Tableaux and orbit harmonics quotients for finite transformation monoids
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915696445751296 |
|---|---|
| author | Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| author_facet | Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| contents | We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$ of the partial transformation monoid on an $n$-element set that contains the symmetric group. To achieve this, we introduce and study a functor from the category of rational representations of the monoid of $n \times n$ matrices to the category of finite dimensional representations of $\mathcal{M}(n)$. We establish two branching rules. Our main results describe graded module structures of orbit harmonics quotients for the rook, partial transformation, and full transformation monoids. This yields analogs of the Cauchy decomposition for polynomial rings in $n\times n$ variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tableaux and orbit harmonics quotients for finite transformation monoids Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia Representation Theory Combinatorics 05E10, 20M30 We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$ of the partial transformation monoid on an $n$-element set that contains the symmetric group. To achieve this, we introduce and study a functor from the category of rational representations of the monoid of $n \times n$ matrices to the category of finite dimensional representations of $\mathcal{M}(n)$. We establish two branching rules. Our main results describe graded module structures of orbit harmonics quotients for the rook, partial transformation, and full transformation monoids. This yields analogs of the Cauchy decomposition for polynomial rings in $n\times n$ variables. |
| title | Tableaux and orbit harmonics quotients for finite transformation monoids |
| topic | Representation Theory Combinatorics 05E10, 20M30 |
| url | https://arxiv.org/abs/2512.22353 |