The mystery of plethysm coefficients
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908273733533696 |
|---|---|
| author | Colmenarejo, Laura Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike |
| author_facet | Colmenarejo, Laura Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike |
| contents | Composing two representations of the general linear groups gives rise to Littlewood's (outer) plethysm. On the level of characters, this poses the question of finding the Schur expansion of the plethysm of two Schur functions. A combinatorial interpretation for the Schur expansion coefficients of the plethysm of two Schur functions is, in general, still an open problem. We identify a proof technique of combinatorial representation theory, which we call the "$s$-perp trick", and point out several examples in the literature where this idea is used. We use the $s$-perp trick to give algorithms for computing monomial and Schur expansions of symmetric functions. In several special cases, these algorithms are more efficient than those currently implemented in {\sc SageMath}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07258 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The mystery of plethysm coefficients Colmenarejo, Laura Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike Combinatorics 05E05, 05E10 Composing two representations of the general linear groups gives rise to Littlewood's (outer) plethysm. On the level of characters, this poses the question of finding the Schur expansion of the plethysm of two Schur functions. A combinatorial interpretation for the Schur expansion coefficients of the plethysm of two Schur functions is, in general, still an open problem. We identify a proof technique of combinatorial representation theory, which we call the "$s$-perp trick", and point out several examples in the literature where this idea is used. We use the $s$-perp trick to give algorithms for computing monomial and Schur expansions of symmetric functions. In several special cases, these algorithms are more efficient than those currently implemented in {\sc SageMath}. |
| title | The mystery of plethysm coefficients |
| topic | Combinatorics 05E05, 05E10 |
| url | https://arxiv.org/abs/2208.07258 |