Title K-schur functions and Affine Schubert calculus / Thomas Lam [and 5 others]

Published New York : Springer Verlag, [2014] ©2014

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Description 1 online resource Series Fields institute monographs ; volume 33 Fields Institute monographs ; v. 33. Contents 880-01 1. Introduction -- 2. Primer on k-Schur Functions -- 3. Stanley symmetric functions and Peterson algebras -- 4. Affine Schubert calculus 880-01/(S Machine generated contents note: 1. Introduction -- Appendix: Sage -- 2. Primer on A:-Schur Functions -- 1. Background and Notation -- 1.1. Partitions and Cores -- 1.2. Bounded Partitions, Cores, and Affine Grassmannian Elements -- 1.3. Weak Order and Horizontal Chains -- 1.4. Cores and the Strong Order of the Affine Symmetric Group -- 1.5. Symmetric Functions -- 1.6. Schur Functions -- 1.7. Hall-Littlewood Symmetric Functions -- 1.8. Macdonald Symmetric Functions -- 1.9. Empirical Approach to A:-Schur Functions -- 1.10. Notes on References -- 2. From Pieri Rules to k-Schur Functions at T = 1 -- 2.1. Semi-standard Tableaux and a Monomial Expansion of Schur Functions -- 2.2. Weak Tableaux and a Monomial Expansion of Dual k-Schur Functions -- 2.3. Other Realizations -- 2.4. Strong Marked Tableaux and a Monomial Expansion of k-Schur Functions -- 2.5. k-Littlewood-Richardson Coefficients -- 2.6. Notes on References -- 3. Definitions of k-Schur Functions -- 3.1. Atoms as Tableaux -- 3.2. Symmetric Function Operator Definition -- 3.3. Weak Tableaux II -- 3.4. Strong Tableaux II -- 3.5. Notes on References -- 4. Properties of A:-Schur Functions and Their Duals -- 4.1. k-Schur Functions Are Schur Functions When K> 4.2. k-Schur Function Is Schur Positive -- 4.3. At T = 1, the k-Schur Functions Satisfy the k-Pieri Rule -- 4.4. k-Conjugation -- 4.5. k-Schur Functions Form a Basis for Λt(k) -- 4.6. k-Rectangle Property -- 4.7. When T = 1, the Product of k-Schur Functions Is k-Schur Positive -- 4.8. Positively Closed Under Coproduct -- 4.9. Product of a k-Schur and l-Schur Function Is (k + l)-Schur Positive -- 4.10. Branching Property from K to K + 1 -- 4.11. k-Schur Positivity of Macdonald Symmetric Functions -- 5. Directions of Research and Open Problems -- 5.1. k-Murnaghan-Nakayama Rule -- 5.2. Rectangle Generalization at T a Root of Unity -- 5.3. Dual-Basis to s(k)λ[X; t] -- 5.4. Product on At(k) -- 5.5. Representation Theoretic Model of k-Schur Functions -- 5.6. From Pieri to K-Theoretic k-Schur Functions -- 6. Duality Between the Weak and Strong Orders -- 6.1. k-Analogue of the Cauchy Identity -- 6.2. Brief Introduction to Fomin's Growth Diagrams -- 6.3. Affine Insertion -- 6.4. k-Compatible Affine Insertion Algorithm -- 7. k-Shape Poset and a Branching Rule for Expressing k-Schur in (k + 1)-Schur Functions -- 3. Stanley Symmetric Functions and Peterson Algebras -- 1. Stanley Symmetric Functions and Reduced Words -- 1.1. Young Tableaux and Schur Functions -- 1.2. Permutations and Reduced Words -- 1.3. Reduced Words for the Longest Permutation -- 1.4. Stanley Symmetric Function -- 1.5. Code of a Permutation -- 1.6. Fundamental Quasi-symmetric Functions -- 1.7. Exercises -- 2. Edelman-Greene Insertion -- 2.1. Insertion for Reduced Words -- 2.2. Coxeter-Knuth Relations -- 2.3. Exercises and Problems -- 3. Affine Stanley Symmetric Functions -- 3.1. Affine Symmetric Group -- 3.2. Definition -- 3.3. Codes -- 3.4. A(n)and Λ(n) -- 3.5. Affine Schur Functions -- 3.6. Example: The Case of S3 -- 3.7. Exercises and Problems -- 4. Root Systems and Weyl Groups -- 4.1. Notation for Root Systems and Weyl Groups -- 4.2. Affine Weyl Group and Translations -- 5. NilCoxeter Algebra and Fomin-Stanley Construction -- 5.1. NilCoxeter Algebra -- 5.2. Fomin and Stanley's Construction -- 5.3. Conjecture -- 5.4. Exercises and Problems -- 6. Affine NilHecke Ring -- 6.1. Definition of Affine NilHecke Ring -- 6.2. Coproduct -- 6.3. Exercises and Problems -- 7. Peterson's Centralizer Algebras -- 7.1. Peterson Algebra and j-Basis -- 7.2. Sketch Proof of Theorem 7.3 -- 7.3. Exercises and Problems -- 8. (Affine) Fomin-Stanley Algebras -- 8.1. Commutation Definition of Affine Fomin-Stanley Algebra -- 8.2. Noncommutative k-Schur Functions -- 8.3. Cyclically Decreasing Elements -- 8.4. Coproduct -- 8.5. Exercises and Problems -- 9. Finite Fomin-Stanley Subalgebra -- 9.1. Problems -- 10. Geometric Interpretations -- 4. Affine Schubert Calculus -- 1. Introduction -- 2. Root Data -- 2.1. Cartan Data and the Weyl Group -- 2.2. Root Data -- 2.3. Affine Root Data -- 3. NilHecke Ring and Schubert Calculus -- 3.1. NilHecke Ring -- 3.2. Coproduct on A -- 3.3. Duality and the GKM Ring -- 3.4. Multiplication in Λ and Coproduct in A -- 3.5. Forgetting Equivariance -- 3.6. Parabolic Case -- 3.7. Geometric Interpretations -- 4. Affine Grassmannian -- 4.1. Affine Grassmannian as Partial Affine Flags -- 4.2. Small Torus Version of Λaf -- 4.3. Homology of the Affine Grassmannian -- 4.4. Small Torus Affine NilHecke Ring and Peterson Subalgebra -- 4.5. j-Basis -- 4.6. Homology Structure Constants -- 4.7. Peterson's "Quantum Equals Affine" Theorems -- A Appendix Proof of Coalgebra Properties -- B Appendix Small Torus GKM Proofs -- B.1. Small Torus GKM Condition for S12 -- C Appendix Homology of Gr Summary 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 Analysis wiskunde mathematics combinatoriek combinatorics algebraic geometry algebra Mathematics (General) Wiskunde (algemeen) Bibliography Includes bibliographical references Notes Print version record Subject Schur functions. Geometry, Algebraic. Mathematics. Mathematics MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. Mathematics Geometry, Algebraic Schur functions Form Electronic book Author Lam, Thomas, 1980- author. ISBN 9781493906826 1493906828

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