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Full Description
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.
Contents
Part I: Groups.- 1 Symmetries.- 2 Coxeter groups and the 15 puzzle.- 3 Composition series.- 4 Platonic and Archimedean solids and special orthogonal groups.- Part II: Algebras and representation theory.- 5 Non-invertible symmetry.- 6 Representation theory.- Part III: Combinatorics.- 7 Catalan combinatorics within Kazhdan-Lusztig theory.- 8 General Kazhdan—Lusztig theory.- Part IV: Categorification.- 9 The diagrammatic algebra for Sₘ x Sₙ ≤ Sₘ₊ₙ.- 10 Lusztig's conjecture in the diagrammatic algebra H(W,P).- Part V: Group theory versus diagrammatic algebra.- 11 Reformulating Lusztig's and Andersen's conjectures.- 12 Hidden gradings on symmetric groups.- 13 The 푝-Kazhdan-Lusztig theory for Temperley-Lieb algebras.
