SL2 tilting modules in the mixed case

Using the non-semisimple Temperley-Lieb calculus, we study the additive and monoidal structure of the category of tilting modules for $\mathrm{SL}_{2}$ in the mixed case. This simultaneously generalizes the semisimple situation, the case of the complex quantum group at a root of unity, and the algebraic group case in positive characteristic. We describe character formulas and give a presentation of the category of tilting modules as an additive category via a quiver with relations. Turning to the monoidal structure, we describe fusion rules and obtain an explicit recursive description of the appropriate analog of Jones-Wenzl projectors. We also discuss certain theta values, the tensor ideals, mixed Verlinde quotients and the non-degeneracy of the braiding.


Introduction
Let k be a field of characteristic p, fix a non-zero element q ∈ k * , and let K be an algebraically closed field containing k. Tilting modules for SL 2 , the reductive group SL 2 (K) if q = ±1 or Lusztig's divided power quantum group for sl 2 if q = ±1, are among the most well-studied objects in representation theory. In this paper, we use diagrammatic methods to study monoidal categories of tilting modules in the mixed case, i.e. for arbitrary (k, q). As a modern day perspective, the mixed case can be thought of as the culmination of the following cases: (a) The semisimple case, where e.g. k = C and q = 1. (b) The complex quantum group case (at a root of unity), where e.g. k = C and q 2 = 1. (c) The characteristic p case, where e.g. k = F p and q = 1. (d) The (strictly) mixed case, where e.g. k = F p and q 2 = 1.
Tilting modules form a monoidal category, so one can ask questions concerning objects, morphisms, and how these behave under the tensor product. Concentrating on objects and their characters is the classical approach in representation theory. Recently, the focus has shifted to understanding morphisms between tilting modules, especially from a monoidal perspective, which has been driven by work from quantum topology and categorification. A more thorough understanding of the associated diagrammatic and combinatorial model that underpins the behavior of these tilting modules, known as the Temperley-Lieb category, was a key ingredient in recent progress.
In this paper, we let Tilt k,q for arbitrary (k, q) denote the monoidal category obtained by idempotent completion from the Temperley-Lieb category TL k,q , see Remark 2.22. We study Tilt k,q with a focus on the behavior of objects and morphisms with respect to its monoidal structure, a natural progression of previous work [TW19] and [TW20]. The main results of this paper are contained within Sections 3 to 5 and can be summarized as follows.
In Section 3B we define mixed JW projectors E v−1 in TL k,q for v ∈ N and show that they correspond to indecomposable tilting modules T(v − 1) of highest weight v − 1. These idempotents have been constructed independently in [MS21] and they are a simultaneous generalization of the classical Jones-Wenzl projectors [Jo83], [We87], the projectors of Goodman-Wenzl [GW93], and the pJones-Wenzl projectors of Burrull-Libedinsky-Sentinelli [BLS19].
In Section 3C we study morphisms between mixed JW projectors in TL k,q and obtain a presentation of Tilt k,q as an additive category by generators and relations. Specifically, we exhibit Tilt k,q as the category of projective modules for the path algebra of a quiver with relations explicitly described in Theorem 3.25, which can be interpreted as the (semi-infinite) Ringel dual of SL 2 .
In Section 4 we turn to the monoidal structure and study fusion rules for Tilt k,q and their categorified analogs in TL k,q . Classically, fusion rules express the structure constants for the representation ring, i.e. the decomposition multiplicities of tensor products of modules, such as T(v) ⊗ T(w), into indecomposable modules. On the categorified level one is interested in explicitly describing the projection and inclusion maps realizing such decompositions. In the Temperley-Lieb context this means decomposing the tensor products E v ⊗ E w into idempotents that project onto the indecomposable summands predicted by the fusion rule.
A famous example is the recursion for the classical Jones-Wenzl projectors v−1 which witnesses the decomposition T(v − 1) ⊗ T(1) ∼ = T(v) ⊕ T(v − 2) that holds whenever all involved tilting modules are simple. In fact, the Jones-Wenzl recursion (1-1) is often taken as (part of) the definition of the Jones-Wenzl projectors.
In Theorem 4.15 we establish decompositions analogous to (1-1) in the mixed setting of TL k,q . These provide a recursive description of the mixed JW projectors, which appear to be new in this generality, even new when specialized to the positive characteristic or complex quantum group cases, cf. [BDRM19]. As an example, we show an instance going beyond (1-1), which witnesses a decomposition T(v − 1) ⊗ T(1) ∼ = T(v) ⊕ T(v − 2) with summands that need not be simple: (1-2) (Here and throughout the paper we use colored boxes to encode mixed JW projectors corresponding to tilting modules that need not be simple.) The middle part of the rightmost diagram in (1-2) corresponds to a nilpotent endomorphism of T(v − 2). In particular, if T(v − 2) is simple, then the rightmost diagram is zero and we recover (1-1). In general, however, the decompositions provided by Theorem 4.15 are more complex than suggested by the example (1-2). In particular, arbitrary many summands can appear, with multiplicities up to two.
In Section 4C we consider the more general problem of splitting the tensor product of projectors E w−1 ⊗ E x−1 , the first step towards a non-semisimple version of recoupling theory in the sense of Kauffman-Lins [KL94]. Realizing that a solution to this problem is well beyond current reach, we restrict to the case when the tilting modules corresponding to both factors are simple. In Theorem 4.32 we give a complete description of the splitting in certain cases, including the complex quantum group case. Along the way we obtain results that may be of independent interest, including Theorem 4.27 which computes the non-semisimple analogs of theta values: [0] q = 0, [1] q = 1 and δ[n] q = [n + 1] q + [n − 1] q . On the side of tilting module, however, we need δ = q + q −1 , so we decided to use the more standard definition of quantum numbers from the start. We call mchar(k, q) = (p, ) the mixed characteristic, while p and , respectively, are called the characteristic and the quantum characteristic (of the pair (k, q)).
Note that p is a prime number, if finite, but can be any element in N 2 ∪ {∞}. Moreover, for finite the equation [ ] q = (1−q − )(q +1) q−q −1 = 0 implies that q = ±1. Conversely, the order n = ord(q) of the root of unity q, if finite, determines and the signs q and (−q) as follows: The signs q and (−q) will appear in (3-10).

Example 2.3
The examples for (k, q) that the reader should keep in mind are: (a) The integral case, where the pair is Z[v ±1 ], v . Beware that here k is not a field, and we will always treat this case separately. (d) The characteristic p case, where p = < ∞, e.g. (F p , 1) or (F p k , 1). As a word of warning, when we refer to the characteristic p case we always mean q = 1 and not q = −1 since the signs q and (−q) for these two cases are different.
(e) The (strictly) mixed cases are all other cases, i.e. p < ∞, < ∞ with p = . An explicit example is the pair (F 7 , 2) for which the mixed characteristic is (7, 3).
For the rest of this paper, with the exception of concrete examples, we fix a pair (k, q) of mixed characteristic (p, ). The numbers p and will play a crucial role in this paper, e.g. via p -adic expansions: Definition 2.4 Set p (0) = 1, and for i ∈ N let p (i) = p i−1 . For any v ∈ N we write [a j , ..., a 0 ] p, = j i=0 a i p (i) = v with a j = 0. The digits are from the sets a i ∈ {0, ..., p − 1} for i > 0, and a 0 ∈ {0, ..., − 1}. The higher digits are declared to be zero a >j = 0.
The p -adic expansion of a natural number v as defined above is clearly unique: a 0 is uniquely determined as the remainder of v upon division by , and the remaining digits [a j , ..., a 1 ] p are determined by the usual p-adic expansion of the quotient v−a 0 . We also point out that the two digits a j , called the leading digit, and a 0 , the zeroth digit, will play slightly different roles than the other digits.
Remark 2.6 As we will see, p is the crucial number for all digits a i with i > 0, while appears only in connection with the zeroth digit. We will henceforth write p∨ for either p or , depending on the involved digit.
Remark 2. 7 We will repeatedly encounter the law of small primes, losp for short: we see special behavior in cases when relevant digits are (close to) 0 modulo p∨ . For large characteristics such cases are exceptions, while for small ones they are the rule.
For any x ∈ k we will also use quantum factorials and binomials: Here a ∈ N and b ∈ Z, and all of these are elements of k. Note that b a 1 = b a . We recall the quantum Lucas' theorem: Proof. This is folklore, but the first written account might be [Oli65, (1.2.4)]. (Translating from that paper, which uses a non-symmetric definition of quantum numbers often used in combinatorics, to our conventions gives the sign .) Another useful tool for quantum calculations is using a change of variables method: [w]q is well-defined (and non-zero if a k = 0). To see The following is taken from [TW19], but for p -adic expansions.

