Big Varchenko-Gelfand rings and orbit harmonics
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918396463939584 |
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| author | Rhoades, Brendon |
| author_facet | Rhoades, Brendon |
| contents | Let $\mathscr{M}$ be a conditional oriented matroid. We define a graded algebra $\widehat{\mathscr{VG}}_\mathscr{M}$ with vector space dimension given by the number of covectors in $\mathscr{M}$ which admits a distinguished filtration indexed by the poset $\mathscr{L}(\mathscr{M})$ of flats of $\mathscr{M}$. The subquotients of this filtration are isomorphic to graded Varchenko-Gelfand rings of contractions of $\mathscr{M}$, so we call $\widehat{\mathscr{VG}}_\mathscr{M}$ the {\em graded big Varchenko-Gelfand ring of $\mathscr{M}$.} We describe a no broken circuit type basis of $\widehat{\mathscr{VG}}_\mathscr{M}$ and study its equivariant structure under the action of $\mathrm{Aut}(\mathscr{M})$. Our key technique is the orbit harmonics deformation which encodes $\widehat{\mathscr{VG}}_\mathscr{M}$ (as well as the classical Varchenko-Gelfand ring) in terms of a locus of points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Big Varchenko-Gelfand rings and orbit harmonics Rhoades, Brendon Combinatorics Let $\mathscr{M}$ be a conditional oriented matroid. We define a graded algebra $\widehat{\mathscr{VG}}_\mathscr{M}$ with vector space dimension given by the number of covectors in $\mathscr{M}$ which admits a distinguished filtration indexed by the poset $\mathscr{L}(\mathscr{M})$ of flats of $\mathscr{M}$. The subquotients of this filtration are isomorphic to graded Varchenko-Gelfand rings of contractions of $\mathscr{M}$, so we call $\widehat{\mathscr{VG}}_\mathscr{M}$ the {\em graded big Varchenko-Gelfand ring of $\mathscr{M}$.} We describe a no broken circuit type basis of $\widehat{\mathscr{VG}}_\mathscr{M}$ and study its equivariant structure under the action of $\mathrm{Aut}(\mathscr{M})$. Our key technique is the orbit harmonics deformation which encodes $\widehat{\mathscr{VG}}_\mathscr{M}$ (as well as the classical Varchenko-Gelfand ring) in terms of a locus of points. |
| title | Big Varchenko-Gelfand rings and orbit harmonics |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.18602 |