Alphabeta MathSoergel Intersection Forms and Hodge Theory — Examples
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- 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
- 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 Field of Fractions and Localisation
Braid Groups
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
Coxeter Groups
Homological Algebra
- Bounded Bimodule Complexes and Derived Tensor
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Derived Functors
- Ext and Balanced Resolutions
- Graded Bimodules and Tensor Functors
- Homological Gaussian Elimination
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Triangulated Categories
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
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- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
- Topology of ℝ
Representation Theory of Groups
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Symmetric Groups and Symmetric Functions
2 · Logical flowchart
3 · Items
Example
The rank-two Bott-Samelson bimodule BₛBₜ: character, decomposition and the braid relation
Example
The coinvariant algebra of S₃ as the Soergel bimodule of the longest element
Counterexample
Hard Lefschetz fails for a non-dominant degree-two element: the rank-one case
Example
The rank-one Soergel bimodule: trace form, reduced module and hard Lefschetz
Example
Hard Lefschetz and the Hodge-Riemann signs for the coinvariant algebra of S₃
Sources
- Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136)
- Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525)
- Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017)