Definition 2.11
If v = [a j , ..., a 0 ] p, ∈ N has only a single non-zero digit, then v is called an eve. The set of eves is denoted by Eve. If v / ∈ Eve, then the mother m v of v is obtained by setting the rightmost non-zero digit of v to zero.
Assume that v / ∈ Eve has k non-zero, non-leading digits. We will also consider the set ancestors of v, whose size gen v is called the generation of v. By convention, A(v) = ∅ and gen v = 0 for v ∈ Eve. The support ∇supp(v) ⊂ N is the set of the 2 gen v integers of the form w = [a j , ±a j−1 , ..., ±a 0 ] p, .
Note that every v / ∈ Eve has an associated eve m ∞ v . We think of the generation and the ancestry chart as a measure of the complexity of the associated SL 2 modules. For example, in Proposition 3.3 we will see that a tilting module is simple if and only if its ρ-shifted highest weight is an eve.

Example 2.12
In the semisimple case = ∞ every v ∈ N 0 is an eve and has no ancestors. In the complex quantum group case p = ∞, < ∞ every v ∈ N 0 is either an eve or of generation 1. In the other cases the generation can be any number in N 0 . The elements w in the support ∇supp(v) of v ∈ N can be described by the sets of indices of digits of v, which are negated (or "reflected") to obtain an expression for w. To obtain a bijection between elements in ∇supp(v) and sets of indices, we enforce certain admissibility conditions on the latter: Definition 2.13 For S ⊂ N 0 a finite set, we consider partitions S = i S i of S into subsets S i of consecutive integers that we call stretches. For the rest of the definition, we let S = i S i be the coarsest such partition into stretches.
The set S is called down-admissible for v = [a j , ..., a 0 ] p, if: (i) a min(S i ) = 0 for every i, and (ii) if s ∈ S and a s+1 = 0, then s + 1 ∈ S.
Conversely, S is up-admissible for v = [a j , ..., a 0 ] p, if the following conditions are satisfied: (i) a min(S i ) = 0 for every i, and (ii) if s ∈ S and a s+1 = p − 1, then we also have s + 1 ∈ S.
where we extend the digits of v by a h = 0 for h > j if necessary, and r(S) is the biggest integer such that a k = 0. Any down-or up-admissible set S has a unique finest partition into down-or up-admissible sets, each of which consist of consecutive integers and which we call minimal down-respectively up-admissible stretches.
Very often (unless losp applies), the minimal stretches will just be singleton sets {i} specifying a single digit in which we reflect. We also tend to omit the set brackets of down-or up-admissible sets if no confusion can arise, e.g.
For 2B. Tilting modules and their diagrams. Let k ⊂ K denote an algebraically closed field containing k. We use the symbol SL 2 to denote the reductive group SL 2 over K if q = ±1 ∈ K and Lusztig's divided power quantum group (using the conventions from [APW91]) associated to sl 2 for other values of q. We will identify dominant integral weights of SL 2 with N 0 and weights with Z in the usual way. We consider finite-dimensional (left) SL 2 -modules of type 1 over K. These form an abelian, K-linear category fdMod K,q = SL 2 -fdMod K,q , for which we additionally choose a monoidal and a pivotal structure using the comultiplication SL 2 , and the antipode of SL 2 and the analog of the involution ω from [Jan96, Lemma 4.6]. The category fdMod K,q contains four families of highest weight modules of particular interest for our purpose, all parameterized by N 0 .

Remark 2.17
Here and in the following, we write the highest weights of these modules often as v − 1 for v ∈ N. This puts an emphasis on the quantity v, the ρ-shifted highest weight, which will play a greater role than the highest weight itself.
The first two families are formed by the Weyl modules ∆(v − 1) and the dual Weyl modules ∇(v − 1). These do not depend on the mixed characteristic in the sense that they can be defined The other two families of modules are formed by the simple modules L(v − 1) and the indecomposable tilting modules T(v − 1). These modules do not admit a construction independent of the mixed characteristic. Their characters are given by Proposition 3.3 below.
Let Tilt K,q = SL 2 -Tilt K,q be the full subcategory of fdMod K,q whose objects are direct sums of T(v − 1) for v ∈ N. We also write T(z) for z < 0 which is zero, by convention. The category Tilt K,q is additive, idempotent closed, Krull-Schmidt (meaning there is a essential unique decompositions into indecomposables, and an object is indecomposable if and only if its endomorphism ring is local), K-linear, and pivotal (restricting the structures from fdMod K,q to Tilt K,q ). It is the main object under study in this paper and called the category of tilting modules of SL 2 .
Remark 2.18 Classically Tilt K,q would be defined as the full subcategory of fdMod K,q whose objects have Weyl and dual Weyl filtrations, and its closure under tensor product would be a theorem. The above definition is equivalent to the classical one for SL 2 , because the sole fundamental representation (it is T(1)) is tilting and thus all indecomposable tiltings appear as direct summands of tensor powers thereof. This may fail for other types in small characteristic.
Generally these four types of modules (Weyl, dual Weyl, simple, and indecomposable tilting) for a fixed highest weight are distinct from one another. If, however, two are isomorphic e.g.
, then it follows that all four types of modules of the same highest weight are isomorphic. An example is which is the monoidal unit of Tilt K,q and which we denote by 1.

Remark 2.19
Let us comment on the references for the above and some of the material below, using the terminology from Example 2.3. In the semisimple case, Tilt K,q is equivalent to fdMod K,q , is semisimple and has the classical combinatorics of SL 2 (C), which is covered in many textbooks. Otherwise Tilt K,q is non-semisimple and we refer to [APW91] and [And92] in the complex quantum group case, to [Rin91] and [Don93] in the characteristic p case, and to [AK92] as well as [Don98] and [And19] in the mixed case. A summary for tilting modules can also be found in [AST15].
The diagrammatic incarnation of Tilt K,q is sometimes called the Temperley-Lieb category (abbreviated to TL category) and can be defined as follows. Let TL Z[v ±1 ],v denote the Z[v ±1 ]linear category with objects indexed by m ∈ N 0 , and with morphisms from m to n being Z[v ±1 ]-linear combinations of unoriented string diagrams drawn in a horizontal strip R × [0, 1] between m marked points on the lower boundary R × {0} and n marked points on the upper boundary R × {1}, considered up to planar isotopy relative to the boundary and the relation that a circle evaluates to −[2] v . The category TL Z[v ±1 ],v is (strict) monoidal with ⊗ given by horizontal concatenation and admits a (strict) pivotal structure given by the duality maps being cups and caps, and all objects being self-dual.
We write FG = F • G for the composition of morphisms in TL Z[v ±1 ],v , and we read diagrams from bottom to top and left to right, e.g.
] k be the scalar extension and specialization Z[v ±1 ] v → q ∈ k. Recall that K denotes an algebraically closed field containing k. Recall also that T(1) generates Tilt K,q as a monoidal category.

