Introduction to Soergel Bimodules

Ben Elias hails from sunny Massachusetts and is well known for his wit and ability to do diagrammatic calculations like no other. He’s also been known to solve a crossword at the same time as playing a Bach prelude. He is currently a professor at the University of Oregon.Shotaro Makisumi (known to his friends as “Macky”) is originally from Ube, Japan. He studied at Stanford and is currently at Columbia. Despite his successes proving theorems, he is still best known for his extraordinary skills with the Rubik’s Cube.Ulrich Thiel studied in Kaiserslautern, Germany, and after a brief voyage around the globe, recently returned there to take up a professorship. Let’s be honest: he’s the brains behind the operation.Geordie Williamson grew up with the wombats and kangaroos in rural Australia. He completed his PhD under Soergel’s supervision, and after stints in Oxford and Bonn, returned to his furry friends in 2017.All four authors delight in challenging each other to bouts of sumo, before retiring to a traditional meal of chanko nabe.

• - Part I The Classical Theory of Soergel Bimodules. - How to Think About Coxeter Groups. - Reflection Groups and Coxeter Groups. - The Hecke Algebra and Kazhdan–Lusztig Polynomials. - Soergel Bimodules. - The “Classical” Theory of Soergel Bimodules. - Sheaves on Moment Graphs. - Part II Diagrammatic Hecke Category. - How to Draw Monoidal Categories. - Frobenius Extensions and the One-Color Calculus. - The Dihedral Cathedral. - Generators and Relations for Bott–Samelson Bimodules and the Double Leaves Basis. - The Soergel Categorification Theorem. - How to Draw Soergel Bimodules. - Part III Historical Context: Category O and the Kazhdan–Lusztig Conjectures. - Category O and the Kazhdan–Lusztig Conjectures. - Lightning Introduction to Category O. - Soergel’s V Functor and the Kazhdan–Lusztig Conjecture. - Lightning Introduction to Perverse Sheaves. - Part IV The Hodge Theory of Soergel Bimodules - Hodge Theory and Lefschetz Linear Algebra. - The Hodge Theory of Soergel Bimodules. - Rouquier Complexes and Homological Algebra. - Proof of the Hard Lefschetz Theorem. - Part V Special Topics. - Connections to Link Invariants. - Cells and Representations of the Hecke Algebra in Type A. - Categorical Diagonalization. - Singular Soergel Bimodules and Their Diagrammatics. - Koszul Duality I. - Koszul Duality II. - The p-Canonical Basis.

“This book paints the theory of Soergel bimodules as a very active and lively area of current research. The book is very well written and organized. Readers will find a wealth of useful information on the theory and applications of Soergel bimodules, much of which … will be of interest beyond the focus of the book.” (‪Götz Pfeiffer, Mathematical Reviews, September, 2022)