Equivariant log concavity and the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d))$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917293839089664 |
|---|---|
| author | Homan, Benjamin |
| author_facet | Homan, Benjamin |
| contents | Previous work has conjectured that the graded $\mathfrak{S}_n$-representations $H^\bullet(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ are strongly equivariantly log concave, and has proven this conjecture in low degrees. By leveraging the theory of representation stability, we are able instead prove a stronger statement about the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ which implies the original conjecture up to degree 19. We conjecture that this equivariant log concavity-like property holds in all degrees for the $\operatorname{FI^\sharp}$-modules $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivariant log concavity and the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d))$ Homan, Benjamin Combinatorics Representation Theory 05E10 (Primary) 05B35, 55R80 (Secondary) Previous work has conjectured that the graded $\mathfrak{S}_n$-representations $H^\bullet(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ are strongly equivariantly log concave, and has proven this conjecture in low degrees. By leveraging the theory of representation stability, we are able instead prove a stronger statement about the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ which implies the original conjecture up to degree 19. We conjecture that this equivariant log concavity-like property holds in all degrees for the $\operatorname{FI^\sharp}$-modules $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$. |
| title | Equivariant log concavity and the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d))$ |
| topic | Combinatorics Representation Theory 05E10 (Primary) 05B35, 55R80 (Secondary) |
| url | https://arxiv.org/abs/2602.21578 |