Categorifying Hecke algebras at prime roots of unity, part I
aa r X i v : . [ m a t h . R T ] M a y Categorifying Hecke algebras at prime roots of unity, part I
Ben Elias, You QiMay 8, 2020
Abstract
We equip the type A diagrammatic Hecke category with a special derivation, so that after specializationto characteristic p it becomes a p -dg category. We prove that the defining relations of the Hecke algebra aresatisfied in the p -dg Grothendieck group. We conjecture that the p -dg Grothendieck group is isomorphicto the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztigbasis and the p -canonical basis. More precise conjectures will be found in the sequel.Here are some other results contained in this paper. We provide an incomplete proof of the classi-fication of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of theclassification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p -dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence.We prove that no such derivation exists in simply-laced types outside of finite and affine type A . We alsoexamine a particular Bott-Samelson bimodule in type A , which is indecomposable in characteristic butdecomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivialfantastic filtrations in any characteristic, which is the analogue in the p -dg setting of being indecomposable. Contents n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 Computations for n = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 A preview of part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.6 Relation to singular Soergel calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Relation to thick calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.8 Relations to thick calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 n = 8 In order to understand what we do in this paper and why, it helps to know a bit about categorification at aroot of unity, and its most popular techniques.
Definition 1.1.
Let A be an algebra over a ground ring k . A k -linear map d : A → A is called an evendifferential or a derivation if it satisfies the Leibniz rule d ( f g ) = d ( f ) g + f d ( g ) (1.1)for any f, g ∈ A .In his seminal paper [Kho16], Khovanov began the program of “categorification at a root of unity.” Hedefined a p -dg algebra to be a graded algebra A over a field k of characteristic p , equipped with an evendifferential d which is homogeneous of degree , satisfying d p = 0 . A p -dg category is defined similarly.Khovanov defined the derived category of a p -dg category, and observed that its Grothendieck group isnaturally a module over the p -th cyclotomic integers. More precisely, it is a module over O p , which is theextension of Z by a variable ζ for which ζ is a primitive p -th root of unity. To categorify something like thequantum group at a root of unity, one should hunt for an interesting p -dg category.The work of Khovanov-Qi [KQ15] successfully applied this idea to the categorification of quantumgroups. They took the quiver Hecke algebra or KLR algebra [KL11, Rou08] in simply-laced type, whichwas known to categorify the positive half of the quantum group at generic q , and equipped it with a degree derivation d . After specialization to characteristic p , it becomes a p -dg category. They conjectured [KQ15,Conjecture 4.18] that this p -dg quiver Hecke algebra categorifies the positive half of the small quantumgroup at q = ζ . They were able to prove this conjecture for sl , and to prove in general that the definingrelations of the small quantum group hold in the Grothendieck group [KQ15, Theorem 3.35, Theorem 4.14].The conjecture has been proven by Andrew Stephens for small quantum sl [Ste18].Before continuing, let us note a common theme to the construction of p -dg algebras: the differentialis independent of the prime p . Typically there is just one differential defined on an integral form of thealgebra, which satisfies d p = 0 only after specialization to characteristic p . Here is why. A differential graded algebra has a degree one map called the differential which satisfies the super Leibniz rule. For an evendegree map, there is no difference between the ordinary and the super Leibniz rule. antasticfiltrations 3Definition 1.2. An even differential graded algebra , or simply edg-algebra , ( A, d ) is a graded Z -algebra A equipped with an even differential of degree . An edg-algebra is called a gaea (short for G a -equivariantalgebra, or a metaphor for a “global” p -dg algebra) if the operator d ( k ) := d k k ! can be defined over Z .A gaea specializes to a p -dg algebra after changing base from Z to any field of characteristic p , because d p = p ! d ( p ) = 0 . Example 1.3.
Let R = Z [ x , . . . , x n ] be a polynomial ring, graded such that deg( x i ) = 2 , and consider theoperator d = n X i =1 x i ∂∂ x i . (1.2)This is the even differential determined by the Leibniz rule and the equation d ( x i ) = x i (1.3)for all i . We call this the standard differential on the polynomial ring. The operator d ( k ) sends x ℓi to (cid:0) ℓ + k − k (cid:1) x ℓ + ki ,and is well-defined over Z . One can prove that (after specialization to characteristic p ) this p -dg algebra isquasi-isomorphic to the ground field k , so that its p -dg Grothendieck group is just O p .So Khovanov-Qi equipped the quiver Hecke algebra with the structure of a gaea. This illustrates theprinciple that, if you already have an additive categorification of an algebra at generic q , you should tryto equip it with a differential (extending to a gaea) in order to categorify the specialization to any primeroot of unity. As further illustration of this principle, we categorified the entire quantum group of sl in[EQ16a, EQ16b], equipping Lauda’s category U and Khovanov-Lauda-Mackaay-Stosic’s category ˙ U (see[Lau10] and [KLMS12] respectively) with such a differential.In this paper, we equip the diagrammatic Hecke category in finite and affine type A with the structureof a gaea, hoping to categorify the Iwahori Hecke algebra at a root of unity. We also prove a negative resultin any other simply-laced type.Let us note that the differential d on the KLR algebra in type A is, for all effective purposes, induced bythe standard differential on the polynomial ring discussed in Example 1.3. What is more interesting is thatthere is a large family of possible differentials on the KLR algebra, but that only the standard differential(and its dual) gives rise to a p -dg algebra with the correct p -dg Grothendieck group! The same seems to betrue of the diagrammatic Hecke category.To reiterate, constructing a differential on the Hecke category is not the hard part, and it is only thefirst step. The real difficulty lies in computing the p -dg Grothendieck group. Let us explain some of thetechnology involved in understanding p -dg Grothendieck groups. In an additive category, suppose we have a direct sum decomposition X ∼ = M j ∈ I M j for some finite index set I . This produces a relation on the (split) Grothendieck group [ X ] = X [ M j ] . (1.4)If (1.4) is a relation you want then you should prove it by constructing a direct sum decomposition. Inpractical terms, to prove that X ∼ = L M j , one must construct projection maps p j : X → M j and inclusion maps i j : M j → X antasticfiltrations 4 which satisfy the basic axioms p j i j = id M j , (1.5a) p j i k = 0 if j = k, (1.5b) id X = X j ∈ I i j p j . (1.5c)Now consider the exact same scenario, but in a p -dg category. To make a long story short, (1.5) is notenough information to deduce that (1.4) holds in the p -dg Grothendieck group. The object X is not actuallya direct sum of the objects M j because the differential need not preserve the summands. Let us say thismore precisely. The functor Hom( X, − ) is (by definition) a representable left module over the category,where each piece Hom(
X, Y ) is equipped with a differential. Let e j = i j p j ∈ End( X ) . Then the directsummand Hom( X, − ) e j ⊂ Hom( X, − ) need not be preserved by the differential on Hom spaces. It is noteven desirable for Hom( X, − ) e j to be preserved by the differential, as this condition often fails in practice.However, one might hope that the differential acts on the pieces of this decomposition in an “upper-triangular” fashion, which would equip X with a filtration whose subquotients might agree with the M j .This idea is codified in [EQ16a, §
5] in the notion of an
Fc-filtration . This is shorthand for either a fantasticfiltration or a finite cell filtration , your choice. A direct sum decomposition as in (1.5) is said to lift to a Fc-filtration if there exists a partial order on the index set I satisfying p j d ( i k ) = 0 whenever j ≤ k. (1.6)If this can be done, then (1.4) still holds in the p -dg Grothendieck group.The reader new to this theory should think of (1.6) as consisting of two separate statements. The firststatement, p j d ( i k ) = 0 whenever j < k , implies that Hom( X, − ) is filtered with subquotients Hom( X, − ) e j .The second statement, p j d ( i k ) = 0 when j = k , implies that Hom( X, − ) e j and Hom( M j , − ) are isomorphicas p -dg modules (i.e. the natural isomorphism between these functors intertwines the differential). Since Hom( M j , − ) is representable, this implies that Hom( X, − ) e j is cofibrant and compact , which is needed for itto have a symbol in the Grothendieck group in the first place. Remark 1.4.
A simpler idea is that of a dg-filtration on an object X , which is a complete collection of or-thogonal idempotents e j satisfying e j d ( e k ) = 0 for j < k . This equips Hom( X, − ) with a filtration by p -dg submodules which are additive summands. The additional data in an Fc-filtration goes one step fur-ther and proves that the subquotients Hom( X, − ) e j are isomorphic to some known representable modules Hom( M j , − ) . For a dg-filtration, it is not obvious that the subquotients Hom( X, − ) e j will be cofibrant.It is not really important which partial order on I gives rise to (1.6), only that some partial order shouldexist. Practically speaking, the method for determining if a direct sum decomposition lifts to a Fc-filtrationis as follows. Construct an oriented graph Γ I,d with vertex set I , having an edge from k to j labeled bythe degree +2 morphism p j d ( i k ) . Erase all edges with the zero label. If Γ I,d has no oriented cycles (and inparticular, no loops) then it is possible to find a partial order on I satisfying (1.6), and consequently (1.4)holds.Direct sum decompositions are somewhat fluid: there are usually many valid choices for the projectionmaps p = { p j } and inclusion maps i = { i j } . Note that the graph under study depends on both the dif-ferential d and the choice of projection and inclusion maps, so we could denote it by Γ I,d,p,i to emphasizethis point. What is rather amazing is that the Fc-filtration requirement is incredibly good at rigidifying thesituation: while there may be large families of differentials d on a category, and large families of projectionand inclusion maps, there is often a unique (up to symmetry) triple ( d, p, i ) such that Γ I,d,p,i has no cycles!If not unique, it is often severely restrictive.
Remark 1.5.
Note that rescaling the inclusion and projection maps (e.g. multiplying p j an invertible scalar κ , and multiplying i j by κ − , for some j ) will not change the graph, it will only rescale the edge labels. Thisis one symmetry we use freely below.To illustrate this, let us return to the setting of categorified quantum groups. The quantum group hascertain defining relations, such as ef λ = f e λ + [ λ ]1 λ , which are usually categorified in ˙ U by direct omputingtheGrothendieckgroup 5 sum decompositions (for which Lauda wrote down some inclusion and projection maps explicitly). Letus call these the defining direct sum decompositions in ˙ U . If a p -differential d on ˙ U is to give rise to the correctGrothendieck group, then at the least it should be the case that Γ I,d has no cycles for all the defining directsum decompositions (for some choice of projection and inclusion maps). Let us temporarily call such adifferential good .In [EQ16a, EQ16b] we first computed all the possible p -differentials on the category U and ˙ U . Thisproduced a rather large family of differentials. Then in [EQ16a, Proposition 5.14] we computed whichdifferentials in this family were good. That is, for each differential we computed the graphs Γ I,d for allthe defining direct sum decompositions (and all possible inclusion and projection maps), and asked whichgraphs had no cycles. Happily, being good is a strong enough condition to pin down the differential andthe inclusion and projection maps precisely! Ultimately, only two nonzero differentials d and ¯ d were good,and these differentials are intertwined by the duality functor (which flips diagrams upside-down). Thusalready one deduces that there are only two dual p -differentials d and ¯ d which could possibly give rise tothe correct Grothendieck group (caveat: see Remark 1.9), though one still needs to confirm that they dohave the correct Grothendieck group.In summary, that the defining direct sum decompositions lift to Fc-filtrations is typically a very restric-tive property for a differential, and differentials which satisfy it are quite special and interesting. Not onlythat, but the projection and inclusion maps compatible with these differentials should be considered asparticularly nice. Remark 1.6.
Perhaps most intriguingly, the special partial orders on the index sets I induced by the cycle-less graph Γ I,d are new and unfamiliar structures which were invisible before the introduction of the differ-ential. To state a rough moral: categorification replaces structure coefficients (numbers) with multiplicityspaces (vector spaces), while p -dg categorification equips these multiplicity spaces with a filtration, whoseshadow is now combinatorial (an oriented graph). Remark 1.7.
Essentially everything discussed in this section depended only on the differential as definedover Z , but not on the choice of prime p or on the fact that d p = 0 . This is not obvious, as specializing tocharacteristic p could, in theory, eliminate cycles in the graph Γ I,d , but in practice it does not. In other words,fantastic filtrations should really be considered as a theory intrinsic to gaeas. At the current moment, thehomological algebra of gaeas has not been developed to the same degree as p -dg algebras were in [Qi14],so whenever discussing the Grothendieck group we play it safe and talk only about p -dg algebras. A useful tool towards computing the Grothendieck group has been the following result of Qi [Qi14], ananalogue of the positive dg algebra case by [Sch11].
Proposition 1.8.
