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Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry

2021-12-30 Mathematics
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Release : 2021-12-30
Genre : Mathematics
Kind : eBook
Book Rating : 429/5 ( reviews)

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Book Synopsis Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry by : Stuart Margolis

Download or read book Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry written by Stuart Margolis. This book was released on 2021-12-30. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Representation Theory of Finite Monoids

2016-12-09 Mathematics
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Release : 2016-12-09
Genre : Mathematics
Kind : eBook
Book Rating : 324/5 ( reviews)

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Book Synopsis Representation Theory of Finite Monoids by : Benjamin Steinberg

Download or read book Representation Theory of Finite Monoids written by Benjamin Steinberg. This book was released on 2016-12-09. Available in PDF, EPUB and Kindle. Book excerpt: This first text on the subject provides a comprehensive introduction to the representation theory of finite monoids. Carefully worked examples and exercises provide the bells and whistles for graduate accessibility, bringing a broad range of advanced readers to the forefront of research in the area. Highlights of the text include applications to probability theory, symbolic dynamics, and automata theory. Comfort with module theory, a familiarity with ordinary group representation theory, and the basics of Wedderburn theory, are prerequisites for advanced graduate level study. Researchers in algebra, algebraic combinatorics, automata theory, and probability theory, will find this text enriching with its thorough presentation of applications of the theory to these fields. Prior knowledge of semigroup theory is not expected for the diverse readership that may benefit from this exposition. The approach taken in this book is highly module-theoretic and follows the modern flavor of the theory of finite dimensional algebras. The content is divided into 7 parts. Part I consists of 3 preliminary chapters with no prior knowledge beyond group theory assumed. Part II forms the core of the material giving a modern module-theoretic treatment of the Clifford –Munn–Ponizovskii theory of irreducible representations. Part III concerns character theory and the character table of a monoid. Part IV is devoted to the representation theory of inverse monoids and categories and Part V presents the theory of the Rhodes radical with applications to triangularizability. Part VI features 3 chapters devoted to applications to diverse areas of mathematics and forms a high point of the text. The last part, Part VII, is concerned with advanced topics. There are also 3 appendices reviewing finite dimensional algebras, group representation theory, and Möbius inversion.

Bimonoids for Hyperplane Arrangements

2020-03-19 Mathematics
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Release : 2020-03-19
Genre : Mathematics
Kind : eBook
Book Rating : 785/5 ( reviews)

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Book Synopsis Bimonoids for Hyperplane Arrangements by : Marcelo Aguiar

Download or read book Bimonoids for Hyperplane Arrangements written by Marcelo Aguiar. This book was released on 2020-03-19. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this monograph is to develop Hopf theory in a new setting which features centrally a real hyperplane arrangement. The new theory is parallel to the classical theory of connected Hopf algebras, and relates to it when specialized to the braid arrangement. Joyal's theory of combinatorial species, ideas from Tits' theory of buildings, and Rota's work on incidence algebras inspire and find a common expression in this theory. The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid relative to a fixed hyperplane arrangement. They also construct universal bimonoids by using generalizations of the classical notions of shuffle and quasishuffle, and establish the Borel–Hopf, Poincaré–Birkhoff–Witt, and Cartier–Milnor–Moore theorems in this setting. This monograph opens a vast new area of research. It will be of interest to students and researchers working in the areas of hyperplane arrangements, semigroup theory, Hopf algebras, algebraic Lie theory, operads, and category theory.

Topics in Hyperplane Arrangements

2017-11-22 Mathematics
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Release : 2017-11-22
Genre : Mathematics
Kind : eBook
Book Rating : 112/5 ( reviews)

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Book Synopsis Topics in Hyperplane Arrangements by : Marcelo Aguiar

Download or read book Topics in Hyperplane Arrangements written by Marcelo Aguiar. This book was released on 2017-11-22. Available in PDF, EPUB and Kindle. Book excerpt: This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material. Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples. Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra. The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.

Combinatorial Algebraic Topology

2008-01-08 Mathematics
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Release : 2008-01-08
Genre : Mathematics
Kind : eBook
Book Rating : 514/5 ( reviews)

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Book Synopsis Combinatorial Algebraic Topology by : Dimitry Kozlov

Download or read book Combinatorial Algebraic Topology written by Dimitry Kozlov. This book was released on 2008-01-08. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first comprehensive treatment of combinatorial algebraic topology in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms.

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