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The Vámos Matroid Has No Second Symmetric Power
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 dth symmetric power for every positive integer d. He also asked whether the Vámos matroid V8 has a second symmetric power. We answer this question in the negative by proving that V8 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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BibTeX
@misc{nakajima2026vámos,
title = {{The Vámos Matroid Has No Second Symmetric Power}},
author = {Yasuhito Nakajima},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08793},
eprint = {2609.08793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08793},
}
RIS
TY - UNPB
TI - The Vámos Matroid Has No Second Symmetric Power
AU - Yasuhito Nakajima
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08793
UR - https://arxiv.org/abs/2609.08793
AB - 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 dth 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.
ER -