Proposition 2.20
We have a K-linear, pivotal functor which induces an equivalence of K-linear, pivotal categories upon additive idempotent completion.
Proof. This is folklore, the semisimple case dates back to [RTW32], and a proof in general can be found in e. (2-1) There is also a braiding on Tilt K,q , assuming that q has a square root in K, given by the so-called R-matrix, see e.g. [Kas95, Section IX.7]. (We clear the denominators in these formulas by using divided powers, and observe that the expression is well-defined on all finite-dimensional modules without further adjustments.) These two braidings, the only ones we will consider in this paper, are compatible, as can be seen, e.g. by comparing on generating objects: Proposition 2.21 If q has a square root in K, then the functor K,q from Proposition 2.20 is an equivalence of braided categories.

Remark 2.22
Note that Proposition 2.20 allows us to identify the additive Karoubi closure of TL K,q with Tilt K,q . Motivated by this, we will also denote the additive Karoubi closure of TL k,q by Tilt k,q in case of the not necessary algebraically closed field k, and the objects T(v − 1) ∈ Tilt k,q are defined as the images of primitive idempotents under k,q . In this notation the functor k,q will be the universal embedding of TL k,q into its Karoubi closure, and we omit it from the notation.

Remark 2.23
At this point, the reader be warned that Tilt k,q may not be equivalent to the category of tilting modules over k (as defined via (dual) Weyl filtrations), even in semisimple cases, if k is finite, see [BD09, Section 5].
In the semisimple situation of TL k(v),v , the primitive idempotents that are mapped to the indecomposable tilting modules T(v − 1) in T(1) ⊗(v−1) are the well-known Jones-Wenzl projectors. Since T(v−1) is a simple module in this case, we will call these idempotents simple Jones-Wenzl projectors (simple JW projectors for short), also to distinguish them from their non-simple analogs. All we need to know about these projectors is summarized in the following proposition, see e.g. [KL94] for a proof.

Proposition 2.24
For all v ∈ N there exists a unique idempotentẽ v−1 ∈ End TL k(v),v (v − 1), which is invariant under duality (ẽ v−1 ) = (ẽ v−1 ) ↔ =ẽ v−1 (this implies that the following relations hold mirrored as well) and which satisfies: Here we use the usual box notation for these projectors, a number k next to a strand means k parallel strands, and the projectorẽ w−1 in (2-2) respectively the cup or cap in (2-3) can be at arbitrary positions. The idempotentẽ v−1 satisfies the recursion in (1-1) In Definition 2.25 and Convention 2.26 we will define various different bases of morphism spaces in Temperley-Lieb categories. The first example are the integral bases given by sets of crossingless matchings (a.k.a. Temperley-Lieb diagrams) B int v−1,w−1 of v + w − 2 points. These are integral in the sense that they provide isomorphisms Hom ,v a basis of the form B v v−1,w−1 is an Artin-Wedderburn basis since these summands are simple.) We stress that these bases are not unique unless one specifies further properties that these should satisfy. The existence of these bases follows from abstract theory, see [AST18], and all of these are cellular and related by unitriangular basis change matrices. To construct these bases explicitly we can use the light ladder strategy. (The terminology is borrowed from [Eli15] and, in fact, all of the bases from [AST18] are of light ladder type. The light ladder strategy can be seen as a general philosophy how to construct bases.)

Convention 2.26
We will use the light ladder strategy from Definition 2.25 in several different contexts. The associated down and up morphisms are consistently distinguished throughout this paper by the following notation convention.
(a) For G v−1 = id v−1 , which works for any ground ring (in particular for Z[v ±1 ], v ), we obtain the integral bases B int v−1,w−1 for morphism spaces. We reserve the following notation for these morphisms: (b) For G v−1 =ẽ v−1 and working over k(v), v we get the Artin-Wedderburn basis. The associated morphisms will be denoted with tilde symbols: (c) For G v−1 = E v−1 , i.e. for the projectors constructed in Section 3B for non-semisimple situations, we will use capital letters: These are specializations of morphisms that one gets for G v−1 = e v−1 , and we will use an overline in this situation:

Definition 2.27 A family of morphisms
and right-aligned if We draw morphisms from a left-aligned family as boxes with a bar at the left-hand side, and vice versa for right-aligned. Using this notation the two conditions in Definition 2.27 read: Note that left-and right-aligned families of morphisms are always idempotents, by the v = w case of the defining relation.

Remark 2.28
The families of identity morphisms id v−1 are both left-and right-aligned and so are simple JW projectors by (2-2). However, in the mixed case the corresponding projectors E v−1 form a family that is only left-aligned, see Example 3.12. (Of course, there are also right-aligned versions (E v−1 ) ↔ .) This asymmetry will play an important role within our setup. For example, in Definition 2.25 we presented a version of the light ladders strategy that favors left-aligned families of projectors and when discussing fusion rules for morphisms this will play an important roles.

Example 2.29
There are six paths of length four which stay in the positive Weyl chamber, and six corresponding down morphisms (simplified by using (2-2)): (Here we chose to omit drawing zero boxes.) Flipping these pictures bottom to top, and stacking them together whenever that makes sense, one obtains the Artin-Wedderburn basis of End TL k(v),v (4). Using identities instead of the simple JW projectors gives the integral basis of ,v (4), while using the projectors from Definition 3.11 below gives the non-semisimple projector basis of End TL k,q (4).
The (defining) recursion of the simple JW projectors recalled in (1-1) is an expression of the classical fusion rules of SL 2 and it uses morphisms of light ladder type. To define the non-semisimple projectors we use a different approach. But we will see in Section 4B that the non-semisimple projectors still admit a recursive description in terms of morphisms of light ladder type, using the non-semisimple fusion rules in Proposition 4.7.

Additive structure
In this section, we explain the additive structure of the category of tilting modules. Some of results in this section are well-known, others generalize the results from [BLS19] and [TW19]. We also added a few new observations.
3A. Character formulas. The Weyl and dual Weyl modules have classical Weyl characters, i.e.
Each T(v − 1) has a (dual) Weyl filtration and we denote the (dual) Weyl multiplicities by

Remark 3.1
The purpose of the admissibility conditions on finite sets S ⊂ N 0 from Definition 2.13 is so that for v ∈ N we have bijections Moreover, each (dual) Weyl module has a filtration by simple modules, and we denote the corresponding simple multiplicities by ∆(v − 1) : L(w − 1) = ∇(v − 1) : L(w − 1) . These have a similar description as the Weyl multiplicities: One can check that ∇supp(v) is always of order 2 gen v , while Lsupp(v) can be of other sizes as soon as losp applies.
(a) We have Thus, the tilting characters are while the simple characters can be obtained by inverting We have a version of (Brauer-Humphreys or BGG) reciprocity, i.e. if a i = 0, p − 1 for all j > i > 0, then In particular, Proof. The Weyl multiplicities are known, see [Don98, Section 3.4] for the potentially first written account in the mixed case. The simple multiplicities can be obtained by direct calculation using the simple characters in (3-2) below. The reciprocity follows immediately from these.
See Figure 1 for an illustration of T(v − 1) : ∆(w − 1) (compare to [JW17, Figure 1]), and Figure 2 for an illustration of ∆(v − 1) : L(w − 1) . Note that comparing these four illustrations shows why reciprocity only holds up to a certain point. Two remarkable results describing the structure of objects in fdMod K,q are Donkin's (3-1) and Steinberg's (3-2) tensor product formulas, which we recall in the following proposition. Both formulas describe modules of highest weight v − 1 in terms of tensor products of Frobenius-Lusztig twist of modules of lower weight, following the p -adic expansion of v. The ith Frobenius-Lusztig twist will be denoted by ( − ) p (i) . It acts as the Frobenius twist on digits a i for i > 0 and as its quantum analog on the zeroth digit. Furthermore, we will accompany the two famous tensor product formulas with a third one. To this end, we note that we can naively apply ( − ) a i p (i) to weight spaces, although we loose the module structure for a i = 1.

