Soergel Diagrammatics for DihedralGroupsBen EliasSubmitted in partial fulfillment of therequirements for the degree ofDoctor of Philosophyin the Graduate School of Arts and SciencesColumbia University2011c2011Ben EliasAll rights reservedAbstractSoergel Diagrammatics for Dihedral GroupsBen EliasWe give a diagrammatic presentat io n for the category of Soergel bimodules for the dihedralgroup W , finite or infinite. The (two-colored) Temperley-Lieb category is embedded insidethis category as the degree 0 morphisms between color-alternating objects. The indecom-posable Soergel bimodules are the images of Jones-Wenzl projectors. When W is finite, theTemperley-Lieb category must be taken at a n appropriate root of unity, and t he negligibleJones-Wenzl projector yields the Soergel bimo dule for the longest element of W .Contents1 Introduction 11.1 Soergel Bimo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Temperley-Lieb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Background 132.1 The dihedral group and its Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The Soergel categorification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Temperley-Lieb categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4 Main Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Dihedral Diagrammatics: m = ∞ 473.1 The category D(∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Singular Soergel Bimodules: m = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Dihedral Diagrammatics: m < ∞ 684.1 The category Dm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Singular Soergel Bimodules: m < ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Thickening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4 Temperley-Lieb categorifies Temperley-Lieb . . . . . . . . . . . . . . . . . . . . . . . 90iAcknowledgementsAbove all, I thank my advisor, Mikhail Khovanov, who has singlehandedly made math-ematics a much more exciting place for me. His generosity is unbounded. Without hisguidance and his plethora of brilliant ideas, I would not be where I am today.Special thanks are due to Noah Snyder and Geordie Williamson. Both gentlemen haveanswered my questions cheerily, time and again, in fruitful conversations without number.This thesis in particular owes much to their suppo r t .I extend my gratitude to the o ther Columbia faculty who have taught me and helpedme along my path. Thanks go out to Aise Johan de Jong, Michael Thaddeus, and AaronLauda, and to the members of my thesis committee: Sabin Cautis, Rachel Ollivier, MelissaLiu and D mitri Orlov.The acceptable state of my menta l health is due in large part to the wonderful studentsof the Columbia math department, who have made these five years a blast. If I listed youall here, t here would be no room left for lemmas. My friends from college and high schoolplayed no inconsequential part as well. I place my gratitude for you all in a small box, alongwith many o ther sentiments which rhyme (none of which are “platitudes”).One other man has made everything possible. Thank you Terrance Cope. You are therock upo n which Columbia’s math department is built.Finally, my family. Thank you Mom and Dad, Dan and Sarah, for everything, andeverything. Thank you Julius, Dorothy, and Seymour. You remain inspirations to me.iiTo my parentsiii1Chapter 1Introduction1.1 Soergel Bimod ules1.1.1 The construc tionGiven a n additive graded monoidal category C, its additive (i.e. split) Grothendieck group[C] has the natural structure of aZ[v, v−1]-algebra. Multiplication by v corresponds to t hegrading shift {1}. We say that C is a categorification of [C]. When C has the Krull-Schmidtproperty, the ring [C] will have aZ[v, v−1]-basis given by the classes of indecomp osableobjects (up to grading shift). We will use indecomposable as a noun, to refer to an indecom-posable object.Let (W, S) be any Coxeter group, finite or infinite, equipped with a natural reflectionrepresentation h. We are interested in categorifications of H, the associated Hecke algebraof W . When W is a Weyl group, geometric representation theory provides us with a naturalcategorification of H, using equivariant perverse sheaves on the flag variety. This constructiondoes not generalize to an arbitrary Coxeter g r oup, though it does have analogs for affine Weylgroups and other crystallographic Coxeter groups. In the early 1990s, Soergel explored whathappens when one takes the hypercohomology of a semisimple equivariant perverse sheaf onthe flag variety. This will naturally be a graded bimodule over the polynomial ring R =C[h]2(with linear terms graded in degree 2). Examining the properties of the bimodules whichappear, Soergel defined a class of R-bimodules, now called Soergel bimodules. These canbe defined for any Coxeter g roup W (agreeing with the hypercohomology bimodules in theWeyl group case), and they categorify H. In other words, Soergel bimodules are an algebraicreplacement for flag varieties, in situations with no ambient geometry. In a similar fashion,Soergel bimodules are an algebraic replacement for Harish-Chandra bimodules acting onthe BGG category O. We refer the reader to [27] for a purely alg ebraic account of Soergelbimodules, and to numerous other papers [23, 24, 25, 26] for the complete story.Defining Soergel bimodules is a simple matter. Let us call a subset J ⊂ S finitary ifthe corresponding parabolic subgroup WJ⊂ W is finite. The ring R is naturally equippedwith a W -action, and for any finitary J ⊂ S one may t ake the subring RJ⊂ …