Electronic ISBN:  9780821898741 
Product Code:  MEMO/223/1050.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $43.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 223; 2013; 101 ppMSC: 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.

Table of Contents

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


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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