论文标题
地图的存在$ det^{s^3} $
Existence of the Map $det^{S^3}$
论文作者
论文摘要
在本文中，我们显示了非平凡线性映射的存在$ det^{s^3}：v_d^{\ otimes \ binom {3d} {3d} {3}}} \ k $ to k $ to k $ to k $ to属于$ det^{s^3}的属性（\ otimes_ 3D}（v_ {i，j，k}））= 0 $，如果存在$ 1 \ leq x <y <z <z <t \ leq 3d $，这样$ v_ {x，y，y，z} = v_ {x，y，y，t} = v_ {x，y，t} = v_ {x，z，z，z，z，z，z，t} = v_ {y，z {y，z {y，z，t} $。这给出了[10]的猜想的部分答案。作为一个应用程序，我们使用地图$ det^{s^3} $来研究完整的hypergraph $ k^3_ {3d} $的d-partition，其betti编号为零。我们还讨论了映射的代数和组合属性，$ det^{s^r}：v_d^{\ otimes \ binom {rd} {rd}}} \ k $ to概括了确定性的映射，$ det^{s^2} $ from [9]，以及$ det^3} $。
In this paper we show the existence of a nontrivial linear map $det^{S^3}:V_d^{\otimes\binom{3d}{3}}\to k$ with the property that $det^{S^3}(\otimes_{1\leq i<j<k\leq 3d}(v_{i,j,k}))=0$ if there exists $1\leq x<y<z<t\leq 3d$ 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 $det^{S^3}$ to study those d-partitions of the complete hypergraph $K^3_{3d}$ that have zero Betti numbers. We also discuss algebraic and combinatorial properties of a map $det^{S^r}:V_d^{\otimes\binom{rd}{r}}\to k$ which generalizes the determinant map, the map $det^{S^2}$ from [9], and $det^{S^3}$.
