Two-color Soergel calculus and simple transitive 2-representations
TTWO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE -REPRESENTATIONS MARCO MACKAAY AND DANIEL TUBBENHAUER
Abstract.
In this paper we complete the ADE-like classification of simple transitive2-representations of Soergel bimodules in finite dihedral type, under the assumption ofgradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to givean explicit construction of a graded (non-strict) version of all these 2-representations.Moreover, we give simple combinatorial criteria for when two such 2-representations areequivalent and for when their Grothendieck groups give rise to isomorphic representations.Finally, our construction also gives a large class of simple transitive 2-representations ininfinite dihedral type for general bipartite graphs.
Contents
1. Introduction 12. Bipartite graphs and dihedral 2-representations 42.1. Combinatorics of dihedral groups 42.2. Bipartite graphs 52.3. Quivers and categorical representations of dihedral groups 62.4. Strong dihedral 2-representations 103. Graded 2-representations 123.1. Some basics about (graded finitary) 2-categories 123.2. Graded simple transitive 2-representations 144. The two-color Soergel calculus and its 2-action 154.1. Soergel calculus in two colors 154.2. Its 2-representations coming from bipartite graphs 205. Proofs 245.1. The uncategorified story 245.2. The infinite case 265.3. The finite case 305.4. Classification of dihedral 2-representations 32References 351.
Introduction
An essential problem in classical representation theory is the classification of the simplerepresentations of any given algebra, i.e. the parametrization of their isomorphism classesand the explicit construction of a representative of each class.In 2-representation theory, the actions of algebras on vector spaces are replaced by functorialactions of 2-categories on certain additive or abelian 2-categories. The Grothendieck groupof a 2-representation is a classical representation. One can say that the 2-representationdecategorifies to the classical representation, or that the latter is a decategorification ofthe former. Vice versa, one can also say that the 2-representation categorifies the classicalrepresentation to which it decategorifies, or that it is a categorification. Note that, in general,categorifications need not be unique. a r X i v : . [ m a t h . R T ] D ec M. MACKAAY AND D. TUBBENHAUER
Examples are 2-representations of the 2-categories which categorify representations ofquantum groups, due to (Chuang–)Rouquier and Khovanov–Lauda, and 2-representations ofthe 2-category of Soergel bimodules, which categorify representations of Hecke algebras.Mazorchuk–Miemietz [MM16b] defined an appropriate 2-categorical analogue of the sim-ple representations of finite-dimensional algebras, which they called simple transitive 2-representations (of finitary 2-categories). The problem is that their classification is very hardand not well understood in general – except when certain specific conditions are satisfied, asfor Soergel bimodules in type A [MM16b, Theorem 21] for example.The authors of [KMMZ16] studied the so-called small quotient of Soergel bimodules andtheir simple transitive 2-representations, for all finite Coxeter types. These 2-representationsare given by categories on which the bimodules act by endofunctors and the bimodule maps bynatural transformations. Each of these categories is equivalent to the (projective or abelian)module category over the path algebra of a finite quiver, which can be obtained by doublinga certain Dynkin diagram. An almost complete classification was given in [KMMZ16], whichwe now recall.In every finite Coxeter type of rank strictly greater than two, all the simple transitive2-representations are equivalent to Mazorchuk and Miemietz’s categorification of the cellrepresentations of Hecke algebras, the so-called cell 2-representations.The rank two case is more delicate. In type I ( n ), for any n ∈ Z > , there is one cell2-representation of rank one and two higher rank cell 2-representations, which correspond tothe two possible bipartitions of the Dynkin diagram of type A n − . (Here we distinguish abipartition of a given graph from the opposite bipartition.) When n is odd, these exhaust allsimple transitive 2-representations, up to equivalence.When n = 2 ,
4, it was already known that the same holds, see [Zim17]. However, when n is even and greater than four, it was shown in [KMMZ16] that there exist additionalsimple transitive 2-representations which are not equivalent to cell 2-representations. Ifone has n (cid:54)∈ { , , } , then there exist exactly two which correspond to the two possiblebipartitions of a type D n +1 Dynkin diagram.For n = 12 , ,
30, the possible existence of one more additional pair of inequivalentsimple transitive 2-representations was discovered, but not proved (not even conjectured) in[KMMZ16]. Should they exist, their underlying Dynkin diagrams were shown to be of typeE for n = 12, of type E for n = 18, and of type E for n = 30.The simple transitive 2-representations with quivers of type A and D were constructedintrinsically in [KMMZ16]. The ones of type A are equivalent to the aforementioned higherrank cell 2-representations, due to Mazorchuk–Miemietz [MM11], and can be constructedas subquotients of the 2-category of Soergel bimodules, as in any finite Coxeter type. Thetype D simple transitive 2-representations of I ( n ), for n > Z -grading of the Soergel bimodules,but the 2-representations admit a unique compatible grading.In this paper, we construct all graded simple transitive 2-representations of the smallquotient of the Soergel bimodules of type I ( n ) by different means. We use Elias’ [Eli16]diagrammatic version of the latter 2-category, the so-called two-color Soergel calculus. Moreprecisely, given a Dynkin diagram of type A , D or E with a bipartition, we define twodegree-preserving, self-adjoint endofunctors Θ s and Θ t on the module category over thecorresponding quiver – which is a zigzag algebra of type ADE – and, for each generatingdiagram in the two-color Soergel calculus, a natural transformation between composites ofthem such that all diagrammatic relations are preserved.Moreover, we show that two graded simple transitive 2-representations are equivalent ifand only if the corresponding bipartite graphs are isomorphic (as bipartite graphs). Finally, WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 3 we also determine when graded simple transitive 2-representations decategorify to isomorphicrepresentations of the corresponding Hecke algebra, using a purely graph-theoretic property.Let us give some interesting consequences of our results. First, there are two inequivalentgraded simple transitive 2-representations of type E (or E ) for the Soergel bimodules oftype I (12) (or I (30)), which decategorify to isomorphic representations of the associatedHecke algebra. (The two simple transitive 2-representations of type E for the type I (18)Soergel bimodules have non-isomorphic decategorifications.) To the best of our knowledgethese are the first examples of simple transitive 2-representations of the same ( ∞ ) for any bipartite graph, not just theones of ADE type. (Note that this 2-category is not always isomorphic to the 2-category ofSoergel bimodules in type I ( ∞ ), cf. Remark 2.18.) For these 2-representations, all of theabove statements are still valid. Potential further developments.
We hope that our construction will also be helpful for theconstruction of simple transitive 2-representations of other 2-categories, e.g. Soergel bimodulesin other finite Coxeter types with more 2-cells.Furthermore, since our preprint first appeared, we wrote a joint paper with Mazorchuk–Miemietz [MMMT16] in which we explain the relation between the simple transitive 2-representations of the Soergel bimodules in finite dihedral type and those of the semisimplifiedsubquotient of the module category of quantum sl at a root of unity, due to Kirillov–Ostrik[KJO02], [Ost03] and others. This relation is based on Elias’ algebraic quantum Satakeequivalence [Eli16], [Eli17]. There should be a similar story for sl > , and we hope that ourpaper will help to develop it. (See [MMMT18] for some first steps in this direction.)And most of our constructions work over certain integral rings, see (4.1). So perhaps ourwork will also be useful for 2-representation theory in finite characteristic. Structure of the paper. (i) In Section 2 we give the details of our graph-theoretical construction.(ii) In Section 3 we explain some notions concerning 2-categories and 2-representations,focusing on the graded and weak setups (by extending Mazorchuk and Miemietz’sconstructions).(iii) In Section 4 we recall Elias’ two-color Soergel calculus and define our 2-action of it.Hereby we follow – and generalize – ideas from [AT17] and [KS02].(iv) Finally, Section 5 contains all proofs.
Remark 1.1.
We use colors in this paper, but they are not necessary to read the paper andjust a service to the reader. Nevertheless, three colors should be mentioned, i.e. sea-greenand tomato denote the two different generators s and t of the dihedral groups, while darkorchid denotes notions which play the role of a dummy and can be replaced by either ofthe two. Moreover, our notation is designed so that these colors can be distinguished inblack-and-white:(i) The color sea-green is always either accompanied by the symbol s , the associatednotions are underlined or we use • .(ii) The color tomato is always either accompanied by the symbol t , the associated notionsare overlined or we use (cid:4) .(iii) The color dark orchid is always either accompanied by the symbols u , v or w , theassociated notions are neither under- nor overlined or we use (cid:7) . (cid:78) M. MACKAAY AND D. TUBBENHAUER
Acknowledgements.
We especially like to thank Ben Elias, Nick Gurski and VolodymyrMazorchuk for patiently answering our questions, and Geordie Williamson for a very fruitfulblackboard discussion.We also thank Nils Carqueville, Michael Ehrig, Lars Thorge Jensen, Vanessa Miemietz,SageMath, Antonio Sartori and Paul Wedrich for helpful comments and discussions, as well asVolodymyr Mazorchuk for many detailed and very helpful comments on a draft of this paper.Special thanks to the anonymous referee for a careful reading of this paper, for very helpfulsuggestions, for pointing out an error in the first version of this paper, and for improving the“grammar of the world”.M.M. likes to thank the Hausdorff Center for Mathematics in Bonn for sponsoring a researchvisit. D.T. thanks the 2016 European Championship for providing a quiet working atmospherein the mathematical institute during which a big part of his work on this paper was done. M.M. is partially supported by FCT/Portugal through the project UID/MAT/04459/2013.2.
Bipartite graphs and dihedral -representations In this section we state our main results (see Section 2.4), and provide the backgroundneeded to understand them.2.1.
Combinatorics of dihedral groups.
First, we recall some basic notions concerningthe dihedral groups.2.1.1.
The dihedral group and its associated Hecke algebra.
We follow [Eli16] with our con-ventions. LetW n = (cid:104) s, t | s = t = 1 , s n = . . . sts (cid:124) (cid:123)(cid:122) (cid:125) n = . . . tst (cid:124) (cid:123)(cid:122) (cid:125) n = t n (cid:105) and W ∞ = (cid:104) s, t | s = t = 1 (cid:105) be the dihedral groups of order 2 n ∈ Z > and of infinite order respectively, presented as therank two Coxeter groups of type I ( n ) and I ( ∞ ). When no confusion is possible, we writeW for either W n or W ∞ . The two generators s and t are always sea-green s and tomato tcolored. (Throughout, we allow n = 1 which is to be understood by dropping one color, saytomato t, and all notions involving it.)Here and in the following, we denote by s k respectively by t k a sequence of length k ,alternating in s and t with rightmost symbol s respectively t . In general, we write an elementof W as a finite word w = w l · · · w with w k ∈ { s, t } (including the empty word ∅ ). Wesay that the word is reduced if it is equal to s k or t k for some k ∈ Z ≥ . Throughout theremainder of this paper, when we write w ∈ W as a word, we always assume that the word isreduced if not stated otherwise.Moreover, let H n or H ∞ denote the associated Hecke algebra over C (v), with v being anindeterminate. Again, when no confusion is possible, we write H for H n or H ∞ . This algebrahas generators T s and T t , and relations T s = (v − − · T s + v − , T t = (v − − · T t + v − , T s n = T t n . For any w ∈ W, pick a reduced word w l · · · w which represents it. Then the element T w ∈ Hdenotes the product T w l · · · T w and T ∅ = 1. The element T w does not depend on the choiceof the reduced word and { T w | w ∈ W } forms a basis of H. So for v = 1 (when working over C [v , v − ]) we recover C [W].2.1.2. The dihedral Kazhdan–Lusztig basis.
Denote by (cid:96) ( w ) the length of w ∈ W and by ≤ the Bruhat order on W. For any w ∈ W define θ w = v (cid:96) ( w ) · (cid:80) w (cid:48) ≤ w T w (cid:48) , w, w (cid:48) ∈ W . The set { θ w | w ∈ W } forms a basis of H, called the Kazhdan–Lusztig basis . WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 5
For an expression w = w l · · · w for w ∈ W – not necessarily reduced – let θ w = θ w l · · · θ w .This element does depend on the choice of expression for w , even amongst reduced expressions.Choosing one reduced expression for each w ∈ W, we get another basis { θ w = θ w (cid:96) ( w ) · · · θ w | w ∈ W } for H, called the Bott–Samelson basis , cf. Section 4.1. (Note that we write θ w todistinguish the Bott–Samelson from the Kazhdan–Lusztig basis.) Example 2.1.
One has θ s = v( T s + 1), θ t = v( T t + 1), and an easy calculation gives θ sts = θ sts − θ s = θ s θ t θ s − θ s . In general, θ w = θ w ∓ “lower order terms”. (cid:78) For any n ∈ Z > , the group W n has a unique longest element w ( n ) = w = s n = t n . (In case “ n = ∞ ” there is no such w .) In this paper, we only consider categorifications ofH n -representations which are killed by θ w ( n ) = θ w . In fact, the only decategorification of asimple transitive 2-representation of H n which is not killed by θ w is, by [MM16b, Theorem18], the trivial H n -representation, cf. Theorem III.2.1.3. The defining relations satisfied by the Kazhdan–Lusztig basis elements.
