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Diagrammatic Algebra provides the intuition and tools necessary to address some of the key questions in modern representation theory, chief among them Lusztig’s conjecture. This book offers a largely self-contained introduction to diagrammatic algebra, culminating in an explicit and entirely diagrammatic treatment of Geordie Williamson’s explosive torsion counterexamples in full detail. The book begins with an overview of group theory and representation theory: first encountering Coxeter groups through their actions on puzzles, necklaces, and Platonic solids; then building up to non-semisimple representations of Temperley–Lieb and zig-zag algebras; and finally constructing simple representations of binary Schur algebras using the language of coloured Pascal triangles. Next, Kazhdan–Lusztig polynomials are introduced, with their study motivated by their combinatorial properties. The discussion then turns to diagrammatic Hecke categories and their associated p-Kazhdan–Lusztig polynomials, explored in a hands-on manner with numerous examples. The book concludes by showing that the problem of determining the prime divisors of Fibonacci numbers is a special case of the problem of calculating p-Kazhdan–Lusztig polynomials—using only elementary diagrammatic calculations and some manipulation of (5x5)-matrices. Richly illustrated and assuming only undergraduate-level linear algebra, this is a particularly accessible introduction to cutting-edge topics in representation theory. The elementary-yet-modern presentation will also be of interest to experts. Foreword Preface What you will have learnt by the end of this book Groups Algebras and Representation theory Combinatorics Categorification Group theory versus diagrammatic algebra Rings and things Categorification without categories Too much symmetry and not enough group? Further reading Acknowledgements Contents Symmetries Size isn't everything Matrix groups Abstract groups and the Sudoku property Presenting groups Isomorphisms Subgroups and direct products of groups Coxeter groups and the 15-puzzle The symmetric group The length function and the alternating group The 15-puzzle The hyper-octahedral groups Groups of decorated strand diagrams Coxeter groups Composition series Cosets and Lagrange's theorem Simple groups and normal subgroups Homomorphisms The Jordan–Hölder Theorem Platonic and Archimedean solids and special orthogonal groups The platonic solids Group actions The symmetry group of the tetrahedron The symmetry group of the cube The symmetry group of the dodecahedron The classification of 3-dimensional rotational symmetry groups Archimedean solids and their symmetry groups Non-invertible symmetry Algebras The Temperley–Lieb algebras Idempotents and things Gradings and zig-zag algebras Oriented Temperley–Lieb algebras The binary Schur algebra Hidden gradings on Schur algebras in positive characteristic Representation theory Examples of simple modules and representations Temperley–Lieb algebras as the prototypical cellular algebras The smallest non-semisimple Schur algebra The symmetric group on 3 letters Semisimple filtrations of non-semisimple modules Weighted cellular algebras and gradings The grading and idempotent tricks for intersection forms Simple modules of zig-zag algebras Simple modules of the binary Schur algebra Decomposition numbers via highest weight theory Alperin diagrams and submodule structures Truncations, quotients, and saturation. Catalan combinatorics within Kazhdan–Lusztig theory Quantum binomial coefficients The poset of partitions in a rectangle Kazhdan–Lusztig polynomials Tile-partitions, weights, and Bruhat graphs Oriented Temperley–Lieb combinatorics General Kazhdan–Lusztig theory Weak Bruhat graphs of parabolic Coxeter systems Kazhdan–Lusztig polynomials in full generality The strong Bruhat order and combinatorial-invariance The infinite dihedral group Kazhdan–Lusztig combinatorics? The diagrammatic algebra for SmSnSm+n From paths to diagrammatic algebras The one colour relations The multi-colour relations for SmSn Sm+n How to manipulate diagrams Return of the zig-zag algebras Cellularity of (W,P) for (W,P)= (Sm+n, SmSn ) The categorification theorem Lusztig's conjecture in the diagrammatic algebra (W,P) From paths to diagrammatic algebras (again) Multi-colour relations Fun with braids Light leaves and the Kazhdan–Lusztig positivity conjecture Calculating intersection forms Counterexamples to Soergel's conjecture in positive characteristic Picking pairs of permutations Fibonacci numbers as values of intersection forms The base case of the proof An observation and inductive reformulation Counterexamples to ``Soergel's conjecture'' Affinization: from Soergel's to Lusztig's conjecture What might the future hold? Reformulating Lusztig's and Andersen's conjectures The symmetric group and its representation theory Generalised Temperley–Lieb algebras Schur algebras Lusztig's and Andersen's conjectures Hidden gradings on symmetric groups The tableaux-theoretic LLT algorithm Bringing together LLT and Kazhdan–Lusztig polynomials The quiver Hecke algebra The Brundan–Kleshchev isomorphism theorem The Hu–Mathas cellular basis The quiver Temperley–Lieb algebra The p-Kazhdan–Lusztig theory for Temperley–Lieb algebras Graded path combinatorics of the Temperley–Lieb algebra Hyperplane-coloured residue sequences Recolouring the quiver Temperley–Lieb algebra The isomorphism theorem Fork-annihilation in the KLR algebra Fork-spot contraction in the KLR algebra Isotopy in the KLR algebra The monochrome barbell relation in the KLR algebra Bijectivity and bases References Index