Valuative invariants for polymatroids

By Harm Derksen, Alex Fink. 2009

87 citations recorded by the source corpus.

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal{G}$ introduced by the first author, are valuative. In this paper we construct the $\mathbb{Z}$-modules of all $\mathbb{Z}$-valued valuative functions for labelled matroids and polymatroids on a fixed ground set, and their unlabelled counterparts, the $\mathbb{Z}$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal{G}$ is universal for valuative invariants.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/a502c110993f71f4eeb1b0bbd7ae78853a3b457e

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Citations in the corpus

K-classes for matroids and equivariant localization (2010, background): “5a of [6] asserts that the group of valuative matroid invariants is free of rank ( n d ) .”
Hopf Monoids and Generalized Permutahedra (2017, methodology): “[24, 33, 63, 70] For a polytope p in RI, the following conditions are equivalent.”
Stellahedral geometry of matroids (2022, background): “By Remark 7.6, we see that such matroid union are exactly the Schubert matroids of rank Combining Theorem 1.5 with Theorem 7.7 recovers the following result of Derksen and Fink [DF10, Theorem 5.4].”
Algebraic and Geometric Methods in Enumerative Combinatorics (2014, background): “k invariant of Mis G(M) = X S2Cn U (r[S 0;S 1];r[S 1;S 2];:::;r[S n 1;Sn]) in the vector space U spanned by the n r formal symbols U w 1 wn where w 1;:::;w n 2f0;1gand w 1 + + w n= r. Theorem 7.10.5. [DF10] The Derksen{Fink invariant G: Matroids n;r!U is a universal valuative matroid invariant; that”
Matroid Theory for Algebraic Geometers (2014, methodology): “This is the approach taken by Fink and Derksen [26, 27].”