Define recur-sively the integers d kl via: d = 1 , d kl = 0 , unless 0 < k ≤ l and l − k is even ,d kl = d k − l − − d kl − . (2.1)Let [2] v = v + v − . When considering the basis { θ w | w ∈ W } , the defining relations of H –by e.g. [Eli16, Section 2.2] – are θ s θ s = [2] v · θ s , θ t θ t = [2] v · θ t , (2.2) (cid:80) k ∈ Z ≥ d kn · θ s k = (cid:80) k ∈ Z ≥ d kn · θ t k . (2.3)Note that either sum in (2.3) is equal to θ w . Example 2.2.
The first few non-zero numbers from (2.1) are as follows. l k · · · − − −
31 1 −
43 . . .. . .. . . . Hence, if n = l = 3, then (2.3) gives θ s θ t θ s − θ s = θ t θ s θ t − θ t (= θ w ) . (2.4)When “ n = ∞ ”, the only relations are the ones from (2.2). (cid:78) Bipartite graphs.
Next, we recall some basics about bipartite graphs and fix notationwhich we use throughout.
M. MACKAAY AND D. TUBBENHAUER
A reminder on bipartite graphs.
Let G be a connected, unoriented, finite graph withoutloops and with at most one edge between each pair of vertices. Let V = V ( G ) be the set ofvertices of G and assume that the vertices are numbered. If these numbers are divided intotwo disjoint subsets I and J , such that V = S (cid:96) T , S = { i | i ∈ I } , T = { j | j ∈ J } , with no edges connecting vertices within each set, then we call the triple ( G, S , T) a bipartitegraph . When no confusion is possible, we simply write G for a bipartite graph. Note that abipartition of G is the same as a two-coloring of its vertices.Two bipartite graphs ( G, S , T) and ( G (cid:48) , S (cid:48) , T (cid:48) ) are called isomorphic if there is an isomor-phism of graphs between G and G (cid:48) which sends S to S (cid:48) and T to T (cid:48) . Example 2.3.
One crucial example in this paper is the type E graph: G = • (cid:4) • (cid:4) • (cid:4) , G (cid:48) = (cid:4) • (cid:4) • (cid:4) • . These two-colorings give non-isomorphic bipartite graphs of type E . As we will see later,these will give rise to two inequivalent 2-representations categorifying the same H-module,see Example 5.10 (keeping Example 2.4 in mind). (cid:78) We write i j(= j i) in case i and j are connected in G .2.2.2. Adjacency matrices and spectra.
Given any graph G , the adjacency matrix A ( G ) of G is the symmetric | V | × | V | -matrix whose only non-zero entries are A ( G ) i , j = 1 for i j. The spectrum S G of G is the multiset of all eigenvalues of A ( G ) (which are all real), repeatingeach one of them according to its multiplicity.Since we assume G to be bipartite, we can clearly choose an ordering of the vertices (whichwe will always do from now on) such that A ( G ) = (cid:18) AA T (cid:19) (2.5)for some matrix A of size | S | × | T | . Next, recall that S G is a symmetric set (see e.g. [BH12,Proposition 3.4.1]), and the fact from linear algebra that AA T has an eigenvalue α (cid:54) = 0 ⇔ A T A has an eigenvalue α (cid:54) = 0 . (2.6)Thus, the non-zero elements (counting multiplicities) of S G are ±√ α for α as in (2.6).2.2.3. Spectrum-color-equivalence.
Given two bipartite graphs G and G (cid:48) . We call them spectrum-color-equivalent if | S | = | S (cid:48) | , | T | = | T (cid:48) | and S G = S G (cid:48) , and spectrum-color-inequivalent otherwise. This clearly gives rise to an equivalence relation. Example 2.4.
Take the two graphs G and G (cid:48) from Example 2.3. Then G and G (cid:48) arenon-isomorphic as bipartite graphs, but they are spectrum-color-equivalent.More generally, any bipartite graph with | S | = | T | is spectrum-color-equivalent to thebipartite graph with the opposite two-coloring. (cid:78) Quivers and categorical representations of dihedral groups.
Fix a bipartitegraph G . The double quiver Q G associated to G is the oriented graph obtained from G bydoubling each edge and giving opposite orientations to the two resulting edges, called arrows.Such an arrow is denoted by j | i if it starts at i and ends at j, and by i | j if it starts at j andends at i (i.e. we are using the “operator notation”). Definition 2.5.
Two distinct arrows of Q G are called partners if they come from the sameedge in G . (cid:78) WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 7
Thus, each arrow has precisely one partner pointing in the opposite direction.
Example 2.6.
An example of a double-quiver is: G = 1 2 3 (cid:32) Q G = 1 2 3 . In the above example, 2 | | | | (cid:78) As already pointed out, we will see that only bipartite graphs of ADE type give rise to2-representations of Soergel bimodules in finite dihedral type.2.3.1.
The path algebra associated to a bipartite graph.
We work over certain base rings A gen , A n or A G,(cid:126)λ, q , all of which are subrings of C . We will impose some technical conditions onthese which we discuss in Remark 4.1, and the reader should think of these as playing therole of an integral form.If no confusion can arise, we simply write A for either A gen , A n or A G,(cid:126)λ, q . Of course, wecan always extend the scalars to C if necessary.Let P G be the path algebra of Q G over A , such that multiplication on the left is given bypost-composition, on the right by pre-composition. We denote the multiplication by andconsider P G to be graded by the path length.By convention, i and j denote the corresponding paths of length zero. Paths of length oneare in one-to-one correspondence with arrows of Q G , and we call them arrows too. Definition 2.7.
Let Q G denote the quotient algebra obtained from P G by the followingrelations. (cid:66) The two and three steps relations.
The composite of two arrows is zero unless they are partners.(2.QG1) The composite of three arrows is zero.(2.QG2) (cid:66)
All non-zero partner composites are equal.
Assume that i j a , for a = 1 , . . . , b .Then: i | j j | i = · · · = i | j b j b | i = i | i , (2.QG3)where the path i | i, called loop , is defined by the above equations.Similar relations hold with i and j swapped.(Note that the relation (2.QG2) is a consequence of (2.QG1) and (2.QG3) as long as G hastwo or more edges.) We call Q G the type G -quiver algebra . (cid:78) The defining relations of Q G are homogeneous, so Q G inherits the path length grading.Clearly, the primitive idempotents of Q G are the i’s. Example 2.8.
The relations (2.QG1), (2.QG2) and (2.QG3) might be familiar to readerswho know about the so-called zigzag algebras in the spirit of [HK01].Let us give two examples. Fix m ∈ Z ≥ . Let G be a Dynkin graph of type A m or ˜A m − with m , respectively 2 m vertices. Then the associated Q G ’s are of the following form (if weconsider the evident two-coloring of G ).1 2 3 · · · m − − , − · · · (cid:70) (cid:70) (cid:47) (cid:47) (cid:24) (cid:24) (cid:6) (cid:6) (cid:24) (cid:24) (cid:6) (cid:6) (cid:111) (cid:111) (cid:88) (cid:88) (cid:70) (cid:70) (cid:88) (cid:88) . M. MACKAAY AND D. TUBBENHAUER
The defining relations of the associated quiver algebras are of the form(2.QG1) : i+2 | i+1 i+1 | i = 0 = i − | i − − | i , (2.QG3) : i | i+1 i+1 | i = i | i = i | i − − | i , with indices modulo 2 m in the cyclic case.The quiver algebra for the Dynkin graph of type A m will be of importance later on andwe denote it by QA m . Similarly, we denote by Q ˜A m − its cyclic counterpart. (cid:78) In the above and throughout; for m = 0 and m = 1 we let Q G = QA = A andQ G = QA ∼ = A [ X ] / ( X ), by convention. Remark 2.9.
The algebras from Example 2.8 also appear in the context of categorical braidgroup actions, e.g. in [GTW17] and [KS02]. Note that Khovanov–Seidel have the additionalrelation 1 | | (cid:78) Some bimodules.
Let us denote by P i and i P the left and the right ideal of Q G generatedby i, respectively. They are clearly graded Q G -modules. Example 2.10.
We can easily visualize them for QA m with 1 < i < m as P i = i i | i i+1 | ii − | i , i P = i i | i i | i+1i | i − .(2.10)They have basis vectors i , i − | i , i+1 | i , i | i and i , i | i − , i | i+1 , i | i, respectively, of degree 0 , , , A -rank three for any m (cid:54) = 0 , (cid:78) Because the i form a complete set of orthogonal primitive idempotents in Q G , the P i andthe i P are graded, indecomposable projective modules and all graded, indecomposable left –respectively right – Q G -modules are of the form P i { a } , respectively i P { a } , for some shift a ∈ Z . Let ⊗ = ⊗ A be the tensor product over A . Then P i {− } ⊗ i P is clearly a gradedQ G -bimodule. These bimodules will be important in this paper.2.3.3. Endofunctors associated to bipartite graphs.
We denote by G gr = Q G -p Mod gr thecategory of graded projective, (left) Q G -modules, which are free of finite A -rank. Our nextgoal, following [KS02, Section 2] and [AT17, Section 3], is to define endofunctors V i : G gr → G gr , i ∈ G. Let (cid:98) ⊗ = ⊗ Q G be the tensor product over Q G . We define V i ( X ) = P i {− } ⊗ i P (cid:98) ⊗ X, V i ( f ) = ID P i {− }⊗ i P (cid:98) ⊗ f. Here
X, Y ∈ G gr and f ∈ Hom G gr ( X, Y ). (Since P i {− } ⊗ i P is clearly biprojective , i.e.projective both as a left and as a right Q G -module, the functor V i sends graded projectivesto graded projectives.) As in [AT17, Section 3.3] we immediately obtain V i ( P j ) ∼ = P i {− } ⊕ P i { + } , if i = j ,P i , if i j , , otherwise . (2.11) Example 2.11.
By looking at (2.10) one observes that i P (cid:98) ⊗ P i ∼ = A (i) ⊕ A (i | i). Thisisomorphism is given by multiplication of paths. Consequently, the degree two Q G -en-domorphism V i (i | i) of P i {− } ⊕ P i { + } sends the copy P i {− } identically to P i { + } , and iszero elsewhere (recall that i | i i | i = 0). (cid:78) Following [AT17, Section 3.3] we define Θ s = (cid:76) i ∈ G V i , Θ t = (cid:76) j ∈ G V j . WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 9
Example 2.12.
We sum over the graph of type A m as (in case m is odd): • (cid:4) • · · · • (cid:4) • Θ s Θ t Θ s Θ s Θ t Θ s . Note that the opposite two-coloring of G switches Θ s and Θ t . In general, this switch neednot be a natural isomorphism. (cid:78) By using (2.11), one directly checks that Θ s ( P i ) ∼ = (cid:40) P i {− } ⊕ P i { + } , if i ∈ S , (cid:76) j i P j , if i ∈ T , , Θ t ( P i ) ∼ = (cid:40) P i {− } ⊕ P i { + } , if i ∈ T , (cid:76) i j P j , if i ∈ S . (2.12)2.3.4. Dihedral modules associated to bipartite graphs.
Definition 2.13.
Let G v be the C (v)-vector space on the basis { i | i ∈ G } . Define aH ∞ -action on G v via(2.13) θ s · i = [2] v · i , θ s · i = (cid:80) j i j , θ t · i = [2] v · i , θ t · i = (cid:80) i j j . (The reader is encouraged to verify that this gives indeed a H ∞ -module structure.) We denotethe associated algebra homomorphism by G : H ∞ → G v . (cid:78) Let [ G gr ] C (v) = K ⊕ ( G gr ) ⊗ Z [v , v − ] C (v) denote the split Grothendieck group tensored withthe field C (v). (As usual, the grading shift decategorifies to multiplication by v.) If we denoteby [ · ] a class in [ G gr ] C (v) , then we get a so-called weak categorification (not to be confusedwith the weak – in the sense of non-strict – setup that we will meet below), as the followingproposition shows. Proposition 2.14.
The functors Θ s and Θ t decategorify to C (v)-linear endomorphisms on[ G gr ] C (v) , which thus becomes an H ∞ -module. The C (v)-linear map(2.14) ζ G : G v ∼ = −→ [ G gr ] C (v) , ζ G ( i ) = [ P i ] , is an isomorphism of H ∞ -modules, intertwining the actions of θ s and [ Θ s ], and that of θ t and [ Θ t ]. (cid:3) Proposition 2.15.
Let G be of ADE type and n be its Coxeter number, i.e.:(A) For G of type A m we let n = m + 1.(D) For G of type D m we let n = 2 m − G of type E we let n = 12, for G of type E we let n = 18, and G of type E welet n = 30.Then the H ∞ actions on G v and [ G gr ] C (v) descend to H n -actions which are matched by ζ G asin (2.14). (We denote these by G n : H n → G v ).These are the only G (with more than one vertex) and n for which this holds. (cid:3) Remark 2.16.
Proposition 2.15 appears as a special case of [Lus83, Proposition 3.8], andwas rediscovered in [KMMZ16, Sections 5, 6 and 7]. Note hereby that Lusztig proves hisstatement using the combinatorics of cells in Coxeter groups, while [KMMZ16] uses categoricalresults. In this paper, we give an independent proof using spectral graph theory. (cid:78)
Example 2.17.