In the special case when A is a positively graded p -dg algebra , the p -dg Grothendieckgroup is just the specialization at q = ζ of the original Grothendieck group.When A is not positively graded this result is often false, and great caution is required. Most interestingcategorifications are not positively graded. In [EQ16a] we developed Fc-filtrations as part of a game whichmanipulates a p -dg category until, hopefully, we can apply Proposition 1.8. Let us elaborate on this methodin the context of ˙ U , which categorifies the quantum group U q ( sl ) .A common technique in the study of additive and abelian categories is to choose a projective generatorand work instead with its endomorphism ring. Given a collection of self-dual indecomposable objects wecan study the full subcategory P in ˙ U with those objects. Equivalently, letting P be the direct sum of theseself-dual indecomposable objects, we can study the p -dg algebra End( P ) . Now the size of morphism spacesbetween objects in ˙ U is determined by a particular sesquilinear pairing on U q ( sl ) , see [Lau10, § ˙ U categorify Lusztig’s canonical basis, and the pairing of canonicalbasis elements has only non-negative powers of q . Consequently, P is a positively-graded category, i.e., End( P ) is a positively-graded algebra, from which one easily computes both the ordinary and the p -dg A positively-graded p -dg algebra has its grading supported in non-negative degrees, with semisimple degree zero part, and thedifferential is trivial in degree zero. hediagrammaticHeckealgebraanditsdifferential 6 Grothendieck group of P . The task is to relate the p -dg category ˙ U with the p -dg category P (the underlyingadditive categories are Morita equivalent, but the p -dg setting is more subtle).In the additive setting, X is generated by P if it can be expressed as a direct sum of the objects in P . Toshow that a given set of objects is a generator, we need to find enough direct sum decompositions. In the p -dg world we need more; it is sufficient for X to be filtered by objects in P via fantastic filtrations. Thatis, one other major implication of a Fc-filtration as above is that X will lie in the triangulated hull of { M j } ,inside the p -dg derived category. Remark 1.9.
At the moment we do not have results on the necessity of Fc-filtrations. That is, in theory X might be in the triangulated hull of { M j } even when there is no fantastic filtration, or (1.4) might hold,because no technology has been developed to provide an obstruction. However, we know of no exampleswhere this happens.There are other important direct sum decompositions, beyond the defining ones: for example, the idem-potent decomposition known as the Stosic formula in [KLMS12, Theorem 5.6]. The remainder of the argu-ment in [EQ16b] went as follows. • Find enough direct sum decompositions to decompose any object in the category ˙ U as a direct sumof objects which are either indecomposable or contractible. (In this case, the defining decompositionsand the Stosic formula were sufficient.) • Prove that each of these direct sum decompositions lifts to a Fc-filtration, for the good differentials. • Let P be the direct sum of all the non-contractible indecomposable objects. Deduce from the abovethat P generates the p -DG derived category of ˙ U , and that End( P ) is positive, so that the p -dgGrothendieck group of ˙ U agrees with the q = ζ specialization of the ordinary Grothendieck groupof P .This method for computing the p -dg Grothendieck group has many obvious limitations. It worked for ˙ U ( sl ) because the category is relatively simple. We understand completely what all the indecomposableobjects are, and we know enough explicit idempotent decompositions to take an arbitrary object and splitit into indecomposables. In his PhD thesis, Andrew Stephens [Ste18] was able to extend this same methodto categorify the positive half of quantum sl , because the explicit idempotent decompositions were alsoconstructed previously by Stosic [Sto11]. It seems hopeless to extend this method to sl n in general: eventhe canonical basis of the positive half of the quantum group is unknown, so there is little hope of under-standing the indecomposable objects explicitly. Some new techniques are clearly required to make progressbeyond what is currently known. In this paper we initiate the program to categorify (Iwahori-)Hecke algebras at a root of unity, with prelim-inary positive results, useful negative results, and extremely intriguing conjectures.The Hecke algebra in type A is a deformation of the group algebra of the symmetric group. It is cate-gorified by the monoidal category of Soergel bimodules, which are certain bimodules over the polynomialring from Example 1.3. The appropriate integral form of this categorification is the diagrammatic Heckecategory, as introduced by Elias-Khovanov in type A [EK10]. Just as the symmetric group is generated byits simple reflections, the adjacent transpositions s i = ( i, i + 1) , the diagrammatic Hecke category is gen-erated by certain objects B i = B s i . Tensors of these objects are commonly called Bott-Samelson bimodulesor objects. The diagrammatic Hecke category encodes morphisms between Bott-Samelson bimodules asplanar diagrams. A basis for these morphism spaces was constructed in [EW16b], called the double leavesbasis .Following the motif from the parallel world of quantum groups, if we want to categorify the Heckealgebra at a prime root of unity, we should equip the diagrammatic Hecke category with the structure ofa gaea. In this paper, we only examine simply-laced type. As in the outline of [EQ16b], we first computeall possible differentials on the category, obtaining a large family of even differentials. Then, for each ofthe defining idempotent decompositions, we compute the corresponding graph and determine for which hediagrammaticHeckealgebraanditsdifferential 7 differentials the graph has no cycles. We temporarily call such a differential good . Once again, this constraintis enough to pin down the differential precisely: only two dual differentials ( d and ¯ d ) are good, and couldpossibly induce the correct p -dg Grothendieck group. We explicitly check all possible choices of projectionand inclusion maps for each of the defining idempotent decompositions. Remark 1.10.
Some of the defining idempotent decompositions require dg filtrations rather than Fc-filtrations,because the summands in question are not Bott-Samelson bimodules and thus not pre-existing objects inthe diagrammatic Hecke category. In fact, they require a slight generalization which mixes the concepts ofa dg filtration and a Fc-filtration, see § Remark 1.11.
A differential on the Hecke category induces a differential on its polynomial ring, which forthe Elias-Khovanov version of the diagrammatic category is assumed to be k [ x , . . . , x n ] with its standardaction of S n . This polynomial ring has a standard differential, where d ( x i ) = x i . In our classification we donot assume that this is the differential on the polynomial ring. Instead, we prove that when the differentialis good, the p -dg polynomial ring is forced to be isomorphic to the standard one. Moreover, we prove thatHecke category, if constructed instead using the ( n − -dimensional reflection representation of S n , doesnot admit a good differential! Remark 1.12.
There are a number of other compatibilities one might desire from a differential. The first isthat it is compatible with the categorical Schur-Weyl duality of Mackaay-Stosic-Vaz [MSV13], which givesa functor from the Hecke category of S n to the categorification of quantum gl n . The second is that thesingular Hecke 2-category, which contains the Hecke category as an endomorphism category, should havea p -differential which restricts to our chosen differential. The third is that the thick calculus of the firstauthor [Eli16a] should have a p -differential which restricts to our chosen differential. All these are satisfiedby the good differential; we briefly discuss some of these compatibilities in this paper.In practice, the classification of differentials is a three-step process. By the Leibniz rule, any differentialon an algebra is determined by its action on the generators. A generator is sent to some morphism ofdegree higher, living in a finite-dimensional morphism space, so we can specify this morphism by anumber of parameters (its coefficients in the double leaves basis). Now we impose three constraints on theseparameters. The first is that the relations are preserved by the differential; this ensures that the differentialis well-defined. The second is that the divided power d ( k ) is well-defined over Z ; this need only be checkedon the generators, since d ( k ) ( f g ) = X i + j = k d ( i ) ( f ) d ( j ) ( g ) (1.7)by the Leibniz rule. The third is that the defining idempotent decompositions should have no cycles. Onlythe first two constraints need to be checked when classifying general differentials rather than good differ-entials.As noted above, our main result is a classification of all good differentials on the Hecke category (up toobject-fixing isomorphism): there are only two, d and ¯ d . We also go most of the way towards classifyingthe general differential, although we do not quite finish the job. The general differential has many morenon-zero parameters than the good differential. For example, a general differential applied to the 4-valentvertex gives a sum of three diagrams with particular coefficients, while a good differential applied to the4-valent vertex is zero. Consequently, it is much easier to check that the good differential satisfies all therelations of the diagrammatic Hecke category than it is to check the general case. We are able to checkevery relation in the general case except the most complicated one, the so-called Zamolodchikov relationassociated to parabolic subgroups of type A . Perhaps it is only laziness which prevents us from finishingthis calculation, although it is a surprisingly thorny one. The Hecke category was generalized to all Coxeter groups by Elias-Williamson in [EW16b]. The compu-tations done here also have implications for possible differentials in other types and for other realizations(see [EW16b, § A (our results are sufficient to classify the good The interested reader is welcome to finish this calculation for us and write an appendix! That said, we are not sure why anyoneshould care about the complete family of differentials anyway; the good differentials seem at the moment like the only interestingones. heGrothendieckgroup? 8 differentials in affine type A as well). Work in progress of the first author and Lars Thorge Jensen is explor-ing differentials in some non-simply-laced types, using a realization which is central extension of the rootrealization.Let us reiterate that the good differential has already been discovered in some sense, in the algebraiccontext of Soergel bimodules. Good differentials on the nilHecke algebra has been studied in the work ofBeliakova-Cooper [BC18] and Kitchloo [Kit13]. In a different direction Khovanov and Rozansky in [KR16]construct an action of the positive half of the Witt algebra W + on all Bott-Samelson bimodules and oncertain complexes thereof. This led to an action of W + on triply graded knot homology (just as predecessorsBeliakova-Cooper and Kitchloo defined an action of the Steenrod algebra on the characteristic p versions).The W + action equips Bott-Samelson bimodules with a differential. This differential on the objects (theBott-Samelson bimodules) can be used in standard fashion to construct a differential on morphism spacesbetween objects. A generator of the W + action gives rise to our good differential. Again, the existence of thisdifferential is no surprise at this point. Our paper has a different set of goals: to prove the key homologicalproperties of d (i.e. it is good), to prove the uniqueness of d (the lack of other good differentials), andto examine other types, en route to computing the p -dg Grothendieck group of the diagrammatic Heckecategory. Having classified the good differentials, the next step is to compute the p -dg Grothendieck group of theElias-Khovanov category for these differentials. Now it is clear that the methods of [EQ16b] will no longersuffice, for several reasons which we now discuss.Indecomposable objects B w in the Hecke category (up to isomorphism and grading shift) are in bijectionwith elements w in the symmetric group S n , and they appear as direct summands inside B i ⊗ · · · ⊗ B i d whenever s i · · · s i d is a reduced expression for w . Aside from these facts, the indecomposable objects B w are extremely mysterious. Unlike the case of quantum sl , the size and structure of the indecomposableobject B w depends on the characteristic of the base field! We write p B w to indicate the dependence of thisobject on the characteristic. The smallest example where the size of p B w depends on p occurs when p = 2 for 28 elements w ∈ S , but examples become ever more frequent as n grows larger.In characteristic zero the indecomposable objects B w categorify the Kazhdan-Lusztig basis, by resultsof Soergel [Soe90]. The size of morphism spaces is determined by a sesquilinear form on the Hecke algebra,and Kazhdan-Lusztig basis elements pair positively, so the endomorphism ring of L B w is positivelygraded. When p is large relative to n , p B w will “agree” with B w for all w , and continue to categorifythe Kazhdan-Lusztig basis. Note that, as proven by Williamson [Wil17], the prime p must grow at leastexponentially with n for this statement to hold! When p is small relative to n , the the endomorphism ringof L p B w need not be positively graded. This is the first nail in the coffin.To decompose an arbitrary object in the Hecke category into indecomposables, we would need to beable to find the idempotent projecting to B w inside a reduced expression, and would need to be able todecompose B w ⊗ B s for each w ∈ S n and each simple reflection s . Essentially nothing is known about theseidempotents, and it is an incredibly difficult open problem to study them! Without an explicit idempotentdecomposition, there are currently no tools which could prove that some decomposition lifts to an Fc-filtration or a dg filtration. This is the second nail in the coffin.For all these reasons, we are forced to abandon the previous methods in p -dg theory and search forsomething new, which will be the focus of the sequel to this paper.Aside from the method of [EQ16b], there are only a few other tools in the literature which can be usedto compute the p -dg Grothendieck group. Recently, the second author and Sussan [QS18] have devel-oped the notion of a p -dg cellular algebra and a p -dg quasi-hereditary algebra, and proven that their p -dgGrothendieck groups are q = ζ specializations of the ordinary Grothendieck group. However, their con-cepts only apply to cellular algebras over the base field k , rather than cellular algebras over other rings. Inthe lingo, their technology works for quasi-hereditary algebras but not for affine quasi-hereditary algebrasin the sense of Kleschchev [Kle15]. The double leaves basis is a cellular basis, but with base ring R (thepolynomial ring in n variables), which itself has a nontrivial differential. One might hope that the tech-niques of Qi-Sussan can be adapted to the more general setting, but this is not an easy adaptation. Still, the omputationsforsmall n cellular structure on the Hecke category is one of the most powerful weapons in the arsenal, and we expectthat any successful approach will use it.Despite the lack of tools, we still believe the end result. Conjecture 1.13.
The diagrammatic Hecke category associated to S n , when equipped with a good differen-tial (either d or ¯ d ) and specialized to characteristic p , has p -dg Grothendieck group isomorphic to the Heckealgebra at the appropriate root of unity. n The methods of [EQ16b] do suffice for small values of n , where we can compute all the idempotent decom-positions by hand, and where the size of the indecomposable objects does not depend on the characteristic.For each w ∈ S n and simple reflection s with ws > w , the direct summands of B w ⊗ B s all have the form B z for various z ∈ S n . We let I w,s ⊂ S n denote the set of such z (with multiplicity). If the graph Γ I w,s ,d hasno cycles, then the decomposition B w B s ∼ = M z ∈ I w,s B z (1.8)is fantastically filtered. Theorem 1.14. If n ≤ , w ∈ S n and s ∈ S , then Γ I w,s ,d has no cycles. As a consequence, any Bott-Samelsonsplits into indecomposable objects via a fantastic filtration, and Conjecture 1.13 holds for n ≤ .The proof of this theorem is by direct, straightforward, and exhaustive computation, and we have cho-sen not to write it up. Preliminary calculations have verified that this theorem continues to hold true forinteresting examples with n = 5 as well.Already one observes some mysterious phenomena. The graph Γ I w,s ,d , when it has no cycles, induces aspecial partial order ≺ on I w,s . In order to prove that Γ I w,s ,d has no cycles in general, it would help to knowin advance what this partial order will be.For example, let n = 3 and let s = (12) and t = (23) inside S . In the additive setting we know that B st ⊗ B s ∼ = B sts ⊕ B s , so that d induces a partial order on the set { s, sts } . This partial order happens to be s ≺ sts . Meanwhile, B ts ⊗ B t ∼ = B sts ⊕ B t , with partial order sts ≺ t . Already it is clear that ≺ is not determined by the Bruhat order. One might stillpray that it is governed by some kind of lexicographic order or convex order on roots, but one would bedisappointed.Letting u = (34) inside S , we can compute various other partial orders, including us ≺ usts from I ust,s , (1.9a) usts ≺ stsuts ≺ sutu from I sutsu,t , (1.9b) sutu ≺ su from I sut,u . (1.9c)There are no cycles in any of these individual sets, but there is a cycle if the sets I w,s are all placed togther!In other words, there is no partial order on S n which restricts to the partial order on each I w,s . This makesthe partial orders on I w,s especially mysterious. Remark 1.15.
