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If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional. The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, \"birdtracks\" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as \"negative dimensional\" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

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.

In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or \"form\" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a \"residually pseudo-split\" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.

The theory of hypergroups is a rapidly developing area of mathematics due to its diverse applications in different areas like probability, harmonic analysis, etc. This book exhibits the use of functional equations and spectral synthesis in the theory of hypergroups. It also presents the fruitful consequences of this delicate \"marriage\" where the methods of spectral analysis and synthesis can provide an efficient tool in characterization problems of function classes on hypergroups.This book is written for the interested reader who has open eyes for both functional equations and hypergroups, and who dares to enter a new world of ideas, a new world of methods - and, sometimes, a new world of unexpected difficulties.Sample Chapter(s)Introduction (261 KB)Contents:IntroductionPolynomial Hypergroups in One VariablePolynomial Hypergroups in Several VariablesSturm-Liouville HypergroupsTwo-Point Support HypergroupsSpectral Analysis and Synthesis on Polynomial HypergroupsSpectral Analysis and Synthesis on Sturm-Liouville HypergroupsMoment Problems on HypergroupsSpecial Functional Equations on HypergroupsDifference Equations on Polynomial HypergroupsStability Problems on HypergroupsReadership: Researchers and post-graduate students working in hypergroups.

The following problem has been known since the 80s. Let$\\Gamma$be an Abelian group of order$m$(denoted$|\\Gamma|=m$ ), and let$t$and$\\{m_i\\}_{i=1}^{t}$ , be positive integers such that$\\sum_{i=1}^t m_i=m-1$ . Determine when$\\Gamma^*=\\Gamma\\setminus\\{0\\}$ , the set of non-zero elements of$\\Gamma$ , can be partitioned into disjoint subsets$\\{S_i\\}_{i=1}^{t}$such that$|S_i|=m_i$and$\\sum_{s\\in S_i}s=0$for every$1 \\leq i \\leq t$ . Such a subset partition is called a zero-sum partition.$|I(\\Gamma)|\\neq 1$ , where$I(\\Gamma)$is the set of involutions in$\\Gamma$ , is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of$m_i\\geq 4$for every$1 \\leq i \\leq t$ , is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.

The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules. In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

The following problem has been known since the 80's. Let$\\Gamma$be an Abelian group of order$m$(denoted$|\\Gamma|=m$ ), and let$t$and$m_i$ ,$1 \\leq i \\leq t$ , be positive integers such that$\\sum_{i=1}^t m_i=m-1$ . Determine when$\\Gamma^*=\\Gamma\\setminus\\{0\\}$ , the set of non-zero elements of$\\Gamma$ , can be partitioned into disjoint subsets$S_i$ ,$1 \\leq i \\leq t$ , such that$|S_i|=m_i$and$\\sum_{s\\in S_i}s=0$for every$i$ ,$1 \\leq i \\leq t$ . It is easy to check that$m_i\\geq 2$(for every$i$ ,$1 \\leq i \\leq t$ ) and$|I(\\Gamma)|\\neq 1$are necessary conditions for the existence of such partitions, where$I(\\Gamma)$is the set of involutions of$\\Gamma$ . It was proved that the condition$m_i\\geq 2$is sufficient if and only if$|I(\\Gamma)|\\in\\{0,3\\}$ . For other groups (i.e., for which$|I(\\Gamma)|\\neq 3$and$|I(\\Gamma)|>1$ ), only the case of any group$\\Gamma$with$\\Gamma\\cong(Z_2)^n$for some positive integer$n$has been analyzed completely so far, and it was shown independently by several authors that$m_i\\geq 3$is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if$|\\Gamma|$is large enough and$|I(\\Gamma)|>1$ , then$m_i\\geq 4$is sufficient. In this paper we generalize this result for every Abelian group of order$2^n$ . Namely, we show that the condition$m_i\\geq 3$is sufficient for$\\Gamma$such that$|I(\\Gamma)|>1$and$|\\Gamma|=2^n$ , for every positive integer$n$ . We also present some applications of this result to graph magic- and anti-magic-type labelings.