Tropical matroid Schubert varieties and the graded Möbius algebra

Abstract: We introduce tropical matroid Schubert varieties, a tropical analogue of arrangement Schubert varieties associated with realisable matroids. We prove that the tropical cohomology ring of the tropical matroid Schubert variety associated to any matroid $M$ is isomorphic to the graded Möbius algebra $\operatorname{B}<sup>\bullet(M)$. This yields a geometric model for $\operatorname{B}<sup>\bullet(M)$, extending the geometric setting of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones.

• The paper introduces a geometric model constructing tropical matroid Schubert varieties for any matroid, including non-realizable cases.
• It demonstrates that the tropical cohomology of Y_M concentrates on the diagonal via a degenerate spectral sequence, ensuring pure Hodge–Tate structures.
• A canonical isomorphism is established between the tropical cohomology ring and the graded Möbius algebra, linking combinatorial invariants with geometric structures.

Tropical Matroid Schubert Varieties and the Graded Möbius Algebra

Introduction

The paper "Tropical matroid Schubert varieties and the graded Möbius algebra" (2604.00750) addresses the longstanding problem of connecting key algebraic and combinatorial invariants of matroids with geometric models in tropical geometry. Specifically, the work introduces the construction of tropical matroid Schubert varieties $Y_M$ for arbitrary matroids $M$, defines their stratifications and tropical cohomology, and proves a canonical isomorphism between the tropical cohomology ring of $Y_M$ and the graded Möbius algebra $\mathrm{B}^\bullet(M)$. This identification generalizes previous results on arrangement Schubert varieties associated to realizable matroids and extends them to the non-realizable context where previously no geometric model was available.

Background and Motivation

Arrangement Schubert varieties $Y_{\mathcal{A}}$ for complex hyperplane arrangements $\mathcal{A}$ (realizable matroids) have played a central role in establishing deep links between algebraic topology, combinatorics, and algebraic geometry. Notably, the top-heavy conjecture, proved for realizable matroids by Huh–Wang, relates the cohomology ring of $Y_{\mathcal{A}}$ to the combinatorial invariants of the associated matroid [Huh–Wang, Acta Math. 2017]. The cellular structure and cohomology ring of $Y_{\mathcal{A}}$ provide a direct link to Whitney numbers of the matroid and their generating functions. For non-realizable matroids, Braden–Huh–Matherne–Proudfoot–Wang introduced, via algebraic and combinatorial methods, analogues of $H^\bullet(Y_{\mathcal{A}})$ (the graded Möbius algebra $\mathrm{B}^\bullet(M)$) and intersection cohomology analogues, but without a satisfactory geometric model.

The tropical viewpoint, specifically via Bergman and augmented Bergman fans, has furnished powerful tools for encoding matroidal structures in polyhedral and fan-theoretic settings, but the cohomological implications of such spaces—especially in the non-realizable case—remained incompletely understood.

Main Constructions and Results

Definition and Structure of $M$0

For any matroid $M$1 of rank $M$2 on ground set $M$3, the tropical matroid Schubert variety $M$4 is defined as the closure of the support of the augmented Bergman fan $M$5 (denoted $M$6) within the tropical toric variety $M$7. This space is shown to possess a compatible extended polyhedral structure, making it an extended polyhedral space in the sense of Mikhalkin, Itenberg–Mikhalkin–Zharkov, and others.

The stratification of $M$8 is thoroughly analyzed: $M$9 admits a stratification indexed by "admissible pairs" $Y_M$0 where $Y_M$1 is an independent set and $Y_M$2 is a flat with $Y_M$3. The closure of each stratum is modeled on the support of the augmented Bergman fan of the matroid minor $Y_M$4. This hierarchical structure generalizes the geometric content of arrangement Schubert varieties, naturally encoding all matroidal data, including failed realizability, loops, and parallel elements.

