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Affine Insertion and Pieri Rules for the Affine Grassmannian.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2010Copyright date: ©2010Edition: 1st edDescription: 1 online resource (103 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470405915
Subject(s): Genre/Form: Additional physical formats: Print version:: Affine Insertion and Pieri Rules for the Affine GrassmannianDDC classification:
  • 516/.4
LOC classification:
  • QA3 -- .A345 2010eb
Online resources:
Contents:
Intro -- Contents -- Introduction -- Chapter 1. Schubert Bases of Gr and Symmetric Functions -- 1.1. Symmetric functions -- 1.2. Schubert bases of Gr -- 1.3. Schubert basis of the affine flag variety -- Chapter 2. Strong Tableaux -- 2.1. n as a Coxeter group -- 2.2. Fixing a maximal parabolic subgroup -- 2.3. Strong order and strong tableaux -- 2.4. Strong Schur functions -- Chapter 3. Weak Tableaux -- 3.1. Cyclically decreasing permutations and weak tableaux -- 3.2. Weak Schur functions -- 3.3. Properties of weak strips -- 3.4. Commutation of weak strips and strong covers -- Chapter 4. Affine Insertion and Affine Pieri -- 4.1. The local rule u,v -- 4.2. The affine insertion bijection u,v -- 4.3. Pieri rules for the affine Grassmannian -- 4.4. Conjectured Pieri rule for the affine flag variety -- 4.5. Geometric interpretation of strong Schur functions -- Chapter 5. The Local Rule u,v -- 5.1. Internal insertion at a marked strong cover -- 5.2. Definition of u,v -- 5.3. Proofs for the local rule -- Chapter 6. Reverse Local Rule -- 6.1. Reverse insertion at a cover -- 6.2. The reverse local rule -- 6.3. Proofs for the reverse insertion -- Chapter 7. Bijectivity -- 7.1. External insertion -- 7.2. Case A (commuting case) -- 7.3. Case B (bumping case) -- 7.4. Case C (replacement bump) -- Chapter 8. Grassmannian Elements, Cores, and Bounded Partitions -- 8.1. Translation elements -- 8.2. The action of n on partitions -- 8.3. Cores and the coroot lattice -- 8.4. Grassmannian elements and the coroot lattice -- 8.5. Bijection from cores to bounded partitions -- 8.6. k-conjugate -- 8.7. From Grassmannian elements to bounded partitions -- Chapter 9. Strong and Weak Tableaux Using Cores -- 9.1. Weak tableaux on cores are k-tableaux -- 9.2. Strong tableaux on cores -- 9.3. Monomial expansion of t-dependent k-Schur functions.
9.4. Enumeration of standard strong and weak tableaux -- Chapter 10. Affine Insertion in Terms of Cores -- 10.1. Internal insertion for cores -- 10.2. External insertion for cores (Case X) -- 10.3. An example -- 10.4. Standard case -- 10.5. Coincidence with RSK as n -- 10.6. The bijection for n = 3 and m = 4 -- Bibliography.
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Intro -- Contents -- Introduction -- Chapter 1. Schubert Bases of Gr and Symmetric Functions -- 1.1. Symmetric functions -- 1.2. Schubert bases of Gr -- 1.3. Schubert basis of the affine flag variety -- Chapter 2. Strong Tableaux -- 2.1. n as a Coxeter group -- 2.2. Fixing a maximal parabolic subgroup -- 2.3. Strong order and strong tableaux -- 2.4. Strong Schur functions -- Chapter 3. Weak Tableaux -- 3.1. Cyclically decreasing permutations and weak tableaux -- 3.2. Weak Schur functions -- 3.3. Properties of weak strips -- 3.4. Commutation of weak strips and strong covers -- Chapter 4. Affine Insertion and Affine Pieri -- 4.1. The local rule u,v -- 4.2. The affine insertion bijection u,v -- 4.3. Pieri rules for the affine Grassmannian -- 4.4. Conjectured Pieri rule for the affine flag variety -- 4.5. Geometric interpretation of strong Schur functions -- Chapter 5. The Local Rule u,v -- 5.1. Internal insertion at a marked strong cover -- 5.2. Definition of u,v -- 5.3. Proofs for the local rule -- Chapter 6. Reverse Local Rule -- 6.1. Reverse insertion at a cover -- 6.2. The reverse local rule -- 6.3. Proofs for the reverse insertion -- Chapter 7. Bijectivity -- 7.1. External insertion -- 7.2. Case A (commuting case) -- 7.3. Case B (bumping case) -- 7.4. Case C (replacement bump) -- Chapter 8. Grassmannian Elements, Cores, and Bounded Partitions -- 8.1. Translation elements -- 8.2. The action of n on partitions -- 8.3. Cores and the coroot lattice -- 8.4. Grassmannian elements and the coroot lattice -- 8.5. Bijection from cores to bounded partitions -- 8.6. k-conjugate -- 8.7. From Grassmannian elements to bounded partitions -- Chapter 9. Strong and Weak Tableaux Using Cores -- 9.1. Weak tableaux on cores are k-tableaux -- 9.2. Strong tableaux on cores -- 9.3. Monomial expansion of t-dependent k-Schur functions.

9.4. Enumeration of standard strong and weak tableaux -- Chapter 10. Affine Insertion in Terms of Cores -- 10.1. Internal insertion for cores -- 10.2. External insertion for cores (Case X) -- 10.3. An example -- 10.4. Standard case -- 10.5. Coincidence with RSK as n -- 10.6. The bijection for n = 3 and m = 4 -- Bibliography.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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