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"Symmetry group"
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The Representation Theory of the Increasing Monoid
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation
category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish
properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous
connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle
algebras.
eBook
Symmetry in Signals: A New Insight
Symmetry is a fundamental property of many natural systems, which is observable through signals. In most out-of-equilibrium complex dynamic systems, the observed signals are asymmetric. However, for certain operating modes, some systems have demonstrated a resurgence of symmetry in their signals. Research has naturally focused on examining time invariance to quantify this symmetry. Measures based on the statistical and harmonic properties of signals have been proposed, but most of them focused on harmonic distortion without explicitly measuring symmetry. This paper introduces a new mathematical framework based on group theory for analyzing signal symmetry beyond time invariance. It presents new indicators to evaluate different types of symmetry in non-stochastic symmetric signals. Both periodic and non-periodic symmetric signals are analyzed to formalize the problem. The study raises critical questions about the completeness of symmetry in signals and proposes a new classification for periodic and non-periodic signals that goes beyond the traditional classification based on Fourier coefficients. Furthermore, new measures such as “symmetrometry” and “distorsymmetry” are introduced to quantify symmetry. These measures outperform traditional indicators like Total Harmonic Distortion (THD) and provide a more accurate measurement of symmetry in complex signals from applications where duty cycle plays a major role.
Journal Article
What Is the Symmetry Group of a d-Psub.II Discrete Painleve Equation?
The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper, we argue against the commonly held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception by using a specific example of a d-P[sub.II] equation, which corresponds to a half-translation on the root lattice dual to its surface-type root lattice but becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways, first by a brute force calculation, and then through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example, which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.
Journal Article
Mumford-tate groups and domains
Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers.
Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.
eBook
Symmetry Breaking: One-Point Theorem
Symmetry breaking is crucial in many areas of physics, mathematics, biology, and engineering. We investigate the symmetry of regular convex polygons, non-convex regular polygons (stars), and symmetric Jordan curves/domains. We demonstrate that removing a single point from the boundary of regular convex and non-convex polygons and symmetrical Jordan curves reduces the symmetry group of the polygon to the trivial C1 group when the point does not belong to the axis of symmetry of the polygon. The same is true for solid and open 2D regular convex polygons and symmetric Jordan curves. The only exception is a circle. Removing a single point from the boundary of a circle creates a curve characterized by the C2 group. The symmetry of circles is reduced to the trivial C1 group by removing a triad of non-symmetrical points. The same is true for a solid circle. The “effort” necessary to break the symmetry of a circle is maximal. A 3D generalization of the theorem is exemplified. Thus, the classification of symmetrical curves following the minimal number of points necessary to break their symmetry becomes possible. The demonstrated theorem shows that the symmetry group action on curves and domains becomes trivial when an asymmetric perturbation is introduced, when the curve is not a circle. An informational interpretation of the demonstrated theorem, which is related to the Landauer principle, is provided.
Journal Article
Probabilistic Measure of Symmetry Stability
Symmetry is a fundamental principle in mathematics, physics, and biology, where it governs structure and invariance. Classical symmetry analysis focuses on exact group-theoretic descriptions, but rarely addresses how robust a symmetric configuration is to perturbations. In this work, we introduce a probabilistic framework for quantifying the stability of finite point-set symmetries under random deletions. Specifically, given a finite set of points with a prescribed nontrivial symmetry group, we define the probability PN that removing N points reduces the symmetry to the trivial group C1. The complementary quantity SN=1−PN serves as a measure of symmetry stability, providing a robustness profile of the configuration. We calculate SN explicitly for representative families of symmetric point sets, including linear arrays, polygons, polyhedra, directed necklace of points, and crystallographic unit cells. Our results demonstrate unexpected behaviors: the regular hexagon loses symmetry with a probability of 0.6 under the removal of three vertices, while cubes and tetrahedra exhibit the maximal robustness (SN=1) for all admissible N. We further introduce a Shannon entropy of symmetry stability, which quantifies the overall uncertainty of symmetry breaking across all deletion sizes. This framework extends classical symmetry studies by incorporating randomness, linking group theory with probabilistic combinatorics, and suggesting applications ranging from crystallography to defect tolerance in physical systems.
Journal Article
A comparative study of LQU and LQFI in general qubit-qutrit axially symmetric states
We derive the compact closed forms of local quantum uncertainty (LQU) and local quantum Fisher information (LQFI) for hybrid qubit-qutrit axially symmetric (AS) states. This allows us to study the quantum correlations in detail and present some essentially novel results for spin-(1/2, 1) systems, the Hamiltonian of which contains ten independent types of physically important parameters. As an application of the derived formulas, we study the behavior of these two quantum correlation measures at thermal equilibrium. New features are observed in their behavior that are important for quantum information processing. Specifically,
cascades
of sudden changes in the behavior of LQU and LQFI are found with a smooth change in temperature or interaction parameters. Interestingly, in some cases, sudden transitions are observed in the behavior of LQU but not in LQFI, and vice versa. Moreover, our compact formulas open a way to apply them to other problems, for instance, when investigating the environmental effects on quantum correlations in open systems.
Journal Article
Strong Gelfand Pairs of the Symplectic Group Spsub.4 Where q Is Even
A strong Gelfand pair (G,H) is a finite group G together with a subgroup H such that every irreducible character of H induces to a multiplicity-free character of G . We classify the strong Gelfand pairs of the symplectic groups Sp[sub.4] (q) for even q .
Journal Article