Tropical Cohomology and its Concentration

The tropical cohomology groups $Y_M$5, defined via the polyhedral structure, are proven to be concentrated on the diagonal, i.e., $Y_M$6 for $Y_M$7. This reflects the purity of the cohomology in the Hodge–Tate sense known from the theory of arrangement Schubert varieties and pure Hodge structures, and is established via a careful spectral sequence analysis. The paper introduces a filtration on $Y_M$8 by the "rank of the stratum," leading to a spectral sequence whose degeneration allows direct computation of the tropical cohomology.

Canonical Isomorphism with the Graded Möbius Algebra

A central result is the construction of an explicit isomorphism:

$Y_M$9

where the degree $\mathrm{B}^\bullet(M)$0 cohomology is mapped to the degree $\mathrm{B}^\bullet(M)$1 part of $\mathrm{B}^\bullet(M)$2. This isomorphism respects the algebra structures: the cup product in tropical cohomology is compatible with the product in the graded Möbius algebra. The proof leverages a comparison with the Chow ring of the canonical compactification of $\mathrm{B}^\bullet(M)$3, identifying tropical cohomology with the Chow ring and then matching it to the combinatorial presentation of $\mathrm{B}^\bullet(M)$4 as a subalgebra generated by pullbacks from the ambient toric structure.

Extension Beyond Realizable Matroids

Even in the realizable scenario, this construction is notable. However, the crucial advance is the geometric realization of $\mathrm{B}^\bullet(M)$5 for non-realizable matroids via tropical geometry. Here, $\mathrm{B}^\bullet(M)$6 serves as the first known geometric model extending the role previously played by $\mathrm{B}^\bullet(M)$7 in the realizable case. In the realizable setting, it is shown that $\mathrm{B}^\bullet(M)$8 coincides (up to extended tropicalization) with the Schubert variety $\mathrm{B}^\bullet(M)$9.

Other Structural and Technical Advances

• Stratification and Spectral Sequences: The explicit description of both the stratification and the spectral sequence degenerating on the diagonal is tied to combinatorial identities involving Whitney numbers and the $Y_{\mathcal{A}}$0-vector of the independence complex.
• Product and Functoriality Properties: The construction is shown to be compatible with direct sum decomposition of matroids, corresponding to product decompositions of $Y_{\mathcal{A}}$1 and the associated cohomology algebra.
• Examples and Explicit Descriptions: The paper provides detailed examples, showing how the construction detects matroidal phenomena including loops and parallelism which are lost in the lattice of flats.
• Implications for Tropical Comparison Theorems: The results suggest that tropical compactifications not regular at infinity (in the sense of Mikhalkin–Zharkov) can still encode deep cohomological data, indicating an avenue for further exploration of singular compactifications in tropical geometry.

Implications and Future Directions

The construction of $Y_{\mathcal{A}}$2 provides a uniform geometric setting for matroid invariants, independent of realizability, and establishes the graded Möbius algebra as the cohomological invariant of this space within tropical geometry. This realization enables new strategies to study combinatorial Hodge theory from a geometric perspective. The construction may also motivate development of further tropical analogues of perverse and intersection cohomology theories for matroids, with potential impact on the study of Kazhdan–Lusztig polynomials, positroid varieties, and representation-theoretic combinatorics related to matroids.

There is further potential for investigating the role of these tropical varieties in moduli space compactifications, matroid moduli, and relations to the geometry of non-realizable combinatorial geometries, especially in light of the functorial and stratified properties established.

Conclusion

This paper delivers a comprehensive geometric model for the graded Möbius algebra of any matroid using the tropical matroid Schubert variety $Y_{\mathcal{A}}$3. Through careful construction and analysis, it generalizes and extends geometric–algebraic connections known for complex hyperplane arrangements to the entire class of matroids, offering new tools for the interplay between tropical geometry, matroid theory, and algebraic combinatorics. The algebraic, combinatorial, and geometric machinery developed here sets the stage for future explorations in tropical and non-Archimedean geometry, combinatorial Hodge theory, and the study of intersection cohomology for combinatorial geometries.