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"Symmetry groups"
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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
Rock blocks
Consider representation theory associated to symmetric groups, or to Hecke algebras in type A, or to $q$-Schur algebras, or to finite general linear groups in non-describing characteristic. Rock blocks are certain combinatorially defined blocks appearing in such a representation theory, first observed by R. Rouquier. Rock blocks are much more symmetric than general blocks, and every block is derived equivalent to a Rock block. Motivated by a theorem of J. Chuang and R. Kessar in the case of symmetric group blocks of abelian defect, the author pursues a structure theorem for these blocks.
eBook
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
Symmetries of simple A T-algebras
Let A be a unital simple AT -algebra of real rank zero. Given an order two automorphism h W K[iota] (A) [right arrow] K[iota] (A), we show that there is an order two automorphism [alpha]: A [right arrow] A such that [[alpha].sub.*0] = id, [[alpha].sub.*1] = h and the action of [Z.sub.2] generated by [alpha] has the tracial Rokhlin property. Consequently, C*(A, [Z.sub.2], [alpha]) is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus, our main result can be considered the [Z.sub.2]-action analogue of the Lin--Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case. Keywords. AT algebra, symmetries, real rank zero, tracial Rokhlin property.
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
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
Universality of Dynamical Symmetries in Chaotic Maps
Identifying signs of regularity and uncovering dynamical symmetries in complex and chaotic systems is crucial both for practical applications and for enhancing our understanding of complex dynamics. Recent approaches have quantified temporal correlations in time series, revealing hidden, approximate dynamical symmetries that provide insight into the systems under study. In this paper, we explore universality patterns in the dynamics of chaotic maps using combinations of complexity quantifiers. We also apply a recently introduced technique that projects dynamical symmetries into a “symmetry space”, providing an intuitive and visual depiction of these symmetries. Our approach unifies and extends previous results and, more importantly, offers a meaningful interpretation of universality by linking it with dynamical symmetries and their transitions.
Journal Article