Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917496148197376 |
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| author | Gui, Tao |
| author_facet | Gui, Tao |
| contents | We derive a graded character formula for the action of any finite group on the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan, which is given by an equivariant version of the classical h-polynomial. This gives the graded character formula for the representation of the group on the cohomology of the associated toric variety when the fan is rational. As an application, we use a navel tool, which we called hybrid fan, to compute the Poincaré polynomial of the invariants of the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan under a finite reflection group action. This implies that the Poincaré polynomial of the quotient of a compact toric orbifold by any finite reflection group is equal to the Poincaré polynomial of the compact toric orbifold associated to the hybrid fan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15061 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials Gui, Tao Combinatorics Algebraic Geometry Representation Theory Primary 13A50, Secondary 14M25, 20F55, 52B05, 52B15 We derive a graded character formula for the action of any finite group on the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan, which is given by an equivariant version of the classical h-polynomial. This gives the graded character formula for the representation of the group on the cohomology of the associated toric variety when the fan is rational. As an application, we use a navel tool, which we called hybrid fan, to compute the Poincaré polynomial of the invariants of the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan under a finite reflection group action. This implies that the Poincaré polynomial of the quotient of a compact toric orbifold by any finite reflection group is equal to the Poincaré polynomial of the compact toric orbifold associated to the hybrid fan. |
| title | Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials |
| topic | Combinatorics Algebraic Geometry Representation Theory Primary 13A50, Secondary 14M25, 20F55, 52B05, 52B15 |
| url | https://arxiv.org/abs/2605.15061 |