This cycle us → usts → stsuts → sutu → us (1.10)is not the only one. One can replace usts with stsu , or sutu with utus , to obtain another cycle. In addition,there is a cycle tsu → sts → tstut → tut → tsu, (1.11)and one can replace tstut with either tutst or with t . We are working in characteristic zero, or assuming that the object B w and its direct sum decomposition agrees with the character-istic zero setting in the Grothendieck group. This holds when n < in any characteristic. In both cases, ¯ d induces the opposite partial order. omputationsfor n = 8 n = 8 In finite characteristic for larger values of n , as already noted, the decomposition of tensor products intoindecomposable objects is different from in characteristic zero. Some summands which split in charac-teristic zero will instead get “glued together.” This does not change the Grothendieck group itself! TheGrothendieck group of the diagrammatic Hecke category is the Hecke algebra in any characteristic. Whatit does change is the basis of the Grothendieck group given by the symbols of indecomposable objects. Incharacteristic p , one defines p B w as the top summand of the Bott-Samelson object associated to a reducedexpression of w , i.e. the unique direct summand of this Bott-Samelson which is not a direct summand ofany shorter Bott-Samelson . These p B w enumerate the isomorphism classes of indecomposable objects upto grading shift. The basis [ B w ] = b w is the usual Kazhdan-Lusztig basis, also called the -canonical basis ,while [ p B w ] = p b w is now called the p -canonical basis .Let us give a concrete example, in the symmetric group S . Let X denote the Bott-Samelson bimoduleassociated to the sequence of simple reflections ( s , s , s , s , s , s , s , s , s , s , s , s , s , s ) , which is a reduced expression for an element w ∈ S . Let y = s s s s s s , which is the longest elementof a parabolic subgroup. In characteristic zero (and any odd finite characteristic), there is a direct sumdecomposition X ∼ = B w ⊕ B y . (1.12)However, in characteristic , X is indecomposable. What is happening is that the integral form of the Heckecategory contains morphisms p : X → B y and i : B y → X such that p ◦ i = 2id B y . These morphisms p and i span their respective Hom spaces over Z . Now change base to a field. If is invertible then ip is anidempotent projecting to B y . If one can not construct any splitting of i or p ; consequently, X = B w isindecomposable, and on the Grothendieck group, b w = b w + b y .In similar fashion, it need not be the case that Γ I w,s ,d has no cycles in order for Conjecture 1.13 to hold!Let us call an object p -dg indecomposable if it does not have any idempotents fitting into a dg-filtration. Arefinement of Conjecture 1.13 might say that a Bott-Samelson associated to a reduced expression (for w ) hasan Fc-filtration with a unique p -dg indecomposable summand d B w not appearing inside any Fc-filtrationof a shorter Bott-Samelson. In theory, these objects d B w might descend to a basis of the p -dg Grothedieckgroup, the d -canonical basis . The p -dg Grothendieck group might still be the Hecke algebra, even if the d -canonical basis is an unexpected one. For example, what if the graph for the decomposition (1.12) hascycles? Then X has no idempotents fitting into a dg-filtration, and X = d B w is p -dg indecomposable.Morally speaking, this is no worse than having X be indecomposable in characteristic .We did not bring up this specific example for no reason. Theorem 1.16.
The decomposition (1.12) is not Fc-filtered in any characteristic. If i and p are the morphismsdiscussed above, then d ( i ) = 0 , d ( p ) = 0 , and d ( p ) i = 0 (so the graph has both a cycle and a loop).The proof is by direct and nasty computation, and we do not write it up. In § p , d ( p ) , i , d ( i ) , and d ( p ) i .This theorem is very surprising, because it implies that (if Conjecture 1.13 is true) the d -canonical basisof the Hecke algebra is different from both the -canonical basis and the p -canonical basis! For S in char-acteristic p > , the p -canonical basis agrees with the -canonical basis, but the d -canonical basis contains d b w = b w + b y . Remark 1.17.
Having just emphasized that it is not important that every direct sum decomposition is anFc-filtration, we wish to re-emphasize that the defining direct sum decompositions (like B s B s ∼ = B s ( − ⊕ B s (1) ) must be Fc-filtrations, or the ring structure on the p -dg Grothendieck group would be incorrect. One can prove that it is independent (up to isomorphism) of the choice of reduced expression. Thanks to Lars Thorge Jensen for his computer calculations and his help finding this and other accessible examples. This decom-position was verified by his programs. previewofpartII 11
As noted above, new techniques are required to compute the p -dg Grothendieck group of the Hecke cat-egory, and to understand Fc-filtrations in the absence of explicit decompositions. In the next paper, weintroduce some new techniques (some of them conjectural) which we hope will fit the bill. Here is a quickpreview.First, we introduce what we call the counterdifferential , a new structure which exists in both the Heckeand the quantum group settings. This is a derivation z of degree − (satisfying the Leibniz rule), giving anew gaea structure. Since most of the generating morphisms live in the minimal nonzero degree of theirrespective Hom spaces, z will kill these generators. So z is determined by what it does to the polynomialring R = k [ x , . . . , x n ] , where it sends x k for each k . In the sequel we will prove the following basicstructural results. • Letting h denote the degree operator (which acts on morphisms of degree k by multiplication by thescalar k ), the triple ( d, h, z ) acts as an sl triple, making all Hom spaces into sl representations. • The triple ( ¯ d, h, z ) is also an sl triple. • Each Hom space is a free R -module with its double leaves basis, as noted above. The R -span of anydouble leaf is preserved by z . Moreover, there is a partial order on the set of double leaves such that d sends each double leaf to the R -span of double leaves which are weakly lower in the partial order.Thus the double leaves basis equips all Hom spaces with a very particular kind of filtration.One should think of the sl representations which appear in the Hom spaces as roughly being filtered bycoverma modules. After all, the polynomial ring in one variable k [ x ] is precisely the coVerma module ∇ (0) (the polynomial ring in multiple variables is more complicated). Hom spaces are large infinite-dimensionalsubmodules, but may also have small finite-dimensional submodules. For example, the k -span of the iden-tity inside R is a finite-dimensional sl subrepresentation. We propose that it is no accident that the (one-dimensional) finite part of R agrees precisely with the span inside R of the units!To give another example, whenever s ∈ S is a simple reflection, Hom( B s B s B s B s , B s ) is a free R -module of graded rank q ( q + q − ) . Note that B s B s B s B s splits into shifted copies of B s , with gradedrank ( q + q − ) , so one should expect to find projection maps inside Hom( B s B s B s B s , B s ) . Amazingly,for any fantastic filtration picking out these summands, the span of the projection maps will be an -dimensional sl subrepresentation of Hom( B s B s B s B s , B s ) , and this is the maximal finite-dimensional partof that infinite-dimensional representation.More generally, we conjecture the following algorithm to find all the projection maps in a fantastic fil-tration on a Bott-Samelson bimodule X . Consider the sl representation V = L w ∈ W Hom(
X, B w ) of allmorphisms to indecomposable objects (there are ways of modeling this representation without needing tounderstand the indecomposable objects). Find the maximal finite-dimensional subrepresentation F ; thisshould be spanned by projection maps. Then, take the quotient of V by J · F , where J is the Jacobsonradical of the category. Call this quotient V . Now repeat, finding the maximal finite-dimensional subrep-resentation of V , and so forth. Not only do we conjecture that sufficiently many projection maps can befound inside these finite-dimensional subquotients, and that this will produce a fantastic filtration on X ,but also that the existence of a fantastic filtration on X is equivalent to the effectiveness of this procedure.Further details will await in the sequel. Remark 1.18.
One should think of this conjecture as a new kind of Hodge theory. The relative hard Lef-schetz theorem (see [EW16a, Theorem 1.2] for the theorem in this context) implies (in characteristic zero)that multiplicity spaces (appropriately defined) of an indecomposable summand in a Bott-Samelson bi-module satisfy the hard Lefschetz property with respect to the appropriate Lefschetz operator, meaningthat they are finite-dimensional sl -representations. Meanwhile, we are stating that the entire Hom spacehas the structure of an sl representation (with a very different raising operator), whose finite-dimensional sl -subrepresentation is related to the multiplicity space. Note that d is not a Lefschetz operator; it is morelike an “infinitessimal Lefschetz operator.” Moreover, we also conjecture some positivity properties for d ,analogous to the relative Hodge-Riemann bilinear relations. The first author was supported by NSF CAREER grant DMS-1553032, NSF FRGgrant DMS-1800498, and by the Sloan Foundation. The second author was supported by the NSF grantDMS-1947532 when working on this paper. Many thanks go to Lars Thorge Jensen for proofreading aprevious version of this manuscript and finding numerous bugs, as well as for useful conversations, andfor his lovely computer programs, which helped us to find good examples to study in S . Let us state the end result of our computations. The rest of the paper will comprise the proof of these results.We will not review the diagrammatic Hecke category here. See [EW16b] for details. We fix a Coxetersystem ( W, S ) with a realization, having polynomial ring R with an action of W . The degrees in R aredoubled, so that the simple roots { α s } s ∈ S have degree . To avoid potential confusion, we refer to thesimple roots and other homogeneous polynomials of the same degree as linear polynomials (rather thandegree polynomials). Since R is the endomorphism ring of the monoidal identity, a differential on thediagrammatic Hecke category induces a differential on R . One should not confuse the differential d withthe divided difference operators ∂ s associated to each s ∈ S , see (3.1). In the pictures below, blue represents s , red represents some t with m st = 3 , and green represents some u with m su = 2 . Theorem 2.1.
Let d be an even differential of degree +2 on the diagrammatic Hecke category in simplylaced type. Then there exist linear polynomials g s , ¯ g s ∈ R for each s ∈ S , such that the differential isdefined on the generating diagrams by the following formulas. d (cid:16) (cid:17) = g s (2.1a) d (cid:16) (cid:17) = ¯ g s (2.1b) d = − gs (2.1c) d = − ¯ gs (2.1d) d = − gu + gs + gu − gs (2.1e) d (cid:18) (cid:19) = A + B + C + D + f . (2.1f)In this final formula, we have A = − ∂ s ( g t ) , (2.2a) B = − ∂ t (¯ g s ) , (2.2b) C = ∂ t (¯ g t ) − ∂ t ( g s ) − ∂ s ( g t ) , (2.2c) D = ∂ s ( g s ) − ∂ s (¯ g t ) − ∂ t (¯ g s ) , (2.2d) f = g s − g t − ( ∂ s ( g s ) + ∂ t ( g s )) α s + ( ∂ t ( g t ) + ∂ s ( g t )) α t . (2.2e)These formulas, together with a differential on R , determine the differential on the category. Let z s = g s + ¯ g s ∈ R. (2.3)Then the differential on R satisfies the following properties: d ( w ( f )) = w ( d ( f )) for all w ∈ W, f ∈ R, (2.4a) d ( α s ) = α s z s , (2.4b) z s ∈ R s (2.4c) z s ∈ R u when m su = 2 , (2.4d) s ( α t ) ∂ s ( z t ) = ∂ s ( α t )( z s − z t ) when m st > . (2.4e)To state the converse, let us call the data of an even differential on R and a collection of linear poly-nomials g s , ¯ g s satisfying (2.4) by the name of a potential differential . Then a potential differential induces adifferential on the diagrammatic Hecke category via the formulas (2.1) if and only if the A Zamolodchikovrelation is sent to zero.In other words, we did not check the A Zamolodchikov relation, and are unsure whether it is sent tozero by any potential differential, or if there are additional requirements to be met. We suspect there are noadditional requirements.
Definition 2.2.
Let us call a differential d on the Hecke category good if the defining idempotent decompo-sitions can be lifted to fantastic filtrations.We will make this more precise later. In type A the defining decompositions lift the relations b s b s = vb s + v − b s , (2.5a) b s b u = b u b s if m su = 2 , (2.5b) b s b t b s − b s = b t b s b t − b t if m st = 3 (2.5c)in the Hecke algebra. Note that the zero differential is good. Theorem 2.3.
