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The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions

Thomas Lam University of Michigan, Ann Arbor, MI
Luc Lapointe Universidad de Talca, Talca, Chile
Jennifer Morse Drexel University, Philadelphia, PA
Mark Shimozono Virginia Polytechnic Institute and State University, Blacksburg, VA
Available Formats:
Electronic ISBN: 978-0-8218-9874-1
Product Code: MEMO/223/1050.E
List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $43.20 Click above image for expanded view The Poset of$k$-Shapes and Branching Rules for$k$-Schur Functions Thomas Lam University of Michigan, Ann Arbor, MI Luc Lapointe Universidad de Talca, Talca, Chile Jennifer Morse Drexel University, Philadelphia, PA Mark Shimozono Virginia Polytechnic Institute and State University, Blacksburg, VA Available Formats:  Electronic ISBN: 978-0-8218-9874-1 Product Code: MEMO/223/1050.E  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$43.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 2232013; 101 pp
MSC: Primary 05; Secondary 14;

The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. The authors obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, the authors give a formula for the Schur expansion of an ungraded $k$-Schur function.

• Chapters
• 1. Introduction
• 2. The poset of $k$-shapes
• 3. Equivalence of paths in the poset of $k$-shapes
• 4. Strips and tableaux for $k$-shapes
• 5. Pushout of strips and row moves
• 6. Pushout of strips and column moves
• 7. Pushout sequences
• 8. Pushouts of equivalent paths are equivalent
• 9. Pullbacks
• A. Tables of branching polynomials
• Requests

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Volume: 2232013; 101 pp
MSC: Primary 05; Secondary 14;

The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. The authors obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, the authors give a formula for the Schur expansion of an ungraded $k$-Schur function.

• Chapters
• 1. Introduction
• 2. The poset of $k$-shapes
• 3. Equivalence of paths in the poset of $k$-shapes
• 4. Strips and tableaux for $k$-shapes
• 5. Pushout of strips and row moves
• 6. Pushout of strips and column moves
• 7. Pushout sequences
• 8. Pushouts of equivalent paths are equivalent
• 9. Pullbacks
• A. Tables of branching polynomials
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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