Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910937933414400 |
|---|---|
| author | Zemel, Shaul |
| author_facet | Zemel, Shaul |
| contents | We consider actions, similar to those of Haglund, Rhoades, and Shimozono on ordered partitions, and their basis in terms of the higher Specht polynomials of Ariki, Terasoma, and Yamada, as carried out by Gillespie and Rhoades. By allowing empty sets and working with multi-sets and weak partitions as indices, we obtain a decomposition of the action of $S_{n}$ on homogeneous polynomials of degree $d$ into irreducible representations, in a way that lifts a formula of Stanley. By considering generalized higher Specht polynomials, we obtain yet another such decomposition, lifting another formula involving Kostka numbers. We also investigate several operations on both types of representations, which are based on normalizations of the generalized higher Specht polynomials that allow for defining their stable versions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups Zemel, Shaul Combinatorics Representation Theory 20C30, 05E05, 05E10 We consider actions, similar to those of Haglund, Rhoades, and Shimozono on ordered partitions, and their basis in terms of the higher Specht polynomials of Ariki, Terasoma, and Yamada, as carried out by Gillespie and Rhoades. By allowing empty sets and working with multi-sets and weak partitions as indices, we obtain a decomposition of the action of $S_{n}$ on homogeneous polynomials of degree $d$ into irreducible representations, in a way that lifts a formula of Stanley. By considering generalized higher Specht polynomials, we obtain yet another such decomposition, lifting another formula involving Kostka numbers. We also investigate several operations on both types of representations, which are based on normalizations of the generalized higher Specht polynomials that allow for defining their stable versions. |
| title | Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups |
| topic | Combinatorics Representation Theory 20C30, 05E05, 05E10 |
| url | https://arxiv.org/abs/2505.07098 |