Alphabeta MathBruhat Decomposition and Flags over Finite Fields
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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
2 · Logical flowchart
3 · Items
Definition
Standard subgroups of finite general linear groups
Theorem
Complete flags are G/B
Definition
Compositions, partial flags, and standard parabolics
Theorem
Block Levi decomposition of standard parabolics
Definition
Permutation Weyl group and inversion length
Lemma
Triangular elimination produces a pivot permutation
Lemma
Southwest rank matrices determine Bruhat cells
Theorem
Bruhat decomposition of GLₙ over a finite field
Theorem
Relative position classifies pairs of complete flags
Proposition
Cardinality of a finite Bruhat cell
Definition
Harish-Chandra induction and restriction for finite general linear groups
Lemma
Finite-group invariants are exact when the group order is invertible
Theorem
Harish-Chandra induction is left adjoint to restriction
Lemma
Parabolic double cosets and block permutations
Lemma
Unipotent double-coset biset splitting
Definition
Ordered partitions and coordinate parabolics
Theorem
Parabolic Mackey formula for finite GLₙ
Theorem
Transitivity and parabolic independence of Harish-Chandra induction
Definition
Cuspidal representations and Harish-Chandra series
Theorem
Existence and uniqueness of cuspidal support
Sources
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5, Lemma 4.7 and Example 8.4, printed pp. 18 and 30
- Jay Taylor, Finite Reductive Groups - Exercise 4.28 and Definition 5.2, printed pp. 38-39 and 42
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5 and Example 8.4, printed pp. 18 and 30
- Jay Taylor, Finite Reductive Groups - Exercise 4.28, printed pp. 38-39
- 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 - Example 8.4(a) and Proposition 8.1, printed pp. 30 and 29
- Jay Taylor, Finite Reductive Groups - Section 5.1, printed p. 42
- Jay Taylor, Finite Reductive Groups - Sections 4.7 and 5.3, printed pp. 38-39 and 46
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5 and Lemma 4.7, printed p. 18
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5, printed p. 18
- 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
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Proposition 9.4(ii), printed p. 35
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Proposition 9.4(i), printed p. 35
- Jay Taylor, Finite Reductive Groups - Exercise 5.3, printed p. 42
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 4.5 and Lemma 4.7, printed p. 18; Example 8.4, p. 30; Lemma 9.9, printed p. 40
- Jay Taylor, Finite Reductive Groups - Bruhat section pp. 37-39; Harish-Chandra section pp. 41-43
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Lemma 9.11 and its proof, printed p. 39
- Jay Taylor, Finite Reductive Groups - Harish-Chandra section, printed pp. 41-43
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Example 8.4 and the proof of Lemma 9.9, printed pp. 30 and 40
- Jay Taylor, Finite Reductive Groups - Sections 3.5, 4.7 and 5.2, printed pp. 37-39 and 41-43
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Theorem 9.6 and its proof, printed pp. 36-37; Lemma 9.10, printed pp. 38-39; Lemma 9.11, printed p. 39
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Proposition 9.5 and Theorem 10.1, printed pp. 36 and 41
- Jay Taylor, Finite Reductive Groups - Lemma 5.5, printed p. 43
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Definition 10.2, Lemma 10.3 and Definition 10.5, printed pp. 41-42
- Jay Taylor, Finite Reductive Groups - Definition 5.7, printed p. 43
- Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Lemma 10.3 and Proposition 10.6, printed pp. 41-43
- Jay Taylor, Finite Reductive Groups - Definition 5.7 and Proposition 5.9, printed pp. 43-44