Alphabeta MathBruhat Decomposition and Flags over Finite Fields — Examples
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Characters and the Orthogonality Relations
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Induced Representations, Frobenius Reciprocity and Applications
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Coxeter Groups
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
Representation Theory of Groups
2 · Logical flowchart
3 · Items
Example
GL₁ and the trivial parabolic endpoints
Example
Flags and Bruhat cells for GL₂(F_q)
Example
The six relative positions of GL₃ flags
Example
Grassmannians as maximal parabolic quotients
Example
Parabolic induction of the trivial module as flag functions
Sources
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5, Definition 9.2 and Definition 10.2, printed pp. 18 and 38-41
- Jay Taylor, Finite Reductive Groups - Definition 5.7 and Proposition 5.9, printed pp. 43-44
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5, printed p. 18
- Jay Taylor, Finite Reductive Groups - Section 3.5, printed pp. 37-39
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 8.4(a), printed p. 30
- Jay Taylor, Finite Reductive Groups - Sections 3.5 and 4.7, printed pp. 37-39
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Definition 9.2 and Remark 9.3, printed p. 35
- Jay Taylor, Finite Reductive Groups - Definition 5.2, printed p. 42