The Vámos Matroid Has No Second Symmetric Power

Source abstract

Symmetric powers of matroids were introduced by Lovász 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 Vámos 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.