Write A gr (3) = QA -p Mod gr and ˜A gr (3) = Q ˜A -p Mod gr . Then [ Θ s ] and[ Θ t ] act on [ A gr (3)] C (v) and [ ˜A gr (3)] C (v) via[ Θ s ] = [2] v v
10 0 0 , [ Θ t ] = v (in type A ) , [ Θ s ] = [2] v v , [ Θ t ] = v
01 1 0 [2] v (in type ˜A ) . (These are written on the bases { [ P ] , [ P ] , [ P ] } and { [ P ] , [ P ] , [ P ] , [ P ] } .)These matrices give [ A gr (3)] C (v) and [ ˜A gr (3)] C (v) the structure of an H ∞ -module. Addi-tionally, the relation from (2.3) holds in type A since we have[ Θ s ][ Θ t ][ Θ s ][ Θ t ] − · [ Θ s ][ Θ t ] = [ Θ t ][ Θ s ][ Θ t ][ Θ s ] − · [ Θ t ][ Θ s ] . This shows that [ A gr (3)] C (v) has the structure of an H -module. (cid:78) Strong dihedral -representations. In Section 3 we will explain how to adapt Mar-zorchuk and Miemietz’s definition of finitary 2-categories, their 2-representations and relatednotions (see e.g. [MM11], [MM16a] or [MM16b]) to our setup. At this point of the paper itis enough to roughly see them as a “higher analog” of graded, finite-dimensional algebrasand their graded, finite-dimensional representations.By using Proposition 2.14, we can identify [ G gr ] C (v) with G v as H-modules. The nexttheorem says that the functorial action of Θ s and Θ t on G gr can be extended to a 2-representation of D ∞ and/or D n , the 2-categories – defined by generating 2-morphisms andrelations – given by the two-color Soergel calculi, due to Elias [Eli16]. (We will recall themin Section 4.1.)We use the same base ring A to define D ∞ and D n as we did for the quiver algebras inSection 2.3. (More details will be given in Section 4.1.) Elias’ construction of D ∞ requiresthe choice of an invertible element q in the base ring (which is ultimately embedded in thecomplex numbers). When q is not a root of unity, D ∞ is equivalent to the 2-category ofSoergel bimodules in type I ( ∞ ). When q is a complex, primitive 2 n th root of unity, thisis no longer true – cf. Remark 2.18 – but D n is equivalent to the 2-category of Soergelbimodules in type I ( n ). Theorem I. ( Dihedral -actions. )(a) For any bipartite graph G , there is at least one value of q ∈ C − { } for which thereexists an additive, degree-preserving, A -linear, weak 2-functor (defined in Section 4.2) G ∞ : D (cid:63) ∞ → p End (cid:63) ( G gr ) . (b) If G is as in (A), (D) or (E) and q is a complex, primitive 2 n th root of unity, then G ∞ gives rise to an additive, degree-preserving, A -linear, weak 2-functor G n : D (cid:63)n → p End (cid:63) ( G gr )such that the following diagram commutes Kar ( D n ) [ · ] C (v) (cid:15) (cid:15) (cid:101) G n (cid:47) (cid:47) p End ( G gr ) [ · ] C (v) (cid:15) (cid:15) H n G n (cid:47) (cid:47) End(G v ) . These are the only (non-trivial) G ’s and n ’s for which this holds. (cid:3) Here
Kar denotes the Karoubi envelope, (cid:101) · the lift of a functor to the Karoubi envelopeand p End denotes the 2-category of biprojective endofunctors, see Example 3.2. We willexplain the meaning of (cid:63) in the next section.
WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 11
Remark 2.18.
In ADE type, we need q in the Soergel calculus to be a complex, primitiveroot of unity. In that case – as already remarked – D ∞ does not quite categorify the Heckealgebra H ∞ , see [Eli16, Remarks 5.31 and 5.32]. However, if one allows infinite ADE typegraphs, then one can work with a generic parameter. The blueprint example is the infinitetype A graph as considered in [AT17], for example. (cid:78) For the following two theorems we switch to C for our ground field, and we keep theparameter q fixed. Theorem II. ( Equivalences and isomorphisms. )(a) All 2-representations as in Theorem I are graded simple transitive 2-representationsof
Kar ( D ∞ ) (cid:63) respectively of Kar ( D n ) (cid:63) .(b) Two 2-representations as in Theorem I are equivalent if and only if their bipartitegraphs are isomorphic.(c) Two 2-representations as in Theorem I decategorify to isomorphic H-modules if andonly if their bipartite graphs are spectrum-color-equivalent.(d) All 2-representations as in Theorem I factor through graded simple transitive 2-representations of Kar ( D f ∞ ) (cid:63) respectively of Kar ( D f n ) (cid:63) . (cid:3) Hereby f means that we work over the coinvariant algebra.We stress that (c) of Theorem II holds for the analogs from Proposition 2.14 and Proposi-tion 2.15 as well.With these theorems we can complete the classification from [KMMZ16], where rank meansthe rank on the level of the Grothendieck groups (for n = 1 cf. Example 3.8). Theorem III. ( Classification. ) There is a bijection between the equivalence classes ofgraded simple transitive 2-representations of D (cid:63)n (of rank >
1) and the isomorphism classesof bipartite graphs as in (A), (D) or (E) with Coxeter number n , for n ∈ Z > .This completes the classification from [KMMZ16], cf. Remark 5.11. Example 2.19.
For n = 8, there are four G ’s of type (A), (D) or (E) which are non-isomorphicas bipartite graphs: | S | = 4 , | T | = 3 : • (cid:4) • (cid:4) • (cid:4) • , | S | = 3 , | T | = 4 : (cid:4) • (cid:4) • (cid:4) • (cid:4) , | S | = 2 , | T | = 3 : • (cid:4) • (cid:4)(cid:4) , | S | = 3 , | T | = 2 : (cid:4) • (cid:4) •• . (cid:78) In general, we get the following complete list of equivalence classes of higher rank gradedsimple transitive 2-representations of D (cid:63)n , for n ∈ Z > :(i) For odd n , there is just one, since the opposite coloring of a type A bipartite graphgives an isomorphic bipartite graph in this case.(ii) For even n (cid:54)∈ { , , , , } , there are four, since the opposite colorings of A and Dgraphs give non-isomorphic bipartite graphs.(iii) For n ∈ { , } , there are two, since the corresponding type D graphs are of type A inthese cases.(iv) For n ∈ { , , } , there are six, since there are two additional non-isomorphicbipartite graphs of type E in each case. Example 2.20.
The spectrum is a full graph invariant for type ADE graphs (see e.g. (5.1)).Thus, in order to check if two inequivalent graded simple transitive 2-representations of D (cid:63)n decategorify to isomorphic H n -modules, we only need to compare their two-colorings. Outsidetypes E and E nothing interesting happens. But the two two-colorings in Example 2.3 giveinequivalent bipartite graphs which are spectrum-color-equivalent. Therefore, the correspond-ing graded simple transitive 2-representations of D (cid:63) are inequivalent, but decategorify toisomorphic H -modules (see also Example 5.10). The same holds for the two type E gradedsimple transitive 2-representations of D (cid:63) .In the infinite case the story is more delicate. As stated above, for all bipartite graphs, i.e.not necessarily of ADE type, we can construct graded simple transitive 2-representations of D (cid:63) ∞ , cf. Section 4.2.5. By a classical result of Schwenk [Sch73], “almost all” trees are notdetermined by their spectrum – in the sense that there are other, non-isomorphic trees withthe same spectrum. However, a lot of them will be spectrum-color-equivalent. Thus, alreadyfor trees there are plenty of examples of inequivalent graded simple transitive 2-representationsof D (cid:63) ∞ which decategorify to isomorphic H ∞ -modules, e.g. G = • (cid:4) • (cid:4) • (cid:4) • (cid:4) • (cid:4) (cid:4) ,G (cid:48) = (cid:4) • (cid:4) • (cid:4) • (cid:4)(cid:4) • • (cid:4) . (cid:78) Graded -representations For us, Mazorchuk and Miemietz’s setting (see e.g. [MM11], [MM16a] or [MM16b]) withfinitary 2-categories and strict 2-representations is too restrictive.The two-color Soergel calculus is defined over the graded algebra of polynomials on thegeometric representation of W ( the polynomial algebra , for short), which is finite-dimensionalin each degree, but infinite-dimensional as a whole. In contrast, Mazorchuk–Miemietz alwayswork over the so-called coinvariant algebra , which is a finite-dimensional quotient of thepolynomial algebra. This quotient inherits a grading, but they do not use it.Further, they consider strict 2-representations. We will define our 2-representations usingthe two-color Soergel calculus over the polynomial algebra and then prove that they descendto a quotient defined over the coinvariant algebra. This is done by prescribing the image ofeach 1-morphism and each generating 2-morphism, and checking the diagrammatic relations.On the level of 1-morphisms these 2-representations sometimes preserve composition only upto natural 2-isomorphisms, i.e. our 2-representations are given by weak 2-functors (also callednon-strict or pseudo). Fortunately, every weak 2-representation in our sense can be strictified(see Remark 3.9) and e.g. the classification results from [KMMZ16] remain true in our setup.In the following abstract setup we work over a field K for simplicity.3.1. Some basics about (graded finitary) -categories. We use strict 2-categories –which we simply call 2-categories – and bicategories. We use strict and weak 2-functors,which are carefully specified in each case.Let C (cid:63) be an additive, graded, K -linear 2-category, i.e. a category enriched over thecategory of additive, graded, K -linear categories. We always assume that C (cid:63) has finitely WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 13 many objects, up to equivalence, and that its 2-morphism spaces are locally finite, i.e.finite-dimensional in each degree with the grading bounded from below.Moreover, in our setup, the 1-morphisms admit grading shifts. Let X { a } denote a given1-morphism X shifted a ∈ Z degrees such that the identity 2-morphism on X gives rise to ahomogeneous 2-isomorphism X ⇒ X { a } of degree a . In general, the 2-morphisms in C (cid:63) are K -linear combinations of homogeneous ones, where a homogeneous 2-morphism from X to Y of degree d becomes homogeneous of degree d + b − a when seen as a 2-morphism from X { a } to Y { b } .There is a 2-subcategory C of C (cid:63) which has the same objects and 1-morphisms, but onlycontains the degree-preserving 2-morphisms. It still has the degree shifts on 1-morphisms,but the 2-morphism spaces are no longer graded. In C the 1-morphisms X { a } and X { b } arein general only isomorphic for a = b .One can recover C (cid:63) from C , because (for any 1-morphisms X, Y in C (cid:63) or C ):2Hom C (cid:63) ( X, Y ) = (cid:76) a ∈ Z C ( X { a } , Y ) . We also assume that, for each pair of objects x, y , the hom-category Hom C ( x, y ) isidempotent complete and Krull-Schmidt. (In the diagrammatic Soergel 2-categories, wetherefore have to take the Karoubi envelope of each hom-category.) We also assume that theidentity 1-morphisms are indecomposable.If the split Grothendieck group of C has finite rank over Z [v , v − ], where v correspondsto the degree shift { } , we say that C (cid:63) is graded finitary . If, additionally, the 2-morphismspaces are finite-dimensional, we say that C (cid:63) is graded -finitary .If the split Grothendieck group of C has countably infinite Z [v , v − ]-rank, we say that C (cid:63) is graded locally finitary . If, additionally, the 2-morphism spaces are finite-dimensional, wesay that C (cid:63) is graded locally -finitary .We also use graded -finitary categories, whose 1-morphism spaces are graded and finite-dimensional. Example 3.1.
The Soergel bimodules of any Coxeter type, when really defined usingbimodules over the polynomial algebra R, form a bicategory. This is easy to see, e.g. R ⊗ R Ris only isomorphic to R and not equal to it. Moreover, it is not a small 2-category, e.g. theisomorphism class of the polynomial algebra R in this 2-category is not a set. Luckily, anybicategory is weakly equivalent to a 2-category – as follows from a strictification theorem dueto Mac Lane (see e.g. [ML98, Section XI.3]) for monoidal categories and to B´enabou [B´en67]for bicategories. (Alternatively, see [Lei98, Theorem 2.3].) And in our case, the 2-categoriesare small.However, for our purposes we need a concrete version of such 2-categories. Thus, we usethe Karoubi envelope of the two-color Soergel calculi D n and D ∞ , respectively, which werecall in Section 4.1. As we will see in Proposition 4.12, Kar ( D n ) (cid:63) and Kar ( D ∞ ) (cid:63) aregraded finitary and graded locally finitary 2-categories.The two-color Soergel calculi are defined over the polynomial algebra, but admit quotientsto 2-categories defined over the coinvariant algebra, denoted by D f n and D f ∞ . We explain thismore carefully later – in the proof of part (iv) of Theorem II. By Proposition 4.12 and byconstruction, Kar ( D f n ) (cid:63) and Kar ( D f ∞ ) (cid:63) are graded 2-finitary and graded locally 2-finitary2-categories, respectively. (cid:78) Example 3.2.
Let A be a graded, finite-dimensional algebra. Then its category of gradedprojective, finite-dimensional (left) A-modules A-p
Mod gr is an additive, graded 1-finitary, K -linear category which is idempotent complete and Krull-Schmidt. We view it as beingsmall by taking equivalence classes of A-modules.We call a finite-dimensional A-bimodule X biprojective if it is projective as a left and as aright A-module (but not necessarily as an A-bimodule). Given a graded biprojective, finite-dimensional A-bimodule X . Then X ⊗ A − gives rise to an exact endofunctor of A-p Mod gr . An endofunctor of A-p
Mod gr is called biprojective , if it is isomorphic to a direct summand ofa finite direct sum of endofunctors of the form X ⊗ A − , where X is a biprojective A-bimodule.Let p End (A-p
Mod gr ) be the 2-category with the unique object A-p Mod gr , whose 1-morphisms are biprojective endofunctors on A-p Mod gr , and whose 2-morphisms are degree-preserving natural transformations between these. Hereby recall that the Godement productinduces the horizontal composition ◦ h via: F , G : X → Y, H , I : Y → Z, f : F ⇒ G , g : H ⇒ I , g ◦ h f : HF ⇒ IG , ( g ◦ h f ) X = g G ( X ) ◦ H ( f X ) = I ( f X ) ◦ g F ( X ) . (cid:78) (3.1) Remark 3.3.
