Representation Theory of Symmetric Groups

DescriptionRepresentation Theory of Symmetric Groups is the most up-to-date abstract algebra book on the subject of symmetric groups and representation theory. Utilizing new research and results, this book can be studied from a combinatorial, algorithmic or algebraic viewpoint.This book is an excellent way of introducing today’s students to representation theory of the symmetric groups, namely classical theory. From there, the book explains how the theory can be extended to other related combinatorial algebras like the Iwahori-Hecke algebra.In a clear and concise manner, the author presents the case that most calculations on symmetric group can be performed by utilizing appropriate algebras of functions. Thus, the book explains how some Hopf algebras (symmetric functions and generalizations) can be used to encode most of the combinatorial properties of the representations of symmetric groups.Overall, the book is an innovative introduction to representation theory of symmetric groups for graduate students and researchers seeking new ways of thought.Table of ContentsSymmetric groups and symmetric functionsRepresentations of finite groups and semisimple algebrasFinite groups and their representationsCharacters and constructions on representationsThe non-commutative Fourier transformSemisimple algebras and modulesThe double commutant theorySymmetric functions and the Frobenius-Schur isomorphismConjugacy classes of the symmetric groupsThe five bases of the algebra of symmetric functionsThe structure of graded self-adjoint Hopf algebraThe Frobenius-Schur isomorphismThe Schur-Weyl dualityCombinatorics of partitions and tableauxPieri rules and Murnaghan-Nakayama formulaThe Robinson-Schensted-Knuth algorithmConstruction of the irreducible representationsThe hook-length formulaII Hecke algebras and their representationsHecke algebras and the Brauer-Cartan theoryCoxeter presentation of symmetric groupsRepresentation theory of algebrasBrauer-Cartan deformation theoryStructure of generic and specialised Hecke algebrasPolynomial construction of the q-Specht modulesCharacters and dualities for Hecke algebrasQuantum groups and their Hopf algebra structureRepresentation theory of the quantum groupsJimbo-Schur-Weyl dualityIwahori-Hecke dualityHall-Littlewood polynomials and characters of Hecke algebrasRepresentations of the Hecke algebras specialised at q = 0Non-commutative symmetric functionsQuasi-symmetric functionsThe Hecke-Frobenius-Schur isomorphismsIII Observables of partitionsThe Ivanov-Kerov algebra of observablesThe algebra of partial permutationsCoordinates of Young diagrams and their momentsChange of basis in the algebra of observablesObservables and topology of Young diagramsThe Jucys-Murphy elementsThe Gelfand-Tsetlin subalgebra of the symmetric group algebraJucys-Murphy elements acting on the Gelfand-Tsetlin basisObservables as symmetric functions of the contentsSymmetric groups and free probabilityIntroduction to free probabilityFree cumulants of Young diagramsTransition measures and Jucys-Murphy elementsThe algebra of admissible set partitionsThe Stanley-Féray formula and Kerov polynomialsNew observables of Young diagramsThe Stanley-Féray formula for characters of symmetric groupsCombinatorics of the Kerov polynomialsIV Models of random Young diagramsRepresentations of the infinite symmetric groupHarmonic analysis on the Young graph and extremal charactersThe bi-infinite symmetric group and the Olshanski semigroupClassification of the admissible representationsSpherical representations and the GNS constructionAsymptotics of central measuresFree quasi-symmetric functionsCombinatorics of central measuresGaussian behavior of the observablesAsymptotics of Plancherel and Schur-Weyl measuresThe Plancherel and Schur-Weyl modelsLimit shapes of large random Young diagramsKerov’s central limit theorem for charactersAppendixA Representation theory of semisimple Lie algebrasNilpotent, solvable and semisimple algebrasRoot system of a semisimple complex algebraThe highest weight theoryReviews“The book will be most useful as a reference for researchers…I believe it is a valuable contribution to the literature on the symmetric group and related algebras.”– Mark J. Wildon, Mathematical Reviews, March 2018