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arXiv · 2609.08793

The V\'amos Matroid Has No Second Symmetric Power

Yasuhito Nakajima
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Abstract

Symmetric powers of matroids were introduced by Lov\'asz and Mason. The subject has recently attracted renewed interest through its connection with the realizability of tropical linear spaces by tropical ideals. Anderson proved that the tropical linear space associated with a matroid $M$ is the variety of a tropical ideal if and only if $M$ has a $d$th symmetric power for every positive integer $d$. He also asked whether the V\'amos matroid $V_8$ has a second symmetric power. We answer this question in the negative by proving that $V_8$ has no second symmetric power. By Anderson's characterization, the associated tropical linear space cannot be realized as the variety of a tropical ideal.

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Yasuhito Nakajima. 2026-09-08. The V\'amos Matroid Has No Second Symmetric Power. https://arxiv.org/abs/2609.08793

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