Tropical Linear Spaces, Valuated Matroids, and the Tropical Grassmannian¶
Job: develop the tropical linear spaces / valuated matroids (Dress–Wenzl axioms), the tropical Grassmannian (Speyer–Sturmfels \(\mathrm{trop}(\mathrm{Gr}(2,n))\cong\) tree space), the Dressian vs the realizable (the gap for \(r\ge3\)), the tropical Stiefel / Plücker / matroidal subdivisions, and the tropical Schubert calculus — with \(\mathrm{trop}(\mathrm{Gr}(2,n))\) the tree space as the worked example.
Core¶
Tropical linear spaces and valuated matroids¶
A tropical linear space \(L\subset\mathbb{T}^n\) (the tropicalization of a linear subspace, or the intrinsic tropical linear space). The valuated matroid \((E,\mu)\): a matroid \(M\) on the ground set \(E\), with a valuation \(\mu:E^r\to\mathbb{R}\cup\{\infty\}\) on the bases, satisfying the Dress–Wenzl axioms (the tropical / valuated Plücker relations).
| Tropical linear space / valuated matroid | Description |
|---|---|
| Tropical linear space \(L\subset\mathbb{T}^n\) | the tropicalization of a linear subspace (or the intrinsic tropical linear space) |
| Valuated matroid \((E,\mu)\) | a matroid \(M\) on \(E\) with a valuation \(\mu:E^r\to\mathbb{R}\cup\{\infty\}\) on the bases |
| Dress–Wenzl axioms | the valuated-matroid axioms (the tropical Plücker relations, the valuated Plücker) |
| Plücker / tropicalization | \(L=\mathrm{trop}(V(I))\) for a linear ideal \(I\) (the Plücker ideal), \(\mu_B=\mathrm{val}(p_B)\) the Plücker valuations |
The tropical Grassmannian and the Speyer–Sturmfels theorem¶
The tropical Grassmannian \(\mathrm{trop}(\mathrm{Gr}(r,n))\) (the tropicalization of the Plücker variety, the valuated matroids realizable over the valued field).
Theorem (Speyer–Sturmfels). \(\mathrm{trop}(\mathrm{Gr}(2,n))\cong\mathrm{Dr}(2,n)\) (the tropical Grassmannian of rank-2 / the space of phylogenetic trees, the tree space / the Billera–Holmes–Vogtmann). For \(r\ge3\), \(\mathrm{trop}(\mathrm{Gr}(r,n))\subsetneq\mathrm{Dr}(r,n)\) (the realizable is a proper subset of the Dressian, the gap).
| Tropical Grassmannian | Description |
|---|---|
| \(\mathrm{trop}(\mathrm{Gr}(r,n))\) | the tropicalization of the Plücker variety (the realizable valuated matroids) |
| Speyer–Sturmfels (\(r=2\)) | \(\mathrm{trop}(\mathrm{Gr}(2,n))\cong\mathrm{Dr}(2,n)=\mathcal{T}_n\) (the tree space, all rank-2 realizable) |
| Dressian \(\mathrm{Dr}(r,n)\) | the space of valuated matroids (the abstract, not necessarily realizable) |
| Higher rank (\(r\ge3\)) | \(\mathrm{trop}(\mathrm{Gr}(r,n))\subsetneq\mathrm{Dr}(r,n)\) (the gap, the non-realizable valuated matroids) |
The Dressian and the realizability¶
The Dressian \(\mathrm{Dr}(r,n)\) (the space of valuated matroids, the tropical Plücker vectors satisfying the valuated-matroid axioms — the abstract, not necessarily realizable). The realizability problem: which valuated matroids are realizable (i.e. \(\mathrm{trop}(\mathrm{Gr}(r,n))\subset\mathrm{Dr}(r,n)\), the realizable as a subset).
Theorem (Speyer). For \(r=2\) all valuated matroids are realizable (\(\mathrm{trop}(\mathrm{Gr}(2,n))=\mathrm{Dr}(2,n)\), the tree space), but for \(r\ge3\) there are non-realizable valuated matroids (the gap between the Dressian and the tropical Grassmannian).
The tropical Stiefel map and the tropical Plücker¶
| Tropical Stiefel / Plücker | Description |
|---|---|
| Tropical Stiefel map | the tropicalization of the Stiefel map: \(A\in K^{r\times n}\), \(\mathrm{trop}(A)\in\mathbb{T}^{r\times n}\), \(\mu_B=\mathrm{val}(\det A_B)\) |
| Tropical Plücker relations | the valuated-matroid axioms (the tropical Plücker vector satisfying the valuated Plücker relations) |
| Matroidal subdivisions | the valuated matroid as the regular subdivision of the hypersimplex \(\Delta(r,n)\) |
| Secondary polytope | the tropical / the secondary polytope connection (the regular subdivisions of the hypersimplex) |
The tropical linear spaces and the matroid theory¶
| Tropical linear algebra / matroids | Description |
|---|---|
| Matroid valuations / tropical linear algebra | the tropical determinant / tropical rank, the tropical linear independence, the Kapranov–Ziegler / the Dress |
| Matroid stratification | the Schubert / the matroid strata, the tropicalization: the tropical Schubert calculus |
| Combinatorics connection | the matroids / the tropical linear spaces as the combinatorial shadow of the linear algebra (discrete math) |
See it¶
\(\mathrm{trop}(\mathrm{Gr}(2,n))\) the space of phylogenetic trees (the Billera–Holmes–Vogtmann tree space). The rank-2 tropical Grassmannian \(\mathrm{trop}(\mathrm{Gr}(2,n))\cong\mathrm{Dr}(2,n)=\mathcal{T}_n\) (the space of phylogenetic trees on \(n\) leaves, the tree space). The \(\mathcal{T}_n\) is the polyhedral complex: the vertices are the trivalent trees (the binary phylogenetic trees, the combinatorial types), the edges are the rotations / the NNI (the nearest-neighbor interchange), and the points are the weighted trees (the internal edge lengths \(\in\mathbb{R}_{\ge0}\)). The Plücker valuations \(\mu_{ij}=\mathrm{val}(p_{ij})\) (the valuations of the \(2\times 2\) Plücker coordinates) are the pairwise distances in the tree (the four-point condition / the tropical Plücker: \(\mu_{ij}+\mu_{kl}\le\max(\mu_{ik}+\mu_{jl},\mu_{il}+\mu_{jk})\) for all \(\{i,j,k,l\}\), the tree metric condition). The key point: the rank-2 tropical Grassmannian is the tree space (the phylogenetic trees), the tropical Plücker / the four-point condition characterizes the tree metrics, and the \(\mathrm{trop}(\mathrm{Gr}(2,n))\) is the moduli of the metric trees — the prototypical tropical moduli / the combinatorial-to-algebraic.