All diagrammatic 2-categories which appear in this paper, e.g. D n as in Sec-tion 4.1, are, by definition, strict. In contrast, p End (A-p
Mod gr ), introduced in Example 3.2,is a bicategory. Still, we always view it as being strict by using the B´enabou–Mac Lanecoherence theorem. (cid:78) Graded simple transitive -representations. Our next goal is to define a gradedand weak version of certain 2-representations due to Mazorchuk–Miemietz, see e.g. [MM16a]or [MM16b] where more details can be found.3.2.1. 2 -representations: definitions.
Let A f gr be the 2-category whose objects are additive,graded 1-finitary, K -linear (small) categories; 1-morphisms are additive, degree-preserving, K -linear functors; 2-morphisms are homogeneous degree-zero natural transformations. Forexample, Q G -p Mod gr is an object of A f gr . Definition 3.4.
Let C (cid:63) be a graded (locally) (2-)finitary 2-category. Then a graded -finitary,weak -representation of C (cid:63) is an additive, K -linear, weak 2-functor M : C (cid:63) → (cid:16) A f gr (cid:17) (cid:63) which preserves degrees and commutes with shifts as follows. For any indecomposable1-morphism F in C (cid:63) and any indecomposable object X ∈ (cid:96) x M ( x ) (note that objects in (cid:96) x M ( x ) are 1-morphisms in the target 2-category) we have M ( F { a } )( X { b } ) = M ( F )( X ) { a + b } , a , b ∈ Z . (cid:78) These form a bicategory, whose 1-morphisms are weak natural transformations, withdegree-zero structural 2-isomorphisms across the squares in their definition, and whose2-morphisms are degree-zero modifications.
Remark 3.5.
We require a graded 2-finitary, weak 2-representation to be a weak 2-functorwhich preserves degrees. Such a 2-functor restricts to, and is uniquely determined by, anadditive, K -linear, weak 2-functor M : C → A f gr . We will use both weak 2-functors almostinterchangeably. (cid:78) Mazorchuk–Miemietz [MM16b] defined simple transitive, strict 2-representations, whichare a categorical analogue of simple representations of finite-dimensional algebras. Theirdefinition remains (almost) unchanged in the our setting.
Definition 3.6.
We say that a graded 2-finitary, weak 2-representation M of a graded(locally) (2-)finitary 2-category C (cid:63) is transitive if for any two indecomposable objects X, Y in (cid:96) x M ( x ) there exists a 1-morphism F in C (cid:63) such that Y is isomorphic to a graded directsummand of M ( F )( X ). It is called graded simple transitive if, additionally, (cid:96) x M ( x ) has nonon-zero proper C (cid:63) -invariant ideals. (cid:78) (We say “graded simple transitive” and omit the “weak”, cf. Remark 3.9.) Remark 3.7.
Suppose that Y { a } is isomorphic to a direct summand of M ( F )( X ), for some a ∈ Z . Then Y is isomorphic to a direct summand of M ( F {− a } )( X ). (cid:78) WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 15
Example 3.8.
Let D A denote the one-color (say sea-green s ) Soergel calculus of Coxetertype A . This is defined as D n in Section 4.1, but dropping the second color (say tomato t )and the relations in which it is involved. This 2-category has one object (which we do notspecify here). The 1-morphisms are formal direct sums of finite words (“tensor products”) ofshifts of s , and 2-morphisms are degree zero Soergel diagrams. The 1-morphisms might notbe indecomposable, e.g. we have ss ∼ = s {− } ⊕ s { + } . In particular, D A is an additive, K -linear 2-category and D A ∼ = Kar ( D A ) is idempotent complete and Krull-Schmidt. Finally, Kar ( D A ) (cid:63) is a graded finitary 2-category.Now, consider the coinvariant algebra C +A = D of the Weyl group of type A , the so-calleddual numbers D ∼ = K [ X ] / ( X ) – with X of degree two. Then the 2-category D fA which isdefined over the coinvariant algebra, is a quotient of D A . The (cid:63) of its Karoubi envelope Kar ( D fA ) (cid:63) is a graded 2-finitary 2-category.We can define a graded 2-finitary, weak 2-representation of Kar ( D A ) (cid:63) on the category(D-p Mod gr ) (cid:63) , by sending the empty word ∅ to the endofunctor D ⊗ D − and s to theendofunctor D ⊗ D ⊗ D − . Note that D and D ⊗ D are graded biprojective, D-bimodules.This 2-representation is simple transitive, see [MM17, Section 3.4], and also graded. Byconstruction, it descends to
Kar ( D fA ) (cid:63) .In the setup of bipartite graphs, this 2-representation is given by a graph with one sea-green s colored vertex • . One can check that there is no other graded simple transitive2-representation for the corresponding small quotient. (cid:78) -representations: strictifications. Let C be a small 2-category, Cat the 2-categoryof small categories and F w : C → Cat any weak 2-functor. By [Pow89, Section 4.2] this2-functor can be strictified: There exists a strict 2-functor F s : C → Cat and a weak natural2-isomorphism f : F w ⇒ F s . Moreover, given two such weak 2-functors F w and G w , the2-categories of weak – respectively strict – natural transformations and modifications between F w and G w , respectively between F s and G s , are equivalent by conjugation with f and g .A close inspection of Power’s arguments shows that the same holds when Cat is replacedby A f gr , as long as C is graded (locally) (2-)finitary with finitely many objects.(We thank Nick Gurski for explaining to us the relevance of 2-monads for strictificationand giving us some pointers to the literature.) Remark 3.9.
Note that the two-color Soergel calculi (and their Karoubi envelopes), in thefinite and the infinite case, are 2-categories with finitely many objects. Thus, by the above,the (graded simple transitive) weak 2-representations in this paper are weakly equivalent to(graded simple transitive) strict 2-representations. (cid:78) The two-color Soergel calculus and its -action Next, we construct the 2-representations from Theorem I.4.1.
Soergel calculus in two colors.
Fix n ∈ Z > or “ n = ∞ ”. We recall Elias’ two-colorSoergel calculi D (cid:63)n and D (cid:63) ∞ , as defined in [Eli16, Sections 5 and 6]. We write D (cid:63) if noconfusion can arise, and also use “unstarred” versions D of these. Remark 4.1.
Let us now fix the technical conditions defining our ground rings.For weak categorifications, as in Section 2.3.4, we can work over A gen = Z [q , q − ] for afixed (generic) q ∈ C − { } .For strong categorifications, as in Section 2.4, we have to adjoin the inverse of somequantum integers as well. For G of ADE type with Coxeter number n > n th root of unity and let A n = Z [ / , q , q − , / [2] q , . . . , / [ n − q ] . (4.1) For n = 1 and n = 2 we let A n = Z [ / ] and Z [ / , ±√− z ] q = q z − + q z − + · · · + q − z + q − z , z ∈ Z > , [0] q = 0 . For more general bipartite graphs we use A G,(cid:126)λ, q as in Section 4.2.5, which is obtained from Z by adjoining a finite number of complex numbers and their inverses. To be precise, weadjoin the (4.BF2)-weightings as in Section 4.2.3. (cid:78) The definition of those calculi depends on a fixed parameter q ∈ C − { } , which is as inSection 2.3 and Remark 4.1. Our ground ring is again the appropriate A .4.1.1. Soergel diagrams.
We consider the following generating 2-morphisms. ss deg = 0 , tt deg = 0 , s • deg = 1 , s • deg = 1 , t (cid:4) deg = 1 , t (cid:4) deg = 1 , ss s deg = − , ss s deg = − , tt t deg = − , tt t deg = − . (4. D gen1)(Recall that the objects s and t correspond to θ s and θ t of H, and not to the Coxetergenerators s and t of W, see e.g. [Eli16, Theorems 5.29 and 6.24].)We give these 2-morphisms the indicated degrees, and call them identities , ( end and start ) dots , and trivalent vertices ( split and merge ) respectively.All generators in (4. D gen1) are independent of n . The following degree-zero generatordepends on n and is called the 2 n -vertex : n (cid:122) (cid:125)(cid:124) (cid:123) v s t s t s tu t s t s t s · · ·· · · deg = 0 , v tu s · · ·· · · n shorthand . (4. D gen2)Here u is either s or t , and v is the opposite, depending on n . The color inverted ( s (cid:29) t )version of the 2 n -vertex in (4. D gen2) is our last generator. Throughout the paper, we willuse the shorthand for the 2 n -vertices as in (4. D gen2).The vertical composition g ◦ v f is given by gluing diagram g on top of diagram f (incase the colors match), the horizontal composition g ◦ h f by putting g to the left of f . Ourconventions are best illustrated in an example. Example 4.2.
For instance, in case n = 3: s t st s t •• ss ◦ v (cid:4)(cid:4) tt ◦ h s t st s t = s t s •• s (cid:4)(cid:4) t t s t : tstt → ssts. Here we have also indicated our reading conventions for Soergel diagrams. (cid:78)
WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 17
Definition 4.3.
Given two words w, w (cid:48) in the symbols s and t , a Soergel diagram from w to w (cid:48) is a diagram ◦ v - ◦ h -generated by the generators from (4. D gen1) and (4. D gen2) such thatthe outgoing edges correspond color-wise to the entries of w , w (cid:48) .The degree of a Soergel diagram is, by definition, the sum of the degrees of its generators.(The Soergel diagram ∅ : ∅ → ∅ is, by convention, of degree zero.) (cid:78) The top and bottom of a Soergel diagram correspond to direct sums (we do not assumethat words in s and t are reduced) of so-called Bott–Samelson bimodules , which categorifyBott–Samelson basis elements θ w , for w ∈ W, cf. Section 2.1.2. In general, the Bott–Samelsonbimodules do not coincide with the indecomposable Soergel bimodules, which categorify theKazhdan–Lusztig basis elements θ w . Example 4.4.
The following shorthand notations and their color inversion ( s (cid:29) t ) s s • = s s , s s • = s s , define cup and cap Soergel diagrams, which are of degree zero. (cid:78) “Dihedral Jones–Wenzl projectors”. From now on we write id for Soergel diagramsconsisting of just identity strands.
Definition 4.5.
For k ∈ Z ≥ , we define JW sk to be the formal A -linear combination ofSoergel diagram obtained as follows. Set JW s = ∅ , JW s = id andJW sk = ssttssuuvvuuvv · · ·· · · JW sk − + [ k − q [ k − q · (cid:7)(cid:7) stsuvuv stsuvuv · · ·· · ·· · · JW sk − JW sk − . (4.4)Similarly, we define JW tk . Note that for any k ∈ Z ≥ , JW sk is a A -linear combination ofdegree zero Soergel diagrams. (cid:78) If q is not a root of unity or q is equal to ±
1, then JW uk is well-defined for any k ∈ Z ≥ (ifone works in e.g. C ). If q is a complex, primitive 2 n th root of unity, then JW uk (with recursionas in Definition 4.5) is well-defined for 0 ≤ k ≤ n , and we can work over A n . Moreover, werequire q to be a complex, primitive 2 n th root of unity in the proof of Proposition 4.12 inorder to make JW un − rotationally invariant, see e.g. [Eli16, before Proposition 1.2]. Remark 4.6.
Formula (4.4) comes from Wenzl’s formula for the Jones–Wenzl projectors inthe Temperley–Lieb algebra, see [Wen87]. See also [AT17, Definition 4.12]. (cid:78)
Remark 4.7.
From JW uk we obtain a certain diagram JW uk . Since we do not need it veryoften in this paper, we refer to [Eli16, below Definition 5.14] for its definition. For ourpurposes it is enough to know that any 2-functor maps JW uk to zero if and only if it mapsJW uk to zero. The picture to keep in mind is: ssttssuuvvuu · · ·· · · JW sk JW sk using the convention that “open” strings are to be closed with dots. This is almost a definitionof JW sk , given the relations in the Soergel calculus. (cid:78) Example 4.8.
With three strands we have the following:JW s = ss tt ss + / [2] q · (cid:4)(cid:4) ss tt ss (cid:32) JW s = • • ts st + / [2] q · (cid:4) (cid:4) ts st . More examples can be found in e.g. [Eli16, Examples 5.16 and 5.17]. (cid:78)
The two-color Soergel calculus.
Recall that we have fixed n ∈ Z > or “ n = ∞ ”. In thefirst case, let q be any primitive 2 n th root of unity. In the second case, let q be any non-zerocomplex number. Definition 4.9.
