Littlewood-Richardson coefficients

Definition

Littlewood-Richardson coefficients are numerical values that arise in the representation theory of symmetric groups and play a crucial role in the study of flag varieties and Schubert calculus. These coefficients count the ways to decompose the product of two Schur functions into a sum of other Schur functions, effectively linking combinatorial properties with algebraic geometry. Their significance extends beyond counting; they also provide valuable insights into the structure of vector bundles on flag varieties and their cohomological properties.

5 Must Know Facts For Your Next Test

1. The Littlewood-Richardson coefficients can be computed using various combinatorial techniques, such as diagrams or tableaux, which visually represent the partitions involved.
2. These coefficients satisfy several important symmetry properties, including the fact that they are zero unless certain conditions on the partitions are met, making them useful in many algebraic contexts.
3. Littlewood-Richardson coefficients play a critical role in calculating intersection numbers on flag varieties, which is essential for understanding their geometric properties.
4. They are closely related to the concept of GL_n (general linear group) representations and help determine how different representations combine.
5. The coefficients have applications in diverse areas such as algebraic geometry, combinatorics, and mathematical physics, showcasing their broad relevance.

Review Questions

• How do Littlewood-Richardson coefficients relate to Schur functions and what is their combinatorial interpretation?
  • Littlewood-Richardson coefficients relate to Schur functions by expressing how the product of two Schur functions can be decomposed into a sum of other Schur functions, each multiplied by a corresponding coefficient. Combinatorially, these coefficients can be interpreted using Young tableaux, which are filled diagrams that represent partitions. By counting specific ways to fill these tableaux, we can determine the value of the Littlewood-Richardson coefficient for given partitions.
• Discuss the significance of Littlewood-Richardson coefficients in the context of flag varieties and their intersection theory.
  • In the context of flag varieties, Littlewood-Richardson coefficients are essential for computing intersection numbers, which describe how subvarieties intersect within these geometric spaces. This is important because it provides insights into the geometry of flag varieties and their cohomological characteristics. The intersection theory facilitated by these coefficients helps us understand the behavior of vector bundles over flag varieties and their applications in algebraic geometry.
• Evaluate how Littlewood-Richardson coefficients contribute to our understanding of representation theory and its applications across different fields.
  • Littlewood-Richardson coefficients significantly contribute to representation theory by revealing how different representations of groups combine and interact. They encapsulate essential information about decomposition patterns in representation theory, which is applicable in various fields such as algebraic geometry, combinatorics, and mathematical physics. Their broad applicability demonstrates how deep algebraic concepts can provide critical insights into other areas of mathematics, facilitating advancements across disciplines.

Related terms

Schur functions: Symmetric functions that can be indexed by partitions, representing characters of symmetric groups and playing a key role in the representation theory.

Flag varieties: Geometric spaces that parameterize nested sequences of vector spaces, serving as important examples in algebraic geometry and representation theory.

Cohomology:

A mathematical tool used to study topological spaces through algebraic means, providing insights into the structure and properties of various spaces, including flag varieties.