Bruhat Decomposition

Definition

Bruhat decomposition is a way to express a semisimple algebraic group as a disjoint union of double cosets, which reflects the group’s structure in relation to its Borel subgroups and Weyl groups. This decomposition is fundamental in understanding the geometric properties of flag varieties and their associated representations, connecting algebraic groups to geometric realizations.

5 Must Know Facts For Your Next Test

1. The Bruhat decomposition expresses a semisimple algebraic group as a disjoint union of double cosets formed by Borel subgroups and representatives from the Weyl group.
2. Each double coset corresponds to a unique set of points in the associated flag variety, showcasing how these mathematical structures are interconnected.
3. The decomposition has important implications in representation theory, particularly in understanding the irreducible representations of semisimple groups.
4. Bruhat decomposition helps in constructing cell decompositions of flag varieties, allowing for a better understanding of their topology and geometry.
5. The concept also extends to the study of Schubert varieties, which are closures of certain orbits under the action of the Borel subgroup, highlighting further connections to algebraic geometry.

Review Questions

• How does Bruhat decomposition relate semisimple algebraic groups to their Borel subgroups and Weyl groups?
  • Bruhat decomposition provides a structured way to break down semisimple algebraic groups into manageable pieces by using Borel subgroups and Weyl groups. It describes the group as a disjoint union of double cosets, where each coset corresponds to an orbit under the action of a Borel subgroup. This relationship illustrates how these subgroups act on the group itself and highlights their significance in both representation theory and geometric realizations.
• Discuss the role of Bruhat decomposition in the study of flag varieties and their topological properties.
  • Bruhat decomposition plays a crucial role in understanding flag varieties by providing a cell decomposition that reflects their geometric structure. Each double coset corresponds to specific orbits within the flag variety, allowing for insights into its topology. This decomposition aids in analyzing how different subspaces interact, contributing to our overall comprehension of the variety's geometric features and their applications in representation theory.
• Evaluate the impact of Bruhat decomposition on representation theory and its significance in modern algebraic geometry.
  • Bruhat decomposition significantly impacts representation theory by providing a clear framework for studying irreducible representations of semisimple groups through their interactions with Borel subgroups and Weyl groups. This relationship allows mathematicians to classify representations based on geometric properties. Additionally, its relevance extends to modern algebraic geometry through applications in Schubert calculus and cell decompositions, linking abstract algebraic concepts to concrete geometric interpretations.

Related terms

Borel Subgroup: A maximal connected solvable subgroup of a linear algebraic group, which plays a key role in the structure of the group and its representation theory.

Flag Variety: A geometric object that parameterizes nested chains of vector subspaces, often associated with representations of algebraic groups.

Weyl Group: A group that describes symmetries of a root system, crucial for understanding the representation theory of semisimple Lie algebras and groups.