Denote by D n the additive closure of the A -linear 2-category, determinedby the following data:(i) There is one (not further specified) object.(ii) The 1-morphisms are formal shifts of finite words w in the symbols s and t . (Tosimplify notation, we often omit these shifts in the diagrams.)(iii) The 2-morphism space 2Hom D n ( w { a } , w (cid:48) { b } ) is the A -linear span of all Soergeldiagrams from w to w (cid:48) of degree b − a , quotiented by the relations from (4.EH) to(4.2 n v3).(iv) Vertical composition ◦ v is induced by the vertical gluing of Soergel diagrams, whilethe horizontal composition ◦ h is induced by putting diagrams next to each otherhorizontally. (Using the same reading conventions as above.)For the usual Eckmann–Hilton relation for 2-morphisms to hold, we have to impose thefar-commutativity relation (here f, g are two arbitrary Soergel diagrams): w l w (cid:48) l (cid:48) w k +1 w (cid:48) k (cid:48) +1 w k w (cid:48) k (cid:48) w w (cid:48) · · · · · ·· · · · · · g f = w l w (cid:48) l (cid:48) w k +1 w (cid:48) k (cid:48) +1 w k w (cid:48) k (cid:48) w w (cid:48) · · · · · ·· · · · · · g f . (4.EH)The other relations among Soergel diagrams are the following. First, the Frobenius relations(including the horizontal mirror of (4.Fr2)):(4.Fr1) ss ss = ss ss = ss ss , (4.Fr2) s s • = ss = ss • . Then the needle relation (including its horizontal mirror): s = 0 . (4.Ne) WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 19
The next relations, still independent of n , are the barbell forcing relations:(4.BF1) ss •• = 2 · ss •• − ss •• , (4.BF2’) ss (cid:4)(cid:4) = ss (cid:4)(cid:4) + [2] q · ss •• − [2] q · ss •• . Finally, three relations which depend on n (including the horizontal mirror of the displayedones, the versions for even n , and all possibilities for the position of the dot in (4.2 n v2)):(4.2 n v1) t t ts s · · ·· · · n = t t ts s · · ·· · ·· · · n n , (4.2 n v2) t tt t (cid:4) s ss ss ···· ···· n = t tt ts ss ss ···· ···· JW un , (4.2 n v3) sst t · · · · · ·· · ·· · · n = t ts s · · ·· · · n = s s tt · · ·· · · · · ·· · · n . Moreover, all of the relations listed above, one- and two-colored, exist in two versions, i.e.the displayed ones and their color inverted ( s (cid:29) t ) counterparts. (cid:78) The 2-category D ∞ is defined similarly, but using only the generators and relations whichare independent of n .Let D (cid:63) denote the 2-category obtained from D as explained in Section 3.1. Since therelations (4.EH) to (4.2 n v2) are homogeneous with respect to the degrees of the Soergeldiagrams, the 2-category D (cid:63) is additive, graded and A -linear. Example 4.10.
Isotopy relations such as “zigzag relations” and other as e.g. s s = ss = s s , s • = s • = s • , ss s = ss s = sss are consequences of the Frobenius relations from (4.Fr1) and (4.Fr2). However, the isotopyrelations in (4.2 n v3) are not consequences of the Frobenius relations. (cid:78) Remark 4.11.
When 2 is invertible it is not hard to show that (4.BF2’) can actually bereplaced by 2 · ss (cid:4)(cid:4) − ss (cid:4)(cid:4) = − [2] q · ss •• − ss •• . (4.BF2)We use this equivalent relation later in Section 4.2.3. (cid:78) The following is a direct consequence of [Eli16, Theorems 5.29 and 6.24]. To be consistentwith our conventions from Section 3, we switch to a field K containing A . Proposition 4.12.
The 2-categories
Kar ( D n ) (cid:63) and Kar ( D ∞ ) (cid:63) are graded finitary andgraded locally finitary, respectively. (cid:4) Its -representations coming from bipartite graphs. The purpose of the presentsubsection is to define the weak 2-functors G : D (cid:63) → p End (cid:63) ( G gr )(in our usual convention, we write G for either G ∞ or G n ) from Theorem I, for any givenbipartite graph G . As we will see in Section 5.2, the 2-functor G is well-defined only forcertain values of q ∈ C − { } , which depend on G .Now, G sends the unique object of D (cid:63) to G gr . In Section 2.3 we already defined G on1-morphisms: G ( s ) = Θ s , G ( t ) = Θ t . Next, we define G on 2-morphisms, which we first do for D (cid:63) ∞ and then for D (cid:63)n .Moreover, we start with G ’s of ADE type (our main interest), and then discuss thegeneralization to arbitrary bipartite graphs.4.2.1. Assignment in the infinite case.
Fix q ∈ C − { } . We have to assign natural transfor-mations to the generators from (4. D gen1). Recall that the functors Θ s and Θ t are givenby tensoring with the sum of the Q G -bimodules P i {− } ⊗ i P over all sea-green i and overall tomato j colored vertices of G , respectively. Thus, Q G -bimodule maps between tensorproducts of the Q G -bimodules P i {− } ⊗ i P induce natural transformations between thecorresponding composites of Θ s and Θ t . (This assignment is not strict, cf. Example 3.1.)Hence, we first specify a Q G -bimodule map for each of the generating Soergel diagrams andthen check that our assignment preserves the relations (4.EH) to (4.BF2) of the two-colorSoergel calculus.To understand our assignment below recall that, for all vertices i ∈ G , there are free A -modules i P i given by i P i = i P (cid:98) ⊗ P i ∼ = A (i) ⊕ A (i | i) , (4.15)where the isomorphism, which we fix, is given by y (cid:98) ⊗ x (cid:55)→ y x . We will use (4.15) strategicallybelow. Moreover, there are graded Q G -bimodules P i {− } ⊗ i P ∼ = (cid:18) A (i) ⊕ A (i | i) ⊕ (cid:76) i j A (j | i) (cid:19) {− }⊗ (cid:18) A (i) ⊕ A (i | i) ⊕ (cid:76) i j A (i | j) (cid:19) , j P i = j P (cid:98) ⊗ P i ∼ = A (j | i) , which easily follows from (2.10) and the same isomorphism as before, respectively. Hereby werecall that the left action is given by post-composition, and the right action by pre-compositionof paths.In the following we only give some of the Q G -bimodule maps. The other ones can beobtained from these via color inversion ( s (cid:29) t ) and interchanging S and T.Some of the maps below are weighted sums. The weights λ i are invertible elements of A ,which depend on G , and will be defined in Definition 4.16.In the assignment below, we fix i ∈ G , while j always means a vertex of G connected to i.We then give each Q G -bimodule map only on certain basis elements. The rest of the mapis determined by the fact that it is supposed to be a Q G -bimodule map. (We will check inLemma 5.6 that our assignments are indeed Q G -bimodule maps.) WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 21
In order to get started, let x i ∈ P i and i y ∈ i P , and let us write (cid:76) i = (cid:76) i ∈ G etc. for short,and notations of the form x i ⊗ i y indicate that we take the corresponding entries from thedirect sums. (We extend the below A -linearly.) Identity generators.
To these we assign the corresponding identity maps.
Dots.
We choose the following Q G -bimodule maps. s • G (cid:32) (cid:40)(cid:76) i P i {− } ⊗ i P → Q G,x i ⊗ i y (cid:55)→ x i i y, (4.d1) s • G (cid:32) Q G → (cid:76) i P i {− } ⊗ i P, i (cid:55)→ λ i · (i ⊗ i | i + i | i ⊗ i) , j (cid:55)→ (cid:80) i j ( λ i · j | i ⊗ i | j) . (4.d2) Trivalent vertices.
Using the identification from (4.15), we pick: ss s G (cid:32) (cid:40)(cid:76) i P i {− } ⊗ i P → (cid:76) i P i {− } ⊗ i P i {− } ⊗ i P,x i ⊗ i y (cid:55)→ x i ⊗ i ⊗ i y, (4.t1) ss s G (cid:32) (cid:76) i P i {− } ⊗ i P i {− } ⊗ i P → (cid:76) i P i {− } ⊗ i P,x i ⊗ i ⊗ i y (cid:55)→ ,x i ⊗ i | i ⊗ i y (cid:55)→ λ − · ( x i ⊗ i y ) . (4.t2) Tomato t colored diagrams. We inverted the colors ( s (cid:29) t ), including λ i (cid:29) λ j .Our assignments above extend to any 2-morphism in D (cid:63) ∞ which is written as a horizontaland vertical composite of the generators (using (3.1)). Example 4.13.
Let us denote by d the natural transformation induced by the Q G -bimodulemap from (4.d1). Consider the following two natural transformations. sss • G (cid:32) id ◦ h d : Θ s Θ s ⇒ Θ s , s ss • G (cid:32) d ◦ h id : Θ s Θ s ⇒ Θ s . Now – by the above – we have using (4.15) (which we will do silently from now on) id ◦ h d (cid:33) (cid:40)(cid:76) i P i {− } ⊗ i P i {− } ⊗ i P → (cid:76) i P i {− } ⊗ i P,x i ⊗ i y (cid:48) x (cid:48) i ⊗ i y (cid:55)→ x i ⊗ (cid:16) i y (cid:48) x (cid:48) i (cid:17) i y. d ◦ h id (cid:33) (cid:40)(cid:76) i P i {− } ⊗ i P i {− } ⊗ i P → (cid:76) i P i {− } ⊗ i P,x i ⊗ i y (cid:48) x (cid:48) i ⊗ i y (cid:55)→ x i (cid:16) i y (cid:48) x (cid:48) i (cid:17) ⊗ i y. Hereby x i , x (cid:48) i ∈ P i and i y, i y (cid:48) ∈ i P , as before. (cid:78) Assignment in the finite case.
Let G be of ADE type and q be – as usual – a complex,primitive 2 n th root of unity. We continue to write G , whose definition we need to completeon the 2 n -valent vertices. To this end, recall Θ s n and Θ t n from Section 2.1.2 n -valent vertices. We assign the zero maps: v tu s · · ·· · · n G (cid:32) , v su t · · ·· · · n G (cid:32) . (4.2 n v)These give rise to the zero natural transformations between Θ s n and Θ t n , and between Θ t n and Θ s n , respectively Again, we extend everything horizontally (using (3.1)).4.2.3. The weighting.
Fix any q ∈ C − { } . Recall that V = S (cid:96) T denotes the two-colorededge set of G . Definition 4.14. A weighting (cid:126)λ of G is an assignment V → C − { } , i (cid:55)→ λ i , j (cid:55)→ λ j . The scalars λ i are called weights .Fixing an ordering of the vertices of G as in Section 2.2.2 we can write (cid:126)λ = ( λ i , . . . , λ i | S | , λ j , . . . , λ j | T | ) . Then the triple (
G, (cid:126)λ, q) is called a (4.BF2) -weighting if A ( G ) (cid:126)λ = − [2] q · (cid:126)λ, (4.21)where A ( G ) is the adjacency matrix of G (in the evident ordering). (cid:78) Note that (4.21) is equivalent to − [2] q · λ i = (cid:80) i j λ j , for all i ∈ G. (4.22) Remark 4.15.
Let us explain where the condition (4.22) comes from. Recall that we assumethat 2 is invertible, and we can replace (4.BF2’) by (4.BF2), cf. Remark 4.11.Thus, in order for the corresponding sums of (5.2) and (5.5) to work out (these appear inthe proof that (4.BF2) holds in the 2-representation), we need precisely condition (4.22) tobe satisfied. Hence, the name (4.BF2)-weighting. (cid:78)
Definition 4.16.
Let G be of ADE type and n be its Coxeter number, where we use theconventions from (A), (D) and (E). Let q be the usual (fixed) complex, primitive 2 n -root ofunity.We define (4.BF2)-weightings for these G ’s as follows. The scalars λ i do not depend onthe two-coloring, and are given next to the vertices. • • • · · · • • • +[1] q − [2] q +[3] q ∓ [ m − q ± [ m − q ∓ [ m ] q , (A m ) • • · · · • • •• +[1] q − [2] q ∓ [ m − q ± [ m − q ∓ [ m − / ∓ [ m − / , (D m ) WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 23 • • • • •• +[1] q − [2] q − [3] q / [2] q +[3] q − [2] q +[1] q , (E ) • • • • • •• +[1] q − [2] q +[3] q +[4] q / [2] q − [4] q +[6] q / [2] q − [4] q / [3] q , (E ) • • • • • • •• +[1] q − [2] q +[3] q − [4] q − [5] q / [2] q +[5] q − [7] q / [2] q +[5] q / [3] q . (E )This defines the triple ( G, (cid:126)λ, q). (cid:78)
Remark 4.17.
These λ i ’s can be found solving the eigenvector problem for the adjacencymatrix of G , cf. (4.21). The weights are actually the entries of the Perron–Frobeniuseigenvector – normalized to have the entry 1 at a fixed vertex – for the smallest eigenvalue.(Recall hereby that the spectrum of a bipartite graph is a symmetric set. Hence, there is aunique smallest eigenvalue. See also in the proof of Lemma 4.21.)The reader familiar with [KJO02] might also note that the weights above are equal tothe quantum numbers associated to the vertices of G in [KJO02, Section 6], after divisionby the quantum number associated to the left-most vertex (and adding signs to match ourconventions). (cid:78) It remains to prove well-definedness, which we will do in Section 5, i.e. we will show thatthe 2-functor G is well-defined if and only if the chosen weighting is (4.BF2).4.2.4. “Uniqueness” of the -functor G . For G of ADE type, the assignment from above isessentially unique: Lemma 4.18.
Let G be of ADE type and fix q to be a complex, primitive 2 n th root of unity.For any additive, degree-preserving, C -linear, weak 2-functor H : D (cid:63) ∞ → p End (cid:63) ( G gr )which agrees with G on 1-morphisms, there exist scalars τ , υ ∈ C − { } such that H ( f ) = τ d st − t me · υ d end − t sp · G ( f ) , for any homogeneous 2-morphism f built from d st start dots, t me merges, d end end dots and t sp splits.In particular, H is equivalent to G . (cid:3) Lemma 4.19.
