Partitions of the complete hypergraph $K_6^3$ and a determinant-like function

Abstract

In this paper, we introduce a determinant-like map $\mathrm{det}^{S^3}$ and study some of its properties. For this, we define a graded vector space $\Lambda ^{S^3}_V$ that has similar properties with the exterior algebra $\Lambda _V$ and the exterior GSC-operad $\Lambda ^{S^2}_V$ from Staic. When $\mathrm{dim}(V_2)=2$, we show that $\mathrm{dim}_k(\Lambda ^{S^3}_{V_2}[6])=1$, which gives the existence and uniqueness of the map $\mathrm{det}^{S^3}$. We also give an explicit formula for $\mathrm{det}^{S^3}$ as a sum over certain 2-partitions of the complete hypergraph $K_6^3$.

Appendices

Appendix A: An explicit formula for $\mathrm{det}^{S^3}$

It follows from Lemma 4.5 that the map $\mathrm{det}^{S^3}$ is invariant under the action of $SL_2(k)$ on $V_2$. From the general results of invariant theory (see [9]), it follows that $\mathrm{det}^{S^3}(\otimes _{1\le i<j<k\le 6} (v_{i,j,k}))$ can be written as sum of product of determinants of two by two matrices with columns consisting of the vectors $v_{i,j,k}$. In this section, we give an explicit formula for $\mathrm{det}^{S^3}$. This is also an alternative proof for the existence of the map $\mathrm{det}^{S^3}$.

For two vectors $u=ae_1+be_2$ and $v=ce_1+de_2$, we denote by [u, v] the determinant of the matrix that has the first column equal to u and the second column equal to v, that is,

$$\begin{aligned}{}[u,v]=ad-bc. \end{aligned}$$

Proposition 5.1

Let $v_{i,j,k}\in V_2$ for all $1\le i<j<k\le 6$, then we have

Proof

We denote by $B(\otimes _{1\le i<j<k\le 6} (v_{i,j,k}))$ the right-hand side of the above equality. In order to prove $B=\mathrm{det}^{S^3}$, one can use the universality property of the $\mathrm{det}^{S^3}$ map. It is easy to check that $B(\omega _{{\mathcal {P}}^{(1)}})=1$.

Next, we want to show that if $\otimes _{1\le i<j<k\le 6}(v_{i,j,k})\in V_2^{\otimes 20}$ such that there exist $1\le x< y<z<t\le 6$ with the property that $v_{x,y,z}=v_{x,y,t}=v_{x,z,t}=v_{y,z,t}$, then $B(\otimes _{1\le i<j<k\le 6}(v_{i,j,k}))=0$.

First, notice that B is invariant under the action of the subgroup $G\subseteq S_6$ that is generated by (2, 3), (3, 4), (4, 5), and (5, 6). Because of this fact, it is enough to check the universality property for $(x,y,z,t)=(1,2,3,4)$, and for $(x,y,z,t)=(2,3,4,5)$.

Case I If $(x,y,z,t)=(1,2,3,4)$, then $v_{1,2,3}=v_{1,2,4}=v_{1,3,4}=v_{2,3,4}$. It is easy to see that all the terms in $B(\otimes _{1\le i<j<k\le 6}(v_{i,j,k}))$ are equal to 0; for example, the first term is equal to zero because $[v_{1,2,3},v_{2,3,4}]=0$.

Case II If $(x,y,z,t)=(2,3,4,5)$, then $v_{2,3,4}=v_{2,3,5}=v_{2,4,5}=v_{3,4,5}$. Let $v=v_{2,3,4}=v_{2,3,5}=v_{2,4,5}=v_{3,4,5}$. In this situation, all the terms will cancel in pairs. For example, if we look to the first term and thirty-seventh term, we have the following expressions, respectively:

$$\begin{aligned}&[v_{1,2,3},v] [v_{1,2,4},v] [v_{1,2,5},v][v_{1,2,6},v_{2,6,5}][v_{1,3,4},v_{3,4,6}][v_{1,3,5},v][v_{1,3,6},v_{3,6,2}]\\&\quad [v_{1,4,5},v_{4,5,6}][v_{1,4,6},v_{4,6,2}][v_{1,5,6},v_{5,6,3}], \end{aligned}$$

and

$$\begin{aligned}&-[v_{1,2,3},v] [v_{1,2,4},v] [v_{1,2,5},v][v_{1,2,6},v_{2,6,5}][v_{1,3,4},v_{3,4,6}][v_{1,3,5},v][v_{1,3,6},v_{3,6,2}]\\&\quad [v_{1,4,5},v_{4,5,6}][v_{1,4,6},v_{4,6,2}][v_{1,5,6},v_{5,6,3}]. \end{aligned}$$

