7.5 Tropical Schubert calculus
7 min read•august 20, 2024
blends classical algebraic geometry with tropical methods. It studies intersections of in tropical Grassmannians, using combinatorial tools like and to compute intersection products.
This framework offers a fresh perspective on problems. It simplifies calculations of Gromov-Witten invariants and provides insights into , bridging classical and tropical approaches in algebraic geometry and mathematical physics.
Tropical Grassmannians
- Tropical Grassmannians are a key concept in tropical geometry that generalize the notion of Grassmannians from classical algebraic geometry to the tropical setting
- Defined as the of the classical Grassmannian, which is the space of all k-dimensional linear subspaces of an n-dimensional vector space
- Play a crucial role in understanding the combinatorial and geometric properties of and their intersections
Tropical linear spaces
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- Tropical linear spaces are analogous to linear subspaces in classical geometry, but are defined using the tropical semiring operations of maximum and addition
- Can be represented as the intersection of tropical hyperplanes, which are defined by tropical linear forms
- Have a unique decomposition as the Minkowski sum of tropical line segments, which allows for a combinatorial description of their structure
Tropical determinants
- are a generalization of the classical determinant to the tropical semiring, obtained by replacing addition with maximum and multiplication with addition
- Used to characterize the singularities and degeneracies of tropical linear spaces and matrices
- Can be computed efficiently using combinatorial methods, such as the Hungarian algorithm for assignment problems
Tropical Plücker coordinates
- are a set of projective coordinates that describe the embedding of the into a tropical projective space
- Obtained by tropicalizing the classical Plücker coordinates, which are the maximal minors of a matrix representing a linear subspace
- Satisfy the tropical Plücker relations, a set of inequalities that characterize the image of the tropical Grassmannian embedding
Tropical Schubert varieties
- Tropical Schubert varieties are a class of subvarieties of the tropical Grassmannian that are defined using combinatorial data, such as permutations or Young tableaux
- Arise as the tropicalization of classical Schubert varieties, which are important objects in the study of flag varieties and intersection theory
- Play a key role in and the computation of tropical intersection numbers
Schubert cells
- are the building blocks of Schubert varieties, and are defined as the set of linear subspaces that have a specific relative position with respect to a fixed flag
- In the tropical setting, Schubert cells are defined using the tropical Plücker coordinates and a set of tropical incidence relations
- The dimension of a tropical Schubert cell is given by the length of the corresponding permutation or the size of the corresponding Young diagram
Tropicalization of Schubert varieties
- Tropicalization is a process that transforms a classical algebraic variety into a tropical variety by replacing the underlying field with the tropical semiring
- The tropicalization of a classical Schubert variety is a tropical Schubert variety, which inherits many of the combinatorial and geometric properties of its classical counterpart
- Tropicalization allows for the use of combinatorial techniques, such as polyhedral geometry and discrete convexity, to study Schubert varieties and their intersections
Tropical Schubert cycles
- are the fundamental classes of tropical Schubert varieties, and play a role analogous to that of Schubert cycles in classical intersection theory
- Can be represented as weighted polyhedral complexes, where the weights are given by the tropical intersection multiplicities
- The set of tropical Schubert cycles forms a basis for the tropical cohomology ring of the tropical Grassmannian, which is a tropical analogue of the classical cohomology ring
Tropical intersections
- are a key concept in tropical geometry that describe the intersection of tropical subvarieties, such as tropical linear spaces or tropical Schubert varieties
- Unlike classical intersections, tropical intersections are not always transverse and may have higher-dimensional components
- The multiplicity of a tropical intersection can be computed using combinatorial methods, such as the tropical Bernstein-Kushnirenko theorem or the
Stable intersections
- are a class of tropical intersections that are well-behaved and have expected dimension, similar to transverse intersections in classical geometry
- Can be characterized using the tropical Plücker relations and a set of combinatorial conditions on the corresponding tropical linear spaces or Schubert varieties
- The stable intersection of two tropical Schubert varieties is a tropical cycle that represents their intersection product in the tropical cohomology ring
Tropical Bézout's theorem
- is a tropical analogue of the classical Bézout's theorem, which relates the degree of the intersection of two subvarieties to their individual degrees
- States that the degree of the stable intersection of two tropical hypersurfaces is equal to the product of their degrees, where the degree is defined as the maximum of the coefficients in the defining the hypersurface
