Arbitrary Group Research Articles

Unitarity constraints on large multiplets of arbitrary gauge groups

We impose partial-wave unitarity on 2→2 tree-level scattering processes to derive constraints on the dimensions of large scalar and fermionic multiplets of arbitrary gauge groups. We apply our results to scalar and fermionic extensions of the Standard Model, and also to the Grand Unified Theories (GUTs) based on the groups SU(5), SO(10), and E6. We find scenarios within the latter two GUTs that violate the unitarity condition; this may require a reevaluation of the validity of perturbation theory in those scenarios.

Product structure and regularity theorem for totally nonnegative flag varieties

AbstractThe totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$ J -total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the $J$ J -totally nonnegative flag variety has a cellular decomposition into totally positive $J$ J -Richardson varieties. Moreover, each totally positive $J$ J -Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive $J$ J -Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the $J$ J -total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of $U^{-}$ U − for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

The cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups

The cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.

On the odd order subgroups of solvable linear groups

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group.

Electric fields near undulating dielectric membranes.

Dielectric interfaces are crucial to the behavior of charged membranes, from graphene to synthetic and biological lipid bilayers. Understanding electrolyte behavior near these interfaces remains a challenge, especially in the case of rough dielectric surfaces. A lack of analytical solutions consigns this problem to numerical treatments. We report an analytic method for determining electrostatic potentials near curved dielectric membranes in a two-dimensional periodic "slab" geometry using a periodic summation of Green's functions. This method is amenable to simulating arbitrary groups of charges near surfaces with two-dimensional deformations. We concentrate on one-dimensional undulations. We show that increasing membrane undulation increases the asymmetry of interfacial charge distributions due to preferential ionic repulsion from troughs. In the limit of thick membranes, we recover results mimicking those for electrolytes near a single interface. Our work demonstrates that rough surfaces generate charge patterns in electrolytes of charged molecules or mixed-valence ions.

Gain total graphs and their spectra via G-phases and group representations

We introduce a definition for the total graph of a gain graph (Γ,ψ) on a group G by using G-phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice of the G-phase associated with ψ. More precisely, we consider the left-right multiplication action of G on the space of all G-phases, proving that its orbits consist of sets of G-phases inducing the same switching equivalence class of gain functions on Γ or, equivalently, on its total graphs. Moreover, we prove that the stabilizers of this action are isomorphic to the centralizers of the sets of gains of closed walks in the associated gain graphs. Our construction is consistent with the existing notions of total graph for signed graphs and, in analogy with the signed case, we are able to explicitly compute the spectrum of the total graph of a regular gain graph over an arbitrary group G, where the spectrum is defined by means of a unitary representation of the gain group G.

Presenting new artificial intelligence techniques for finding optimum performance of solar chimney power plant

The various applications of solar energy have progressed significantly over recent decades. In this paper, finding the optimum performance criteria of a SCPP is studied. The mathematical modelling of 1-D flow is written and the optimum performance is determined for each group of geometric and environmental variables including the angle of collector roof (β), chimney divergence (α) and ambient wind velocity (Va ). In addition, different artificial intelligence techniques including regression tree, genetic programming and a novel method named trended regression tree are applied to find optimum performance conditions at any arbitrary group of α, β and Va . Results indicate that maximum power output (Pt ) and associated mass flow rate ( m ˙ ) increase with increasing chimney divergent angle and wind velocity at a constant collector roof angle. Comparing the obtained results of artificial intelligence techniques shows that trended regression tree method predicts the optimum conditions of SCPP better than other methods. Genetic programming is used to present fitted algebraic functions on Pt and m ˙ and the ratio of turbine pressure drop to pressure potential (PR). Results demonstrate the algebraic relation of PR estimates the PR accurately at each group of variables.

Profinite version of the Beckmann-Black problem

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 ( K ) of some finite regular Galois extension F / K ( T ) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group.

Realizing finite groups as automizers

Abstract It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.

Units, zero-divisors and idempotents in rings graded by torsion-free groups

Abstract The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

Transcendence on algebraic groups

Abstract In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and p-adic fields, respectively.

