The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions
Author | : Thomas Lam |
Publisher | : American Mathematical Soc. |
Total Pages | : 113 |
Release | : 2013-04-22 |
ISBN-10 | : 9780821872949 |
ISBN-13 | : 082187294X |
Rating | : 4/5 (94X Downloads) |
Download or read book The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions written by Thomas Lam and published by American Mathematical Soc.. This book was released on 2013-04-22 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.