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Wedge-to-symmetric independence implication

Prove that if a graph G on n vertices has its edge set E(G) independent in the wedge power matroid W_n(r, 0), then E(G) is independent in the symmetric power matroid S_n(r−1, 0).

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Background

Using the dualities established in the paper, the authors reformulate a special case of a problem posed by Kalai and conjectured by Crespo and Ruiz-Santos into a concrete matroidal implication between independence in wedge and symmetric power matroids.

They report verification for small ranks (r ≤ 6) but leave the general result open, positioning it as a direct combinatorial-algebraic question within the framework introduced.

References

Using Theorem~\ref{thm:equivalence}, a special case of *{Problem 3} which is given as a conjecture in *{Conjecture 4.3} becomes the following. Let $G$ be a graph on $n$ vertices, and suppose that $E(G)$ is independent in $\mathrm{W}_n(r, 0)$. Then $E(G)$ is independent in $\mathrm{S}_n(r-1, 0)$. We checked this for $r \le 6$.

— Rigidity matroids and linear algebraic matroids with applications to matrix completion and tensor codes (2405.00778 - Brakensiek et al., 1 May 2024) in Example, Dualities subsection