Tropical Linear Spaces: Combinatorics & Geometry

• Tropical linear spaces are piecewise-linear analogs of classical subspaces defined via tropical Plücker relations and closely linked to valuated matroids.
• They exhibit rich polyhedral fan structures, tropical convexity, and intrinsic metric properties that support applications in algebraic geometry, phylogenetics, and statistics.
• Advanced operations such as tropical images, stable sums, and tropical PCA enable effective optimization and analysis in high-dimensional and combinatorial contexts.

A tropical linear space is a piecewise-linear generalization of classical linear subspaces, defined over idempotent semifields such as the tropical semifield $\mathbb{T}=\mathbb{R}\cup\{-\infty\}$ with tropical addition $a\oplus b = \max(a,b)$ and multiplication $a\otimes b = a+b$. Tropical linear spaces are the polyhedral fans or complexes cut out by the tropicalization of Plücker relations, carrying rich combinatorial, algebraic, and metric structures. Their foundational aspect is the equivalence between valuated matroids, their associated tropical Plücker vectors, and the geometry of tropical convexity and hyperplane intersections. Tropical linear spaces arise in tropical geometry, combinatorics, algebraic and analytic geometry, with widespread applications in metric geometry, statistics, moduli, and phylogenetics.

1. Combinatorial Foundations: Valuated Matroids and Tropical Plücker Relations

Tropical linear spaces are parameterized by tropical Plücker vectors $w\in \mathbb{T}^{\binom{E}{d}}$, indexed by $d$-element subsets of a finite ground set $E$. A tropical Plücker vector of rank $d$ satisfies the tropical Plücker (bend) relations: for every $J\in\binom{E}{d+1}$ and $K\in\binom{E}{d-1}$,

$$\max_{i\in J\setminus K}(w_{J-\{i\}} + w_{K\cup\{i\}})$$

is attained at least twice. These relations encode the tropical version of the classical Grassmann-Plücker equations and guarantee the existence of an underlying matroid $M$, whose bases are those $B$ with $w_B \neq -\infty$ (Mundinger, 2018, Miura et al., 2021).

Given $w$, the tropical linear space $L_w$ is the intersection of tropical hyperplanes $H_J(x) = \max_{i\in J} (w_{J-i} + x_i)$ for all $(d+1)$-subsets $J$. The resulting space $L_w \subseteq \mathbb{T}^E$ is a polyhedral complex of pure dimension, invariant under tropical scaling by the all-ones vector, and admits the structure of a tropical projective torus $\mathbb{T}^n / \mathbb{R}\cdot(1,\dots,1)$.

This link between valuated matroids, tropical Plücker vectors, and tropical linear spaces is categorical: every tropical linear space is determined, up to translation, by a valuated matroid satisfying the Plücker relations (Hampe, 2015).

2. Tropical Convexity and Fan Structures

A tropical linear space $L$ is tropically convex, meaning for $x,y \in L$ and scalars $\lambda,\mu \in \mathbb{T}$, the tropical combination $(\lambda \otimes x) \oplus (\mu \otimes y)$ lies in $L$ (Hampe, 2015, Lin et al., 2015). The converse holds: any tropical variety that is tropically convex is the support of a valuated matroid by Hampe's equivalence theorem.

Explicitly, tropical convexity is induced by the underlying matroidal fan structure. In particular, the Bergman fan of a matroid $M$—the case $w\equiv 0$—is a pure polyhedral fan supported on

$$B(M) = \{ x \in \mathbb{R}^n : \forall C \text{ circuit}, \max_{i\in C} x_i \text{ attained at least twice} \}$$

and its rays are indexed by the cyclic flats or singleton elements of $M$ (Rincón, 2011). Refined fan decompositions such as the cyclic Bergman fan, nested set fan, and fine subdivision further stratify $L$ by combinatorial data from the matroid (Rincón, 2011).

Each local cone associated to a basis $B$ is homeomorphic to $\mathbb{R}^d$ via a piecewise-linear map, exploiting the fundamental circuits of $B$ in $M$ (Rincón, 2012).

3. Algebraic Realization: Tropicalization and Limit Spaces

Tropical linear spaces are realized as tropicalizations of classical linear subspaces over fields of Puiseux or Laurent series (Mundinger, 2018, Battistella et al., 2023). For generic lifts $\Delta$ of a matrix $A$, the tropical image $\text{tropim}_A(L)$ of a classical linear space $\Lambda \subset K^n$ coincides with the tropicalization of the image $\Delta \Lambda$, provided the residue field has large cardinality.

The universal viewpoint identifies the space of seminorms (Goldman–Iwahori space) $\mathcal{S}(V)$ as the inverse limit of all tropicalized linear embeddings $\PP^r \hookrightarrow \PP^n$. $\mathcal{S}(V)$ is canonically homeomorphic to the tropical linear space of the universal realizable valuated matroid (Battistella et al., 2023). These limit spaces provide the analytic and combinatorial compactifications of buildings for groups like $\text{PGL}$.

