Categorifying Valuative Invariants of Polyhedra and Matroids via Categorical Convolution

Conceitos Básicos

This paper introduces the concept of "categorical valuative invariants" for polyhedra and matroids, which elevates traditional numerical invariants to exact sequences in additive categories, offering a deeper understanding of valuativity and enabling computations that respect matroid symmetries.

Resumo

Elias, B., Miyata, D., Proudfoot, N., & Vecchi, L. (2024). Categorical valuative invariants of polyhedra and matroids. arXiv preprint arXiv:2401.06869.

This paper aims to "categorify" various valuative invariants of matroids, such as the Poincaré polynomial, Chow polynomial, and Kazhdan–Lusztig polynomial, by lifting them from numerical values to richer structures within additive categories. This approach seeks to provide a more insightful explanation for the valuative relations observed among these invariants and to incorporate matroid symmetries into the analysis.

Principais Insights Extraídos De

Categorical valuative invariants of polyhedra and matroids

by Ben Elias, D... às arxiv.org 10-22-2024

https://arxiv.org/pdf/2401.06869.pdf

Perguntas Mais Profundas

How might this categorical framework be extended to other combinatorial objects beyond polyhedra and matroids?

This categorical framework, centered around valuative functors and categorical valuative invariants, holds promising potential for extension to other combinatorial structures beyond polyhedra and matroids. Here are some avenues for exploration:

Generalized permutohedra: Polyhedra and matroids are closely linked to generalized permutohedra. Extending the framework to this broader class could leverage existing results and provide a unified perspective.
Posets and Lattices: The notion of weak maps in matroids has analogues in poset and lattice theory. One could investigate valuative invariants for these structures, potentially focusing on functors to categories of representations of their automorphism groups.
Simplicial Complexes: Simplicial complexes are fundamental objects in combinatorial topology. Defining suitable notions of "valuative equivalence" and "weak maps" for simplicial complexes could pave the way for studying valuative functors in this context.
Graphs and Hypergraphs: Graphs and hypergraphs admit various geometric realizations, some of which might be amenable to the framework of valuative functors. Exploring connections with graph polynomials and invariants like the chromatic polynomial could be fruitful.
Key challenges in extending the framework lie in:

Identifying appropriate notions of valuative equivalence and weak maps for the specific combinatorial objects.
Constructing interesting examples of valuative functors and understanding their properties.
Investigating whether a universal valuative category analogous to V(E) exists for these structures.

Could there be alternative categorical constructions that yield different insights into valuative invariants?

While the paper focuses on a specific categorical framework, alternative constructions could offer complementary insights into valuative invariants. Here are some possibilities:

Derived Categories: Instead of focusing on split-exact sequences, one could work in the derived category of an abelian category. This might allow for the study of more refined invariants and potentially connect with derived algebraic geometry.
Higher Categories: Generalizing from ordinary categories to higher categories could provide a framework for studying "higher" valuative invariants. This might involve considering functors to categories of chain complexes or spectra.
Enriched Categories: Enriching the categories of polyhedra or matroids over a different category, such as the category of vector spaces or chain complexes, could lead to new invariants. This might be particularly relevant for studying equivariant versions of valuative invariants.
Sheaf-Theoretic Constructions: Viewing polyhedra or matroids as spaces and considering sheaves on these spaces could provide a different perspective on valuative invariants. This might connect with ideas from topological data analysis.
Exploring these alternative constructions could lead to:

New and potentially more powerful categorical valuative invariants.
Deeper connections between valuative invariants and other areas of mathematics, such as homological algebra, higher category theory, and sheaf theory.
A more comprehensive understanding of the structure of valuative equivalences.

What are the implications of this work for understanding the connections between combinatorics and other areas of mathematics, such as algebraic geometry or representation theory?

This work strengthens the bridge between combinatorics and other mathematical realms, particularly algebraic geometry and representation theory, by providing a categorical lens through which to view valuative invariants:
Algebraic Geometry:

Motivic Invariants: Valuative invariants are closely related to motivic invariants in algebraic geometry. The categorical framework might offer new tools for studying these invariants and their relationships with combinatorial structures.
Geometric Realizations: The paper's focus on polyhedra hints at deeper connections with toric varieties and other geometric objects. Categorical valuative invariants could provide insights into the geometry of these spaces.
Intersection Theory: Valuative properties are fundamental in intersection theory. The categorical framework might lead to new perspectives on intersection-theoretic questions arising from combinatorial settings.
Representation Theory:

Equivariant Invariants: The paper emphasizes the role of symmetry groups and equivariant invariants. This framework could be further developed to study representations of these groups arising from combinatorial objects.
Categorification of Representations: The categorical lifts of valuative invariants can be seen as a form of categorification of representations. This perspective might lead to new connections between combinatorial representation theory and other categorification phenomena.
Kazhdan-Lusztig Theory: The paper's treatment of Kazhdan-Lusztig polynomials suggests deeper links with geometric representation theory. The categorical framework could provide new tools for studying these polynomials and their generalizations.
Overall, this work highlights the power of categorical methods in revealing hidden structures and connections within combinatorics and its interactions with other mathematical disciplines. It lays the groundwork for further exploration of valuative invariants and their significance in a broader mathematical context.