The rank-nullity ring of a matroid
Tara Fife, Eline Mannino, Felipe Rincón
TL;DR
This work defines the rank-nullity ring $R^*(M)$ as a subring of the permutahedral Chow ring $A^*(U_{n,n})$ that naturally contains the tautological Chern classes of a matroid, enabling a combinatorial handle on matroidal tautological data. It provides explicit combinatorial formulas for the Chern classes $c_k(S_M)$ and $c_k(Q_M)$, and proves these lie in $R^*(M)$; the ring is generated by the $y_{i,j}$-type generators encoding rank and nullity information. In the uniform matroid case, the authors show $R^*(U_{r,n})=A^*(U_{n,n})^{S_n}$, give a concrete $\mathbb{Z}$-basis and Hilbert function $\mathrm{HF}(d)=\binom{n-1}{d}$, and supply a Gröbner basis for the relation ideal, illustrating the rich symmetry and invariant structure. The paper also connects tautological Chern data with Chern-Schwartz-MacPherson cycles via duality on the Bergman fan, and analyzes Hilbert function behavior across general matroids, including monotonicity results and open questions about unimodality and log-concavity. Overall, the results furnish explicit combinatorial tools to study matroid tautological data through invariant theory, Bergman geometry, and Gröbner-basis techniques, with clear structural insights for the uniform case and groundwork for broader classes of matroids.
Abstract
We introduce the rank-nullity ring of a matroid $M$, which is a subring of the Chow ring of the permutahedral toric variety. This subring contains the tautological Chern classes of $M$, a fact we deduce from a highly symmetric formula for these classes. When the matroid $M$ is a uniform matroid, the rank-nullity ring coincides with the subring of $S_n$-invariants of the Chow ring of the permutahedral toric variety. In this case, we compute its Hilbert function explicitly and provide a Gröbner basis for the ideal of relations among its generators.