- Can be generalized to the intersection of higher-dimensional tropical subvarieties using the tropical mixed volume, a combinatorial invariant that captures the complexity of the intersection
Tropical Schubert calculus
- Tropical Schubert calculus is the study of the intersection theory of tropical Schubert varieties and the computation of tropical intersection numbers
- Involves the use of combinatorial tools, such as tropical tableaux and Littlewood-Richardson coefficients, to describe the intersection product of tropical Schubert cycles
- Allows for the computation of classical intersection numbers, such as Gromov-Witten invariants, using tropical geometry and the correspondence between tropical and classical Schubert calculus
Tropical Littlewood-Richardson rule
- The tropical Littlewood-Richardson rule is a combinatorial formula for computing the intersection product of two tropical Schubert cycles in the tropical Grassmannian
- Generalizes the classical Littlewood-Richardson rule, which describes the structure coefficients of the cohomology ring of the Grassmannian, to the tropical setting
- Involves the enumeration of tropical tableaux, which are combinatorial objects that encode the intersection multiplicities of tropical Schubert varieties
Tropical tableaux
- Tropical tableaux are combinatorial objects that generalize the notion of semistandard Young tableaux to the tropical setting
- Defined using the tropical semiring operations and a set of filling conditions that ensure compatibility with the tropical Plücker relations
- The weight of a tropical tableau is a vector that encodes the intersection multiplicity of the corresponding tropical Schubert varieties
Littlewood-Richardson coefficients
- Littlewood-Richardson coefficients are the structure coefficients of the cohomology ring of the Grassmannian, and describe the intersection product of Schubert cycles
- In the tropical setting, Littlewood-Richardson coefficients are computed using the tropical Littlewood-Richardson rule and the enumeration of tropical tableaux
- Satisfy a set of combinatorial identities, such as the Horn inequalities and the Knutson-Tao saturation conjecture, which have important applications in representation theory and algebraic combinatorics
Computing tropical intersections
- involves the use of combinatorial and algorithmic techniques to determine the structure and multiplicity of the intersection of tropical subvarieties
- Can be reduced to the enumeration of tropical tableaux or the solution of tropical linear programming problems, which can be solved efficiently using methods from discrete optimization
- The tropical intersection product is a powerful tool for studying the geometry and combinatorics of tropical varieties, and has applications in various areas of mathematics, such as algebraic geometry, combinatorics, and mathematical physics
Applications of tropical Schubert calculus
- Tropical Schubert calculus has numerous applications in various areas of mathematics, including algebraic geometry, combinatorics, and mathematical physics
- Provides a framework for studying the intersection theory of algebraic varieties and the computation of enumerative invariants, such as Gromov-Witten invariants and Hurwitz numbers
- Offers a new perspective on classical problems in algebraic combinatorics, such as the Horn conjecture and the Knutson-Tao saturation conjecture, by translating them into the language of tropical geometry
Tropical enumerative geometry
- Tropical enumerative geometry is the study of counting problems in algebraic geometry using techniques from tropical geometry and combinatorics
- Involves the computation of tropical intersection numbers, such as the number of curves of a given degree and genus passing through a set of points or the number of lines on a quintic threefold
- Tropical methods often provide a simpler and more combinatorial approach to enumerative problems, compared to classical techniques such as Gromov-Witten theory or Donaldson-Thomas theory
Tropical Gromov-Witten invariants
- are tropical analogues of the classical Gromov-Witten invariants, which count the number of curves of a given degree and genus in a smooth projective variety
- Defined using tropical moduli spaces of stable maps and the intersection theory of tropical Schubert varieties
- Can be computed using tropical Schubert calculus and the correspondence between tropical and classical Gromov-Witten theory, which provides a new approach to the study of mirror symmetry and the enumerative geometry of Calabi-Yau manifolds
Tropical mirror symmetry
- is a tropical analogue of the classical mirror symmetry conjecture, which relates the Gromov-Witten invariants of a Calabi-Yau manifold to the Hodge numbers of its mirror dual
- Involves the study of tropical Landau-Ginzburg models and the correspondence between tropical and classical enumerative invariants
- Tropical methods have led to new proofs and generalizations of classical mirror symmetry results, such as the Gross-Siebert program and the Gammage-Shende conjecture, and have provided new insights into the structure of mirror families and the geometry of Calabi-Yau manifolds
Key Terms to Review (34)
Bernd Sturmfels: Bernd Sturmfels is a prominent mathematician known for his contributions to algebraic geometry, combinatorial geometry, and tropical geometry. His work has been influential in developing new mathematical theories and methods, particularly in understanding the connections between algebraic varieties and combinatorial structures.