Any additive, degree-preserving, C -linear, weak 2-functor H : D (cid:63)n → p End (cid:63) ( G gr )which agrees with G on 1-morphisms sends the 2 n -valent vertices to zero. (cid:3) More general bipartite graphs.
We will now briefly discuss (4.BF2)-weightings forgraphs which are not of ADE type.We first give two examples:
Example 4.20.
Here are two examples for G ’s being not of ADE type: • • • ••• +[1] q − [1] q +[1] q − [1] q +[1] q − [1] q , • •• •• •• •• +[1] q +[1] q +[1] q +[1] q − [2] q − [2] q − [2] q − [2] q +[3] q . The first weighting is (4.BF2) if and only if − [2] q · ± [1] q = 2 · ∓ [1] q , i.e. if and only if q = 1.The second weighting is (4.BF2) if and only if − [4] q + 3 · [2] q = 0 and all involved weightsare non-zero. This happens if and only if q ∈ { / · (1 ± √ , − / · (1 ± √ } . (cid:78) In general, we will prove the following result.
Lemma 4.21.
For any bipartite graph G , there is at least one value of q ∈ C − { } suchthat a (4.BF2)-weighting exists. (cid:3) Note that Lemma 4.21 is not immediate, since we need all weights to be invertible.5.
Proofs
Finally, we give the proofs of all statements.5.1.
The uncategorified story.
Fix a bipartite graph G . The following lemma followsfrom the fact that the P i {− } ⊗ i P ’s are graded biprojective Q G -bimodules. Lemma 5.1.
The functors Θ s and Θ t are degree-preserving and biprojective. (cid:4) Lemma 5.2.
The functors Θ s and Θ t are additive, A -linear and self-adjoint. (cid:3) Proof.
We only treat the case of Θ s here, the other is similar and omitted.Up to the self-adjointness of Θ s the statement is clear. Moreover, we claim that s s G (cid:32) i : ID ⇒ Θ s Θ s , s s G (cid:32) e : Θ s Θ s ⇒ ID , form the unit respectively counit of the self-adjunction for Θ s . Next, there are two things tocheck. Namely that Θ s is left adjoint to itself, and that it is right adjoint to itself, i.e. wehave to show that id Θ s = e Θ s ◦ v Θ s i , id Θ s = Θ s e ◦ v i Θ s . Recalling the definitions of cup and cap from Example 4.4, we see that graphically this is the“zigzag relation” from Example 4.10. Hence, the claim of self-adjointness follows from thewell-definedness of G , which we show later in Section 5.2. (cid:4) Alternatively, after checking that the underlying quiver algebra is weakly symmetric andself-injective – which can be done by e.g. copying [HK01, Proposition 1] – one could also use[MM11, Section 7.3] to prove Lemma 5.2 abstractly. In contrast, our proof above fixes thenatural transformations realizing the adjunctions.
WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 25
Proof of Proposition 2.14.
First, by using Lemma 5.2, we see that the two functors Θ s and Θ t are exact and descent to the Grothendieck group.Next – using (2.12) – we get for i ∈ S that Θ s Θ s ( P i ) ∼ = P i {− } ⊕ P i ⊕ P i ⊕ P i { + } ∼ = Θ s ( P i ) {− } ⊕ Θ s ( P i ) { + } , Θ t Θ t ( P i ) ∼ = (cid:76) i j P j {− } ⊕ P j { + } ∼ = Θ t ( P i ) {− } ⊕ Θ t ( P i ) { + } . We get a similar result for P j , with j ∈ T. Therefore, we get the following natural isomorphismsof degree-preserving functors: Θ s Θ s ∼ = Θ s {− } ⊕ Θ s { + } , Θ t Θ t ∼ = Θ t {− } ⊕ Θ t { + } . Since these are the defining relations of H ∞ from (2.2), we see that [ G gr ] C (v) is indeed anH ∞ -module. Next, choosing the evident basis of [ G gr ] C (v) given by the [ P i ]’s and using (2.12),one can see that [ Θ s ] and [ Θ t ] act as in (2.13). This shows that ζ G is an H ∞ -homomorphism.That ζ G is bijective is clear. (cid:4) Proof of Proposition 2.15.
The claim basically follows by observing that the scalars from(2.1) for the defining relations of the θ s and θ t generators of H are the coefficients of the(normalized) Chebyshev polynomials (of the second kind) given by˜ U = 1 , ˜ U = X, ˜ U k +1 = X ˜ U k − ˜ U k − for k ∈ Z ≥ . To give a few more details: Given a polynomial in X , one can obtain a non-commutativepolynomial in two variables – say θ s and θ t – by replacing X k with an alternating string . . . θ s θ t θ s of length k (always having θ s to the right). We write ˜ U k ( θ s , θ t ) for the non-commutative polynomial obtained from ˜ U k in this way. Then (2.1) implies that θ s k is˜ U k ( θ s , θ t ).Now, if a representation of H ∞ factors through H n and is annihilated by θ w , then˜ U n − ( θ s , θ t ) = 0 = ˜ U n − ( θ t , θ s ) . It is not hard to deduce from this that the eigenvalues of the matrices associated to θ s and θ t in the representations from Definition 2.13 (these are – up to base change – of theform (cid:16) [2] | S | v A (cid:17) respectively (cid:16) A T [2] | T | v (cid:17) ) have to be (multi)subsets of the (multi)subset ofeigenvalues of X ˜ U n − . This follows from (2.6) and the observation stated after it.The (normalized) Chebyshev polynomials have only real roots which are given by S A m ⊂ ] − ,
2[ in (5.1) below. Now, it follows from [Smi70, Theorem 2] that the largest eigenvalue of AA T or A T A is strictly less than 4 if and only if G is as in (A), (D) or (E). (See also [BH12,Theorem 3.1.3] for a more recent proof using Perron–Frobenius theory.) This shows – keeping(2.6) in mind – that only a G as in (A), (D) or (E) (if non-trivial) can give a well-definedaction of H n .Moreover, we have the following spectra: S A m = (cid:8) (cid:0) kπ/m +1 (cid:1)(cid:12)(cid:12) k = 1 , . . . , m (cid:9) ,S D m = (cid:8) (cid:0) kπ/ m − (cid:1)(cid:12)(cid:12) k = 1 , , , . . . , m − , m − , m − (cid:9) ∪ { } ,S E = (cid:8) (cid:0) kπ/ (cid:1)(cid:12)(cid:12) k = 1 , , , , , (cid:9) ,S E = (cid:8) (cid:0) kπ/ (cid:1)(cid:12)(cid:12) k = 1 , , , , , , (cid:9) ,S E = (cid:8) (cid:0) kπ/ (cid:1)(cid:12)(cid:12) k = 1 , , , , , , , (cid:9) . (5.1)This list is known, see e.g. [BH12, Section 3.1.1]. The type A m spectrum is the (multi)set ofroots of the polynomial ˜ U m . The fact that all ADE type graphs give well-defined actions ofH n then follows easily using these spectra. In case θ w does not act as zero the explicit form of the matrices associated to θ s and θ t immediately shows that any well-defined action of H n has to be trivial, i.e. both – θ s and θ t –have to act as zero. (cid:4) Remark 5.3.
In Example 2.17 we have seen that the largest eigenvalue of AA T and A T A in type ˜A was 4. This is true for all affine types ˜A, ˜D and ˜E and these are also the onlygraphs with this property, see e.g. [BH12, Section 3.1.1]. (cid:78) The following is now a direct consequence of Proposition 2.15. (Hereby [ Θ w ] correspondsto θ w under ζ G from (2.14).) Corollary 5.4.
In the setup from Section 2.3: [ Θ w ] = 0 if and only if G is as in (A), (D)or (E) (if non-trivial). (cid:4) We finish this section with the proof of Lemma 4.21, because its proof is very much in thespirit of the proof of Proposition 2.15 above.
Proof of Lemma 4.21.
First note that Perron–Frobenius theory guarantees an eigenvector (cid:126)λ α of A ( G ) with strictly positive entries. Moreover, the corresponding (so-called) Perron–Frobenius eigenvalue α is strictly positive.Thus, after letting q to be a solution of the equation − [2] q = α , we get a solution to (4.21),i.e. a (cid:126)λ α without zero entries and a q ∈ C − { } such that (4.21) holds. (cid:4) Remark 5.5.
Note that the proof of Lemma 4.21 is constructive and can be used to produce(4.BF2)-weightings for any given bipartite graph.However, for ADE type graphs the Perron–Frobenius eigenvalue does not give q = exp( πi/n ),but rather q = − exp( πi/n ). For example, for n = 3 and type A the Perron–Frobeniuseigenvalue gives q = − exp( πi/ ), for which − [2] q = 1, rather than q = exp( πi/ ), for which − [2] q = − (cid:78) Since the Kazhdan–Lusztig basis elements of H ∞ are categorified by indecomposable 1-morphisms in Kar ( D (cid:63) ∞ ), see [Eli16, Theorem 5.29], we get a stronger version of Corollary 5.4by Theorem I later on. Namely, the matrices associated to [ Θ w ] have non-negative entriesfor all w ∈ W ∞ if and only if G is not of type ADE. The “if” part of this statement followsfrom Corollary 5.4 and the Chebyshev recursion; the “only if” part needs Theorem I. (Notehereby that for G being of type ADE we need the quantum parameter to be a root of unityto have a well-defined 2-functor and [Eli16, Theorem 5.29] does not apply anymore. Hence,there is no contradiction to Theorem I.)5.2. The infinite case.
First, we need to check that the maps specified in Section 4.2 areactually Q G -bimodule maps and give rise to natural transformations. Lemma 5.6.
The maps from (4.d1) to (4.t2) are Q G -bimodule maps. (cid:3) Proof.
The map from (4.d1) is the (scaled) multiplication map and thus, a Q G -bimodulemap. The two maps from (4.t1) and (4.t2) clearly intertwine the left and right action of Q G since their definition only involves the middle tensor factors in a non-trivial way. That themap from (4.d2) is a Q G -bimodule map can be checked via a case-by-case calculation. Weillustrate this in an example. To this end, let us denote the sea-green version of it by d ∗ .Then, recalling (2.QG3), we get d ∗ (j | i) = d ∗ (j | i i) = j | i d ∗ (i) = λ i · j | i (i ⊗ i | i + i | i ⊗ i) = λ i · j | i ⊗ i | i , d ∗ (j | i) = d ∗ (j j | i) = d ∗ (j) j | i = ⊕ i j (cid:48) λ i · (j (cid:48) | i ⊗ i | j (cid:48) ) j | i (2.QG3) = λ i · j | i ⊗ i | i . The remaining cases can be checked verbatim. (cid:4)
WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 27
The next lemmas follow directly from the definitions respectively via direct computation,and the proofs are omitted.
Lemma 5.7.
The maps from (4.d1) to (4.t2) induce 2-morphisms in p
End (cid:63) ( G gr ). Moreover,the extension of the local assignment to arbitrary Soergel diagrams is consistent with the2-structure of p End (cid:63) ( G gr ). (cid:4) Lemma 5.8.
The weightings from Definition 4.16 are (4.BF2)-weightings. (cid:4)
Proof of Theorem I, part (a).
We first note that, by Lemma 5.1, Lemma 5.6 and Lemma 5.7,the weak 2-functor G , if well-defined, is between the stated 2-categories.Furthermore, if G is well-defined, then it extends uniquely to the Karoubi envelope by e.g.[Bor94, Proposition 6.5.9], and the corresponding diagram will commute by Proposition 2.14(note that [ G gr ] C (v) ∼ = G v ). Moreover, G – if well-defined – clearly preserves all additionalstructures. Thus, it remains to check that G is well-defined which amounts to checking that G preserves the relations of D (cid:63) .We start with the far-commutativity (4.EH), which is just the interchange law (il) inp End (cid:63) ( G gr ). Denote by id , f and g the images under G of the Soergel diagrams in question.Then (by our conventions how to apply G to arbitrary Soergel diagrams): G w l w (cid:48) l (cid:48) w k +1 w (cid:48) k (cid:48) +1 w k w (cid:48) k (cid:48) w w (cid:48) · · · · · ·· · · · · · g f id id ◦ h ◦ h ◦ v = ( id ◦ h f ) ◦ ( g ◦ h id ) il = ( id ◦ g ) ◦ h ( f ◦ id )= ( g ◦ id ) ◦ h ( id ◦ f ) il = ( g ◦ h id ) ◦ ( id ◦ h f ) = G w l w (cid:48) l (cid:48) w k +1 w (cid:48) k (cid:48) +1 w k w (cid:48) k (cid:48) w w (cid:48) · · · · · ·· · · · · · g f id id ◦ h ◦ h ◦ v . Next, we check that the other relations hold. There are several cases depending on whetheri is in S or T, as well as on the color of the involved Soergel diagrams. We do some of themand leave the other (completely similar) cases to the reader. We also omit to indicate thesources and targets of the maps. As usual, we write x i ∈ P i , i y ∈ i P and i z i ∈ i P i . We alsocalculate the assignments on the corresponding direct summands only. The Frobenius relation (4.Fr1) . We get the Q G -bimodule maps ss ss G (cid:32) (cid:40) x i ⊗ i ⊗ i y (cid:55)→ ,x i ⊗ i | i ⊗ i y (cid:55)→ λ − · x i ⊗ i ⊗ i y, ss s ss G (cid:32) x i ⊗ i z i ⊗ i y (cid:55)→ x i ⊗ i ⊗ i z i ⊗ i y, ss sss G (cid:32) (cid:40) x i ⊗ i z i ⊗ i ⊗ i y (cid:55)→ ,x i ⊗ i z i ⊗ i | i ⊗ i y (cid:55)→ λ − · x i ⊗ i z i ⊗ i y, ss sss G (cid:32) x i ⊗ i z i ⊗ i y (cid:55)→ x i ⊗ i z i ⊗ i ⊗ i y, ss s ss G (cid:32) (cid:40) x i ⊗ i ⊗ i z i ⊗ i y (cid:55)→ ,x i ⊗ i | i ⊗ i z i ⊗ i y (cid:55)→ λ − · x i ⊗ i z i ⊗ i y. These compose as claimed.
