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Soergel Diagrammatics for Dihedral Groups Ben Elias Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences Columbia University 2011 c 2011 Ben Elias All rights reserved Abstract Soergel Diagrammatics for Dihedral Groups Ben Elias We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W finite or infinite The two colored Temperley Lieb category is embedded inside this category as the degree 0 morphisms between color alternating objects The indecomposable Soergel bimodules are the images of Jones Wenzl projectors When W is finite the Temperley Lieb category must be taken at an appropriate root of unity and the negligible Jones Wenzl projector yields the Soergel bimodule for the longest element of W Contents 1 Introduction 1 1 1 Soergel Bimodules 1 1 2 Temperley Lieb 6 1 3 Structure of the paper 12 2 Background 2 1 The dihedral group and its Hecke 2 2 The Soergel categorification 2 3 Temperley Lieb categories 2 4 Main Techniques algebra 13 13 23 35 43 3 Dihedral Diagrammatics m 47 3 1 The category D 48 3 2 Singular Soergel Bimodules m 61 4 Dihedral Diagrammatics m 4 1 The category Dm 4 2 Singular Soergel Bimodules m 4 3 Thickening 4 4 Temperley Lieb categorifies Temperley Lieb i 68 69 74 87 90 Acknowledgements Above all I thank my advisor Mikhail Khovanov who has singlehandedly made mathematics a much more exciting place for me His generosity is unbounded Without his guidance 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 have answered my questions cheerily time and again in fruitful conversations without number This thesis in particular owes much to their support I extend my gratitude to the other Columbia faculty who have taught me and helped me along my path Thanks go out to Aise Johan de Jong Michael Thaddeus and Aaron Lauda and to the members of my thesis committee Sabin Cautis Rachel Ollivier Melissa Liu and Dmitri Orlov The acceptable state of my mental health is due in large part to the wonderful students of the Columbia math department who have made these five years a blast If I listed you all here there would be no room left for lemmas My friends from college and high school played no inconsequential part as well I place my gratitude for you all in a small box along with many other sentiments which rhyme none of which are platitudes One other man has made everything possible Thank you Terrance Cope You are the rock upon which Columbia s math department is built Finally my family Thank you Mom and Dad Dan and Sarah for everything and everything Thank you Julius Dorothy and Seymour You remain inspirations to me ii To my parents iii 1 Chapter 1 Introduction 1 1 1 1 1 Soergel Bimodules The construction Given an additive graded monoidal category C its additive i e split Grothendieck group C has the natural structure of a Z v v 1 algebra Multiplication by v corresponds to the grading shift 1 We say that C is a categorification of C When C has the Krull Schmidt property the ring C will have a Z v v 1 basis given by the classes of indecomposable objects up to grading shift We will use indecomposable as a noun to refer to an indecomposable object Let W S be any Coxeter group finite or infinite equipped with a natural reflection representation h We are interested in categorifications of H the associated Hecke algebra of W When W is a Weyl group geometric representation theory provides us with a natural categorification of H using equivariant perverse sheaves on the flag variety This construction does not generalize to an arbitrary Coxeter group though it does have analogs for affine Weyl groups and other crystallographic Coxeter groups In the early 1990s Soergel explored what happens when one takes the hypercohomology of a semisimple equivariant perverse sheaf on the 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 which appear Soergel defined a class of R bimodules now called Soergel bimodules These can be defined for any Coxeter group W agreeing with the hypercohomology bimodules in the Weyl group case and they categorify H In other words Soergel bimodules are an algebraic replacement for flag varieties in situations with no ambient geometry In a similar fashion Soergel bimodules are an algebraic replacement for Harish Chandra bimodules acting on the BGG category O We refer the reader to 27 for a purely algebraic account of Soergel bimodules 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 if the corresponding parabolic subgroup WJ W is finite The ring R is naturally equipped with a W action and for any finitary J S one may take the subring RJ R of invariants def under WJ When s S is a simple reflection we define the bimodule Bs R Rs R 1 Tensor products of various bimodules Bs and their direct sums and grading shifts will form the additive category BBS of Bott Samelson bimodules so named because they are obtained geometrically from Bott Samelson resolutions Including all direct summands we get the category B of Soergel bimodules One may also wish to consider the category BgBS of def generalized Bott Samelson bimodules which is generated by the bimodules BJ R RJ R l J Here l J indicates the length of the longest element wJ of the finite subgroup WJ Though not immediately obvious it is true that BgBS B When W is a finite group every subset of S is finitary and the set of all rings RJ for all subsets forms a Frobenius hypercube in the sense of 9 This is to say that whenever J I S every ring extension RI RJ is actually a Frobenius extension and that these extensions are mutually compatible in some sense This has immediate applications to understanding the morphisms in B and implies that the category can be depicted diagrammatically Using diagrammatics to study categorification was pioneered by Khovanov and Lauda 18 19 We will be examining the case of the dihedral group in this paper In this case S s t consists of two elements and the group W Wm is determined by a single number m ms t 3 which is either or a natural number 2 1 1 2 General subtlety dihedral simplicity Soergel proves in 27 that there is one indecomposable Soergel bimodule Bw for each element w W