Tropical Plücker Relations

• Tropical Plücker relations are polynomial equalities and inequalities that tropicalize classical Plücker relations to define combinatorial analogues of Grassmannians and valuated matroids.
• They extend to symplectic and flag varieties with modified conditions ensuring the minimum in tropical sums is attained at least twice, influencing incidence and isotropy in geometry.
• These relations underpin connections between tropical geometry, matroid theory, and scheme-theoretic tropicalization, prompting further study into tropical bases and Coxeter matroid generalizations.

The tropical Plücker relations are a class of polynomial equalities and inequalities arising in tropical geometry as combinatorial shadow of the classical Plücker relations for Grassmannian-type varieties. In the tropical setting, these relations serve as scheme-theoretic or combinatorial equations defining tropical analogues of Grassmannians, flag varieties, and, more generally, moduli of linear or isotropic subspaces. They underlie the structure of tropical linear spaces, valuated matroids, and their generalizations, and play a central role in the paper of tropical algebraic geometry, matroid theory, and the tropicalization of incidence varieties. The tropical Plücker relations have been extended and adapted to encode isotropy conditions (as for the symplectic Grassmannian), incidence conditions (as in flag Dressians), and mixed area constraints in geometric combinatorics.

1. Definition and Formulation of Tropical Plücker Relations

The starting point for tropical Plücker relations is the classical quadratic Plücker relations among the Plücker coordinates $(X_I)$ (with $I$ a $k$-element subset of $[n]$) that define the classical Grassmannian $\operatorname{Gr}(k, n)$. Under tropicalization, addition is replaced by the min or max operation (depending on convention), and relations are interpreted as conditions on the minimum (or maximum) being achieved at least twice.

The standard tropical Plücker relations for a family $(\mu_J)_{J \in \binom{[n]}{k}}$ are, for every $S \in \binom{[n]}{k-1}$ and $T \in \binom{[n]}{k+1}$ with $S \subset T$:

$$\min_{i \in T \setminus S} \left\{ \mu_{T \setminus i} + \mu_{S \cup \{i\}} \right\} \text{ is achieved at least twice}.$$

In the symplectic setting, linear symplectic relations define additional tropicalizations. For $S \in \binom{[2n]}{k-2}$:

$$\min_{i} \mu_{S i \overline{i}} \quad \text{is achieved at least twice}$$

where the minimization is over all $i$ such that $\{i, \overline{i}\} \cap S = \emptyset$.

Table: Archetypal Tropical Plücker and Symplectic Relations

Relation type	Index sets	Equation/Condition
Tropical Plücker	$S \in \binom{[n]}{k-1},\, T \in \binom{[n]}{k+1}$	$$\min_{i \in T \setminus S} \left\{ \mu_{T \setminus i}+\mu_{S \cup \{i\}} \right\}$$ min attained at least twice
Tropical symplectic	$S \in \binom{[2n]}{k-2}$, $i \in [n]$, $\overline{i}$	$\min_{i} \mu_{S i \overline{i}}$ min attained at least twice (as above)

These tropical relations algebraically define the tropical Grassmannian or its symplectic/type B–C analogues as prevarieties or tropical varieties.

2. Tropical Plücker Relations and the Tropical Grassmannian

The tropical Grassmannian is the subset of $\mathbb{R}^{\binom{n}{k}}$ defined by the tropical Plücker relations. In this context, tropical Plücker vectors—i.e., vectors satisfying all relations—correspond to valuated matroids (Fink et al., 2013). The set cut out by the tropical Plücker relations is known as the Dressian, and the true tropical Grassmannian coincides with this prevariety in type $A$ (classical Grassmannian of vector spaces).

When a vector of tropical Plücker coordinates arises as the image of a matrix under the tropical Stiefel map (assigning to a matrix the valuations of its maximal minors), the coordinates always satisfy the tropical Plücker relations (Fink et al., 2013, Miura et al., 2021). However, not every tropical Plücker vector arises this way, encapsulating the difference between general valuated matroids and those of "Stiefel type".

3. Tropical Plücker Relations in the Symplectic Grassmannian

For the symplectic (type C) Grassmannian, the defining ideal in Plücker coordinates is generated by the usual Plücker quadratics together with symplectic linear relations. The tropicalization yields the corresponding "tropical symplectic Grassmannian":

• The tropical Plücker relations are given as above.
• The tropical symplectic relations are, for $S \in \binom{[2n]}{k-2}$:

$$\min_{i} \mu_{S i \overline{i}} \text{ attained at least twice},\quad (\{i,\overline{i}\}\cap S=\emptyset)$$

Critically, the system formed by these relations is only a tropical basis (i.e., defines the tropical symplectic Grassmannian as a variety) in ranks $k \leq 2$ (Balla et al., 2021). For $k\geq 3$, there exist valuated matroids that satisfy all tropical Plücker and symplectic relations yet are not in the image of actual tropicalized symplectic subspaces; in these cases, additional relations are necessary. For instance, in $Sp_2(3,6)$, the set is cut out by tropical Plücker, symplectic relations, and a single further relation.