The Frobenius relation (4.Fr2) . We get (keeping (2.QG3) in mind which kills a lot of terms): sss • G (cid:32) x i ⊗ i y (cid:55)→ λ i · (cid:0) x i ⊗ i y i ⊗ i | i + x i ⊗ i y i | i ⊗ i (cid:1) + (cid:80) i j ( λ i · x i ⊗ i y j | i ⊗ i | j) , s ss • G (cid:32) x i ⊗ i y (cid:55)→ λ i · (cid:0) i ⊗ i | i x i ⊗ i y + i | i ⊗ i x i ⊗ i y (cid:1) + (cid:80) i j ( λ i · j | i ⊗ i | j x i ⊗ i y ) . The remaining two assignments were already calculated in Example 4.13. These composewith the assignments for the trivalent vertices from (4.t1) and (4.t2) as claimed. (This relieson (2.QG3).) Let us stress that the (4.BF2)-weighting scalars cancel, since the assignmentsfrom (4.d2) and (4.t2) are multiplied by inverse scalars.
The needle relation (4.Ne) . Directly by composing (4.t2) and (4.t1) (in this order).
The first barbell forcing relation (4.BF1) . Using (4.d1) and (4.d2), we get: •• G (cid:32) i (cid:55)→ λ i · · i | i , j (cid:55)→ − [2] q · λ j · j | j , i | i and i | j and j | i and j | j (cid:55)→ , (5.2)where we used (4.22) to replace (cid:80) i j λ i by − [2] q · λ j . Next, we have ss •• G (cid:32) x i ⊗ i y (cid:55)→ λ i · (cid:0) ( x i i) ⊗ (i | i i y ) + ( x i i | i) ⊗ (i i y ) (cid:1) + (cid:80) i j ( λ i · ( x i j | i) ⊗ (i | j i y )) . (5.3)Therefore – by (5.2) – we have ss •• G (cid:32) x i ⊗ i y (cid:55)→ λ i · (cid:0) · (i | i x i ) ⊗ i y (cid:1) − [2] q · λ j · ((j | j x i ) ⊗ i y ) , ss •• G (cid:32) x i ⊗ i y (cid:55)→ λ i · (cid:0) · x i ⊗ ( i y i | i) (cid:1) − [2] q · λ j · ( x i ⊗ ( i y j | j)) . (5.4)Note that, for all x i ∈ P i and all i y ∈ i P , we have j | j x i = 0 = i y j | j. This holds since allpaths in P i start in i and all paths in i P end in i, and j | j composes only with j to a non-zeroelement. Moreover, we also have x i j | i = 0 = i | j i y . In total, all the terms involving tomato tvanish. For the rest one checks directly on all possible x i , i y that the claimed equation holds. WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 29
For example, in case x i = i and i y = i | i one gets λ i · · i | i ⊗ i | i from the sum of the two mapsin (5.4), and λ i · i | i ⊗ i | i from the map in (5.3). The second barbell forcing relation (4.BF2) . Now the scaling will be crucial.Similarly to (5.2) we get (cid:4)(cid:4) G (cid:32) j (cid:55)→ λ j · · j | j , i (cid:55)→ − [2] q · λ i · i | i , j | j and j | i and i | j and i | i (cid:55)→ , (5.5)The two not yet computed assignments are ss (cid:4)(cid:4) G (cid:32) x i ⊗ i y (cid:55)→ λ j · (cid:0) · (j | j x i ) ⊗ i y (cid:1) − [2] q · λ i · ((i | i x i ) ⊗ i y ) , ss (cid:4)(cid:4) G (cid:32) x i ⊗ i y (cid:55)→ λ j · (cid:0) · x i ⊗ ( i y j | j) (cid:1) − [2] q · λ i · ( x i ⊗ ( i y i | i)) . (5.6)As before we have j | j x i = 0 = i y j | j and the factors with the 2 die. That the claimedequation holds can be verified by a case-by-case check. (Hereby we stress that Lemma 5.8comes crucially into the game since it ensures that the scalars add up as they should.)All together this shows that G is well-defined. (cid:4) Note that in the proof above we never used that G is of ADE type, but rather that we havea (4.BF2)-weighting. Thus, the above goes through – mutatis mutandis – for any bipartitegraph with a (4.BF2)-weighting.Now we switch to C since we use some spectral theory below. Proof of Lemma 4.18.
Assume that we have fixed G of ADE type with the correspondingroot of unity q, its double-quiver algebra Q G and its module category G gr . Assume also thatwe have a (well-defined) weak 2-functor H : D (cid:63) ∞ → p End (cid:63) ( G gr ) which on 1-morphisms isequal to the weak 2-functor G from Section 4.2.Next, recall the bases of the left and the right Q G -modules P i and i P , as exemplified in(2.10). Using these bases, one easily sees that the image of the first of the two trivalentvertices has to be given by (using the notational conventions from above) ss s H (cid:32) x i ⊗ i y (cid:55)→ λ i ( ) · x i ⊗ i ⊗ i y, for some scalar λ i ( ) ∈ C . This follows directly for degree reasons and the fact that theassignment should be a Q G -bimodule map.For the same reasons, the first of the two dots has to be mapped to s • H (cid:32) x i ⊗ i y (cid:55)→ λ i ( ) · x i i y, for some scalar λ i ( ) ∈ C . Now, under the assumption that H is well-defined, the secondFrobenius relation (4.Fr2) holds, which implies that both scalars have to be invertible andsatisfy λ i ( ) = ( λ i ( )) − . Using the same reasoning, (and (4.Ne)), we see that the image of the other two generatorshas to be given by ss s H (cid:32) (cid:40) x i ⊗ i ⊗ i y (cid:55)→ ,x i ⊗ i | i ⊗ i y (cid:55)→ λ i ( ) · x i ⊗ i y. s • i H (cid:32) (cid:40) i (cid:55)→ λ i ( ) · (i ⊗ i | i + i | i ⊗ i) , j (cid:55)→ (cid:80) i j ( λ i ( ) · j | i ⊗ i | j) , for some invertible scalars λ i ( ) , λ i ( ) ∈ C satisfying λ i ( ) = ( λ i ( )) − .Similarly for tomato t , where we get four invertible complex scalars which satisfy λ j ( ) =( λ j ( )) − and λ j ( ) = ( λ j ( )) − .It remains to check that there is no choice for the weighting.To this end, we write λ i ( ) = λ i ( ) · λ i ( ) and λ j ( ) = λ j ( ) · λ j ( ). As in the proof ofTheorem I, part (a), we get •• H (cid:32) i (cid:55)→ λ i ( ) · · i | i , j (cid:55)→ ( (cid:80) i j λ i ( )) · j | j , i | i and i | j and j | i and j | j (cid:55)→ , (cid:4)(cid:4) H (cid:32) j (cid:55)→ λ j ( ) · · j | j , i (cid:55)→ ( (cid:80) j i λ j ( )) · i | i , j | j and j | i and i | j and i | i (cid:55)→ . This gives the scaled versions of (5.3) and (5.6).Using the assumption that H preserves the second barbell forcing relation (4.BF2), we canrelate the λ i ( )’s and the λ j ( )’s. Since relation (4.BF2) holds, we see that an equation ofthe form (4.22) must be satisfied for the λ i ( )’s and the λ j ( )’s. Spectral theory shows thatthe eigenspace of A ( G ) for the eigenvalue − [2] q is one-dimensional for ADE type graphs andthe corresponding roots of unity, cf. (5.1). This means that our chosen (4.BF2)-weightingfrom Definition 4.16 is unique up to a scalar ς with ς = λ i ( ) · λ i − = λ j ( ) · λ j − , for all i , j ∈ G. Observe that this amounts to saying that the corresponding products do not depend on thevertices.The first part of the claim now follows by letting τ be ς divided by the value of the enddot λ i ( ) and υ be ς divided by the value of the start dot λ i ( ) (for any vertex).Clearly, any consistent rescaling of the Q G -bimodule maps associated to the generating2-morphisms of D (cid:63) ∞ gives rise to an equivalence of 2-representations. (cid:4) Note that the one-dimensionality of the eigenspace of A ( G ) for the eigenvalue − [2] q failsin general for bipartite graphs. Consequently, Lemma 4.18 is not always true for bipartitegraphs with a given (4.BF2)-weighting.5.3. The finite case.
We again use the setup from Section 4.2.
Proof of Theorem I, part (b).
By statement (a) of Theorem I – and the evident analog ofLemma 5.6 for the maps from (4.2 n v) – it only remains to check three extra relations, i.e.the relations (4.2 n v1) to (4.2 n v3) need to be preserved under the functor G n . The relations WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 31 (4.2 n v1) and (4.2 n v3) are clearly preserved, and we need to prove that (4.2 n v2) is preservedif and only if G is of ADE type. “Only if ” . This is clear by Proposition 2.15, since otherwise [ G gr ] C (v) would inherit thestructure of an H n -module. “If ” . Note that the JW uk are well-defined for k ∈ { , . . . , n } , both when q is generic and whenq is a complex, primitive 2 n th root of unity.Let us assume that G is of ADE type. Below we show that G n (JW un ) = 0, when q is ourfixed complex, primitive 2 n th root of unity. This implies, by definition, that (4.2 n v2) ispreserved under G n .We have (see e.g. [Kho05, Section 3] and [Eli16, Section 2.2])(5.7) [ V l ] = (cid:80) kl =0 d k +1 l +1 · [ V ⊗ k ] , ≤ k ≤ l, in the Grothendieck ring of U q ( sl )- Mod (the finite-dimensional type 1 modules of quantum sl ) for q not a root of unity or generic. Here the V l ’s are the simple objects of highest weight l , and the d kl ’s are as in (2.1).The same holds in the semisimplified quotient category U q ( sl )- Mod s , when q is a complex,primitive 2 n th root of unity as long as l ≤ n . (See e.g. [And03, Theorem 3.1] for a versionthat works for even roots of unity. See also e.g. [Saw06, Section 5] for the semisimplifiedquotient.) Recall also that [ V n ] = 0 holds in [U q ( sl )- Mod s ] C (v) .Note that the sign of d k +1 l +1 alternates as k runs from 0 to l . Khovanov [Kho05, Theorem 1]showed that the right-hand side of (5.7) is the Euler characteristic of a complex C ∗ l : C l → C l → · · · → C l /2 l , for l even , C ∗ l : C l → C l → · · · → C l − l , for l odd , which is acyclic in positive cohomological degree and H ( C ∗ l ) ∼ = V l . More explicitly, C kl = (cid:16) V ⊗ ( l − k )1 (cid:17) ⊕ d k +1 l +1 , and the differential is defined as a direct sum obtained by applying the evaluation map V ⊗ V → C , multiplied by a suitable sign, to each neighboring tensor pair V ⊗ V in eachcopy of V ⊗ ( l − k )1 .Note that Khovanov proves his theorem for q = 1 and mentions that the proof also worksfor generic q. However, his proof follows by induction from the fact that V ⊗ V l ∼ = V l − ⊗ V l +1 ,which is also true in our case (by a special case of the so-called linkage principle) as long as0 ≤ l ≤ n − q ( sl )- Mod s is a semisimple abelian category, its bounded derived category isalso semisimple [GM03, Section III.3] and equivalent to (cid:76) Z U q ( sl )- Mod s by the homologyfunctor. Moreover, the bounded derived category is equivalent to the homotopy categoryof bounded complexes, because all objects are projective, see e.g. [Wei94, Theorem 10.4.8].Thus, C ∗ l is equivalent to V l (seen as a complex concentrated in homological degree zero).By [Eli16, Proposition 1.2], this implies that, for any 0 ≤ l ≤ n , the object ( u l , JW ul ) in Kar ( D ) is homotopy equivalent to a complex D ∗ l ( u ) with D kl ( u ) = u ⊕ d k +1 l +1 l , for u = s, t. Since this holds in the finite and infinite case alike, the definitions of the complex D ∗ l ( u )and the homotopy equivalence with ( u l , JW ul ) do not use any 2 n -vertices. For the samereason, the defining properties of the homotopy equivalence follow from the relations in D which do not involve 2 n -vertices.Since we already proved that G n preserves all the relations which do not involve 2 n -vertices,we see that G n ( D ∗ n ( u )) is a complex and that it is homotopy equivalent to G n ( u n , JW un ). The latter has Euler characteristic equal to zero, by Proposition 2.15. Therefore, G n (JW un ) = 0,which is what we had to prove. (cid:4) Example 5.9.
