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The Free m-Cone of a Matroid and Its \({\mathcal {G}}\)-Invariant

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Abstract

For a matroid, its configuration determines its \({\mathcal {G}}\)-invariant. Few examples are known of pairs of matroids with the same \({\mathcal {G}}\)-invariant but different configurations. In order to produce new examples, we introduce the free m-cone \(Q_m(M)\) of a loopless matroid M, where m is a positive integer. We show that the \({\mathcal {G}}\)-invariant of M determines the \({\mathcal {G}}\)-invariant of \(Q_m(M)\), and that the configuration of \(Q_m(M)\) determines M; so if M and N are nonisomorphic and have the same \({\mathcal {G}}\)-invariant, then \(Q_m(M)\) and \(Q_m(N)\) have the same \({\mathcal {G}}\)-invariant but different configurations. We prove analogous results for several variants of the free m-cone. We also define a new matroid invariant of M, and show that it determines the Tutte polynomial of \(Q_m(M)\).

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Acknowledgements

We thank Joseph Kung for valuable observations and comments on the topics of this paper. We thank both referees for their careful reading of the manuscript, their valuable comments, and for pointing out a gap in the original proof of Theorem 3.8.

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Authors and Affiliations

  1. Department of Mathematics, The George Washington University, Washington, DC, 20052, USA

    Joseph E. Bonin & Kevin Long

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  1. Joseph E. Bonin
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  2. Kevin Long
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Correspondence to Joseph E. Bonin.

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Communicated by Peter Nelson.

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Bonin, J.E., Long, K. The Free m-Cone of a Matroid and Its \({\mathcal {G}}\)-Invariant. Ann. Comb. 26, 1021–1039 (2022). https://doi.org/10.1007/s00026-022-00606-2

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