Proposition 3.4 Let
(a) We have where the monoidal product runs over all non-leading digits of v. Thus, where the monoidal product runs over all (non- We have an isomorphism of Z-graded vector spaces where the monoidal product runs over all non-zero and non-leading digits of v. Thus, can be realized as an isomorphism of SL 2 -modules if all non-zero digits a i are equal to 1. In this case T(v − 1) is a tensor product of simple modules.
Proof. For the tensor product formulas (3-1) and (3-2) see [And19, Proposition 5.2] (to be precise, the above is [TW20, Proposition 4.7] adjusted to mixed characteristic) and [AK92, Theorem 1.10] for the mixed versions. We will give a diagrammatic proof of the (apparently new) character formula in (c) in Proposition 3.23 below. For the final statement, by (c), it suffices to observe that a i =0 T(1) (a i p (i) ) is simple by (3-2), which implies that the right-hand side of (3-3) is tilting by the mixed characteristic analog of [BEO20, Lemma 3.3].   Note that (d) of Proposition 3.4 implies a remarkable appearance of losp: Corollary 3.6 All indecomposable tilting modules are tensor products of simple modules in characteristic p = 2.
The tilting modules T(v − 1) for v ∈ Eve will also be called eves. By Proposition 3.3 these are the only simple tilting modules, i.e. T(v − 1) ∼ = L(w − 1) if and only of v = w ∈ Eve. The prime eves are those where v = p (i) , and they play special role, see e.g. Theorem 5.1.
3B. Non-semisimple projectors. In order to define the projectors E v−1 , we need a few notions. Crucial will be certain down and up morphisms that are defined in the same spirit as those in Definition 2.25, but with an emphasis on good compositional properties.

Definition 3.7 Fix a left-aligned family of morphisms
Then we define morphisms in TL k,q as follows.
The box represents the morphisms G v −1 , and we will consider the three variations with their corresponding notation (meaning For the final equation we have used that the morphisms G v−1 form a left-aligned family. The corresponding upwards version of these morphisms are defined by υ S = δ S . We will also use the case S = ∅ for which all involved operations are identities.
For simplicity of notation we often only indicate the number of strands at the beginning or end of a composite of such morphisms, since the other numbers are then determined.
Definition 3.8 Suppose that S = {s k > · · · > s 1 > s 0 } is down-admissible for v and S = {s l > · · · > s 1 > s 0 } is up-admissible for v. Then we define simple trapezes and loops , which we also define for the other three variations from Convention 2.26, with the appropriate adjustment of notation.
Note that in all cases loops carry an idempotent G v[S]−1 in the center and down and up morphisms carry this idempotent on their thin end.
Remark 3.9 Using Convention 2.26, we can give an alternative description of the simple trapezes. If S is down-admissible for v = [a j , ..., a 0 ] p, , then we define a sign sequence We get S =d(π S (v)), S =ũ(π S (v)).
Note the subtle, but important difference thatd(π S (v)) includes an idempotentẽ v[S]−1 on the left, whiled S does not. As a consequence, composites of morphisms of typed(π) are automatically zero, while the morphisms of typed S can be composed in interesting ways. Remarkably, this distinction disappears when considering analogs of such morphisms built from mixed projectors, see Proposition 3.22.

Example 3.10
For v = [a, b, c] p, we have:

Definition 3.11 For
The choice of name for e v−1 is because the associated tilting module is a direct sum of simple tilting modules, i.e.
Example 3.12 By construction, (e v−1 ) = e v−1 . However, (e v−1 ) ↔ = e v−1 in general (we will address this in Lemma 3.16), as can be seen by which is an example in characteristic p = 3.

Remark 3.13
Further concrete examples for projector expansions (3-4) can be found in [TW19, Examples 2.20 and 2.23]. Relative to the treatment there, we allow the following two generalizations. First, the p-adic expansions should be replaced by the p -adic expansions, and p k therein by p (k) . Second, all coefficients use quantum numbers instead of integers.
The following is reproduced from [TW19, Section 3B], adjusting the scalars.

Lemma 3.15 The semisimple p JW projectors can be expanded as
where a s is the first non-zero digit of v. As a consequence, for any ancestor m j v of v, we have Proof. With the properties listed in Lemma 3.14, the proof follows verbatim as in [TW19, Lemma 2.24] and [BLS19, Proposition 3.3].
The next statement of this section enables us to related the left and right version of the p JW projectors. Let v = [a j , ..., a 0 ] p, , as usual, and let S v−1 denote the symmetric group on v − 1 letters. Assuming the existence of square roots, we can use (2-1) to define g = g(v − 1) to be the positive braid lift of the longest element of S v−1 (the positive half twist, a Garside element), and r = r(a j , ..., a 0 ) to be the positive braid lift of a shortest coset representative for for an example which fixes conventions.

Lemma 3.16
Assume that v has a square root in k(v), we have Proof. Using the integral basis B int v−1,w−1 of crossingless matchings, it is easy to see that the conjugation of f ∈ B int v−1,w−1 with the half twist g is precisely the operation ( − ) . This implies that the left and the right semisimple p JW projectors are related by conjugation with g.
Next, we use that JW projectors absorb crossings up to scalars, i.e. v−1 which are direct consequences of (2-1) and (2-3). Using this, we see that crossings of the half twist g are absorbed up to a scalar unless they reorder the bundles of strands corresponding to the digits of v. After removing all such crossings, the remaining braid is r. (To see this use (3-6) inductively, the case of v ∈ Eve being clear.) By symmetry, the same works for g −1 , but with exactly the opposite scalars, so the scalars balance out.
Example 3.17 Continuing Example 3.12 we have By coefficients of a morphism in Hom TL k(v),v (v − 1, w − 1) we always mean with respect to the integral basis. Let F p be the prime field of k, with F ∞ = Q. The question we would like to address next is whether the denominators of these coefficients are non-zero after specialization to (k, q). With respect to the below definition the example to keep in mind is: Thus, when specializing to (F 7 , 2) the quantum number [63] v vanishes of order 3, but for (C, exp(πi/3)) it only vanishes of order 1.
The semisimple p JW projectors e v−1 are defined over F p (v), but the algorithm to construct them generates coefficient that we can view as elements of Q(v), which we will below.
The p -adic valuation ν p, : Moreover, for a morphism D ∈ Hom TL Q(v),v (v − 1, w − 1) we let ν p, (D) be the entry-wise minimum of the p -adic valuations of the coefficients of D when expressed in terms of the integral basis of Hom . Note that ν p, (D) 0 implies that we can specialize to (k, q) and obtain a well-defined morphism sp p, (D) ∈ Hom TL k,q (v − 1, w − 1).
Theorem 3. 19 We have ν p, (e v−1 ) 0. Thus, is a well-defined idempotent whose coefficients are elements of In particular, under the equivalence induced by K,q , see Proposition 2.20, the image of the Proof. Note that e v−1 has the correct character, namely is an idempotent and it absorbs the tensor product e mv−1 ⊗id v−mv of the idempotent for the mother with extra strands. Now we claim there is exactly one idempotent in End Tilt k(v),v T(1) ⊗(v−1) with this property and the correct character. To see that this is true let us denote by T(m v − 1) ∈ Tilt k(v),v the direct sum of Weyl modules with the correct character.
,v contains each Weyl factor of T(v − 1) exactly once, see Lemma 4.5, so there is exactly one idempotent in End Tilt k(v),v T(1) ⊗(v−1) with the correct character and absorption property, and the claim follows. Now let L be the localization of k[v] at the ideal (v − q). In other words, the elements of L are rational functions in v whose denominators do not have a zero at q. Note that there are specialization maps and functors To prove ν p, (e v−1 ) 0, we show that e v−1 can be lifted to End Tilt L,q T(1) ⊗(v−1) and its specialization to k projects to T(v − 1). To this end, we use induction over the ancestry of v, with the case of v ∈ Eve being clear since T(v − 1) ∼ = ∆(v − 1) in these cases. So let e mv be liftable and let l v →v (e mv ) be its lift. Induction implies .
projecting to this summand, which absorbs the idempotent corresponding to the mother tensor product with strands. By idempotent lifting, cf.
[Lam01, Theorem 21.31], we can pull E back to Tilt L,q giving us another projector l v →q (E ).
Pushing this forward gives a projector F v →v l v →q (E ) in the semisimple case with the correct character and absorption property. However, as we have seen, such a projector is unique and thus, has to be e v−1 . Hence, we get ν p, (e v−1 ) 0 and l v →q (E ) is a lift of e v−1 . Thus, we can specialize e v−1 to E v−1 = E , and the claims about the coefficients and E v−1 = id T(v−1) follow by construction of e v−1 . The Krull-Schmidt property then follows inductively as the above constructs all highest weight projectors.