Check¶
Verify the four-point condition for the tropical Plücker of \(\mathrm{Gr}(2,n)\), and explain the tree metric.
The tropical Plücker for \(\mathrm{Gr}(2,n)\): the Plücker coordinates \(p_{ij}\) (the \(2\times 2\) minors), the valuations \(\mu_{ij}=\mathrm{val}(p_{ij})\). The valuated Plücker relation (the four-point condition): for all \(\{i,j,k,l\}\), \(\mu_{ij}+\mu_{kl}\le\max(\mu_{ik}+\mu_{jl},\mu_{il}+\mu_{jk})\) (the tropical / the min-plus version of the Plücker relation \(p_{ij}p_{kl}-p_{ik}p_{jl}+p_{il}p_{jk}=0\)). The tree metric: a metric \(d_{ij}\) on \(\{1,\dots,n\}\) is a tree metric (arises from a weighted tree) iff the four-point condition holds: \(d_{ij}+d_{kl}\le\max(d_{ik}+d_{jl},d_{il}+d_{jk})\) for all \(\{i,j,k,l\}\) (the Pachter–Speyer / the Buneman condition). So the tropical Plücker \(\mu_{ij}\) is a tree metric, and the \(\mathrm{trop}(\mathrm{Gr}(2,n))\) is the tree space.
Explain the gap between the Dressian \(\mathrm{Dr}(r,n)\) and the tropical Grassmannian \(\mathrm{trop}(\mathrm{Gr}(r,n))\) for \(r\ge3\), and identify the obstruction.
The Dressian \(\mathrm{Dr}(r,n)\) is the space of all valuated matroids (the abstract, satisfying the Dress–Wenzl axioms / the valuated Plücker). The tropical Grassmannian \(\mathrm{trop}(\mathrm{Gr}(r,n))\) is the realizable subset (the valuated matroids arising from a linear subspace over the valued field, \(\mu_B=\mathrm{val}(p_B)\) for the Plücker coordinates). For \(r=2\), all valuated matroids are realizable (the Speyer–Sturmfels, the tree space), so \(\mathrm{trop}(\mathrm{Gr}(2,n))=\mathrm{Dr}(2,n)\). For \(r\ge3\), there are non-realizable valuated matroids (the Speyer, the gap): some valuated matroids satisfy the Dress–Wenzl axioms but do not arise from any linear subspace over the valued field (the realizability obstruction — the non-realizable matroids, the same obstruction as the classical matroid realizability, the tropical version). The gap is the measure of the non-realizable valuated matroids, the tropical Grassmannian a proper subset of the Dressian.
Pitfalls¶
- The \(\mathrm{trop}(\mathrm{Gr}(2,n))=\mathrm{Dr}(2,n)\) (the Speyer–Sturmfels, all rank-2 valuated matroids realizable), but for \(r\ge3\) the gap (the Speyer, the non-realizable valuated matroids) — the rank-2 is the special / the tree space, the higher-rank is the general / the gap.
- The Dressian is the abstract (the valuated-matroid axioms), the tropical Grassmannian is the realizable (the Plücker from a linear subspace) — the two are different for \(r\ge3\).
- The matroidal subdivisions (the valuated matroid as the regular subdivision of the hypersimplex \(\Delta(r,n)\)) is the polytope-theoretic interpretation, the secondary polytope connection — the valuated matroid is the regular subdivision, not the abstract matroid.
- The tropical Stiefel map (the tropicalization of the Stiefel map) gives the realizable valuated matroids (the Plücker from the matrix), not the abstract Dressian.
- The tropical Schubert calculus (the tropicalization of the Schubert strata) is the matroid stratification, the combinatorial shadow — the connection to the discrete math / matroids is the combinatorial aspect, not the full algebraic.
Refs¶
- Speyer–Sturmfels, "The tropical Grassmannian" (J. Algebra 2004); Speyer, "Tropical linear spaces" (SIAM J. Disc. Math. 2008)
- Dress–Wenzl, "Matroids with coefficients" (1990); Fink–Rincón, "Tropical Grassmannians and Plücker functions" (2014)
- Billera–Holmes–Vogtmann, "Geometry of the space of phylogenetic trees" (Adv. Appl. Math. 2001); Pachter–Speyer, "Reconstructing trees from subtree weights" (2004)
- Herrmann–Jensen–Joswig–Sturmfels, "How to draw the Dressian" (2009); the secondary polytope / the matroidal subdivisions
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