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Ikenmeyer, Christian: Geometric complexity theory, tensor rank, and Littlewood-Richardson coefficients. 2012
Inhalt
Introduction
Main Results
Outline and Further Results
Part I: Geometric Complexity Theory
Part II: Littlewood-Richardson Coefficients
I Geometric Complexity Theory
Preliminaries: Geometric Complexity Measures
Circuits and Algebraic Complexity Theory
Completeness and Reduction
Approximating Polynomials
Complexity of Bilinear Maps
Tensor Rank and Border Rank
Summary and Unifying Notation
Preliminaries: The Flip via Obstructions
Classical Algebraic Geometry
Linear Algebraic Groups and Polynomial Obstructions
Representation Theoretic Obstructions
Coordinate Rings of Orbits
Geometric Invariant Theory
Peter-Weyl Theorem
Preliminaries: Classical Representation Theory
Young Tableaux
Explicit Highest Weight Vectors
Polarization, Restitution, and Projections
Schur-Weyl Duality and Highest Weight Vectors
Plethysm Coefficients
Plethysm Coefficients and Weight Spaces
Kronecker Coefficients
Littlewood-Richardson Coefficients
Coordinate Rings of Orbits
Stabilizers
Tensors
Polynomials
Branching Rules
Unit Tensor
Determinant, Permanent and Matrix Multiplication
Inheritance
Generic Tensors
Stability and Exponent of Regularity
Representation Theoretic Results
Kernel of the Foulkes-Howe Map
Even Partitions in Plethysms
Nonvanishing of Symmetric Kronecker Coefficients
Moment Polytopes
Asymptotic Result
Obstruction Designs
Set Partitions
Obstruction Designs
Symmetric Obstruction Designs
Reduced Kronecker Coefficients
Explicit Obstructions
Some Computations
Orbit-wise Upper Bounds
Regular Determinant Function
m x m Matrix Multiplication
Vanishing on the Unit Tensor Orbit
Evaluation at the Matrix Multiplication Tensor
Further Results
Orbit-wise Upper Bound Proof
2 x 2 Matrix Multiplication
Some Negative Results
SL-obstructions
Cones and Saturated Semigroups
II Littlewood-Richardson coefficients
Hive Flows
Flow Description of LR Coefficients
Flows on Digraphs
Flows on the Honeycomb Graph G
Hives and Hive Flows
Properties of Hive Flows
The Support of Flows on G
The Graph of Capacity Achieving Integral Hive Flows
The Residual Digraph
Turnpaths and Turncycles
Flatspaces
The Rerouting Theorem
Algorithms
A First Max-flow Algorithm
A Polynomial Time Decision Algorithm
Enumerating Hive Flows
The Neighbourhood Generator
A First Approach
Bypassing the Secure Extension Problem
Proofs
Proof of the Rerouting Theorem
Canonical Turnpaths in Convex Sets
Parallelograms
Trapezoids, Pentagons, and Hexagons
Triangles
Proof of the Shortest Turncycle Theorem
Special Rhombi
Rigid and Critical Rhombi
Proof of the Connectedness Theorem
Proof of the King-Tollu-Toumazet Conjecture
Proof of Proposition 12.4.4
Proof of Key Lemma 12.4.5
Proof of Lemma 12.4.11
Appendix
Calculations
Obstruction Candidates
List of Algorithms
Bibliography
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