Smith theory and cyclic base change functoriality

Abstract Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $\mathbf {Z}/p\mathbf {Z}$ -extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along $\mathbf {Z}/p\mathbf {Z}$ -extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod p spherical Hecke algebras, in a joint appendix with Gus Lonergan.

Minimal Group Procedures and Outcomes

Research using the Minimal Group Paradigm has demonstrated the power of arbitrary group membership to produce prejudice and discrimination on a variety of measures. Despite the continued prominence of this paradigm in the social and behavioral sciences, the relative efficacy of minimal group induction procedures and methodological variations in producing intergroup biases remains largely unknown. The present research compared the effects of minimal group induction procedures across multiple measures of discrimination and both implicit and explicit measures of evaluation and identification. We tested six induction procedures and other methodological variations, such as manipulations designed to increase the meaning of the groups or to undermine assumed reciprocity in the allocation tasks. Regardless of procedural manipulation, participants demonstrated bias in favor of their minimal ingroup on most outcome measures. However, the magnitude of the minimal group effect varied somewhat according to outcome, induction, and other methodological features.

Entanglement asymmetry in the ordered phase of many-body systems: the Ising field theory

Global symmetries of quantum many-body systems can be spontaneously broken. Whenever this mechanism happens, the ground state is degenerate and one encounters an ordered phase. In this study, our objective is to investigate this phenomenon by examining the entanglement asymmetry of a specific region. This quantity, which has recently been introduced in the context of U(1) symmetry breaking, is extended to encompass arbitrary finite groups G. We also establish a field theoretic framework in the replica theory using twist operators. We explicitly demonstrate our construction in the ordered phase of the Ising field theory in 1+1 dimensions, where a ℤ2 symmetry is spontaneously broken, and we employ a form factor bootstrap approach to characterise a family of composite twist fields. Analytical predictions are provided for the entanglement asymmetry of an interval in the Ising model as the length of the interval becomes large. We also propose a general conjecture relating the entanglement asymmetry and the number of degenerate vacua, expected to be valid for a large class of states, and we prove it explicitly in some cases.

COUNTING DISCRETE, LEVEL-, QUATERNIONIC AUTOMORPHIC REPRESENTATIONS ON

Abstract Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm {GL}_2$ . Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$ , eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level- $1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$ . The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series. We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

On the difference graph of power graphs of finite groups

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.

Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

Consequences of arbitrary binning the midpoint category in survey data: an illustration with student satisfaction in the National Student Survey

ABSTRACT Arbitrary placing cut-offs in data, i.e. binning, is recognised as poor statistical practice. We explore the consequences of using arbitrary cut-offs in two large datasets, the National Student Survey (2019 and 2022). These are nationwide surveys aimed at capturing student satisfaction amongst UK undergraduates. For these survey data, it is common to group the responses to the question on student satisfaction on a five point Likert scale into ‘% satisfied’ based on two categories. These % satisfied are then further used in metrics. We examine the consequences of using three rather than two categories for the rankings of courses and institutions, as well as the consequences of excluding the midpoint from the calculations. Across all courses, grouping the midpoint with satisfied leads to a median shift of 8.40% and 11.41% in satisfaction for 2019 and 2022, respectively. Excluding the midpoint from the calculations leads to a median shift of 4.20% and 5.70% in satisfaction for 2019 and 2022, respectively. While the overall stability of the rankings is largely preserved, individual courses or institutions exhibit sizeable shifts. Depending on the analysis, the most extreme shifts for courses in rankings are between 13 and 79 ranks, for institutions between 24 and 416 ranks. Our analysis thus illustrates the potentially profound consequences of arbitrarily grouping categories for individual institutions and courses. We offer some recommendations on how this issue can be addressed but primarily we caution against the reliance on arbitrary grouping of response categories in survey data such as the NSS.

Identifying 1-rectifiable measures in Carnot groups

Abstract We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of analyst’s traveling salesman theorem, which characterizes the subsets of rectifiable curves in R 2 {{\mathbb{R}}}^{2} (P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15), in R n {{\mathbb{R}}}^{n} (K. Okikiolu, Characterization of subsets of rectifiable curves in R n {{\bf{R}}}^{n} , J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. Li) in terms of local geometric least-squares data called Jones’ β \beta -numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n {{\mathbb{R}}}^{n} that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.