A differential is good (in simply laced type) if and only if it satisfies the following properties.1. These equations hold. d ( g s ) = g s , (2.6a) ¯ g s = s ( g s ) , (2.6b) u ( g s ) = g s whenever m su = 2 , (2.6c) st ( g s ) = g t whenever m st = 3 . (2.6d)This last equation implies that if g s = 0 for some s , then g t = ¯ g t = 0 for all t in the same connectedcomponent of the Coxeter graph.2. If g s = 0 then g s / ∈ R s .3. If g s = 0 and m st = 3 then exactly one of these two possibilities holds. We encode which one holdsusing an orientation on the corresponding edge in the Coxeter graph.(a) g t is fixed by s , and g s = t ( g t ) is fixed by sts . We orient the edge from t to s .(b) g s is fixed by t , and g t = s ( g s ) is fixed by sts . We orient the edge from s to t .4. The orientation is consistent in that, for any parabolic subgroup of type A , the middle vertex is neithera source nor a sink.Finally, a potential differential which satisfies the properties listed in this theorem will send the A Zamolod-chikov relation to zero, so it does induce a differential on the diagrammatic Hecke category.
Corollary 2.4.
The only connected simply-laced Coxeter groups which admit a consistent orientation havetype A n , type ˜ A n , or type A ∞ , and they each have precisely two consistent orientations. Proof.
There is no way to consistently orient the D Coxeter graph, which is contained inside any connectedsimply-laced Coxeter group aside from those listed above. When the differential is good, the formula for the differential simplifies. The scalar κ = ∂ s ( g s ) is inde-pendent of the choice of s in a connected component of the Coxeter graph. We have d = 0 , (2.7a) d (cid:18) (cid:19) = 0 if t → s, (2.7b) d (cid:18) (cid:19) = κ (cid:18) − (cid:19) if s → t, (2.7c)and can deduce that d (cid:16) (cid:17) = − κ , d (cid:16) (cid:17) = κ . (2.7d)Finally, we prove that there are only two good differentials up to equivalence. Theorem 2.5.
Up to an automorphism of the Hecke category in type A n , if there is a good derivation thenwe can assume that R contains the polynomial ring k [ x , . . . , x n ] with its usual S n action and differential d ( x i ) = x i , and we can assume that either g ( i,i +1) = x i for all i with κ = 1 , or that g ( i,i +1) = x i +1 for all i with κ = − . An exception is when k has characteristic , in which case it is possible that one may need toimpose the relation P x i = 0 . We start by fixing a realization of a Coxeter system ( W, S ) . For more on realizations, see [EW16b, Section3.1]. Later we will assume that the Coxeter system is simply-laced, but not at first.Recall that ∂ s : R → R denotes the Demazure operator of a simple reflection s ∈ S , which is defined bythe formula ∂ s ( f ) α s = f − s ( f ) , (3.1)and satisfies the twisted Leibniz rule ∂ s ( f g ) = ∂ s ( f ) g + s ( f ) ∂ s ( g ) . (3.2)The Cartan matrix encodes the values of ∂ s ( α t ) for various simple reflections.We always assume that Demazure surjectivity holds, see [EW16b, Assumption 3.7]. The implication isthat for each s ∈ S there exists some linear polynomial ̟ s such that ∂ s ( ̟ s ) = 1 . (3.3)For example, if m st = 3 we could set ̟ s = − α t . We do not assume that ̟ s is a fundamental weight (i.e.that ∂ t ( ̟ s ) = 0 for t = s ).We also assume (c.f. [EWS16, Definition 1.7]) that ( ⋆ ) : the linear terms in R s and the linear terms in R t together span the linear terms in R , whenever s = t. This rules out the degenerate possibility that the fixed hyperplanes of s and t coincide. Both Demazure sur-jectivity and ( ⋆ ) are requirements for the Hecke category to behave optimally (though they are sometimesreplaced by related assumptions, like reflection faithfulness).Let us try to define a differential on the Hecke category, by defining it as generally as possible, and thendetermining what constraints are imposed by the fact that it must preserve the relations of the category.We will simultaneously determine what additional constraints are imposed if we want the differential tobe good. We do not assume that our differential is invariant under the symmetries of the Hecke category(vertical and horizontal flips, rotation by 180 degrees, Dynkin diagram automorphisms) . It will turn out that any differential must be invariant under horizontal flip, but not typically under the other symmetries. The reader interested in type A can fix n ≥ and work with the standard realization associated to S n .The base ring R has the form R = k [ x , . . . , x n ] (3.4)with its usual action of S n , and deg x i = 2 . There is a standard differential on the polynomial ring R , namely d ( x i ) = x i (3.5)for all ≤ i ≤ n . We will not assume that this is the induced differential on R , although this will eventuallybe a consequence of our computations.Below, s will be an arbitrary simple reflection and will be drawn using the color blue, u will be distantfrom s and drawn as green, and t will be either be another arbitrary simple reflection, or will be adjacent to s ( m st = 3 ) and drawn using the color red. One of the generating morphisms in the diagrammatic Hecke category is the s -colored enddot , a mor-phism B s → of degree +1 . The enddot generates Hom( B s , ) as a free rank module over R . Thus forany differential we have d ( ) = g s (4.1)for some linear polynomial g s ∈ R .Consider the s -colored startdot , a morphism → B s of degree +1 . For similar reasons we have d ( ) = ¯ g s (4.2)for some linear polynomial ¯ g s ∈ R . Remark 4.1.
We will eventually see that the differential on the entire category is determined by the valuesof g s and g ′ s .Now we compute that d ( ) = ( g s + ¯ g s ) . (4.3)Since the barbell is equal to multiplication by α s = α s , we deduce that d ( α s ) = α s ( g s + ¯ g s ) . (4.4)It turns out that the linear polynomial g s + ¯ g s is more intrinsic than either g s or ¯ g s . Henceforth we write z s = g s + ¯ g s , (4.5)so that d ( α s ) = α s z s (4.6)and d ( ) = ( z s − g s ) . (4.7)If the differential on R is known, then z s is determined by (4.6). Conversely, the differential on R mustbe such that d ( α s ) is a multiple of α s , which is not true of the most general differential. otsandpolynomials 16Remark 4.2. When s = ( i, i + 1) we have α s = x i − x i +1 . The standard differential satisfies d ( x i − x i +1 ) = x i − x i +1 = ( x i − x i +1 )( x i + x i +1 ) (4.8)and thus z s = x i + x i +1 . Later on we will deduce that for a good differential one has g s = x i and ¯ g s = x i +1 ,or vice versa (this will be the difference between d and ¯ d ). Again, we will not assume anything about thedifferential on R or z s or g s just yet, but we recommend that the reader verify the formulas below with thesespecializations, as a motivational sanity check.Now we examine the polynomial forcing relations. We have f = ∂ s ( f ) + s ( f ) . (4.9)Applying the differential to both sides we have d ( f ) = z s ∂ s ( f ) + d ( ∂ s ( f )) + d ( s ( f )) . (4.10)We resolve the LHS by applying (4.9) again. In the result, the equality of coefficients of is ∂ s ( d ( f )) = z s ∂ s ( f ) + d ( ∂ s ( f )) , (4.11)and the equality of coefficients of is s ( d ( f )) = d ( s ( f )) . (4.12)By (4.12) we deduce that the differential d commutes with the action of the symmetric group S n . A conse-quence is that d preserves the invariant subring R s , for all simple reflections s .In fact, (4.11) already follows from the fact that d commutes with the symmetric group action. By defi-nition of the Demazure operator we know that α s ∂ s ( f ) = f − s ( f ) , which appeared above as (3.1). Taking the differential of both sides of (3.1) and using (4.12) we get that α s z s ∂ s ( f ) + α s d ( ∂ s ( f )) = d ( f ) − s ( d ( f )) = α s ∂ s ( d ( f )) . Dividing both sides by α s (a non-zero-divisor in R ) we recover (4.11).In the special case when f = α s , (4.11) reduces to ∂ s ( α s z s ) = 2 z s . However, we already knew from the twisted Leibniz rule that ∂ s ( α s z s ) = 2 z s − α s ∂ s ( z s ) from which we deduce that ∂ s ( z s ) = 0 , z s ∈ R s . (4.13)We could have deduced that z s ∈ R s in a different way. Since α s ∈ R s , and R s is preserved by d , we seethat d ( α s ) = 2 α s z s ∈ R s , which implies that z s ∈ R s . We have brought up this alternative method because we can use it for otherpurposes as well. For example, suppose that m su = 2 , so that s and u are distant simple reflections. Since α s ∈ R u , we must have d ( α s ) ∈ R u , from which we deduce that z s ∈ R u when m su = 2 . (4.14) We will often use without mention the fact that d kills any scalar multiple of the identity, and hence any polynomial of degreezero. This is a consequence of the Leibniz rule for differentials. rivalentvertices 17 Now consider the case when f = α t for some s = t , so that ∂ s ( f ) is some scalar. Then (4.11) reduces to ∂ s ( α t z t ) = ∂ s ( α t ) z s . (4.15)Applying the twisted Leibniz rule, we get s ( α t ) ∂ s ( z t ) = ∂ s ( α t )( z s − z t ) . (4.16)Both sides are zero when m st = 2 . When m st > , so that ∂ s ( α t ) is nonzero, we deduce that z s − z t isproportional to the root s ( α t ) .In the special case when m st = 3 we find that ( α s + α t ) ∂ s ( z t ) = z t − z s . (4.17)Applying ∂ t to both sides of the equation, we get ∂ s ( z t ) = − ∂ t ( z s ) . (4.18) Remark 4.3.
The first major consequence of (4.16) is that one need not have an incredibly large realization inorder to find a p -differential d . If the Coxeter graph is connected, then the subspace spanned by { α s , z s } s ∈ S is at most one dimension higher than the subspace spanned by the roots. One should think that there is arealization spanned by the simple roots and a single new element z , and that each z s can be written as alinear combination of the roots and z . Let R ′ denote the subring of R generated by the roots and z . Then,so long as d ( z ) ∈ R ′ , our computations will imply that the Hecke category, when defined over R ′ , will bepreserved by the differential. In the standard setup for S n , we might let z = x , for example. Remark 4.4.
