second-step lectotypes, and one epitype are designated.

Subelliptic sharp Gårding inequality on compact Lie groups

Subelliptic sharp Gårding inequality on compact Lie groups

Almost gauge-invariant states and the ground state of Yang-Mills theory

We consider the problem of the explicit description of the gauge-invariant subspace of pure lattice gauge theories in the Hamiltonian formulation, where the gauge group is either a compact Lie group or a finite group. The latter case is particularly interesting for quantum simulation. A basis of states where configurations are grouped according to their holonomies is shown to have several advantages over other descriptions. Using this basis, we compute some properties of interest for some non-Abelian finite groups on small lattices, and in particular we examine the question of whether a certain ansatz introduced long ago is a good approximation for the ground state. Published by the American Physical Society 2024

Complexity of non-abelian cut-and-project sets of polytopal type I: special homogeneous Lie groups

Abstract The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like $r^{{\mathrm {homdim}}(G) \dim (H)}$ . Further, we generalize the concept of acceptance domains to locally compact second countable groups.

Some Remarks on Projective Representations of Compact Groups and Frames

Some Remarks on Projective Representations of Compact Groups and Frames