In principle one could also check by hand that G n (JW un ) = 0 holds. This is adaunting task in general, but let us do a small case which is quite illustrative. Fix the typeA graph with vertices 1 and 2, and the weighting is λ = [1] q = 1 and λ = − [2] q = −
1. Inthis case n = 3 and q is the complex, primitive 6th root of unity q = exp( πi/ ). After a smallcalculation one gets (cid:4)(cid:4) ss tt ss G n (cid:32) λ − λ · ID X , •• tt ss tt G n (cid:32) λ λ − · ID Y . Hereby X and Y are the corresponding tensor products for the boundary sequences. Hence,with our weighting, we get G (JW s ) = 0 and G (JW t ) = 0, since the two summands in theirexpressions in Example 4.8 cancel. (cid:78) Proof of Lemma 4.19.
If a 2-representation of D (cid:63)n agrees with our G on 1-morphisms, thenits decategorification has to be isomorphic to [ G ]. Therefore, it has to kill [ Θ w ]. But Θ w corresponds to the idempotent defined as the image of the 2-colored version of the“Jones–Wenzl projector” JW un , so that has to be sent to zero by the above. This implies thatthe 2 n -valent vertex has to be sent to zero as well by [Eli16, (6.16)]. (cid:4) Classification of dihedral -representations. We work over C in Theorem II. Proof of Theorem II.
We have to prove the statements (a), (b), (c) and (d).
Proof of statement (a) . By Proposition 4.12,
Kar ( D n ) (cid:63) and Kar ( D ∞ ) (cid:63) are graded (locally)finitary, and thus, we see that we actually get graded finitary, weak 2-representations inTheorem I.Hence, we only need to show that G is simple transitive. Transitivity follows from theconnectivity of G . For each P i and P j , apply an alternating sequence Θ alt of Θ s and Θ t ,starting in the opposite color of i ∈ G , to P i of length determined by the minimal pathconnecting i and j in the graph G . Then P j { a } will be a summand of Θ alt ( P i ) for some shift a ∈ Z . This, by Remark 3.7, shows transitivity.It remains to show that there are no non-trivial ideals. This is clearly true in case G hasonly one vertex. Otherwise, fix one sea-green s colored vertex i. Let us restrict the 2-actionto Θ s on the additive category generated by P i . This category is equivalent to the categoryof graded D-modules for D = C [ X ] / ( X ). Hence, the 2-action of Θ s is given by tensoringwith the biprojective D-bimodule D ⊗ D, see e.g. [MM17, Section 3.3]. Thus, the non-identityendomorphism i | i on P i cannot belong to any ideal which is stable under the 2-action anddoes not contain any identity morphism (this follows from [MM17, Proposition 2]). This,of course, holds for any vertex i, i.e. none of the maps i | i are in such an ideal. Now, ifi | j : P j → P i (or j | i : P i → P j ) would be contained in such an ideal, then, because it is a 2-ideal,composition with j | i : P i → P j (or i | j : P j → P i ) would imply that the result i | i : P i → P i (orj | j : P j → P j ) belongs to the ideal, which is a contradiction. Proof of statement (b) . (Below we use (cid:48) as a notation for anything related to G (cid:48) . Similarly inthe rest of the whole proof.) First assume that G and G (cid:48) are isomorphic as bipartite graphsby some map f : G → G (cid:48) . Such an isomorphism induces a reordering of the indecomposablesof G gr and ( G (cid:48) ) gr via P i (cid:55)→ P (cid:48) f(i) and P i (cid:55)→ P (cid:48) f(i) . Thus, we can define a functor F : G gr → ( G (cid:48) ) gr , F ( P i ) = P (cid:48) f(i) , WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 33 which maps each morphism in G gr to the evident ( f -reordered) morphism in ( G (cid:48) ) gr . This iswell-defined since f is an isomorphism of bipartite graphs (in particular, for each i ∈ G theimage f (i) ∈ G (cid:48) has precisely the same two-step connectivity neighborhood). This functoris clearly structure preserving, and it is an equivalence since for all i ∈ G the valencies arepreserved (which shows that the functor induces isomorphisms between the finite-dimensionalhom-spaces). Thus, F is a structure preserving equivalence of categories.By construction, Θ s is given by tensoring with sums of Q G -bimodules of the form P i {− }⊗ i P , while Θ (cid:48) s is given by tensoring with sums of Q G -bimodules of the form P (cid:48) i {− } ⊗ i P (cid:48) .Thus, there is a commuting diagram G gr Θ s (cid:47) (cid:47) F (cid:15) (cid:15) G gr F (cid:15) (cid:15) ( G (cid:48) ) gr Θ (cid:48) s (cid:47) (cid:47) ( G (cid:48) ) gr , which similarly exists for Θ t and Θ (cid:48) t as well. Hence, we have Θ (cid:48) s = F Θ s F − and Θ (cid:48) t = F Θ t F − . This means that G and G (cid:48) agree on 1- and on 2-morphisms of D (cid:63) up to F -conjugation. Thus, this induces a (degree-zero) modification between (cid:101) G and (cid:101) G (cid:48) , which is a2-isomorphism.Vice versa, any equivalence between (cid:101) G and (cid:101) G (cid:48) sends indecomposables to indecomposables,so it descends to an intertwining isomorphism of the form[ G gr ] C (v) ∼ = −→ [( G (cid:48) ) gr ] C (v) , [ P i ] (cid:55)→ [ P (cid:48) i (cid:48) ]between the Grothendieck groups. Such an isomorphism can neither send a [ P i ] to a [ P (cid:48) i (cid:48) ] nora [ P i ] to a [ P (cid:48) i (cid:48) ], since this will not intertwine the action of [ Θ s ] and [ Θ t ] to [ Θ (cid:48) s ] and [ Θ (cid:48) t ].In particular, the number of sea-green i and tomato j colored vertices has to be the samefor G and G (cid:48) . Further, for the same reasons, if such an isomorphism sends [ P i ] and [ P j ] to[ P (cid:48) i (cid:48) ] and [ P (cid:48) j (cid:48) ], and i , j are connected via an edge in G , then i (cid:48) , j (cid:48) are connected via an edge in G (cid:48) . In total, such a permutation gives rise to an isomorphism of bipartite graphs f : G → G (cid:48) defined via f (i) = i (cid:48) . Proof of statement (c) . First – by using traces – one can see that two bipartite graphs G and G (cid:48) which are spectrum-color-inequivalent give non-isomorphic H-modules. Precisely, forall k ∈ Z ≥ , we have (top: even k ; bottom: odd k ):[ Θ s k ] = ( AA T ) k − · (cid:18) AA T A [2] v (cid:19) , [ Θ t k ] = ( A T A ) k − · (cid:18) A T [2] v A T A (cid:19) , [ Θ s k ] = ( AA T ) k − · (cid:18) [2] v A (cid:19) , [ Θ t k ] = ( A T A ) k − · (cid:18) A T [2] v (cid:19) . (5.8)(Where Θ s k and Θ t k mean the evident analog of the notation from Section 2.1.) Here A isas in (2.5) for A ( G ). This works completely analogously for [ Θ (cid:48) s k ] and [ Θ (cid:48) t k ] as well, whereone uses A (cid:48) – coming from A ( G (cid:48) ) – instead of A .If | S | (cid:54) = | S (cid:48) | or | T | (cid:54) = | T (cid:48) | , then the bottom row in (5.8) produces different traces, so therepresentations are non-isomorphic.If S G (cid:54) = S G (cid:48) , then one also gets non-isomorphic representations. To see this, note that thespectrum of the matrices in the top row associated to [ Θ s k ] and [ Θ t k ], is that of ( AA T ) k +2 / and ( A T A ) k +2 / , for all k ∈ Z ≥ . A similar observation holds for the spectrum of [ Θ (cid:48) s k ] and[ Θ (cid:48) t k ]. So these spectra consist of powers of the elements of S G and S G (cid:48) , respectively. Sincethe spectrum is an invariant of the representation, the result follows. Next, let us assume that A ( G ) and A ( G (cid:48) ) have the same spectra. Thus, by (2.5), thereexists a singular value decomposition of the form A = U Σ V ∗ , A (cid:48) = W Σ X ∗ with unitary, complex-valued matrices U, V, W, X of the appropriate sizes. (Note that wework over C (v), but A ( G ) and A ( G (cid:48) ) are integral matrices. Hence, their singular valuedecompositions exist.) Thus, (cid:18) W U ∗ XV ∗ (cid:19) [ Θ s ] (cid:18) U W ∗ V X ∗ (cid:19) = (cid:18) W U ∗ XV ∗ (cid:19)(cid:18) [2] v A (cid:19)(cid:18) U W ∗ V X ∗ (cid:19) = (cid:18) [2] v A (cid:48) (cid:19) = [ Θ (cid:48) s ] , and similarly – using the same matrices for conjugation – for [ Θ t ] and [ Θ (cid:48) t ]. (Hereby we havealso used that | S | = | S (cid:48) | and | T | = | T (cid:48) | .)This shows that there is a change of basis such that [ Θ s ] is sent to [ Θ (cid:48) s ] and [ Θ t ] is sent to[ Θ (cid:48) t ], showing that the underlying H-modules are in fact isomorphic. Proof of statement (d) . By using the relations of D (cid:63) one easily sees that2End D (cid:63) ( ∅ ) ∼ = C [ •• , (cid:4)(cid:4) ] = R , deg( •• ) = deg( (cid:4)(cid:4) ) = 2 , where •• , (cid:4)(cid:4) are the “floating barbells”, see [Eli16, Corollary 5.20 and Lemma 6.23].It follows from the barbell forcing relations (4.BF1) and (4.BF2) that the 2-hom spaces in D (cid:63) are graded R-bimodules with a generating set given by Soergel diagrams without “floatingcomponents”. Hence, D (cid:63) is defined over the polynomial algebra R, see [Eli16, Proposition5.19 and Proposition 6.22].Now fix n ∈ Z > (the case n = 1 is discussed in Example 3.8). Let R W n ⊂ R denote thesubalgebra of invariant elements. The coinvariant algebra is defined asC +W n = R /I + , where I + is the ideal generated by the elements in R W n which are homogeneous of positivedegree. Define z = •• •• − [2] q · •• (cid:4)(cid:4) + (cid:4)(cid:4) (cid:4)(cid:4) ∈ C [ •• , (cid:4)(cid:4) ] ,Z = (cid:81) w ∈ W n w (2 · •• + [2] q · (cid:4)(cid:4) ) ∈ C [ •• , (cid:4)(cid:4) ] , where s ( •• ) = •• , t ( •• ) = •• − [2] q · (cid:4)(cid:4) , s ( (cid:4)(cid:4) ) = (cid:4)(cid:4) − [2] q · •• , t ( (cid:4)(cid:4) ) = (cid:4)(cid:4) . (Note that the element Z does not make sense for “ n = ∞ ”.) By [Eli16, Claim 3.23], thesubalgebra R W n is isomorphic to C [ z, Z ] ⊂ R.Thus, in order to check that our weak 2-functors G n descend from D (cid:63)n to ( D f n ) ∗ , we haveto check that G n ( z ) = 0 = G n ( Z ).First note that any product of two or more barbells is zero, becausei | i i | i = j | j i | i = i | i j | j = j | j j | j = 0 . By (5.2) and (5.5), this implies that G n ( z ) = 0.The same argument shows that G n ( Z ) = 0, e.g. we already have(2 · •• + [2] q · (cid:4)(cid:4) ) s (2 · •• + [2] q · (cid:4)(cid:4) ) = (2 · •• + [2] q · (cid:4)(cid:4) )((2 − [2] ) · •• + [2] q · (cid:4)(cid:4) ) = 0 . When “ n = ∞ ”, one has R W ∞ ∼ = C [ z ]. This case therefore follows in the same way as inthe finite case. (cid:4) WO-COLOR SOERGEL CALCULUS AND SIMPLE TRANSITIVE 2-REPRESENTATIONS 35
Example 5.10.
Note that the proof of (c) of Theorem II is effective in the sense that onecan explicitly compute the change of basis matrix via the singular value decomposition. Forthe type E situation from Example 2.3 one gets for instance U W ∗ = 12 √ · √ − −√ − − − −√ − √ − , V X ∗ = ( U W ∗ ) T , (using the ordering of the vertices from left to right, sea-green • before tomato (cid:4) ) which givesthe (highly non-integral) change of basis matrix. (cid:78) Proof of Theorem III.
By Theorem II it remains to rule out the case that there are gradedsimple transitive 2-representations which are not as in Theorem I.In order to rule these out: By (d) of Theorem II and Remark 3.9, any graded simpletransitive 2-representation of
Kar ( D n ) (cid:63) would give the corresponding simple transitive2-representation of Kar ( D f n ). The underlying quiver of such a 2-representation is of ADEtype and their action on the level of the Grothendieck groups is fixed up to change of basis,see [KMMZ16, Sections 6 and 7]. This structure on the level of 1-morphisms is preservedunder strictification. By combining Lemma 4.18 and Lemma 4.19 (“uniqueness of higherstructure”), these are equivalent to the ones from Theorem I. Indeed, we can define adegree-preserving autoequivalence of G gr : on objects it is the identity; on morphisms it isgiven by multiplying any homogeneous Q G -bimodule map by (suitable) powers of τ and υ .By definition, this autoequivalence intertwines the 2-representation with the τ , υ -scaling andthe one from Theorem I. (cid:4) Note that the proof of Theorem III relies on Lemma 4.18, whose proof (as given in thispaper) uses the grading in an essential way.
Remark 5.11.
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