Computing tropical intersections: Computing tropical intersections refers to the process of finding the intersection points of tropical varieties, which are geometric objects defined in tropical geometry. This involves determining where these varieties meet in a way that respects their piecewise linear structure, often utilizing valuations and combinatorial methods. This concept is crucial for applying tropical techniques to problems in algebraic geometry, particularly in the context of Schubert calculus.
Enumerative Geometry: Enumerative geometry is a branch of mathematics that focuses on counting the number of geometric figures that satisfy certain conditions. It plays a vital role in understanding how different geometric configurations relate to algebraic geometry and can be extended to tropical geometry by examining how these counting problems manifest in the tropical setting. This area connects combinatorial aspects with geometric properties, particularly through the use of schemes and their intersection theory.
Giorgio Tenaglia: Giorgio Tenaglia is a mathematician known for his contributions to tropical geometry, particularly in the field of tropical Schubert calculus. His work often focuses on the intersection of combinatorial and algebraic geometry, providing insights into how tropical techniques can be applied to classical problems in Schubert calculus.
Littlewood-Richardson Coefficients: Littlewood-Richardson coefficients are numerical values that appear in the context of representation theory and algebraic combinatorics, particularly related to the multiplication of Schur functions. They count the number of ways to express a product of two Schur functions as a sum of other Schur functions, connecting directly to the structure of the symmetric group and algebraic geometry concepts like Schubert varieties.
Max-plus algebra: Max-plus algebra is a mathematical framework that extends conventional algebra by defining operations using maximum and addition, rather than traditional addition and multiplication. In this system, the sum of two elements is their maximum, while the product of two elements is the standard sum of those elements. This unique approach allows for the modeling of various optimization problems and facilitates the study of tropical geometry, connecting with diverse areas such as geometry, combinatorics, and linear algebra.
Mirror Symmetry: Mirror symmetry is a phenomenon in mathematics, particularly in algebraic geometry and string theory, where two different geometric structures can yield equivalent physical theories or mathematical properties. This concept connects various areas such as complex geometry and tropical geometry, highlighting deep relationships between seemingly unrelated geometrical entities.
Piecewise Linear Structure: A piecewise linear structure is a mathematical framework where functions or equations are defined by multiple linear segments, each applicable to specific intervals of the input variable. This concept allows for the representation of more complex relationships that can change behavior at certain points, enabling a clearer understanding of tropical equations and calculations in algebraic geometry. Such structures facilitate the analysis of intersection properties and geometric configurations, particularly in the context of tropical Schubert calculus.
Schubert Cells: Schubert cells are specific subvarieties within the Grassmannian manifold that arise in Schubert calculus, characterized by their combinatorial structure and defined using geometric conditions on linear subspaces. These cells correspond to the intersection of certain Schubert varieties, allowing for a systematic way to study problems in intersection theory and geometry through their associated tropically interpreted forms.
Stable Intersections: Stable intersections refer to the behavior of tropical varieties at their intersection points, particularly focusing on ensuring that these intersections do not have extraneous components and are 'stable' under certain perturbations. This concept helps in understanding how various tropical objects intersect in a controlled manner, which is crucial for applying results like Bézout's theorem, computing intersection products, and analyzing Schubert calculus in tropical geometry.
Tropical Bézout's theorem: Tropical Bézout's theorem is a fundamental result in tropical geometry that provides a formula for calculating the number of intersection points of two tropical varieties, considering their degrees. It connects algebraic geometry with tropical geometry by showing how the intersection number is related to the combinatorial structure of these varieties, allowing for insights into more complex geometric scenarios.
Tropical Convexity: Tropical convexity refers to a geometric structure that arises in tropical geometry, where the classical notions of convex sets and convex hulls are redefined using the tropical semiring. This concept allows for the study of combinatorial and algebraic properties of sets defined over the tropical numbers, enhancing our understanding of tropical equations, hypersurfaces, and halfspaces.