Remark 3.20
The crucial ingredient in the proof of Theorem 3.19 and, even more fundamentally, the construction of the projectors e v−1 are the tilting characters χ T(v−1) , a certain numerical datum. There are two other ways to get equivalent numerical data: First, one could use the Soergel category for affine type A 1 and the p-Kazhdan-Lusztig basis as in [BLS19]. A second method is to calculate the simple multiplicities within the projective cover of the trivial Temperley-Lieb module, as done in [MS21] (which is a follow-up of [Spe20] where the decomposition numbers of the Temperley-Lieb algebra are computed). However, it might be that only the tilting characters approach generalizes beyond SL 2 , e.g. see [Soe97] and [Soe98] for the complex quantum group case, very explicitly worked out in [Str97]. The same method as above should cover this case, so we decided to spell out the argument above.
Diagrammatically, the three types of projectors are distinguished as follows: (3-8) The middle and the rightmost projector have the same character, but e v−1 corresponds to a direct sum of simple tilting modules in the semisimple setting, cf. (3-5), and E v−1 corresponds to the indecomposable T(v − 1). We will use the middle projectors to deduce properties of the right projectors. Moreover, as illustrated in (3-8), we also use white boxes for eves to indicate that these satisfy the same diagrammatic properties as the simple JW projectors.

Remark 3.21
We warn the reader that the projectorsẽ v−1 descend to well-defined morphisms in TL k,q if and only if v ∈ Eve. However, scalar multiplies ofẽ v−1 may descend to TL k,q even if v / ∈ Eve. For example, in characteristic p = 2 the projectorẽ 2 is not well-defined but 2 · 2 = 2 · + 1 2 · = 2 · + can be seen as a morphism in TL Z[v ±1 ],v and specializes to a well-defined and non-zero morphism in TL k,q (indeed, a nilpotent endomorphism of E 2 ). So the white boxes need to be treated with care whenever one works in (k, q).
We stress again that the non-semisimple projectors do not have a left-right-symmetry, and their properties do not have such a symmetry either. For the remainder of the paper, each cup and cap in the illustrations is a parallel bundle of cups and caps, depending on S respectively S , or a plain number. (We also omit to illustrate these if no confusion can arise.) We have the following generalizations of Proposition 2.24, called classical absorption, nonclassical absorption, shortening and partial trace.
where the small box is labeled by a v,S −1 for a v,S being the youngest ancestor of v for which all digits indexed by elements of S are zero.
where the product runs over all non-zero digits of w, and m (A special case of this is the trace down to the empty diagram.) Proof. All except the final statement can be shown as in [TW19, Propositions 3.11, 3.13 and 3.14]. The final statement follows by using (2-4) and observing that the projector after taking partial trace satisfiesẽ v−k = e v−k and the zero obtained by (2-4) annihilates it.
The projectors e v−1 typically do not form a right-aligned family. Example 3.12 gives a counterexample to right-aligned absorption of id 1 ⊗ e 2 into e 3 .
Recall the definition of the categorical dimension dim C of objects in a pivotal category C.
, where the product runs over all non-zero and non-leading digits of v.
Proof. The categorical dimension in TL k,q is given by closing pictures in the usual way, and for the first equality we calculate Note that the categorical dimension of T(v − 1) is an element of the underlying field, but interpreted in N 0 [v ±1 ] we obtain the character χ T(v−1) .
3C. Tilting modules as an additive category. Let us define a (locally unital) k-algebra via Let Proj-Z k,q denote the category of finitely generated, projective (right) Z k,q -modules. By construction we obtain, as instance of Ringel duality (semi-infinite in the sense of [BS18]), that is an equivalence of additive, k-linear categories, sending indecomposable tiltings to indecomposable projectives. Let us describe Z k,q explicitly.
By construction, morphisms in Hom Tilt k,q T(v − 1), T(w − 1) are given by flanking TL morphisms with E v−1 from the bottom and with E w−1 from the top, and the primitive idempotents (which are local units) are the E v−1 for v ∈ N. Other morphisms, called mixed trapezes and loops, are diagrammatically given by the analog of Definition 3.8: if S and S be down-and up-admissible for v, respectively, and assume that S and S are minimal admissible stretches of consecutive integers then we define where the last equality uses Proposition 3.22.(b). These are the generators of Z k,q , and (up to losp) the respective minimal stretches are singleton sets S = {i}, reflecting along the ith digit. The corresponding S-labeled cups and caps in (3-9) consist of a i p (i) parallel strands. Finally, note that these morphisms can be defined more generally for any down-and up-admissible stretches, but then their diagrammatic incarnation can involve multiple stretch-labeled cups and caps.
To describe the relations between expressions in the generating morphisms, we will use the same scalars (depending on the digits) as in [TW19, Section 3A], namely (3-10) In fact, as we will see later, these scalars can be seen as (inverses of higher order) local intersection forms in the language of [Eli15].

