7.1 Tropical Plücker vectors

Tropical Plücker vectors are a key concept in tropical geometry, representing tropical linear spaces. They're defined by coordinates in tropical projective space that satisfy specific relations, allowing us to study tropical linear spaces algebraically.

These vectors have unique properties that set them apart from classical Plücker vectors. They have non-negative coordinates and satisfy tropical Grassmann-Plücker relations, which use min and plus operations instead of addition and multiplication.

Tropical Plücker vectors

• Represent tropical linear spaces, a key concept in tropical geometry that generalizes classical linear spaces
• Vectors defined by a set of tropical coordinates subject to certain relations and constraints
• Enable studying properties and structures of tropical linear spaces algebraically

Definition of tropical Plücker vectors

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• Vectors in tropical projective space $\mathbb{TP}^n$ with coordinates satisfying tropical Grassmann-Plücker relations
• Coordinates are elements of the tropical semiring $(\mathbb{R} \cup \{\infty\}, \oplus, \odot)$
  • $a \oplus b = \min(a, b)$ and $a \odot b = a + b$
• Each tropical Plücker vector corresponds to a unique tropical linear space

Coordinates of tropical Plücker vectors

• Indexed by subsets of $\{0, 1, \ldots, n\}$ of a fixed size $d$
• Coordinate $p_I$ associated with subset $I = \{i_1, \ldots, i_d\}$ represents the "tropical determinant" of the corresponding $d \times d$ submatrix
• Coordinates satisfy exchange relations derived from tropical Grassmann-Plücker relations

Representing tropical linear spaces

• Tropical linear spaces of dimension $d$ in $\mathbb{TP}^n$ are in bijection with tropical Plücker vectors of appropriate size
• Each tropical linear space can be represented by a unique tropical Plücker vector
• Allows studying properties of tropical linear spaces using vector coordinates and relations

Properties of tropical Plücker vectors

• Exhibit unique characteristics that distinguish them from classical Plücker vectors
• Satisfy specific relations and constraints that characterize tropical linear spaces
• Reveal connections between tropical geometry and combinatorics

Non-negativity of coordinates

• Coordinates of tropical Plücker vectors are non-negative, i.e., in $\mathbb{R} \cup \{\infty\}$
• Follows from the definition of tropical determinant as the minimum weight of a perfect matching
• Contrasts with classical Plücker vectors, which can have negative coordinates

Tropical Grassmann-Plücker relations

• Tropical analogue of classical Grassmann-Plücker relations that characterize Plücker vectors
• Involve the tropical sum $\oplus$ and tropical product $\odot$ of Plücker coordinates
• Example: For $I, J \subset \{0, 1, \ldots, n\}$ with $|I| = |J| = d$, $p_I \odot p_J = \bigoplus_{k \in J \setminus I} p_{I \cup \{k\}} \odot p_{J \setminus \{k\}}$

Characterizing tropical linear spaces

• Tropical Plücker vectors completely characterize tropical linear spaces
• A vector in $\mathbb{TP}^{\binom{n+1}{d+1}-1}$ is a tropical Plücker vector if and only if its coordinates satisfy tropical Grassmann-Plücker relations
• Allows determining if a given vector corresponds to a tropical linear space

Tropical Plücker vectors vs classical Plücker vectors

• Share some analogies with classical Plücker vectors but also exhibit distinct properties
• Differences stem from the use of tropical arithmetic and the resulting algebraic structure
• Comparing the two reveals insights into the unique aspects of tropical geometry

Analogies between tropical and classical

• Both represent linear spaces in their respective geometric settings
• Coordinates indexed by subsets and satisfy Grassmann-Plücker relations
• Plücker vectors completely characterize linear spaces in both cases

Differences in behavior and properties

• Tropical Plücker coordinates are non-negative, while classical ones can be negative
• Tropical Grassmann-Plücker relations involve min and plus operations, classical ones use addition and multiplication
• Tropical linear spaces exhibit distinct combinatorial properties and connections to optimization

Applications of tropical Plücker vectors

• Arise in various fields beyond pure mathematics, showcasing the utility of tropical geometry
• Enable modeling and solving problems in areas such as biology, optimization, and phylogenetics
• Provide new perspectives and techniques for tackling computational challenges

Modeling in computational biology

• Tropical linear spaces can model biological systems and processes
• Example: Representing gene expression data or evolutionary relationships
• Tropical Plücker vectors provide a compact and algebraically tractable representation

Solving optimization problems

• Tropical geometry offers tools for solving certain classes of optimization problems
• Tropical linear programming can be formulated using tropical Plücker vectors
• Exploits the piecewise-linear structure of tropical objects to develop efficient algorithms

Connections to phylogenetics

• Phylogenetic trees can be viewed as tropical linear spaces
• Tropical Grassmannians, defined using tropical Plücker vectors, relate to tree spaces in phylogenetics
• Allows applying tropical geometric techniques to study evolutionary relationships and diversity

Computational aspects of tropical Plücker vectors

• Computing and manipulating tropical Plücker vectors efficiently is crucial for applications
• Involves developing algorithms, analyzing their complexity, and implementing software tools
• Enables practical use of tropical Plücker vectors in various domains

Algorithms for computing tropical Plücker coordinates

• Efficient algorithms exist for computing tropical Plücker coordinates from a given tropical linear space
• Example: Tropical Cramer's rule, which expresses coordinates using tropical determinants
• Exploit the combinatorial structure and sparsity of tropical objects to achieve good performance

Complexity of tropical Plücker vector operations

• Analyzing the computational complexity of operations on tropical Plücker vectors
• Includes addition, multiplication, and checking tropical Grassmann-Plücker relations
• Often leads to faster algorithms compared to classical counterparts due to the simpler tropical arithmetic

Software implementations and libraries

• Several software libraries implement tropical Plücker vectors and related algorithms
• Example: Polymake, a comprehensive software for polyhedral and tropical geometry
• Provide efficient data structures and methods for computing and manipulating tropical Plücker vectors
• Enable practical applications and exploration of tropical geometric objects