4. Metrics, Projection, and Tropical Principal Components

Tropical linear spaces admit extrinsic metric structures. The tropical distance on $\mathbb{T}^n / \mathbb{R}\cdot 1$ is defined by

$$d_{\text{trop}}(x, y) = \max_{i < j} |(x_i - y_i) - (x_j - y_j)|$$

(Lin et al., 2015, Yoshida et al., 2017). Projection onto a tropical linear space $L(A)$ is performed using the "Blue Rule," a formula involving the tropical Plücker coordinates of $A$ to find $\pi_{L(A)}(x)$ (Miura et al., 2021, Yoshida et al., 2017).

These metric properties facilitate tropical PCA, regression, and centroid computation: one seeks $L$ minimizing the sum of tropical distances over a data cloud, yielding best-fit Stiefel tropical linear spaces in the sense of piecewise-linear optimization (Miura et al., 2021, Yoshida et al., 2017). Tropical PCA is applied to Gaussian mixtures, tree metrics, and high-dimensional biological datasets.

5. Tropical Images and Operations: Stable Sums and Linear Maps

The set-theoretic image of a tropical linear space under a matrix $A$ is generally not tropical linear, as closure under tropical addition and the bend relations may fail (Mundinger, 2018). The tropical image construction $\text{tropim}_A(L)$ repairs this defect, producing a minimal tropical linear space containing $A L$, described via tropical exterior algebra and explicit max-plus Cauchy–Binet formulas.

Stable sum operations combine tropical linear spaces by direct sum followed by tropical image under an addition matrix. The stable sum $L_{w_1} \text{ st } L_{w_2}$ is a tropical linear space whose Plücker vector is the wedge product $w_1 \wedge w_2$, generalizing classical matroid union (Mundinger, 2018). This framework unifies Stiefel spaces, tropical modifications, and various sum/intersection constructions.

6. Polyhedral and Subdivision Perspectives

Matroid subdivisions of the hypersimplex, their mixed subdivisions, and dual tight spans encode the polyhedral geometry of tropical linear spaces (Hampe et al., 2016, Fink et al., 2013). Each subdivision corresponds to a regular matroidal or mixed subdivision, with tropical linear spaces arising as the dual complexes to these subdivisions. Tight spans, canonical face lattices, and Ganter's closure enumeration provide output-sensitive algorithms for their computation.

Local tropical linear spaces $L_B$ are dual to mixed subdivisions of Cayley sums of simplices, each piece homeomorphic to Euclidean space, allowing sharp bounds for the $f$-vectors of tropical linear spaces and supporting Speyer's conjecture (Rincón, 2012).

7. Incidence, Moduli, and Advanced Structures

The incidence geometry of tropical linear spaces diverges from classical linear spaces. The moduli of codimension-1 tropical subspaces is tropically convex; for the Dressian $D(d,n)$, only 3-term tropical incidence equations cut out many important loci (Wang, 2024). Some classical properties, such as submodularity of the matroid quotients, fail for tropical linear spaces with large ground sets. Adjoint structures for tropical linear spaces generalize classical adjoints and restore certain incidence properties.

Lorentzian polynomials and M-convex functions interact richly with tropical linear spaces, defining new cones of proper position and informing the tropicalization of incidence varieties (Wang, 2024). Real tropical linear spaces tracked in the signed Goldman–Iwahori space further refine the polyhedral structure by sign data, leading to oriented matroid and phase enrichments in tropical geometry (Kuehn et al., 2024).

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Summary Table: Key Constructions

Concept	Definition/Role	Reference
Tropical Plücker vector	Max-plus vector satisfying bend relations	(Mundinger, 2018)
Bergman fan	Matroid fan where all valuations zero	(Rincón, 2011)
Tropical image $\text{tropim}_A(L)$	Minimal tropical linear space containing $A L$	(Mundinger, 2018)
Stiefel tropical space	Image of tropical Stiefel map, via minors	(Fink et al., 2013)
Dressian $D(d,n)$	Moduli space of tropical linear spaces	(Wang, 2024)
Tropical convexity	Closure under tropical linear combinations	(Hampe, 2015)
Local tropical spaces	Dual to mixed subdivisions, homeomorphic to $\mathbb{R}^d$	(Rincón, 2012)

• (Mundinger, 2018): The image of a tropical linear space
• (Miura et al., 2021): Plücker Coordinates of the best-fit Stiefel Tropical Linear Space to a Mixture of Gaussian Distributions
• (Hampe, 2015): Tropical linear spaces and tropical convexity
• (Battistella et al., 2023): Buildings, valuated matroids, and tropical linear spaces
• (Lin et al., 2015): Convexity in Tree Spaces
• (Wang, 2024): Lorentzian polynomials and the incidence geometry of tropical linear spaces
• (Yoshida et al., 2017): Tropical Principal Component Analysis and its Application to Phylogenetics
• (Kuehn et al., 2024): The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces
• (Hampe et al., 2016): Algorithms for Tight Spans and Tropical Linear Spaces
• (Rincón, 2012): Local tropical linear spaces
• (Dukkipati et al., 2013): Tropical Grassmannian and Tropical Linear Varieties from phylogenetic trees
• (Fink et al., 2013): Stiefel tropical linear spaces
• (Rincón, 2011): Computing Tropical Linear Spaces