Existence of the Map $det^{S^3}$

In this paper we show the existence of a nontrivial linear map $d e t^{S^{3}} : V_{d}^{\otimes (\binom{3 d}{3})} \to k$ with the property that $d e t^{S^{3}} (\otimes_{1 \leq i < j < k \leq 3 d} (v_{i, j, k})) = 0$ if there exists $1 \leq x < y < z < t \leq 3 d$ such that $v_{x, y, z} = v_{x, y, t} = v_{x, z, t} = v_{y, z, t}$ . This gives a partial answer to a conjecture from [10]. As an application, we use the map $d e t^{S^{3}}$ to study those d-partitions of the complete hypergraph $K_{3 d}^{3}$ that have zero Betti numbers. We also discuss algebraic and combinatorial properties of a map $d e t^{S^{r}} : V_{d}^{\otimes (\binom{r d}{r})} \to k$ which generalizes the determinant map, the map $d e t^{S^{2}}$ from [9], and $d e t^{S^{3}}$ .

Existence of the Map $d e t^{S^{3}}$

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Existence of the Map $d e t^{S^{3}}$

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Existence of the Map detS3detS3

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Existence of the Map detS3detS3

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Existence of the Map $d e t^{S^{3}}$

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Existence of the Map $d e t^{S^{3}}$

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