Remark 3.24
The quantum version of f and g given in (4-8) will be relevant in Section 4B, but we do not need quantum numbers to describe f and g for Theorem 3.25. By Proposition 2.8 this is expected: these functions will never be evaluated on the zeroth digit and the quantum Lucas' theorem implies that, up to a sign, the only relevant digit for quantum numbers is the zeroth digit.
We obtain the mixed characteristic version of [TW19, Theorem 3.2]: Theorem 3.25 The algebra Z k,q is generated by E v−1 for v ∈ N, and elements D S E v−1 and U S E v−1 , where S and S denote minimal down-and up-admissible stretches for v, respectively. These generators are subject to the following complete set of relations. (As before, we omit idempotents from the notation if they can be recovered from the given data.) (1) Idempotents.
Here, if the down-admissible hull S, or the smallest minimal down-admissible stretch T with T > S does not exist, then the involved symbols are zero by definition. The elements of the form Any word E w−1 XE v−1 in the generators of Z k,q can be rewritten as a linear combination of basis elements from (Basis) using only the above relations.
Proof. This is analogous to the ten page proof of the characteristic p case in [TW19], but the proofs given therein need some adjustment due to e.g. the appearance of signs in f and g from (3-10). We record the necessary modification to the numerical arguments used in [TW19].
First of all, the scalars λ v,S and partial trace formulas for the various JW projectors now involve fractions of quantum numbers. Moreover, a few signs that have started their lives as −1 = (−1) p i now have to replaced by (−1) p (i) = (−1) when i > 0. This concerns the sign of the fraction of quantum numbers in [TW19, (4-2)] (this replacement leads to the desired interpretation in terms of g), the sign in q from [TW19, (4-8)] should be (−1) p (s) , which balances against the sign of λ w,R in the following display. Further, in the Proof, which caveat for [TW19, Lemma 4.9], the signs (−1) w−u and (−1) w+1−u are to be replaced with (−q) w−u and (−q) w+p (i) −u , which is again compatible with f and g as desired. The vanishing of q from [TW19, (4-4)] follows using a similar argument using quantum numbers. Finally, the zigzag relation are established by an inductive argument based on the case of generation 2, which is proved exactly as outlined in [TW19, Lemma 4.8].
Note that the non-idempotent generators of Z k,q are given by down and up morphisms for minimal stretches, a.k.a. singleton sets if we ignore losp (we will write e.g. D i instead of D {i} for these to simplify notation). By using the relations, e.g. Theorem 3.25.(4), one obtains down and up morphisms for more general stretches.
Example 3.29 One can show the useful relation that which illustrate containment, far-commutativity and zigzag relations. Indeed, the former two types of relations admit diagrammatic proofs.
The (b) All of these bases are unitriangularly equivalent to the crossingless matching bases (with respect to (N 0 , <)).
(c) All of these bases together with (N 0 , <) and ( − ) endow Tilt k(v),v respectively Tilt k,q with the structure of a (strictly object adapted) cellular category.
Proof. Theorem 3.25 shows that b and B give bases of the respective hom-spaces, and the former is unitriangularly equivalent tob, by construction. Moreover,b is unitriangularly equivalent to the crossingless matching basis, and a basis change that is unitriangular with respect to the cell order preserves all structures defining a (strictly object adapted) cellular category.
The basis changes for gen v = 2 with minimal down-admissible stretches S < T are: Finally, we record a useful consequence of Theorem 3.25.Basis.
Lemma 3.32 Suppose that v, w ∈ N are such that ∇supp(v) ∩ ∇supp(w) = ∅. We have In particular, this holds true if the zeroth digit b 0 of w satisfies b 0 = a 0 and b 0 = − a 0 , or b 0 = a 0 = 2 but p > 2 and the parity of the sum of the remaining digits of v and w is different.
Proof. The first part of the statement is clear by Theorem 3.25.Basis. The first condition for when ∇supp(v) ∩ ∇supp(w) = ∅ is immediate from the definitions as the corresponding p -adic expansions of the elements of ∇supp(v) and ∇supp(w) have to agree on the zeroth digit. For the second condition note that b 0 = a 0 = 2 ensures that every element of ∇supp(v) is distinguished from the elements of ∇supp(w) by its zeroth digit or by the parity of the sum of higher digits (here we use that p is odd).
Note that the conditions given at the end of Lemma 3.32 are in general only sufficient to ensure ∇supp(v) ∩ ∇supp(w) = ∅.

Remark 3.33
The statement in Lemma 3.32 is known as a Weyl factor overlap criterion in the theory of tilting modules and follows from Ext-vanishing, see see e.g. [AST18, Section 2B], using the integrality of these statements which follows from [RH03]. One can see Lemma 3.32 as an explicit incarnation of these (general) facts about tilting modules.
3D. More partial trace formulas. The next lemma deals with partial traces that do not reach an ancestor. Hence, they are complementary to part (c) of Proposition 3.22. v = [a j , ..., a k , 0..., 0] p, with k > 0, and a k = 0. Suppose that w = (p − a i )p (i) for some 1 i < k. Then we have v−1

Lemma 3.34 Let
Proof. We compute using projector absorption and shortening: where S = {i, ..., k − 1} and we have used that S is the minimal down-admissible stretch of v + w. The formula for w and x can be proven verbatim.
We can now use the zigzag relation from Theorem 3.25.(6) to simplify D S U S E x−1 further. To use this relation, recall that if the down-admissible hull S, or the smallest minimal down-admissible stretch T with T > S does not exist, then the involved symbols are zero by definition.

Proposition 3.35 Retain notation as in Lemma 3.34. We have
Moreover, the formula holds for w and x .
Proof. By the zigzag relation from Theorem 3.25.
Note that the f-terms in Proposition 3.35 vanish if a k = 1. On the other hand, the g-terms can be zero only if no admissible hull S exists. Hence, the whole partial trace vanishes if and only if a k = a j = 1, i.e. if and only if v = p (k) is a prime eve.
We will state more partial trace formulas in Theorem 4.15 later on.

Monoidal structure
In this section, we study Tilt k,q as a monoidal category. In the semisimple and complex quantum group cases, the results in this section appear throughout the literature.
4A. Fusion rules. We start by recalling the well-known fusion rules for tilting modules lying in the fundamental alcove, in which tilting modules are simple. Note that in the semisimple case, i.e. when = ∞, the fundamental alcove is the whole of N 0 .

Lemma 4.1 For 1 v, w
and v + w − 2 < we have where, for later use, we indicate the p -adic expansions of the occurring terms. (The first direct sum is empty if max(v, w) = .) Proof. The first part is classical; the second part is easy using Proposition 3.3. The summands with highest weights in the fundamental alcove appear in the first direct sum in (4-2). The second direct sum collects all remaining summands. If v + w and have the same parity, then each of the summands are of generation one, otherwise there exists a simple summand T( − 1). Equation (4-1) is the Clebsch-Gordan rule.

Remark 4.3
The second direct sum displayed in (4-2) does not appear for the Verlinde category (as e.g. in [EGNO15, Example 4.10.6]) because these terms are factored out in that category. For example, in Example 4.2 all tilting modules of weight greater or equal 8 would be zero.
Remark 4.4 Below we will use the following character argument. Over (Z[v ±1 ], v), the SL 2 -module ∆(v − 1) ⊗ ∆(w − 1) always has a Weyl filtration whose Weyl factors are given by (4-1) when = ∞ (in which case T(v − 1) ∼ = ∆(w − 1)). Thus, we can determine the Weyl factors of tensor products of tilting modules by using (4-1) together with the tilting characters in Proposition 3.3. We can further collect the computed Weyl factors into indecomposable tilting modules since we know that tensor products of tilting modules decompose into indecomposable tilting modules. We can perform this process recursively by highest weight, using the characters in Proposition 3.3.
We note the following consequence, used in the proof of Theorem 3.19:

Lemma 4.5 For all v, w ∈ N such that v /
∈ Eve, suppose that T(v − 1) : ∆(w − 1) = 1, then Proof. This follows a character argument: we compute the Weyl factors of T(m v − 1) ⊗ T(1) ⊗(v−mv) by repeatedly raising or lowering the corresponding highest weights of these Weyl factors by ±1 since In other words, we multiply the character of T(m v − 1) as it appears in part (a) of

Definition 4.6
We define the tail-length tl(v) of v = [a j , ..., a 0 ] p, to be the minimal k ∈ N 0 such that a i is maximal for all i < k. If a 0 is not maximal, then tl(v) = 0.
The first fusion rule, beyond the fundamental alcove that we address, involves tensoring with the monoidal generator T(1).

