Homomesy of Tropical $T$-systems: Finite Type

Ariana Chin; Pavlo Pylyavskyy

Home/Archive/arXiv

arXiv · 2610.02548

Homomesy of Tropical $T$-systems: Finite Type

Ariana Chin·Pavlo Pylyavskyy

View original source ↗Read source document ↗Export metadata ↓

Abstract

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite cluster mutations whose exchange relations form a discrete dynamical system. For each Dynkin diagram, this system is periodic, as a special case of Zamolodchikov periodicity. In the associated tropical dynamics, every mutation along the belt can be naturally colored red or blue, according to which term attains the maximum in the tropical exchange relation. This gives a red-blue statistic on tropical orbits. We prove that this statistic is homomesic, i.e. that the average numbers of red and blue mutations are independent of the orbit. Moreover, the number of red mutations is given by the number of roots in the associated root system. This resolves a conjecture of the first author. In the process we introduce the Zamolodchikov fan, which is closely related to the positive tropical Grassmannian, and may be of independent interest.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ariana Chin, Pavlo Pylyavskyy. 2026-10-01. Homomesy of Tropical $T$-systems: Finite Type. https://arxiv.org/abs/2610.02548

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hamilton cycles in generalized dihedral Cayley graphs and digraphs

We prove that every connected Cayley digraph on a generalized dihedral group of order at least $4$ has a directed Hamilton cycle. In particular, this confirms a conjecture of Holsztyński and Strube from 1978 for dihedral groups. The key new ingredient is a three-fold sumset covering theorem for the terminal coordinates of Hamilton paths in cubic Haar graphs over abelian groups of odd order, with connection sets minimal subject to connectivity.

math.CO↗

arXiv2026

Some multidimensional Rogers--Ramanujan type identities

With the help of the contour integral method, we derive a parametric reduction formula that transforms a double series into a single series. This formula recovers two results of Uncu and Zudilin as well as two results of Cao and Wang, and it is also connected with an identity due to Berkovich and Warnaar. In addition, we obtain several triple-sum generalizations of Cao and Wang's formulas. As applications, we present a number of multidimensional Rogers--Ramanujan type identities, both with and without parameters.

math.CO↗

arXiv2026

Interaction between skew-representability, tensor products, extension properties, and rank inequalities

Skew-representable matroids form a fundamental class in matroid theory, bridging combinatorics and linear algebra. They play an important role in areas such as coding theory, optimization, and combinatorial geometry, where linear structure is crucial for both theoretical insights and algorithmic applications. Since skew-representability is undecidable even for rank-3 matroids, structural characterizations and explicit certificates of non-skew-representability are particularly interesting. In this paper, we introduce an approach to studying skew-representability and structural properties of matroids and polymatroid functions via tensor products. We characterize skew-representable matroids, as well as matroids representable over skew fields of a prescribed characteristic, in terms of iterated tensor products. In particular, a connected matroid is non-skew-representable if and only if, for some positive integer $k$, no $k$-fold iterated tensor product with $U_{2,3}$ exists. Thus, non-skew-representability admits a finite, computably verifiable matroid-theoretic obstruction; an analogous statement holds when the characteristic is prescribed. We also prove that every rank-3 matroid admits a tensor product with every uniform matroid and give a construction yielding the unique freest tensor product in this setting. Finally, as an application of the tensor product framework, we give a new proof of Ingleton's inequality and, more importantly, derive the first known linear rank inequality for folded skew-representable matroids that does not follow from the common information property.

math.CO↗