Representation Theory of Finite Monoids (Universitext)

December 8, 2020
Representation Theory of Finite Monoids (Universitext)

An entire part of the text is devoted to applications to Markovchains, combinatorics, and automata theoryAccessible to a wide readership of graduate students andresearchers, including non-experts in semigroupsContains exercises, chapter notes, and thoroughly workedexamples——————————This first text on the subject provides a comprehensiveintroduction to the representation theory of finite monoids.Carefully worked examples and exercises provide the bells andwhistles for graduate accessibility, bringing a broad range ofadvanced 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 thebasics of Wedderburn theory, are prerequisites for advancedgraduate level study. Researchers in algebra, algebraiccombinatorics, automata theory, and probability theory, will findthis text enriching with its thorough presentation of applicationsof the theory to these fields.Prior knowledge of semigroup theory is not expected for thediverse readership that may benefit from this exposition. Theapproach taken in this book is highly module-theoretic and followsthe modern flavor of the theory of finite dimensional algebras. Thecontent is divided into 7 parts. Part I consists of 3 preliminarychapters with no prior knowledge beyond group theory assumed. PartII forms the core of the material giving a modern module-theoretictreatment of the Clifford –Munn–Ponizovskii theory of irreduciblerepresentations. Part III concerns character theory and thecharacter table of a monoid. Part IV is devoted to therepresentation theory of inverse monoids and categories and Part Vpresents the theory of the Rhodes radical with applications totriangularizability. Part VI features 3 chapters devoted toapplications to diverse areas of mathematics and forms a high pointof the text. The last part, Part VII, is concerned with advancedtopics. There are also 3 appendices reviewing finite dimensionalalgebras, group representation theory, and Möbius inversion.