This question is related to my last question and is originally motivated by recent advances in quantum physics.

I am looking for combinatorial characterizations of some algebraically defined families of sets, like the well-known Taylor and Zwicker theorem (see, for instance, here) stating that a voting system is weighted iff it is trade robust.

Thus, we are given an even number $n$, a set $V$ of vertices of a complete graph $K_n$, and a weight function $w$ which assigns a complex number to any edge of $K_n$. Then we can extend the function $w$ to any nonempty even-sized subset $U$ of $V$ putting $w(U)=\sum w(e_1)...w(e_k)$, where the sum is taken over all perfect matchings $\{e_1,\dots, e_k\}$ of the subgraph of $K_n$ induced by $U$. It easily implies the sum decomposition $w(U)=\sum_{u\in U\setminus\{v\}} w(\{v,u\})w(U\setminus\{v,u\})$ for any vertex $v\in U$. It is also additionally required that $w(V)\ne 0$. Let $\mathcal S=\{U\subset V: w(U)\ne 0\}$ and $\mathcal K=\{U\subset V: w(U)=0\}$ be the support and the kernel of the weight $w$, respectively.

Question. What are combinatorial characterizations of families $\mathcal S$ of nonempty even-sized subsets of $V$ which are weight supports?

As a partial answer, I found the following necessary combinatorial conditions for a weight support $\mathcal S$.

The first two of them directly follow from the sum decomposition.

Condition 1. For any set $U\in\mathcal S$ with $|U|\ge 4$ and any element $v\in U$ there exists an element $u\in U$ such that both sets $\{v,u\}$ and $U\setminus\{v,u\}$ belong to $\mathcal S$. $\square$

Condition 2. An even-size subset $U$ of $V$ belongs to $\mathcal S$ provided for some element $v\in U$ there exists a unique element $u\in U$ such that both sets $\{v,u\}$ and $U\setminus\{v,u\}$ belong to $\mathcal S$. $\square$

To formulate a more refined condition, for any vertex $v\in V$ put $V(v)=\{u\in V: vu\in \mathcal S\}$. For any distinct vertices $v,v'\in V$ we define an auxiliary graph $H=H(vv')$ with the vertex set $V(v)\cap V(v’)$ such that two vertices $u$ and $u'$ of $H$ are adjacent in $H$, if $\{v,v',u,u'\}\in\mathcal K$.

Condition 3. Each odd cycle in $H$ contains an edge $e\in\mathcal S$. Moreover, if $vv'\in\mathcal K$ then the graph $H$ is bipartite.

Proof. For each vertex $u$ of $H$ put $w'(u)=w(vu)/w(v'u)$. Let $u_1\dots u_k$ be any odd cycle in $H$. Put $u_{k+1}=u_1$. Since for any positive integer $i$ with $1\le i\le k$, the vertices $u_i$ and $u_{i+1}$ are adjacent in $H$, we have $\{v,v',u_{i},u_{i+1}\}\in\mathcal K$, that is $w(vv'u_{i}u_{i+1})=0$. But $$w(vv'u_{i}u_{i+1})=w(vu_{i})w(v'u_{i+1})+w(vu_{i+1})w(v'u_{i})+w(vv')w(u_{i}u_{i+1}).$$

If either $vv'\in\mathcal K$ or $u_{i}u_{i+1}\in\mathcal K$ then the latter summand is zero. It follows $w'(u_{i+1})=-w'(u_i)$. Since $k$ is odd, we obtain a contradiction. $\square$

My last condition involves linear algebra.

For each subset $U$ of $V$ let $\overline{U}$ be its characteristic function, that an element of the linear space $\mathbb R^V$, such that for any $v\in V$, the $v$-th coordinate of $\overline{U}$ equals $1$, if $v\in U$, and $0$, otherwise.

Condition 4. Let $v$ be any vertex of $V$, $u_1,u_2,u_3$ be any distinct vertices of $V(v)$, $U=\{v,u_1,u_2,u_3\}$, $U_v$ be the elment of $\mathbb R^V$ which is equal to $\overline{u_1u_2}+\overline{u_2u_3}+\overline{u_1u_3}$. Then $U\in\mathcal K$ provided there exist a natural number $m$, three-element subsets $U_1,\dots, U_m$ of $V(v)$ such that $\{v\}\cup U_i\in\mathcal K$ for each positive integer $i\le m$, and real numbers $\lambda_1,\dots,\lambda_m$ such that $U_v=\sum_{i=1}^m \lambda_i\overline{U_i}$.

Proof. For each positive integer $i\le m$ let $U_i=\{u_{1i}, u_{2i}, u_{3i}\}$. Then $$0=w(\{v\}\cup U_i)=w(vu_{1i})w(u_{2i} u_{3i})+ w(vu_{2i})w(u_{1i} u_{3i})+ w(vu_{3i})w(u_{1i} u_{3i}).$$ Also $$w(U)=w(vu_1)w(u_2u_3)+ w(vu_2)w(u_1u_3)+w(vu_3)w(u_1u_2).$$ The equality $U_v=\sum_{i=1}^m \lambda_i\overline{U_i}$ implies that $$0=\sum_{i=1}^m \lambda_i\frac{w(\{v\}\cup U_i)}{w(vu_{1i})w(vu_{2i})w(vu_{3i})}=$$ $$\sum_{i=1}^m \lambda_i \left(\frac{w(u_{2i} u_{3i})}{w(vu_{2i})w(vu_{3i})}+\frac{w(u_{1i} u_{3i})}{w(vu_{1i})w(vu_{3i})}+\frac{w(u_{1i} u_{2i})}{w(vu_{1i})w(vu_{2i})}\right)=$$ $$\frac{w(u_{2} u_{3})}{w(vu_{2})w(vu_{3})}+\frac{w(u_{1} u_{3})}{w(vu_{1})w(vu_{3})}+\frac{w(u_{1}u_{2})}{w(vu_{1})w(vu_{2})}=$$ $$\frac{w(U)}{w(vu_{1})w(vu_{2})w(vu_{3})}. \square$$

Thanks.