The van der Waerden Simplicial Complex and its Lefschetz Properties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917375375310848 |
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| author | Ragunathan, Naveena Van Tuyl, Adam |
| author_facet | Ragunathan, Naveena Van Tuyl, Adam |
| contents | The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A(n,k) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A(n,k)$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A(n,3)$ that have the Weak Lefschetz Property. We conjecture that $A(n,k)$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A(n,k)$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The van der Waerden Simplicial Complex and its Lefschetz Properties Ragunathan, Naveena Van Tuyl, Adam Commutative Algebra 13E10, 13F55 The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A(n,k) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A(n,k)$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A(n,3)$ that have the Weak Lefschetz Property. We conjecture that $A(n,k)$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A(n,k)$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair. |
| title | The van der Waerden Simplicial Complex and its Lefschetz Properties |
| topic | Commutative Algebra 13E10, 13F55 |
| url | https://arxiv.org/abs/2603.29978 |