We have imposed many conditions on the elements z s and on the differential, making thesituation seem quite overdetermined, and the reader may already be convinced that in type A the standarddifferential on R = k [ x , . . . , x n ] is the only one which could satisfy them all. This intuition is entirely false.An arbitrary S n -invariant differential on R has the form d ( x i ) = Ax i + Bx i X j = i x j + C X j = i x j + D X j Consider the case when n = 2 and the differential satisfies d ( x ) = x − x x , d ( x ) = x − x x . (4.52)Then d ( x − x ) = ( x − x )( − x − x ) (4.53)so that z s = − ( x + x ) . Meanwhile, if ρ is a primitive third root of unity and g = ρx + ρ x , the readercan verify that d ( g ) = g and g + s ( g ) = − ( x + x ) . elationtosingularSoergelcalculus 22Example 4.8. We have computed that the most general differential on k [ x , x ] which works. It is S -invariant and satisfies d ( x ) = Ax − Cx x + Cx . (4.54)The case C = 0 is the standard differential. Then g = a x + a x satisfies d ( g ) = g and g + s ( g ) = z s , when a and a are distinct roots of the quadratic equation y + y ( C − A ) + C ( C − A ) = 0 . (4.55) A diagrammatic calculus for singular Soergel bimodules (also called the (diagrammatic) Hecke 2-category ) intype A is long-standing work in progress of Elias-Williamson. Diagrammatic calculus for dihedral groupsis due to Elias [Eli16b], where one can also find a review of the background. We will not provide furtherreview here. Let us examine what kind of differentials could exist on the Hecke 2-category, and how theywould restrict to the ordinary Hecke category. We will be brief and only provide summary results.The one-color generators of the Hecke 2-category are oriented cups and caps, as below.They also span their morphism spaces up to the action of R (in the white region). Consequently, anydifferential on the Hecke 2-category must send these diagrams to a multiple of themselves by a linearpolynomial; for the four diagrams pictured we call these linear polynomials g s , ¯ f s , f s , ¯ g s respectively.Checking that the isotopy/biadjunction relations are preserved by the differential will immediately implythat f s = − g s , ¯ f s = − ¯ g s . This then implies (4.1), (4.2), (4.25) and (4.26). Checking the remaining relationsagain gives the same constraints on g s . Remark 4.9. This is a more restrictive and easier way to find a formula for (4.25). In the computation of § f = f = 0 is a consequence of the differential being restricted from the Hecke 2-category,rather than being a consequence of the unit relation.In similar fashion, the fact that d preserves R s is forced upon any differential on the Hecke 2-category,rather than being a consequence. After all, R s is the endomorphism ring of the identity -morphism of theparabolic subset s .The defining idempotent decomposition is the decomposition of R as an ( R s , R s ) -bimodule into twoshifted copies of R s . That this descends to a Fc-filtration gives exactly the same two possibilities as inProposition 4.6, although the computation is quite different. Let s and u be distant. Let us recall the assumption ( ⋆ ) from § 3, which states that all linear polynomialsare in the span of R s and R u . One implication (with Demazure surjectivity) is the existence of a linearpolynomial ̟ s for which ∂ s ( ̟ s ) = 1 and ∂ u ( ̟ s ) = 0 , and similarly for a linear polynomial ̟ u .We now compute the differential applied to the generating -valent vertex , also known as the cross-ing , which has degree .Once again, by examination of the double leaves basis, one can deduce that any degree +2 map B s ⊗ B u → B u ⊗ B s is a linear combination of taking and placing a linear polynomial in one of the fourregions. Here is a useful consequence of ( ⋆ ) which we wish to record. Lemma 5.1. Let f be an arbitrary linear polynomial in R . Then − f + f + − f + f = 0 . (5.1) -valentvertices 23 Proof. By ( ⋆ ) we can write f = g + h where g ∈ R s and h ∈ R u . Take the copies of + f on the right andleft, decompose them as g + h , and slide g across the s -strand and h across the u -strand. Now we have − f + g + h = 0 in both the top and bottom, so the result is zero. Remark 5.2. In fact, an analogous lemma applies to any m st -valent vertex: the alternating sum of placing f in each region is zero. The proof is the same.Now consider an arbitrary degree +2 morphism, which is a linear combination of with variouslinear polynomials. Using the lemma we can remove the polynomial from the leftmost region. Using thesame simplification as in Remark 4.5, we can assume that the polynomial on top is a multiple of ̟ s andthe polynomial on bottom a multiple of ̟ u , by modifying the polynomial on the right. Thus the degree +2 morphism d ( ) has the following form. d = b̟s + a̟u + f . (5.2)One could also slide these polynomials to the left, giving the equivalent statement d = b̟s + a̟u + f . (5.3)Let us check the relation which slides a dot through a crossing. There are actually four such relations,depending on where one puts the dot. First we place the dot in the upper left. = . (5.4)Taking the differential of both sides, we get g u + b̟ s + f + a̟ u = g u . (5.5)In order to break the blue strand with zero coefficient, we need ∂ s ( g u + b̟ s ) = 0 , or in other words b = − ∂ s ( g u ) . (5.6)Then, given that g u + b̟ s ∈ R s , the equality of both sides is equivalent to f + a̟ u + b̟ s + g u = g u , or in other words f + a̟ u + b̟ s = 0 . Similarly, checking the relation with a dot on the upper right gives a = ∂ u ( g s ) . (5.7)Thus we deduce that f = ∂ s ( g u ) ̟ s − ∂ u ( g s ) ̟ u . (5.8)This pins down the differential of the crossing exactly, given the known differential of the dots. Note that if g s ∈ R u and g u ∈ R s then d ( ) = 0 , and otherwise the differential is nonzero.Let us simplify the answer. Adding and subtracting g u to the region on top, the polynomial in thatregion is − g u + ( g u − ∂ s ( g u ) ̟ s ) . (5.9) hecommutingrelationanditsidempotentdecomposition 24 Since ( g u − ∂ s ( g u ) ̟ s ) is s -invariant, it slides through to the rightmost region, leaving − g u behind. Similarly,we can add and subtract g s to the region on bottom, and slide ( ∂ u ( g s ) ̟ u − g s ) through the u -wall to therightmost region. What remains is d = − gu + gs + gu − gs . (5.10)Using Lemma 5.1 we can shuffle around these polynomials to obtain other nice descriptions of d ( ) aswell. For example, d = − gu + gs + gu + − gs . (5.11)Let us check the cyclicity of the -valent vertex. We have d = − gs + gs − gu + ∗ + gs + ¯ gs , (5.12)where ∗ = g u − g s − ¯ g s . We used (5.10) to get four of these terms (which, instead of appearing on thetop, right, and bottom as in (5.10), now appear on the right, bottom, and left respectively because of thetwisting). The remaining four terms came from taking the differential of the blue cap and cup. Now noticethat g s + ¯ g s = z s ∈ R u , so it can be slid through the green strand. This cancels four of the terms, yielding d = − gs + gs − gu + gu . (5.13)But this agrees with the formula (5.10) after swapping the colors s and u , as desired.We claim that checking the relations below is completely straightforward, and we leave it as an exerciseto the reader. • Sliding a trivalent vertex through a crossing. • Crossings are inverse isomorphisms: ◦ = . • The A × A × A Zamolodchikov relation. One of the defining relations of the Hecke algebra is the commuting relation b s b u = b u b s , when m su = 2 .This is lifted by an isomorphism B s B u ∼ = B u B s (5.14)in the Hecke category, and the inverse isomorphisms p and i are given by the 4-valent vertices. In fact, upto rescaling, these are the only choices of isomorphisms.One should think that B s B u has a decomposition with a single term, so that I has size . We still needto check that Γ I,d has no cycles, or in other words, that there is no loop at the single vertex. In otherwords, we need to check that pd ( i ) = 0 in order for the isomorphism to lift to a fantastic filtration, and for [ B s ][ B u ] = [ B u ][ B s ] in the p -dg Grothendieck group. Proposition 5.3. The isomorphism B s B u ∼ = B u B s is an isomorphism of p -DG objects if and only if ◦ d (cid:16) (cid:17) = 0 , (5.15)if and only if g s ∈ R u and g u ∈ R s , if and only if d ( ) = 0 . Proof. The first equivalence is definitional, and the remainder are very straightforward computations. mplicationsintype A A For the standard differential, as deduced in § s = s i = ( i, i + 1) then either g s i = x i or g s i = x i +1 .In either case, g s ∈ R u whenever u is distant from s , as desired. There is very little additional perspective added from considering thick calculus (or singular calculus) fortwo distant colors. For completeness, we include a brief discussion.In the thick calculus, there is an object B s,u corresponding to the parabolic subset { s, u } , which we drawwith an olive strand. There are splitting and merging maps which give inverse isomorphisms between B s,u and the tensor product B s B u , as well as the tensor product B u B s . Here is the drawing of the isomorphism B s,u → B u B s .The differential applied to this map will place a linear polynomial in each of the regions. Using ( ⋆ ) ,there is no need to put a polynomial in the region on top, so the polynomials must go on the left and right. d ( ) = f + f . (5.16)Moreover, by moving polynomials in R su from left to right, we can assume that f = b ′ ̟ s + a ′ ̟ u for somescalars a ′ , b ′ .The inverse isomorphism comes from flipping the diagram upside down, and applying the differentialto the relation stating that they are inverse isomorphisms implies d ( ) = − f + − f . (5.17)For these inverse isomorphisms to give p -dg isomorphisms, it is immediate to compute that f = f = 0 .The isomorphism B s,u → B s B u has a similar picture, with polynomials h and h instead. Composingthe morphisms B s B u → B s,u → B u B s is supposed to give the -valent vertex, and the relations in the thickcalculus are derived from this. So, without any appreciable difference from the computation in the rest ofthis section, we deduce that h − f = f − h = ∂ s ( g u ) ̟ s − ∂ u ( g s ) ̟ u . (5.18)Thus, unlike the ordinary diagrammatic Hecke category, a differential on the thick calculus is not uniquelydetermined by what happens to the dots. One can choose f and f freely, and then h and h are deter-mined by (5.18). However, if the differential is to be good (i.e. it satisfies Proposition 5.3) then f = f = h = h = 0 , and this additional flexibility disappears. Let s and t be adjacent, so that m st = 3 . Our next task is to compute the differential applied to the gener-ating -valent vertex, which has degree . So let us examine for a time the space of degree +2 morphisms B s B t B s → B t B s B t .Inside this space we have six broken 6-valent vertices which we call the 12 o’clock break, the 2 o’clockbreak, etcetera.In fact, these six morphisms only span a four-dimensional subspace. There is a relation + = + , (6.1) reliminaries 26 together with its rotations around the clock. Using this, one can prove that 12, 2, 4, and 6 o’clock form abasis for this four-dimensional subspace.Now consider the larger subspace spanned by the broken 6-valent vertices, and by morphisms obtainedfrom the 6-valent vertex by adding a linear polynomial in some region. Forcing a polynomial from oneregion to another is possible by (4.9) at the cost of breaking some strands. Consequently, this subspace isspanned by the four broken 6-valent vertices above, and by the morphisms of the form f . Meanwhile, there are five double leaves of degree +2 , so the entire space of degree +2 morphisms isspanned by the previous subspace and one more morphism, which we can take to beThus we can assume that d (cid:18) (cid:19) = A + B + C + D + E + f (6.2)for some scalars A, B, C, D, E and some linear polynomial f .At this point, our computations are linear combinations of many very similar diagrams, and it helps tointroduce some new and extremely abusive notation. First, instead of drawing a broken strand, we willmerely draw a strand marked with a coefficient. Thus if A is a scalar then A := A · . (6.3)Second, with the understanding that we are only interested in morphisms of a particular degree, we su-perimpose diagrams rather than adding them together! So, if we knew we were discussing morphisms B s → B s of degree , then f A f is actually shorthand for the sum f + A + f . This notation is horribly abusive (oh, if only my mother could see me now!) but being able to draw alarge linear combination succinctly has benefits both for the page count and for the readability and under-standibility of this paper. For sanity, we will always use the olive color when we use this particular abuseof notation.So, instead of (6.2) we can write the very compact d ( ) = A CDB f + E . (6.4)To give some more examples, here is a rewriting of (6.1). AA = AA . (6.5) inningdownthedifferential 27 Here is a useful equation involving the trivalent vertex. A = A A − Aα s . (6.6)Finally, before we begin the computation proper, let us set up notation for some important scalars. Let κ ss = ∂ s ( g s ) , κ st = ∂ s ( g t ) , κ ts = ∂ t ( g s ) , κ tt = ∂ t ( g t ) , (6.7) ¯ κ ss = ∂ s (¯ g s ) , ¯ κ st = ∂ s (¯ g t ) , ¯ κ ts = ∂ t (¯ g s ) , ¯ κ tt = ∂ t (¯ g t ) . (6.8)Here are some things we know about these scalars. Since g s + ¯ g s ∈ R s and similarly for t , we have κ ss + ¯ κ ss = 0 , κ tt + ¯ κ tt = 0 . (6.9a)Using (4.18) we have κ st + ¯ κ st + κ ts + ¯ κ ts = 0 . (6.9b) We will pin down the coefficients A, B, C, D, E, f by checking the “death by pitchfork” relations, which saythat putting a dot on one input to a 6-valent vertex, and merging its two neighbors with a trivalent vertex,will yield the zero morphism. In this first example, we put the dot at 12 o’clock. = 0 (6.10)Applying the differential to both sides we get QY: Is it just my pdf reader, or it is hard to read for you too? AB CD f ∗ − E . (6.11)The polynomial ∗ is g s − ¯ g t . Now, the contributions of the terms with B , D , and f are all zero, since theyhave a subdiagram with (6.10) inside. The A term just creates a copy of Aα s which is added to ∗ . Then ∗ can be forced to the right using (4.9), and only the term which breaks the strand will survive. Consequently,the result is C + ∂ t ( g s − ¯ g t + Aα s )) − E . (6.12)These two diagrams are linearly independent, from which we conclude that E = 0 , (6.13) C − A + κ ts − ¯ κ tt = 0 . (6.14)Similarly, we can put the pitchfork on bottom, putting the dot on 6 o’clock. An entirely similar argumentshows that ( E = 0 again, and) D − B + ¯ κ st − κ ss = 0 . (6.15) heckingtherelations 28 In general, the two-color Hecke category has an automorphism which flips a diagram upside-down andswaps the colors s and t . This automorphism preserves the 6-valent vertex. The effect of this symmetry willbe to swap g s with ¯ g t , ¯ g s with g t , A with B , C with D , κ st with ¯ κ ts , etcetera, while fixing E and f . Thus thissymmetry interchanges (6.14) with (6.15).Now we put the dot on 10 o’clock, and take the differential. AB CDf ∗ ¯ g s . (6.16)This time the polynomial ∗ is g t − ¯ g s − g s , with contributions coming from the red dot, the blue trivalentvertex, and the blue cup. Since g s + ¯ g s = z s ∈ R s , it can be slid out of this region. Thus the only survivingterms are A + κ st ) . (6.17)From this (and the corresponding computation for 8 o’clock, given by symmetry) we deduce that A = − κ st , B = − ¯ κ ts . (6.18)Then (6.14) and (6.15) give us C = ¯ κ tt − κ ts − κ st , D = κ ss − ¯ κ st − ¯ κ ts . (6.19)Thus we have solved for A, B, C, D, E . If we resolved the 2 o’clock and 4 o’clock relations we would getequations for ∂ s ( f ) and ∂ t ( f ) , but it is easier to check other relations to determine f precisely. Let us check one version of 2-color associativity. d = A AB BC CD Df f − g s (6.20)Now we simplify this morphism. The central red strand, if broken, yields the zero morphism; consequentlyone of the B terms does not contribute, and the − g s term can be forced across this red line to become − t ( g s ) .This polynomial can be forced further across the blue line, breaking this line with a coefficient of − ∂ s ( t ( g s )) ,which is added to the previous coefficient for breaking this line, namely A + D . Note that t ( g s ) = g s − κ ts α t so that ∂ s ( t ( g s )) = κ ss + κ ts . (6.21) heckingtherelations 29 In particular, the line is broken with coefficient A + D − ∂ s ( t ( g s )) = − κ st + κ ss − ¯ κ st − ¯ κ ts − κ ss − κ ts = 0 (6.22)where we used (6.9) to deduce that the coefficient was zero. Thus what remains at the end is d = A BC CD ∗ f (6.23)where the polynomial ∗ is f − st ( g s ) .Meanwhile, we have d = AB CD f − g t = AB C CD f ∗ . (6.24)Here ∗ represents − g t − Cα t , and the second equality followed from (6.6).Now 2-colored associativity is the equality = . (6.25)Applying the differential to both sides, we have a linear combination of terms which look like either side of(6.25) except with one strand broken, and the coefficients on each strand match up perfectly. We also have f in the rightmost region, matching perfectly, and a polynomial in the upper right region, which does notobviously match. So all that is required is for these polynomials to agree, namely, f = − g t − Cα t + st ( g s ) . (6.26)This solves for the polynomial f .To simplify further, note that st ( g s ) = g s − κ ss α s − κ ts ( α s + α t ) so that (using (6.14)) we have f = g s − g t − ( κ ss + κ ts ) α s − (¯ κ tt − κ st ) α t . (6.27)There is another version of 2-colored associativity (which is not a rotation of this one), which can bechecked by flipping upside-down and swapping colors, a symmetry we have previously discussed. Thisyields the equality f = ¯ g t − ¯ g s − (¯ κ tt + ¯ κ st ) α t − ( κ ss − ¯ κ ts ) α s . (6.28)We must confirm that these two equations for f are consistent. Taking the difference, we have g s + ¯ g s − g t − ¯ g t − ( κ ts + ¯ κ ts ) α s + ( κ st + ¯ κ st ) α t heckingtherelations 30 which is equal to z s − z t − ∂ t ( z s ) α s + ∂ s ( z t ) α t . Using (4.18) and (4.17), this is zero as desired.The next relation we should check is when a dot is placed on the 6-valent vertex. We can check whathappens if a blue dot is placed, and determine what happens for a red dot by symmetry. Confirming thatthe differential preserves this relation is extremely tedious but straightforward, and uses only the tricksalready used above, so we leave it to the reader.Finally, we should check the cyclicity relation. We postpone the 3-color relations until the next chapter.First we must discuss the differential of the other 6-valent vertex, from B t B s B t → B s B t B s . In thenext chapter, the 6-valent vertex B s B t B s → B t B s B t will be denoted φ , and the 6-valent vertex B t B s B t → B s B t B s will be denoted ψ .We can deduce many things about ψ by applying the symmetry of the two-color Hecke category whichswaps s and t ; this will swap g s with g t , ¯ g s with ¯ g t , etcetera. Thus we have d = A ′ C ′ D ′ B ′ f ′ , (6.29)where A ′ = − κ ts , B ′ = − ¯ κ st , (6.30a) C ′ = ¯ κ ss − κ ts − κ st , D ′ = κ tt − ¯ κ st − ¯ κ ts , (6.30b) f ′ = g t − g s − ( κ tt + κ st ) α t − (¯ κ ss − κ ts ) α s . (6.30c)Using (6.9) one can see that D ′ = − C, C ′ = − D. (6.31)Now applying the formula (6.4), and taking the differential of cups and caps as well, we get d = C DBA ¯ g s g t − ¯ g t f − gs . (6.32)Our goal will be to simplify this linear combination until it agrees with (6.29), which we do by forcing allthe polynomials to the rightmost region, one strand at a time.Recall that A = − κ st = − ∂ s ( g t ) . If we try to force g t from the leftmost region across its neighboring bluestrand (10 o’clock), we get s ( g t ) on the other side, plus a broken strand with scalar + ∂ s ( g t ) . Thus the 10o’clock break has overall coefficient A + ∂ s ( g t ) = 0 . Similarly, if we force − ¯ g t across its neighboring bluestrand (6 o’clock), we get − s (¯ g t ) on the other side, and break the strand with coefficient − ∂ s (¯ g t ) = − ¯ κ st = hebraidrelationanditsdefiningidempotentdecomposition: abstractions 31 B ′ . Combining these two manipulations we have d = C DBB ′ ¯ g s f − gss ( gt ) − s (¯ gt ) . (6.33)In similar fashion, we continue to force more polynomials to the right. Forcing s ( g t ) across its redneighbor will produce ts ( g t ) on the other side, and break the strand with coefficient ∂ t ( s ( g t )) = κ tt + κ st .Adding this to C , the overall coefficient on the 12 o’clock break will be C + κ tt + κ st = ¯ κ tt + κ tt − κ ts = − κ ts = A ′ . To get the coefficient of the 2 o’clock break, we need to force f − g s + ts ( g t ) across its neighboring bluestrand. In the same way that we deduced (6.27) from (6.26), one can also deduce that f = g s + C ′ α s − ts ( g t ) . (6.34)Hence f − g s + ts ( g t ) = C ′ α s , and it breaks the 2 o’clock strand with coefficient ∂ s ( C ′ α s ) = 2 C ′ . Addingthis to the existing coefficient D = − C ′ , we get an overall coefficient of C ′ .The ultimate coefficient of the 4 o’clock break will be B − ∂ t ( s (¯ g t )) = − ¯ κ ts − ¯ κ tt − ¯ κ st = D ′ . Putting this together, we get d = A ′ C ′ D ′ B ′ ∗ (6.35)where ∗ = ¯ g s + s ( f − g s + ts ( g t )) − ts (¯ g t ) = ¯ g s − C ′ α s − ts (¯ g t ) . (6.36)It remains to show that ∗ = f ′ . We leave this to the reader, having done enough similar computations. The braid relation on the Hecke algebra, reinterpreted in the Kazhdan-Lusztig presentation, is b s b t b s − b s = b t b s b t − b t , (6.37)as both sides are actually descriptions of the Kazhdan-Lusztig basis element b sts . The categorification ofthis statement is the direct sum decompositions B s B t B s ∼ = M ⊕ B s , B t B s B t ∼ = M ′ ⊕ B t , (6.38) hebraidrelationanditsdefiningidempotentdecomposition: abstractions 32 together with the isomorphism M ∼ = M ′ . (6.39)However, M and M ′ are not objects in the diagrammatic Hecke category, but are only objects in the Karoubienvelope, being the image of certain idempotents.In the introduction we discussed the practical way to prove a direct sum decomposition, see (1.5). Anal-ogously, one wishes to prove (6.38) and (6.39) practically, but using only morphisms between Bott-Samelsonbimodules. For sake of brevity let us write X = B s B t B s and Y = B t B s B t . One should provide morphisms i s : B s → X, i t : B t → Y, p s : X → B s , p t : Y → B t ,φ : X → Y, ψ : Y → X, satisfying p s i s = id s , p t i t = id t , (6.40a) φi s = 0 , ψi t = 0 , p s ψ = 0 , p t φ = 0 , (6.40b) id X = ψφ + i s p s , id Y = φψ + i t p t . (6.40c)One should think that the maps φ and ψ pass through the common summand M ∼ = M ′ , so their compositionis the idempotent projecting to this summand. In particular, M is to be identified with the object ( X, ψφ ) inthe Karoubi envelope, and M ′ with ( Y, φψ ) . The map φ induces an isomorphism ¯ φ : ( X, ψφ ) → ( Y, φψ ) and similarly ψ induces the inverse isomorphism ¯ ψ . Now we ask what extra conditions produce the appro-priate relations on the Grothendieck group in the p -dg setting. Remark 6.1. Everything we say in this section, including the main result (Proposition 6.2), is easily adapt-able to the general situation where one has X ∼ = M ⊕ P, Y ∼ = M ′ ⊕ Q, M ∼ = M ′ in some p -dg category, where P and Q are genuine objects, while M and M ′ are only objects in the Karoubienvelope.The identity of X is decomposed as a sum of two orthogonal idempotents e = i s p s and e = ψφ . Inorder for the decomposition X ∼ = M ⊕ B s to be a dg-filtration, we need either d ( e ) e = 0 or d ( e ) e = 0 .Now d ( e ) e = d ( i s p s ) ψφ = i s d ( p s ) ψφ since p s ψ = 0 . Clearly d ( e ) e = 0 if d ( p s ) ψφ = 0 . Conversely, by postcomposing with p s , if d ( e ) e = 0 then d ( p s ) ψφ = 0 . Using a similar argument, precomposing with either ψ or φ , we see that d ( e ) e = 0 ⇐⇒ d ( p s ) ψ = 0 . (6.41)We will use this pre- and post-composition trick several times below.Similarly, d ( e ) e = 0 ⇐⇒ d ( φ ) i s = 0 . (6.42)This seems entirely analogous, but the proof is slightly trickier. Clearly d ( e ) e = ψd ( φ ) i s p s so d ( e ) e = 0 ⇐⇒ ψd ( φ ) i s = 0 ⇐⇒ φψd ( φ ) i s = 0 . As a reminder, this condition will imply that either Hom( X, − ) e or Hom( X, − ) e is preserved by the differential, as a left p -dgmodule over the category, and that the other one is the quotient of Hom( X, − ) by the first. It also implies the analogous condition forright p -dg modules. hebraidrelationanditsdefiningidempotentdecomposition: abstractions 33 But in fact φψ can be replaced with id tst here, since the difference is i t p t d ( φ ) i s , and p t d ( φ ) i s = d ( p t φ ) i s − d ( p t ) φi s = 0 . (6.43)In addition to checking that the decomposition on X is a dg-filtration, we need to confirm that the imageof i s p s has the appropriate p -dg structure, which amounts to checking that d ( p s ) i s = 0 (6.44)as a degree endomorphism of B s .Analogous conditions need to hold for the decomposition of Y .Finally, we need to confirm that M = ( X, ψφ ) and M ′ = ( Y, φψ ) are isomorphic as p -dg modules, withtheir induced differentials. Recall that, if ( X, e ) and ( Y, f ) are two objects in the Karoubi envelope, then theinduced differential ¯ d on Hom(( X, e ) , ( Y, f )) = f Hom( X, Y ) e is ¯ d ( f ξe ) = f d ( f ξe ) e (6.45)where ξ ∈ Hom( X, Y ) and d is the usual differential on Hom( X, Y ) . If ¯ φ and ¯ ψ are to induce inverseisomorphisms of p -dg modules then we need ¯ d ( ¯ φ ) ¯ ψ = 0 (6.46)and (equivalently) ¯ d ( ¯ ψ ) ¯ φ = 0 . (6.47)The first equation unravels to φψd ( φ ) ψφ = 0 which is equivalent by pre- and post-composition to ψd ( φ ) ψ = 0 . (6.48)The second equation is equivalent to φd ( ψ ) φ = 0 . (6.49)To convince the reader that (6.48) and (6.49) are equivalent, let us pre- and post-compose (6.48) with φ .The result is φψd ( φ ) ψφ = φψd ( φψ ) φ − φψφd ( ψ ) φ = φψd ( φψ ) φ − φd ( ψ ) φ. So φd ( φ ) ψ = 0 will imply that φd ( ψ ) φ = 0 so long as φψd ( φψ ) φ = 0 . But since d (id tst ) = 0 , we know that d ( φψ ) = − d ( i t p t ) . Then ψd ( φψ ) φ = − ψd ( i t ) p t φ − ψi t d ( p t ) φ = 0 + 0 = 0 , as desired.Together, all these conditions imply that X is fantastically filtered by M and B s in the Karoubi envelope,they Y is fantastically filtered by M ′ and B t , and that M ∼ = M ′ as p -dg objects. In conclusion, we haveproven the following result. Proposition 6.2. Given maps i s , p s , i t , p t , φ, ψ as in (6.40), then consider the following graph. B s M ′ M B td ( p s ) i s d ( φ ) i s d ( p s ) ψ ψd ( φ ) ψφd ( ψ ) φ d ( p t ) φd ( ψ ) i t d ( p t ) i t (6.50)Erase the edges with zero labels (noting that if one edge between M and M ′ is zero, then so is the other).If the resulting graph has no cycles or loops, then the idempotent decompositions (6.38) and (6.39) lift tofantastic filtrations, the objects M and M ′ are cofibrant, and the braid relation (6.37) holds in the p -dgGrothendieck group. hebraidrelationanditsdefiningidempotentdecomposition: computations 34Remark 6.3. The existence of a fantastic filtration is useful not only because it implies a relation in theGrothendieck group, but also because it implies, e.g., that B s B t B s is in the triangulated hull of B s and M ,and hence in the triangulated hull of { B s , B t , B t B s B t } . Since all the other objects are cofibrant, the 2/3 rulein triangulated categories implies that M is cofibrant. Remark 6.4. One should think of the pair of edges between M and M ′ as being like a loop. There is apartial idempotent completion where we could add a new object Z corresponding to either M or M ′ , andrewrite the idempotent decompositions as X ∼ = B s ⊕ Z, Y ∼ = B t ⊕ Z. Then we could use the ordinary theory of Fc-filtrations to analyze these decompositions. We would get twographs corresponding to the right and left halves of the graph in Proposition 6.2, and in each graph therewould be a loop at Z which corresponds to the edges between