Proposition 4.7
Let v = [a j , ..., a 0 ] p, . We have The proof is given by a character argument and appears below. To warm up, we first comment on the qualitative differences between the cases found in Proposition 4.7. For the sake of exposition, we consider 4. Focusing on the zeroth digit, we see: The generation drop case occurs when the zeroth digit is maximal, in which case additional direct summands may appear. Recall that T(v − 1) has 2 gen v Weyl factors. Under tensoring with T(1) ∼ = ∆(1), most of them produce two new Weyl factors by (4-3). In total, T(v − 1) ⊗ T(1) will have 2 gen v +1 or 2 gen v +1 −1 Weyl factors. Observe that we are guaranteed to find a direct summand T(v) in T(v − 1) ⊗ T(1). Now we have three cases depending on whether the generation increases, stays constant or drops, which are precisely the respective cases a 0 = 0, a 0 ∈ {1, ..., − 2} and a 0 = − 1, as above. In the first case, T(v) exhausts all newly generated Weyl factors, so it is the only summand that appears. In the second case, it exhausts roughly half of all Weyl factors, so one expects a further summand to appear. In the generation drop case, we have gen v+1 = gen v − tl(v). Hence T(v) only accounts for a small proportion of the Weyl modules, and we expect several other tilting summands to appear.
Proof of Proposition 4.7. This is a neat and direct application of (3-3), as we now explain. First we observe that a character argument is enough to verify the formulas. We will use for these character computations, where w = [b j+1 , ..., b 0 ] p, are the relevant p -adic expansions for the summands that appear. Hereby, b i = a i whenever tl(v) < i j and b j+1 = 0 if and only if tl(v) = j + 1.
The cases where we do not have generation drops are almost immediate (and have already been discussed above): The first case where a 0 = 0 follows by observing that the corresponding slot for [2] v a 0 p (0) = [2] v is not occupied in the expression for χ T(v−1) in (4-4), so we end up with an indecomposable tilting module. In the second case, the slot is occupied by a [2] v p (0) , and we calculate that [2] v p (0) [2] v = [2] v 2p (0) + 2, which splits the character into two, one appearing twice. The final case without generation drops, where the slot is now given by [2] a 0 v p (0) for a 0 > 1, can then be proven by calculating The generation drop case follows using the relations These equations imply that for a k / ∈ {0, 1}. Now we collect terms, which gives the claimed summands in order using (4-4). The cases a k ∈ {0, 1} follow by replacing the term [2] v (a k −1)p (k) by either 0 or 2. Observe that in the case p∨ = 2 we obtain the claimed doubling of summands since [2] v (p∨ −2)p (i) = 2. (b) In the case mchar(k, q) = (2, 2) and v = [1, 1, 1, 1] 2,2 , one gets Proposition 4.7 immediately implies the following appearance of losp.
Note the contrast to the semisimple situation, where the multiplicities of 1 in tensor products T(1) ⊗2d are given by the Catalan numbers, which grow exponentially.

Remark 4.10 For any tilting module T the class [T]
acts on the Grothendieck ring of Tilt k,q by multiplication and thus gives rise to an N 0 × N 0 -matrix with entries in N 0 when evaluated on the basis of classes of indecomposable tilting modules. The fusion graph is the (multi)graph with this adjacency matrix. The fusion graph for T(1) has periodic and fractal structures. First, the number of edges going to lower highest weight summands repeat l-periodically in the order {0, 2, 1, ..., 1}. This is the first generation of edges. Secondly, the edges also become longer and repeat with bigger periodicity, depending on p. One can illustrate this fusion graph in many ways in order to highlight its fractal structure. In Figure 3, we illustrate the fusion graph and a Bratteli diagram (which is strictly speaking not a fusion graph as vertices appear multiple times). In both of these cases, the generation drop becomes a long edge.
The fusion rules for eves are as follows. These, together with Lemma 4.1, give a complete collection of fusion rules for simple tilting modules.

Proposition 4.11
Let a, b ∈ N, k, t ∈ N 0 , 1 a, b < p∨ , and ap (k) bp (t) . Let us write x to denote x, if x is odd, and x − 1, if x is even.

(a) For k > t we have
where the potentially non-zero digits appear at positions k and between t and 0.
where the leftmost sum is running over all where the two leftmost sums are running over Note that the case k = t is a variation of (4-5) where the kth and k + 1th digits for the tilting summands that appear are given by the first two digits in (4-1) or (4-2) for T(a − 1) ⊗ T(b − 1) and p∨ , while the other digits are given as in (4-5).
We decompose [ap (k) ] v [bp (t) ] v into a sum of indecomposable tilting characters, presented as in (4-4). Note that [ap (k) ] v is already one of the factors expected in the character of every summand, hence we focus on rewriting [bp (t) ] v . We claim: where we interpret [2] v 0 as 1 and not 2. (This is due to the same reason why we need to take the product over the non-zero digits in (4-4).) In order to verify this, we rewrite the second line as The last equation is a telescoping product when writing [n] x = x n −x −n x−x −1 . For k = t, the same type of argument leaves us with rewriting [ap (k) ] v [b] v p (k) . To this end, we will use the quantum number calculations underlying Lemma 4.1. Let us assume that we are in the case of (4-6), where we calculate where x runs over the same set as in (4-6), and the second equality is (4-1) for v substituted by v p (k) . Note that [xp (k) ] v are terms of the form [m ∞ v ] v in (4-4), so we are done in this case. The case of (4-6) uses the same argument with (4-2) instead of (4-1).

Remark 4.13
Questions about the structure constants of the representation ring have been studied for the finite group SL 2 (F p k ) for a long time. (The connection to our setup is to embed SL 2 (F p k ) into SL 2 (F p k ) via fixed points under the Frobenius twist.) For example, Lemma 4.1 and [EH02a, Lemma 5] is used in [Cra13, Section 3] to find the finite group analog of fusion rules.
Actually, a bit more is true and worthwhile to point out: Recall that for a finite group a module T is algebraic if there exists f ∈ N 0 [X] such that f [T] = 0 holds in the representation ring; otherwise modules are called transcendental, cf. [Alp76]. As shown in [Cra13, Section 3] the tilting module T(1) considered as a SL 2 (F p k )-module is algebraic, and so are eves. Being algebraic is a measure of how complex fusion rules get. This was a motivation to stick to the fusion rules presented above. In fact, this is a rare property for modules of groups of Lie type as [Cra13, Theorem A] shows and most of the time even the vector representation is transcendental, e.g. for all SL a for a > 2.
4B. Categorified fusion rules for tensoring with the vector representation. The fusion rule from Proposition 4.7 describes the multiplicities of indecomposable tilting modules in the tensor product T(v − 1) ⊗ T(1). In this section, we consider the refined problem of describing the morphisms that project onto such summands using the Temperley-Lieb calculus. Specifically, in Theorem 4.15 we will decompose the idempotent E v−1 ⊗ id 1 into a sum of orthogonal, primitive idempotents factoring through E v as well as the other E v−2p (i) predicted by Proposition 4.7. Conversely, such a decomposition can also be read as a recursive description of the mixed JW projector E v in terms of mixed JW projectors of lower order.
For the following definition, we use scalars determined by evaluating the functions g q and f q on digits. On all digits, except for the zeroth one, we use (3-10). For the zeroth digit, we instead use: x if a = 0. (4-8) Armed with this notation, we now define the morphisms that will feature in the decomposition of E v−1 ⊗ id 1 into orthogonal, primitive idempotents. (a) If a 0 = 1 and j = 0, then we consider the morphisms Here the caps and cups are of thickness 1 and are thus admissible. If j = 0, then we declare A 0 v = 0 (the diagram makes no sense in this case since v = 1 < 2). We will also consider the morphism (A 0 v ) reflected along a horizontal line.