So, we get these terms to sum up to zero. The rest of the matching pairs are given in the following table:

One can see that the pairs of terms have opposite signs. So, all the terms will cancel in pairs, which proves our statement. $\square$

Appendix B: Equivalence classes of 2-partitions under the $S_6\times S_2$ Action

As discussed earlier in the paper, there are 184,756 homogeneous 2-partitions of the hypergraph $K_6^3$. Using MATLAB, one can show that 13,644 of them are nontrivial (see Lemma 4.6). We also know that there is an action of the group $S_6\times S_2$ on ${\mathcal {P}}_2^{h,nt}(K_6^3)$. In this section, we present the 20 equivalence classes of ${\mathcal {P}}_2^{h,nt}(K_6^3)$ under the action of $S_6\times S_2$ and the value of $\varepsilon ^{S^3}_2$ on each element in ${\mathcal {P}}_2^{h,nt}(K_6^3)$. All results in this section were obtained using MATLAB.

A summary of this information is presented in the following table, where ${\mathcal {P}}^{(i)}$ for $1\le i\le 20$ are representatives of the 20 equivalence classes:

In Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 and 26, we present 20 partitions that are representatives for the equivalences classes under the action of $S_6\times S_2$. To understand these pictures better, we consider Fig. 7 and give an explicit description of the hyperedges in ${\mathcal {P}}^{(1)}$. Here, we have

$$\begin{aligned} E({\mathcal {H}}^{(1)}_1)= & {} \{\{1,2,3\}, \{1,2,4\},\{1,3,4\},\{1,2,5\}, \{3,4,5\},\{1,5,6\},\{2,4,6\},\\&\{2,5,6\},\{3,4,6\},\{4,5,6\}\} \end{aligned}$$

and

$$\begin{aligned} E({\mathcal {H}}^{(1)}_2)= & {} \{\{1,2,6\},\{1,3,5\}, \{1,3,6\},\{1,4,5\},\{1,4,6\}, \{2,3,4\},\{2,3,5\},\\&\{2,3,6\},\{2,4,5\},\{3,5,6\} \end{aligned}$$

given by which triangles in the picture are shaded and not shaded, respectively. The other representatives can be obtained from the corresponding pictures similarly.

Fig. 7

${\mathcal {P}}^{(1)}$ of orbit size 1440 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(1)})=1$

Fig. 8

${\mathcal {P}}^{(2)}$ of orbit size 240 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(2)})=-1$

Fig. 9

${\mathcal {P}}^{(3)}$ of orbit size 1440 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(3)})=-1$

Fig. 10

${\mathcal {P}}^{(4)}$ of orbit size 360 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(4)})=1$

Fig. 11

${\mathcal {P}}^{(5)}$ of orbit size 1440 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(5)})=-1$

Fig. 12

${\mathcal {P}}^{(6)}$ of orbit size 1440 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(6)})=1$

Fig. 13

${\mathcal {P}}^{(7)}$ of orbit size 720 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(7)})=-1$

Fig. 14

${\mathcal {P}}^{(8)}$ of orbit size 1440 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(8)})=-1$

Fig. 15

${\mathcal {P}}^{(9)}$ of orbit size 1440 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(9)})=1$

Fig. 16

${\mathcal {P}}^{(10)}$ of orbit size 360 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(10)})=1$

Fig. 17

${\mathcal {P}}^{(11)}$ of orbit size 720 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(11)})=-1$

Fig. 18

${\mathcal {P}}^{(12)}$ of orbit size 720 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(12)})=1$

Fig. 19

${\mathcal {P}}^{(13)}$ of orbit size 360 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(13)})=1$

Fig. 20

${\mathcal {P}}^{(14)}$ of orbit size 360 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(14)})=-1$

Fig. 21

${\mathcal {P}}^{(15)}$ of orbit size 72 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(15)})=-1$

Fig. 22

${\mathcal {P}}^{(16)}$ of orbit size 360 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(16)})=1$

Fig. 23

${\mathcal {P}}^{(17)}$ of orbit size 120 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(17 )})=1$

Fig. 24

${\mathcal {P}}^{(18)}$ of orbit size 360 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(18)})=-1$

Fig. 25

${\mathcal {P}}^{(19)}$ of orbit size 240 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(19)})=1$

Fig. 26

${\mathcal {P}}^{(20)}$ of orbit size 12 and sign $\varepsilon ^{S^3}_2({\mathcal {P}}^{(20)})=-4$