Tropical determinants: Tropical determinants are a concept in tropical geometry that extends the idea of classical determinants to a piecewise-linear setting, where addition is replaced by the minimum operation and multiplication is replaced by addition. This new perspective allows for a rethinking of algebraic structures in terms of combinatorial geometry and leads to new insights into various mathematical fields, including discriminants and Schubert calculus.
Tropical Enumerative Geometry: Tropical enumerative geometry studies the solutions to geometric counting problems in the framework of tropical mathematics, which uses piecewise linear structures instead of classical algebraic varieties. This field connects tropical geometry to classical enumerative problems, allowing for new interpretations and computations involving counts of curves, intersection numbers, and more, using tropical methods.
Tropical flag variety: A tropical flag variety is a geometric structure that represents flags, which are sequences of vector subspaces, in a tropical setting. These varieties extend classical concepts of flag varieties into the realm of tropical geometry, where the usual operations like addition and multiplication are replaced by tropical addition (taking the minimum) and tropical multiplication (addition). Tropical flag varieties connect to various important mathematical constructs, including tropical Stiefel manifolds and Schubert calculus.
Tropical Grassmannian: The tropical Grassmannian is a combinatorial object that generalizes the classical Grassmannian to tropical geometry, capturing the essence of linear subspaces in a tropical setting. It arises naturally in various contexts, including the study of tropical polytopes and as a tool for understanding tropical varieties through their Plücker coordinates. This framework also connects deeply with concepts like tropical discriminants and Schubert calculus, providing insights into how different geometrical structures can be analyzed through the lens of tropical algebra.
Tropical Gromov-Witten Invariants: Tropical Gromov-Witten invariants are combinatorial invariants that count the number of certain types of tropical curves in a given tropical variety, serving as a tropical analog to classical Gromov-Witten invariants. These invariants help in understanding the geometry of moduli spaces and their compactifications, linking them to enumerative geometry through various structures like Schubert calculus and mirror symmetry.
Tropical Hypersurface: A tropical hypersurface is a geometric object defined in tropical geometry, typically given by a tropical polynomial equation. These hypersurfaces can be thought of as piecewise linear counterparts of classical algebraic varieties, emerging from the notion of taking the maximum (or minimum) of linear functions. Their structure plays a critical role in various mathematical contexts, including the study of tropical powers and roots, interactions with Bézout's theorem, and the analysis of tropical discriminants.
Tropical Intersection Number: The tropical intersection number is a concept in tropical geometry that quantifies the intersection of two tropical varieties. It is a combinatorial invariant that reflects how many points the varieties intersect, taking into account their respective multiplicities. This number plays a crucial role in understanding the structure and relationships of tropical varieties, especially in applications like the tropical Schubert calculus, where it helps in calculating intersections in projective spaces using a tropical approach.
Tropical Intersections: Tropical intersections refer to the points where tropical varieties meet in the context of tropical geometry, providing a way to study intersection theory using piecewise linear structures. This concept allows for the exploration of how these varieties intersect in a tropical setting, often using polyhedral geometry and combinatorial techniques to represent classical intersection problems in a new light. It plays a crucial role in understanding Schubert calculus within tropical geometry, as it enables the computation of intersection numbers in a more manageable way.
Tropical Linear Spaces: Tropical linear spaces are geometric structures that arise in tropical geometry, where the classical notions of linear algebra are adapted to the tropical semiring. In these spaces, points correspond to vectors, and the tropical operations of addition and multiplication replace traditional arithmetic, leading to unique properties and insights in geometry and algebra.
Tropical Linearity: Tropical linearity refers to a concept in tropical geometry where the usual operations of addition and multiplication are replaced with their tropical counterparts. In this framework, addition is interpreted as taking the maximum of two values, while multiplication is treated as ordinary addition. This new perspective allows us to analyze geometric structures like hypersurfaces, polytopes, and halfspaces in a different light, highlighting the rich combinatorial properties and connections between various mathematical concepts.
Tropical Littlewood-Richardson Rule: The Tropical Littlewood-Richardson Rule is a combinatorial formula used to calculate the structure constants of the tropical version of a semigroup algebra associated with a given set of polytopes. This rule connects the classical Littlewood-Richardson coefficients with tropical geometry, enabling the determination of how these coefficients behave under tropical operations. The rule provides a way to compute intersection numbers in tropical geometry, particularly in relation to the computation of Schubert classes in Grassmannians.