M and M ′ . Now let us compute the graph of Proposition 6.2. We should note that all maps i s , p s , φ , etcetera areuniquely determined up to scalar, living in one-dimensional Hom spaces, so there is only one graph tocompute. The maps φ and ψ are 6-valent vertices, and the remaining maps are pitchforks.First we check the loop at B s . We have d ( p s ) i s = ∗ , (6.51)where ∗ is the polynomial α t ( g t − ¯ g s ) . For this to be zero we need ∗ to be s -invariant, which requires that g t − ¯ g s is proportional to s ( α t ) = α s + α t . This is a new condition! The scalar of proportionality can bedetermined by applying demazure operators. Since ∂ s ( s ( α t )) = ∂ t ( s ( α t )) = 1 , we see that g t − ¯ g s = ( κ st − ¯ κ ss )( α s + α t ) = ( κ tt − ¯ κ ts )( α s + α t ) . (6.52)Checking the loop at B t , we get the analogous equation g s − ¯ g t = ( κ ts − ¯ κ tt )( α s + α t ) = ( κ ss − ¯ κ st )( α s + α t ) . (6.53)Now we check the pair of edges between M and M ′ . We have ψd ( φ ) ψ = AB CD f . (6.54)The A term and the B term vanish. We encourage the reader to confirm that the C term, the D term, andthe f term are linearly independent ! Thus ψd ( φ ) ψ = 0 ⇐⇒ C = D = f = 0 . (6.55) After all, M is supposed to be isomorphic to the indecomposable Soergel bimodule B sts . By the Soergel hom formula End ( B sts ) is spanned by: linear polynomials times the identity (the f term), a degree map factoring through B st (the D term), and a degree map factoring through B ts (the C term). hebraidrelationanditsdefiningidempotentdecomposition: computations 35 Using (6.31), this will imply that C ′ = D ′ = 0 as well (and also f ′ = 0 ).Note that f = − g t − Cα t + st ( g s ) by (6.26), so if f = C = 0 then g t = st ( g s ) . (6.56)If g s = 0 then g t = 0 and vice versa. Recall that Proposition 4.6 left us with two cases for each color. Either g s = ¯ g s = 0 or g s = ¯ g s = s ( g s ) = 0 , and similarly for t . We can now observe that we are in the same casefor both s and t : either g s = g t = 0 , or g s = 0 = g t . In the case where g s = g t = 0 one deduces that thedifferential is also zero on both 6-valent vertices, so it is zero on any diagram with colors in { s, t } . This is avalid solution, but one which makes the computations trivial.Let us restrict our attention to the other case. Thus we assume that ¯ g s = s ( g s ) , ¯ g t = t ( g t ) , g t = st ( g s ) . (6.57)We leave it as an exercise to verify that (6.57) implies that C = D = f = f ′ = 0 as well as (6.52) and(6.53). Thus (6.57) is the only assumption we need. The reader attempting this exercise will be helped bythe formula st∂ s ( h ) = ∂ t ( st ( h )) (6.58)for any polynomial h .Recall also the scalar κ = ∂ s ( g s ) which we used in a previous chapter. Then g t = st ( g s ) and (6.58) implythat κ = ∂ s ( g s ) = ∂ t ( g t ) . (6.59) Remark 6.5. If g t = st ( g s ) then ¯ g t = tst ( g s ) . So it makes sense that g s − ¯ g t = g s − tst ( g s ) is proportional to α s + α t , since tst is a reflection and α s + α t is its root.Now we analyze the two remaining pairs of edges, under the requirements that the loops vanishedabove. We have d ( p t ) φ = ∗ (6.60)where ∗ is the polynomial g s − ¯ g t = g s − tst ( g s ) . This polynomial can be forced out, and only the term whichbreaks the pitchfork will survive, so d ( p t ) φ = 0 ⇐⇒ ∂ t ( g s − ¯ g t ) = 0 . (6.61)However, we have already seen that g s − ¯ g t is proportional to α s + α t , which is not t -invariant. Thus ∂ t ( g s − ¯ g t ) = 0 if and only if g s − ¯ g t = 0 ! Consequently, d ( p t ) φ = 0 ⇐⇒ g s = ¯ g t = sts ( g s ) . (6.62)Meanwhile, d ( ψ ) i t = A ′ B ′ (6.63)Only the B ′ term will contribute, and it will contribute ∂ t ( B ′ α s ) = − B ′ . Recall that B ′ = − ∂ s (¯ g t ) . Using(6.58), we can also observe that − ∂ s (¯ g t ) = ∂ t ( g s ) . (6.64)Then d ( ψ ) i t = 0 ⇐⇒ B ′ = 0 ⇐⇒ ∂ s (¯ g t ) = 0 ⇐⇒ ∂ t ( g s ) = 0 . (6.65) mplicationsinsimplylacedtypes 36 Only one of these two conditions (6.62) and (6.65) need hold, so either g s = ¯ g t = sts ( g s ) or B ′ = 0 = ∂ s (¯ g t ) = ∂ t ( g s ) , but not necessarily both. That is, g s is either fixed by the reflection t or by the reflection sts .If both hold, then g s = 0 .Similarly, either d ( p s ) ψ = 0 or d ( φ ) i s = 0 , from which we deduce that either g t = ¯ g s = sts ( g t ) or B = 0 = ∂ t (¯ g s ) = ∂ s ( g t ) . These two conditions are equivalent to the conditions above, but in the reverseorder: if g s is fixed by sts then g t = st ( g s ) is fixed by s , and if g s is fixed by t then g t is fixed by sts .Let us consider one possibility, where B = 0 , so that g t is fixed by s and g s is fixed by sts . In this case both A = B = 0 , so that the differential kills φ ! We leave the reader to determine that B ′ = − A ′ = ∂ t ( g s ) = 0 .Moreover, ∂ t ( g s ) = ∂ t ( tst ( g s )) = − ∂ t ( st ( g s )) = − ∂ t ( g t ) = − κ. (6.66)Thus, A ′ = κ and B ′ = − κ and our differential satisfies d (cid:18) (cid:19) = 0 , d = κ − κ . (6.67)Analogously, the other possibility, where B ′ = 0 , also satisfies A ′ = 0 and B = − A = ∂ s ( g t ) , and ∂ s ( g t ) = ∂ s ( st ( g s )) = ∂ s ( s ( g s )) = − κ. (6.68)so that A = κ and B = − κ and d (cid:18) (cid:19) = 0 , d = κ − κ . (6.69)These are our two options.Let us summarize. Proposition 6.6. The graph from Proposition 6.2 has no loops or cycles if and only if one of the followingthree possibilities holds.1. g s = g t = 0 , and the differential is zero on every diagram with colors in { s, t } .2. The equations (6.57) hold, and g t is fixed by s , and g s = t ( g t ) is fixed by sts . Then the differential on6-valent vertices is given by (6.67).3. The equations (6.57) hold, and g s is fixed by t , and g t = s ( g s ) is fixed by sts . Then the differential on6-valent vertices is given by (6.69).In other words, any good differential must satisfy one of these three possibilities. Example 6.7. For the standard differential on k [ x , . . . , x n ] , we deduced in § g i = x i or g i = x i +1 . Ere now it seemed we could make this choice for each i independently. However, Proposition 6.6forces the choices of g i and the choice of g i +1 to be related to each other, since g i +1 = s i s i +1 g i . If g i = x i forsome i then g i +1 = x i +1 , and consequently g j = x j for all j . If g i = x i +1 then g i +1 = x i +2 , and consequently g j = x j +1 for all j . Let ( W, S ) be an irreducible Coxeter group with m st ∈ { , } for all s, t ∈ S .If g s = 0 for any s ∈ S , then by Proposition 6.6 we must have g t = 0 for all t in the same connectedcomponent of the Coxeter graph as s . This is the boring case.Otherwise, Proposition 6.6 gives one a dichotomy for each pair s, t ∈ S with m st = 3 . We will encodethis with an orientation on the edges in the Coxeter graph: the edge points from s to t if g s is fixed by t . mplicationsintype A Now suppose that s, t, u ∈ S generate a copy of A inside W , with m su = 2 . Suppose that the s − t edgeis oriented from s to t . Then g s is fixed by t . We also know that g s is fixed by u , thanks to Proposition 5.3.Now, g t = st ( g s ) = s ( g s ) , so that u ( g t ) = us ( g s ) = su ( g s ) = s ( g s ) = g t . Consequently g t is fixed by u , and the t − u edge is oriented from t to u . Similarly, we leave the reader todeduce that when the s − t edge is oriented from t to s , the t − u edge must be oriented from u to t . Inparticular, this implies that every copy of A must be consistently oriented: t is never a source or a sink.This fact is sad because of the following proposition. Proposition 6.8. There is no good differential in any simply laced type outside of finite or affine type A . Proof. Outside of finite and affine type A , the Coxeter graph must have a subgraph of type D . There is noway to orient D such that every copy of A inside is consistent. A We continue to let g i denote g s i when s i = ( i, i + 1) , for ≤ i ≤ n − . As just noted in § k [ x , . . . , x n ] ,and the differential is assumed to be the standard one, with d ( x i ) = x i . We’ve proven that there are onlytwo nonzero good differentials possible which extend the standard differential on the polynomial ring, theone where g i = x i for all i , and the one where g i = x i +1 for all i . Technically we still need to check thethree-color relations, but this is done in the next chapter. We refer to the differential where g i = x i for all i as having the standard orientation , and the other differential as having the reverse orientation . In Manin-Schechtman theory, braid relations s i s i +1 s i → s i +1 s i s i +1 can be given the lexicographic orientation, andthe set of all reduced expressions for a given element becomes a semioriented graph (the commuting braidrelations s i s j = s j s i have no orientation). This graph has a unique source and a unique sink, up to com-muting braid relations. See [Eli16a] for more details. For the standardly oriented good differential, d willkill any 6-valent vertex corresponding to a reverse-oriented braid relation. Vice versa, for the reverse ori-ented good differential, 6-valent vertices corresponding to standardly oriented braid moves are killed. Letus refer to the p -dg category associated to the standard orientation as the standard p -dg diagrammatic Heckecategory .Now we work with an arbitrary realization and its polynomial ring R , but under the assumption thatits diagrammatic Hecke category possesses a good differential. For ease of discussion we will assume theDynkin diagram gets the standard orientation; the other case can be handled similarly, or by duality. Wewill prove that our p -dg category is isomorphic to the standard p -dg diagrammatic Hecke category, perhapsafter extension of the realization.We know that g ∈ R s . We also know that g ∈ R s j for all j ≥ , by Proposition 5.3. Thus g is invariantin all simple reflections but the first, so it is invariant in the parabolic subgroup S × S n − . In other words, g has the same stabilizer in the symmetric group as does x in the standard polynomial ring. Similarly, g i has the same stabilizer as x i for all ≤ i ≤ n − , which one can check in the same way, or can confirm since g i = s i − s i − · · · s s ( g ) . Let us denote s n − ( g n − ) by g n ; it has the same stabilizer as x n .Let κ = ∂ ( g ) , which is also equal to ∂ i ( g i ) for all i by (6.58). Note that, by the usual definition ofDemazure operators, for all ≤ i ≤ n − one has g i − g i +1 = g i − s i ( g i ) = κα i . (6.70)In particular, the span of the g i contains all the simple roots, and has dimension at least n − . It is alsopreserved by the differential, since d ( g i ) = g i . Thus we may as well restrict our realization and our attentionto the span of { g i } ni =1 . Proposition 6.9. There is a ring homomorphism from k [ x , . . . , x n ] to the subring of R generated by g , . . . , g n ,sending x i g i . This homomorphism is S n -equivariant and intertwines the standard differential with thedifferential on R . In fact, it is an isomorphism so long as the characteristic of k is not . If the characteristicis , it is either an isomorphism or is the quotient map by the ideal generated by e = x + . . . + x n . mplicationsintype A Proof. The first two sentences are straightforward, so it remains to show that the homomorphism is anisomorphism. Let I be the kernel of the map. If g , . . . , g n is linearly independent, then they must bealgebraically independent (since they are linear terms inside a polynomial ring), and I = 0 .Suppose to the contrary that there were a linear dependence P a i g i = 0 for some scalars a i . Let y = P i ≥ a i g i , so that a g + a g + y = 0 . Since y is fixed by s , so must be a g + a g . Since g = s ( g ) = g ,this implies that a = a . By similar arguments, a = a , etcetera, and all the scalars a i are equal. Sothe only possible linear dependence relation is a ( g + · · · + g n ) for some nonzero scalar a . Since the linearpolynomials form a free k -module in any realization, a relation of the form a ( g + · · · + g n ) = 0 for a = 0 implies that g + · · · + g n = 0 .Thus if I is nonzero then I contains the symmetric polynomial e = x + . . . + x n . Since there is at mostone linear relation, at least n − of the elements g i are algebraically independent. If I is any bigger than theideal generated by e then this contradicts the algebraic independence of { g , . . . , g n − } .Now I is preserved by the differential, and d ( e ) = P x i = e − e . Thus e ∈ I . Unless thecharacteristic of k is , this gives the desired contradiction. Remark 6.10. Similarly, d ( e k ) = e e k − ( k + 1) e k +1 for all k . Ignoring the case of small primes, any idealpreserved by I which contains e will also contain e k for all ≤ k ≤ n . Thus k [ x , . . . , x n ] /I is finite-dimensional, as a quotient of the coinvariant ring. But its image inside R is infinite dimensional, containingat least a polynomial ring with n − generators. This is a contradiction.The conclusion is that our realization with good differential need not be the standard one with thestandard differential, but it (or the relevant part of it, the subring generated by the g i ) is equivariantlyisomorphic to the standard realization with the standard differential. Example 6.11. There is an S -invariant isomorphism k [ x , x , x ] → k [ y , y , y ] sending x y + y , x y + y , and x y + y . This intertwines the differential, when d ( y i ) = y i + y i y j + y i y k − y j y k for all { i, j, k } = { , , } . So, even when the realization is the standard one, we can not and should not ruleout the possibility that R is equipped with a non-standard differential, because it might be a non-standarddifferential isomorphic to a standard differential.The above proposition showed that the underlying polynomial rings of the standard realization and ourgood realization are isomorphic (outside of the possible characteristic exception), but we still need to showthat the associated Hecke categories are isomorphic categories (via an isomorphism which