4. Interrelations, Tropical Basis Property, and Counterexamples

A prominent phenomenon in the tropical setting is the non-equivalence of various tropical analogues for classical characterizations:

• For $k=1$: All tropical analogues coincide.
• For $k=2$: The Dressian and tropical symplectic Grassmannian coincide; the relations form a tropical basis.
• For $k\geq 3$: The prevariety strictly contains the tropical variety; tropical Plücker and symplectic relations are not a tropical basis. Explicit counterexamples include the graphical matroid of $K_4$ in $SpD(3,6)$ (satisfies all relations) which is not realizable (not in the true $\TSp(3,6)$), and isotropic matroids that do not lie in the Dressian [(Balla et al., 2021), examples].

The relationships between the various sets associated to these structures—prevariety, isotropic region, symplectic Dressian, images of Stiefel maps—are strictly inclusive and mapped out in comprehensive diagrams and theorems, with equivalences breaking for higher rank.

5. Connections with Matroid Theory and Generalizations

The tropical Plücker relations encode valuated matroid data, and the symplectic variants invite connection with type C Coxeter (symplectic) matroids (Balla et al., 2021). Unlike the classical setting, not all symplectic matroids satisfy the tropical symplectic relations, and their basic tropical combinatorial characterization is not fully understood.

Conormal fans of matroids provide a further geometric construction. For a realizable matroid $M$, the tropical linear space associated to the conormal bundle $L_{M \oplus M^*}$ always yields a point in the Lagrangian tropical symplectic Grassmannian. Arbitrary valuated matroids yield isotropic points satisfying the symplectic tropical relations, yet may lack Stiefel presentations with required symmetries.

A key phenomenon documented is that for non-linear relations (such as in the symplectic case), the tropical prevariety and the true tropical variety can differ even in the generic (non-constant coefficient) setting, reflecting a well-known phenomenon in tropical geometry (Balla et al., 2021).

6. Plücker-Type Inequalities, Incidence, and Extensions

Generalizations of tropical Plücker relations appear outside the Grassmannian context as inequalities involving geometric quantities. In arrangements of convex planar sets, "Plücker-type inequalities" for mixed areas take the form

$V_{ij} V_{kl} \leq V_{ik} V_{jl} + V_{il} V_{jk}$

for all disjoint pairs $\{i,j\} \sqcup \{k,l\}$, providing a necessary and, for $n=4$, sufficient description of the possible configuration space of intersection numbers of tropical curves (Averkov et al., 2021).

In the context of incidence and flag varieties, tropical Plücker and explicit incidence relations define the flag Dressian and its subdivisions. For 2-step flag Dressians, the analogues of classical incidence relations are:

$$\operatorname{Trop}\left(\sum_{i \in T \setminus S} x_{S \cup \{i\}} y_{T \setminus \{i\}} \right)$$

with the min attained at least twice, for suitable index sets $S, T$ (Haque, 2012).

These extensions link tropical Plücker relations not only to the combinatorics of linear spaces, but also to incidence, isotropy, and intersection-theoretic phenomena.

7. Scheme-Theoretic Perspective and Blueprint Formulation

Adopting the language of the tropical hyperfield and ordered blueprints leads to a functorial approach in which tropical Plücker relations correspond to "bend relations" (Lorscheid, 2019). Each classical polynomial relation is replaced by a bend locus condition: its tropicalization is defined as the locus where the maximum among the tropical monomials occurs at least twice. From the ordered blueprint perspective, these conditions are encoded as partial order relations, and, upon imposing idempotency ($1+1=1$), as the Giansiracusa bend relations. This connects the combinatorial structure of the tropical Grassmannian to the scheme-theoretic tropicalization via the tropical hyperfield.

8. Open Questions and Directions

The breakdown of the tropical basis property for the symplectic Grassmannian in higher ranks signals the necessity of further investigation into the complete tropical defining equations in type C, the combinatorics of symplectic matroids, and the relationship with matroid subdivisions. Further avenues include the paper of tropical flag varieties, total positivity in symplectic contexts, and the full development of tropical Coxeter matroid theory (Balla et al., 2021). For intersection-theoretic applications, the characterization of the configuration space for $n > 4$ remains unresolved (Averkov et al., 2021). Scheme-theoretic and blueprint-theoretic approaches portend new categorical and structural results linking tropical, combinatorial, and algebraic geometry.