Tropical mirror symmetry: Tropical mirror symmetry refers to a conjectural relationship between two different types of geometric objects: the classical mirrors and their tropical counterparts. This concept suggests that certain invariants of a classical algebraic variety can be computed in a tropical setting, highlighting deep connections between algebraic geometry and combinatorial geometry. The interplay between these two realms is crucial for understanding phenomena like duality in algebraic varieties, leading to insights in both tropical Schubert calculus and the Deligne-Mumford compactification.
Tropical plücker coordinates: Tropical plücker coordinates are a set of coordinates that represent linear subspaces in tropical geometry, providing a way to encode the relationships among these subspaces using tropical mathematics. These coordinates help in understanding intersections and various properties of subspaces, connecting closely to concepts like tropical Grassmannians and the computation of Schubert classes in tropical Schubert calculus.
Tropical polynomial: A tropical polynomial is a function formed using tropical addition and tropical multiplication, typically defined over the tropical semiring, where addition is replaced by taking the minimum (or maximum) and multiplication is replaced by ordinary addition. This unique structure allows for the study of algebraic varieties and geometric concepts in a combinatorial setting, connecting them to other areas like optimization and piecewise linear geometry.
Tropical Schubert calculus: Tropical Schubert calculus is a branch of mathematics that combines the principles of tropical geometry with Schubert calculus, which traditionally deals with intersections of subvarieties in algebraic geometry. This approach allows for the study of intersection problems through a combinatorial lens, focusing on tropical varieties and their relationships to classical Schubert problems. It provides new insights into enumerative geometry by transforming classical problems into simpler, piecewise-linear forms that can be analyzed using tropical methods.
Tropical Schubert Class: A tropical Schubert class is a class in the cohomology of a tropical variety that corresponds to a Schubert cycle in a Grassmannian or flag variety. These classes arise from the study of intersection theory in the context of tropical geometry, where they encode geometric information about how subvarieties intersect. They play a crucial role in generalizing classical Schubert calculus, allowing us to compute intersection numbers using tropical methods.
Tropical Schubert Cycles: Tropical Schubert cycles are geometrical constructs that arise in tropical geometry, specifically within the context of Schubert calculus. They represent the tropicalization of classical Schubert varieties, which are important in algebraic geometry for understanding intersections and enumerative geometry problems. These cycles can be used to compute intersection numbers in a tropical setting, facilitating combinatorial techniques in solving classical geometric questions.
Tropical Schubert varieties: Tropical Schubert varieties are geometric objects that arise in tropical geometry, representing solutions to intersection problems in the context of toric varieties. These varieties provide a tropical analogue of classical Schubert varieties, capturing combinatorial properties of intersections of linear subspaces and their relations to algebraic geometry through piecewise-linear structures. They play a key role in tropical Schubert calculus, linking combinatorial methods with geometric intuition.
Tropical tableaux: Tropical tableaux are combinatorial structures that extend the concept of standard Young tableaux into the realm of tropical geometry. They are used to study the properties of tropical polytopes and relate to problems in tropical Schubert calculus, helping to compute intersections of tropical varieties and provide insights into the enumerative geometry of these objects.
Tropicalization: Tropicalization is the process of translating algebraic varieties and their properties into a piecewise-linear setting using tropical geometry. This allows for the study of complex geometric structures through combinatorial means, enabling a more accessible approach to problems involving algebraic curves and surfaces.
Tropicalization of schubert varieties: The tropicalization of Schubert varieties is the process of transforming classical Schubert varieties, which are intersections of subspaces in a projective space, into their tropical counterparts in tropical geometry. This transformation uses the tropical semiring, where addition is replaced by taking minimums and multiplication by addition, resulting in a piecewise linear structure that preserves combinatorial information about the original varieties. Tropicalization provides powerful tools for solving problems in enumerative geometry by allowing for easier computation and visualization.
Valuation: In the context of tropical geometry, a valuation is a function that assigns a value to elements in a field, capturing information about their geometric properties. This concept plays a crucial role in defining tropical equations and polynomial functions, influencing the structure of curves and surfaces. Valuations allow for the study of algebraic varieties through their tropical counterparts, providing a bridge between classical algebraic geometry and its tropical analogs.