intertwines thedifferential). This is slightly subtle, because the isomorphism k [ x , . . . , x n ] → R sending x i g i need notsend simple roots to simple roots! Instead, α i = x i − x i +1 g i − g i +1 = κα i , (6.71)and roots are rescaled by the scalar κ . Proposition 6.12. Assume that the Hecke category is equipped with a good differential d . The homomor-phism k [ x , . . . , x n ] → R sending x i g i lifts to an functor between the standard p -dg Hecke category andthe version equipped with d . This functor rescales each s -colored enddot by κ , and each s -colored startdotby . Thus α s κα s , as noted above. The merging trivalent vertex is rescaled by , and the splittingtrivalent vertex is rescaled by κ − . The 4- and 6-valent vertices are rescaled by . This functor is an isomor-phism, except in the case of characteristic , when it is either an isomorphism or the kernel is generated bythe polynomial e . Proof. Let us assume that the map k [ x , . . . , x n ] → R is an isomorphism (as it must be outside of charac-teristic ). In [EH] we classify all autoequivalences of the Hecke category which fix the objects B s for each s . Any such automorphism is determined uniquely by how it rescales the start and end dots, and howit affects the remainder of the polynomial ring (beyond the part spanned by roots), and these choices canbe made arbitrarily. So, the functor defined above is an autoequivalence. Checking that it intertwines thedifferential is straightforward. Modifying these results to account for the possible e kernel in characteristic is straightforward. elationstothickcalculus 39 The thick calculus has a new object B s,t (drawn as purple) and several new morphisms: a trivalent vertex B s,t B s → B s,t together with variants thereof; splitters B s,t → B s B t B s and B s,t → B t B s B t , and mergers B s B t B s → B s,t and B t B s B t → B s,t , pictured below.The composition of a splitter and a merger gives the -valent vertex B s B t B s → B t B s B t . The compositionof a merger with a splitter gives the identity map of B s,t .Once again there is not very much new to say about differentials on the thick calculus. Here are thehighlights. • Because all the direct summands of B s B t B s are actually objects in the thick diagrammatic category,one need not worry about the abstractions of § • The differential on the trivalent vertex is forced to be the same as for an ordinary trivalent vertex (e.g.for the map B s,t B s → B s,t , put − ¯ g s in the middle on the bottom). • One can compute the general differential on the splitters and mergers. The computation is just asnasty and thorny as the one above for the general differential of the 6-valent vertices. Just as in § f from (6.4). Also as in § s → t . d = − ¯ g s d = − ¯ g s (6.72a) d = − g s d = − g s (6.72b) d = 0 d = 0 (6.72c) d = κ d = − κ (6.72d) It remains to check the Zamolodchikov relations associated to finite rank 3 Coxeter subgroups. Since weare in simply laced type, there are only three possible rank 3 subgroups: A × A × A , A × A , and A .When the differential is good, the differential kills all -valent vertices, and kills half of the 6-valentvertices. There is a version of each Zamolodchikov relation where the differential kills both sides, and thusthe relation is checked trivially! This is because the Zamolodchikov relations are equalities between twooriented paths in the reduced expression graph. For the reverse-oriented good differential, all orientedpaths are sent to zero by the differential. Rotating the relations by 180 degrees, we obtain an equivalentrelation which is an equality between reverse-oriented paths in the reduced expression graph, and theseare sent to zero by the standardly oriented good differential. Remark 7.1. It is a good exercise for the reader learning diagrammatics to compute directly that the stan-dardly oriented good differential preserves the oriented version of the Zamolodchikov relation (the versionthat it does not just send to zero).We have checked the A × A × A and A × A relations for an arbitrary differential, and they hold; thetechniques required for this check have all been discussed above. We tried to check the A relation for anarbitrary differential, but it was a surprisingly thorny computation, and we gave up. Yes, at least for the good differentials. We did not bother to check the general differential.Consider the graded ring T = Z [ x ] with differential d ( x ) = x . Let M be a free T -module of rank ,generated by the element m , and equip with with an edg-structure (i.e. a degree +2 derivation) where d M ( m ) = ax · m for some a ∈ Z . We call this edg-module M a . Then it is easy to compute that d ( k ) M x ℓ · m = (cid:18) ℓ + k + a − k (cid:19) x ℓ + k · m (8.1)for all ℓ ≥ . Thus the divided powers are defined over Z . Remark 8.1. More generally, if T = k [ x ] for some field of characteristic p , and M a is defined as above forsome a ∈ k , then d p = 0 on M a if and only if a lives in the prime field F p ⊂ k .Now consider the diagrammatic Hecke category over Z , and let θ be a generator. We ask whether d ( k ) ( θ ) lives inside the Z -form as well. Obviously this holds if d ( θ ) = 0 , so we can consider only those differentialswhich do not kill θ .Suppose that θ is an s -colored enddot. Then d ( θ ) = g s θ , and d ( g s ) = g s . Thus we can consider thesub-edg-algebra T = Z [ g s ] ⊂ R . The subset T · θ is closed under the differential, and is isomorphic as anedg module over T to M . In particular, d ( k ) ( θ ) is well-defined over Z . The same argument works for the s -colored startdot.The analogous argument also works for the s -colored trivalent vertices, where T acts by putting a poly-nomial in the appropriate region. This time T · θ is isomorphic to M − instead.Any good differential kills every 4-valent vertex, and one of the two 6-valent vertices. We need onlycheck what happens to the other 6-valent vertex. We do the case when the orientation is s → t .Let α , β , and γ denote the following three diagrams. α = , β = , γ = . Lemma 8.2. The differential acts by the following formulae. d ( α ) = 2 g s α − κγ, (8.2a) d ( β ) = 2¯ g t β + κγ, (8.2b) d ( γ ) = 2( g s + ¯ g t ) γ. (8.2c)Here, multiplication by a polynomial means putting that polynomial in the leftmost region (or rightmostregion, it happens to be equal). Proof. The proof is straightforward. Let us derive the first equality, as the others are similar. Applying thedifferential to α we get a sum of three terms: • The differential applied to the broken strand on top (the pair of dots). This places z s in the top region. • Breaking the top strand again, with a plus sign and a factor of κ . This contributes κα s to the topregion. • Breaking the bottom strand, with a minus sign and a factor of κ . This contributes − κγ .Then one observes that z s + κα s = ( g s + ¯ g s ) + ( g s − ¯ g s ) = 2 g s . Since g s is t -fixed, it slides across the t -coloredstrand to the leftmost or rightmost region. Lemma 8.3. When the orientation is s → t , the differential applied iteratively to the 6-valent vertex φ : B s B t B s → B t B s B t is equal to d k ( φ ) = k ! κ ( g k − s α − ¯ g k − t β − κ ( X a + b = k − g as ¯ g bt ) γ ) (8.3)for any k ≥ . Consequently, d ( k ) is well-defined integrally. Proof. We have d ( φ ) = κ ( α − β ) , and d ( φ ) = κ (2 g s α − κγ − g t β ) . (8.4)This proves the cases k = 1 , . The inductive step is a simple exercise in the Leibniz rule. A helpfulobservation is that whenever d ( x ) = x and d ( y ) = y we have d ( X a + b = m x a y b ) = mx m +1 + ( m − xy X a ′ + b ′ = m − x a ′ y b ′ + my m +1 . (8.5)Putting it all together, we have proven the following result. Theorem 8.4. For any good differential d on the diagrammatic Hecke category in simply-laced type, d ( k ) isdefined integrally. n = 8 Let us work in the diagrammatic Hecke category for S associated to the standard realization with poly-nomial ring R = Z [ x , . . . , x ] . Equip it with the standard differential, where g s i = x i and ¯ g s i = x i +1 and κ = 1 . We also extend this differential to the thicker calculus which includes objects B sts for m st = 3 , using(6.72).Let X denote the Bott-Samelson bimodule associated with the sequence w = ( s , s , s , s , s , s , s , s , s , s , s , s , s , s ) . Let y = s s s s s s . To describe the indecomposable object B y we can most easily use thick calculus, since B y ∼ = B s s s B s s s . Here is are degree map p : X → B y and i : B y → X which span their respective degree Hom spaces. p = i = (9.1)To summarize these maps in words: the four (grayscale) strands with colors in { , , } get dotted off; thefive (red and blue) strands with colors in { , } get merged into B s s s (colored purple) with two thicktrivalent vertices; the five (aqua and green) strands with colors in { , } get merged into B s s s (coloredteal) with two thick trivalent vertices. It is a very fun exercise to compute that pi = 2id B y . (9.2)A computation yields − d ( p ) = x − x +1 +1 (9.3)where again we use our abusive olive-colored sum notation and broken line notation from § − d ( p ) is the sum of three terms with the same underlying diagram as p : one which breaks thesecond strand colored s with coefficient +1 , one which breaks the last strand colored s with coefficient +1 , and one which places the polynomial x − x in the center. Another computation yields + d ( i ) = x − x +1 +1 (9.4)One can actually derive the second computation from the first, using various symmetries: rotation by 180degrees, and the Dynkin diagram automorphism.It is not immediately obvious that d ( p ) = 0 , since it is not expressed as a linear combination of lightleaves with polynomials on the right, but it is not too difficult to justify. For example, the polynomial x isfixed by s i for i ≥ and slides through the entire diagram; none of the other terms can possibly contributea polynomial involving x . Similarly, d ( i ) = 0 as can be seen from the x term.Finally, a computation yields d ( p ) i = x + x − x − x − − . (9.5)Note that the split-merge appearing on the rightmost part of this picture is equal to the identity of B s s s ,but if the s -colored strand is broken (with coefficient − ), it yields a nonzero degree +2 endomorphism of B s s s . Similarly with the leftmost part of the picture and the identity of B s s s .As a consequence of this computation, B y is not a summand of X in any dg-filtration, for any prime!This example shows that the conjectural d -canonical basis does not agree with the p -canonical basis. See § EFERENCES 43 References [BC18] A. Beliakova and B. Cooper. Steenrod structures on categorified quantum groups. Fund. Math. ,241(2):179–207, 2018. arXiv:1304.7152.[EH] Ben Elias and Matthew Hogancamp. Homotopy lifting and conjugation by Rouquier complexes.In preparation.[EK10] Ben Elias and Mikhail Khovanov. Diagrammatics for Soergel categories. Int. J. Math. Math. Sci. ,pages Art. ID 978635, 58, 2010.[Eli16a] Ben Elias. Thicker Soergel calculus in type A . Proc. Lond. Math. Soc. (3) , 112(5):924–978, 2016.[Eli16b] Ben Elias. The two-color Soergel calculus. Compos. Math. , 152(2):327–398, 2016.[EQ16a] Ben Elias and You Qi. An approach to categorification of some small quantum groups II. Adv.Math. , 288:81–151, 2016.[EQ16b] Ben Elias and You Qi. A categorification of quantum sl (2) at prime roots of unity. Adv. Math. ,299:863–930, 2016.[EW16a] Ben Elias and Geordie Williamson. Relative hard Lefschetz for Soergel bimodules. preprint,2016. arXiv 1607.03271.[EW16b] Ben Elias and Geordie Williamson. Soergel calculus. Represent. Theory , 20:295–374, 2016.arXiv:1309.0865.[EWS16] Ben Elias, Geordie Williamson, and Noah Snyder. On cubes of Frobenius extensions. In Repre-sentation theory - current trends and perspectives , pages 171–186. European Mathematical Society,2016. arXiv:1308.5994.[Kho16] Mikhail Khovanov. Hopfological algebra and categorification at a root of unity: the first steps. J. Knot Theory Ramifications Trans. Amer. Math. Soc. , 363(5):2685–2700, 2011.[Kle15] A. S. Kleshchev. Affine highest weight categories and affine quasihereditary algebras. Proc.Lond. Math. Soc. (3) , 110(4):841–882, 2015. arXiv:1405.3328.[KLMS12] Mikhail Khovanov, Aaron D. Lauda, Marco Mackaay, and Marko Stoˇsi´c. Extended graphicalcalculus for categorified quantum sl(2) . Mem. Amer. Math. Soc. , 219(1029):vi+87, 2012.[KQ15] M. Khovanov and Y. Qi. An approach to categorification of some small quantum groups. Quan-tum Topol. , 6(2):185–311, 2015. arXiv:1208.0616.[KR16] M. Khovanov and L. Rozansky. Positive half of the Witt algebra acts on triply graded linkhomology. Quantum Topol. , 7(4):737–795, 2016. arXiv:1305.1642.[Lau10] Aaron D. Lauda. A categorification of quantum sl(2) . Adv. Math. , 225(6):3327–3424, 2010.[MSV13] Marco Mackaay, Marko Stoˇsi´c, and Pedro Vaz. A diagrammatic categorification of the q -Schuralgebra. Quantum Topol. , 4(1):1–75, 2013.[Qi14] You Qi. Hopfological algebra. Compositio Mathematica , 150(01):1–45, 2014.[QS18] Y. Qi and J. Sussan. p-DG cyclotomic nilHecke algebras II. 2018. arXiv:1811.04372.[Rou08] Rapha¨el Rouquier. 2-Kac-Moody algebras. Preprint, 2008. arXiv:0812.5023. EFERENCES 44 [Sch11] O. M. Schn ¨urer. Perfect derived categories of positively graded DG algebras. Applied CategoricalStructures , 19(5):757–782, 2011. arXiv:0809.4782.[Soe90] Wolfgang Soergel. Kategorie O , perverse Garben und Moduln ¨uber den Koinvarianten zurWeylgruppe. J. Amer. Math. Soc. , 3(2):421–445, 1990.[Ste18] Andrew Stephens. A categorification of quantum sl (3) at a prime root of unity . PhD thesis, Universityof Oregon, September 2018.[Sto11] Marko Stosic. Indecomposable 1-morphisms of ˙ U +3 and the canonical basis of U + q ( sl ) . Preprint,2011. arXiv:1105.4458.[Wil17] Geordie Williamson. Schubert calculus and torsion explosion. J. Amer. Math. Soc. , 30(4):1023–1046, 2017. With a joint appendix with Alex Kontorovich and Peter J. McNamara.B. E.: DepartmentofMathematics,UniversityofOregon,Eugene,OR97403,USA email: [email protected] Y. Q.: