Alphabeta MathType-A Soergel Bimodules and Hecke Categorification
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
Braid Groups
Category Theory
Commutative Algebra
Coxeter Groups
Homological Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
The standard type-A reflection realization and its polynomial ring
Lemma
Type-A reduced words and the Coxeter presentation
Definition
The type-A Hecke algebra in Soergel normalization
Lemma
The standard basis of the type-A Hecke algebra and its multiplication rule
Definition
The Soergel bimodule Bᵢ of a simple reflection
Lemma
Soergel generators and Bott–Samelson products are finite free on both sides
Definition
The Bott–Samelson bimodule of a word
Definition
The type-A Soergel category SBimₙ
Definition
Standard graph bimodules, support filtrations and characters
Lemma
Type-A graph-bimodule extension vanishing
Lemma
Bott–Samelson bimodules carry delta and nabla support filtrations
Lemma
The type-A support filtration multiplicities are intrinsic
Lemma
Frobenius biadjunction for the type-A Soergel generators
Lemma
The type-A character recursion under simple Soergel tensoring
Lemma
Special Bott–Samelson Hom formula before reflection localization
Lemma
Type-A top support layers are controlled by reflection localization
Theorem
The type-A Soergel Hom formula
Lemma
The rank-one Soergel bimodule square splits
Lemma
Distant Soergel generators commute
Definition
The rank-two longest type-A Soergel bimodule
Theorem
Rank-two type-A Soergel bimodule decompositions
Definition
The type-A diagrammatic Soergel category and its candidate bimodule functor
Definition
Split Grothendieck rings of the type-A Soergel categories
Lemma
The type-A diagrammatic relations hold for Soergel bimodules
Theorem
Double leaves form graded R-bases of type-A diagrammatic Hom spaces
Theorem
Indecomposable type-A diagrammatic Soergel objects are indexed by permutations and shifts
Theorem
The diagrammatic character is the split K₀ Hecke isomorphism
Theorem
Light leaf maps form bases of type-A Soergel homs to the unit
Theorem
Evaluated double leaves form bases of type-A Soergel bimodule homs
Theorem
The type-A diagrammatic and bimodule Soergel categories are equivalent
Theorem
The split Grothendieck group of the Soergel category is the type-A Hecke algebra
Lemma
The type-A standard character is multiplicative
Sources
- Elias–Williamson, Soergel Calculus, §§3, 5–7
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §§2–5
- Elias–Williamson, Soergel Calculus, §2.1, PDF pp.13–14
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §3, PDF pp.11–13
- Elias–Williamson, Soergel Calculus, §2.1, PDF pp.13–15
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §3.1, PDF pp.12–13
- Libedinsky, Gentle Introduction to Soergel Bimodules I, Lemma 3.1, PDF p.13
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, §§5–6
- Elias–Williamson, Soergel Calculus, §§3.4–3.6
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Lemma 5.8 and §5
- Elias–Williamson, Soergel Calculus, §3.4
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Proposition 5.7(1), 5.9(1), PDF pp.13–16
- Elias–Williamson, Soergel Calculus, §§3.4, 5–7
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Bemerkung 5.5 and 5.12, PDF pp.12, 17
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Proposition 5.10, PDF p.16
- Elias–Williamson, Soergel Calculus, §3.3, PDF pp.23–24
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Propositions 5.7 and 5.9, PDF pp.13–16
- Elias–Williamson, Soergel Calculus, §§2.1, 3.4, PDF pp.13–15, 24–27
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Theorem 5.15 and its proof, PDF p.17
- Elias–Williamson, Soergel Calculus, §3.5
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Notation 6.5, Lemma 6.10, Lemma 6.13 and Satz 6.6, PDF pp.19–21
- Elias–Williamson, Soergel Calculus, §3.6
- Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Theorem 5.15, Lemma 6.13, Satz 6.14, PDF pp.17–22
- Elias–Williamson, Soergel Calculus, §3.5 equation (3.6), PDF p.28
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §4.3, PDF pp.22–26
- Elias–Williamson, Soergel Calculus, §3.4 and p. 6, PDF pp. 5–6, 24–27
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §4.2, PDF pp. 21–24
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §4.3–4.4, PDF pp. 22–26
- Elias–Williamson, Soergel Calculus, §5.1 and p. 5, PDF pp. 5–6, 37–40
- Elias–Khovanov, Diagrammatics for Soergel Categories, Definition 3.8 and §5.1, PDF pp. 27–33, 49–55
- Libedinsky, Sur la catégorie des bimodules de Soergel, §§4.4–4.6 and Théorème 5.1
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §§4–5
- Elias–Williamson, Soergel Calculus, §3.5, PDF pp. 27–29
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §4.1, PDF pp. 20–22
- Elias–Khovanov, Diagrammatics for Soergel Categories, §5.1 with Definition 3.8 and Claim 5.1, PDF pp. 27–33, 49–55
- Elias–Williamson, Soergel Calculus, §5.1, PDF pp. 37–40
- Elias–Williamson, Soergel Calculus, Theorem 6.11, Construction 6.1–6.9, §7 with Proposition 7.6, PDF pp. 58–63, 71–73
- Libedinsky, Sur la catégorie des bimodules de Soergel, §§3–5
- Elias–Williamson, Soergel Calculus, Theorem 6.11, Lemma 6.24, Theorem 6.25, PDF pp. 63–68
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §5
- Elias–Williamson, Soergel Calculus, §6.5, Definition 6.23 with (6.3)–(6.4), Corollaries 6.26–6.27, PDF pp. 65–68
- Elias–Williamson, Soergel Calculus, §6.1 Construction 6.1, §6.2 Proposition 6.12, §6.7 Remark 6.29, PDF pp. 57–63, 69
- Libedinsky, Sur la catégorie des bimodules de Soergel, §6 Definition 6.1 with Lemma 3.3
- Libedinsky, Gentle Introduction to Soergel Bimodules I, §§6.4–6.5, Theorems 6.3–6.4
- Elias–Williamson, Soergel Calculus, §6.2–6.3, Theorem 6.11 with Proposition 6.9, PDF pp. 61–63