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This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.

Schur functions. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- S-functions --- Schur's functions --- Holomorphic functions --- Combinatorics. --- Geometry, algebraic. --- Algebraic topology. --- Algebraic Geometry. --- Algebraic Topology. --- Topology --- Combinatorics --- Algebra --- Mathematical analysis --- Algebraic geometry.

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The theory of Schur-Weyl duality has had a profound influence over many areas of algebra and combinatorics. This text is original in two respects: it discusses affine q-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum Schur-Weyl theory. To begin, various algebraic structures are discussed, including double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of the book investigates the affine quantum Schur-Weyl duality on three levels. This includes the affine quantum Schur-Weyl reciprocity, the bridging role of affine q-Schur algebras between representations of the quantum loop algebras and those of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel-Hall algebra with a proof of the classical case. This text is ideal for researchers in algebra and graduate students who want to master Ringel-Hall algebras and Schur-Weyl duality.

Schur functions. --- Weyl groups. --- Representations of Lie groups. --- Affine algebraic groups. --- Algebraic groups, Affine --- Group schemes (Mathematics) --- Lie groups --- Weyl's groups --- Group theory --- S-functions --- Schur's functions --- Holomorphic functions

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Ordered algebraic structures --- Partially ordered sets. --- Schur functions --- Ensembles partiellement ordonnés --- Fonctions de Schur --- Schur functions. --- 51 <082.1> --- Mathematics--Series --- Ensembles partiellement ordonnés --- Partially ordered sets --- S-functions --- Schur's functions --- Holomorphic functions --- Posets --- Sets, Partially ordered --- Ordered sets

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An Introduction to Quasisymmetric Schur Functions is aimed at researchers and graduate students in algebraic combinatorics. The goal of this monograph is twofold. The first goal is to provide a reference text for the basic theory of Hopf algebras, in particular the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric functions and connections between them. The second goal is to give a survey of results with respect to an exciting new basis of the Hopf algebra of quasisymmetric functions, whose combinatorics is analogous to that of the renowned Schur functions.

Combinatorial analysis. --- Combinatorics. --- Quasisymmetric groups. --- Schur functions. --- Schur functions --- Quasisymmetric groups --- Combinatorial analysis --- Engineering & Applied Sciences --- Computer Science --- Hopf algebras. --- Combinatorics --- Algebras, Hopf --- S-functions --- Schur's functions --- Mathematics. --- Topological groups. --- Lie groups. --- Applied mathematics. --- Engineering mathematics. --- Algorithms. --- Topological Groups, Lie Groups. --- Applications of Mathematics. --- Algebra --- Mathematical analysis --- Algebraic topology --- Holomorphic functions --- Topological Groups. --- Math --- Science --- Groups, Topological --- Continuous groups --- Algorism --- Arithmetic --- Foundations --- Engineering --- Engineering analysis --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Mathematics

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The origins of Schur analysis lie in a 1917 article by Issai Schur in which he constructed a numerical sequence to correspond to a holomorphic contractive function on the unit disk. These sequences are now known as Schur parameter sequences. Schur analysis has grown significantly since its beginnings in the early twentieth century and now encompasses a wide variety of problems related to several classes of holomorphic functions and their matricial generalizations. These problems include interpolation and moment problems as well as Schur parametrization of particular classes of contractive or nonnegative Hermitian block matrices. This book is primarily devoted to topics related to matrix versions of classical interpolation and moment problems. The major themes include Schur analysis of nonnegative Hermitian block Hankel matrices and the construction of Schur-type algorithms. This book also covers a number of recent developments in orthogonal rational matrix functions, matrix-valued Carathéodory functions and maximal weight solutions for particular matricial moment problems on the unit circle.

Differential equations -- Numerical solutions. --- Differential equations. --- Integral equations -- Numerical solutions. --- Interpolation --- Schur functions --- Inverse problems (Differential equations) --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Interpolation. --- Schur functions. --- Moment problems (Mathematics) --- S-functions --- Schur's functions --- Mathematics. --- Functions of complex variables. --- Integral transforms. --- Operational calculus. --- Operator theory. --- Operator Theory. --- Integral Transforms, Operational Calculus. --- Functions of a Complex Variable. --- Calculus, Operational --- Holomorphic functions --- Approximation theory --- Numerical analysis --- Integral Transforms. --- Complex variables --- Elliptic functions --- Functions of real variables --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Functional analysis --- Operational calculus --- Differential equations --- Electric circuits