(b)
For a 0 > 1 we define Here the caps and cups are of thickness a 0 − 1 and are thus admissible. If j = 0, then we use the same formula to define B 0 v , except we omit the second summand.
We now look at the generation drop case. Suppose that v has a tail of maximal digits, that is v = [a j , ..., a i , p − 1, ..., p − 1, − 1] p, with i = 0.
(c) If a i = 1 and i = j, then we define Here the caps and cups have thickness w = p (i) , and are thus admissible. If a i = 1 and i = j, then we declare A i v = 0 (the diagram makes no sense in this case since v < 2w). We will also consider the reflected morphism (A i v ) along a horizontal line. (d) For a i > 1 and i = j we consider Here the caps and cups are of thickness a i p (i) and are thus admissible. If i = j, then we use the same formula to define B i v , except we omit the second summand.
The categorified fusion rule for T(v − 1) ⊗ T(1) is now given by the following theorem.
(a) We have the following decomposition of E v−1 ⊗ id 1 into a sum of orthogonal, primitive idempotents.

Remark 4.17
The fusion rule (1-1) can be used to express classical JW projectors in terms of JW projectors of lower order. Analogously, Theorem 4.15 gives a recursion of p JW projectors in terms of p JW projectors of lower order. This is in contrast to the defining description in (3-6), which uses classical JW projectors.

Remark 4.18
In the complex quantum group case and for v 2 − 2, the fusion rule (4-9) can be deduced from [BDRM19, Lemma 3.2]. The three cases of their rule reflect the trichotomy of a 0 = 0, a 0 = 1, and a 0 > 1.

Remark 4.19
We do not know a good partial trace formula of type (4-10) in the case a i = 1, not even for i = 0 and w = 1. (One can write down a formula using (4-9), of course.) We expect this formula to be more complicated, because it deals with a generation increase (on comparison with the increased complexity of the fusion rule when the generation drops).
Proof of Theorem 4.15. The proof proceeds by induction on v. To do so, we split the statement of the theorem into the following two types of assertions.

F(v) denotes:
The categorified fusion rules are given by (4-9) for v.
The former makes sense for all v 1 and the latter for v 2. We will also write F(< v) to express the assertion that F(w) holds for all 1 w < v, and similarly for PT(< v).
The base cases for the inductive argument is F(1), which is immediate. The induction step will be accomplished by two arguments that we separate into two distinct statements below. Lemma 4.20 shows the implication F(< v) =⇒ PT(v) for all v 2. Lemma 4.21 shows the implication PT(v) =⇒ F(v) for all v 2. Induction then shows that both assertions hold for all relevant values of v.
We now turn to the two lemmas that form the heart of the proof of Theorem 4.15.

Lemma 4.20 We have F(< v) =⇒ PT(v) for all v 2.
Proof. We first consider i = 0, where we have w = 1 and assume a 0 = 0, i.e. we aim to prove To verify this, we will use the fusion rule for v − 1 in reverse to expand the projector E v−1 . If a 0 = 1, then we have E v−1 = E v−2 ⊗ id 1 . The claimed statement follows since the circle value is −[2] q = g q (0) and the second term is zero by definition. If = 2, then we are done. Thus, from now on suppose that > 2.
We consider the case a 0 = 2, where the fusion rule involves The first term in each of the brackets is L 0 v−2 ; the second is (L 0 v−2 ) 2 , which is zero by (3-11). Since g q (1) = −[2] q and f q (1) = −2 the claim follows for a 0 = 2.
Next we consider a 0 ∈ {3, ..., − 1}. Here we use − , (4-12) and it remains to compute the final term. To this end, we will use the fusion rule for v − 2a 0 + 2 on the mini box (using induction), which corresponds to E v−2a 0 +1 . The last digit of its relevant p -adic expansion is an element of {3, ..., − 1}, namely − a 0 + 2. We claim that fusion results in: Here, we have three things to check. To start, the first term in the middle is zero because T(v − 2) and T(v − 2a 0 + 2) do not share any common Weyl factors. To see this, first observe that the relevant zeroth digits are a 0 − 1 and respectively − a 0 + 3. Now the claim follows from the condition in Lemma 3.32 for any odd since one has a 0 − 1 = − a 0 + 3 and a 0 − 1 = a 0 − 3. For even it can happen that a 0 − 1 = − a 0 + 3, namely for a 0 = +4 2 . However, in this case the next digit of v − 1 and v − 2a 0 + 3 differ by one, so Lemma 3.32 also applies. Second, the fusion rule includes a term of the form B 0 v−2a 0 +2 , which is typically a sum of two diagrams (although the second may not appear in some cases). The first diagram combines with the present caps and cups to form The second diagram (if it is present at all) vanishes when it is sandwiched because of the containment relation U 0 U 0 = 0. Third, all possible terms of even lower order arising from the fusion rule become zero when sandwiched. Such terms only arise if Sandwiching a term B i v−2a 0 +2 for i > 0 produces a linear combination of (at most) two diagrams: where w = p (i) . We focus on the common outlined portion. Once we bend up the free strand, this is in fact the morphism D S E v−5 D 0 E v−3 for the stretch S = {0, 1, ..., i − 1}, which vanishes by the containment relation. The terms A i v−2a 0 +2 or (A i v−2a 0 +2 ) for i > 0 only occur when i = tl(v − 2a 0 + 2) and a i = 2. Sandwiching such a term results in a diagram that factors through a morphism from E v−2 to E v−2a 0 +2−2p (i) , and we claim that such morphisms are zero because the corresponding tilting modules have no common Weyl factors. Indeed, the relevant p -adic expansions are v − 1 = [..., a i , 0, 0, ..., 2] p, and v−2a 0 + 3 − 2p (i) = [..., a i − 2, 0, ..., 0] p, , and thus the claim follows from Lemma 3.32.
Finally we consider the case 0 < i tl(v), which, surprisingly, is much easier to prove. Recall that we assume that a i = 0 and aim to prove: To verify this claim, we calculate that v − w + 1 = [a j , ..., a i , 0, ..., 0] p, . In particular, we can use (a slight generalization of) Proposition 3.35 to trace off w − 1 = p (i) − 1 strands and we get v−w where x = [a j , ..., a i − 1, 0, ..., 1] p, (if i = j, the f-term vanishes but the g-term does not because we assume a j > 1). Now we use shortening to compute the full partial trace as: which we pull straight to get the claimed partial trace formula.

Lemma 4.21 We have PT(v) =⇒ F(v) for all v 2.
Proof. We start with a few observations. First, Proposition 4.7 ensures that we know how many orthogonal, primitive idempotents to expect in the categorified fusion rule. Second, by the same arguments as in the proof of Theorem 3.19 (however, it is easier in this case since we only need to tensor with T(1)) the idempotents projecting onto isotypic components are uniquely determined by the property of absorbing E v−1 ⊗ id 1 . These isotypic idempotents are automatically orthogonal because a straightforward computation, using Proposition 3.3 and Lemma 4.1, shows that the isotypic components share no Weyl factors, which implies that there are no non-zero morphisms between them by Lemma 3.32. Combining these observations, it remains to show that the morphisms B i v , A i v , (A i v ) from Definition 4.14 satisfy the absorption property and are indeed idempotents whenever they appear in (4-9). Finally, we also check that Absorption for the remaining cases, namely A i v and (A i v ) with i > 0, can be similarly shown using shortening.
We will now verify the idempotency of the candidate expressions on a case-by-case basis. Idempotency (and orthogonality) of A i v and (A i v ) . We start with the term A i v (which also covers the symmetric case (A i v ) ), which is defined as the sum of the two diagrams in which w = p (i) . Next we compute the pairwise products of X, X , and Y. First we use shortening and absorption of mixed JW projectors to compute Symmetrically, we also have (X ) 2 = X and Y(X ) = Y. Now we claim that all other products are zero, namely (X )X = X(X ) = YX = (X )Y = Y 2 = 0. This can be seen as follows. Up to symmetry, these statements all follow from The equivalence is given by bending, as illustrated. On the right-hand side we undid shortening and translated the caps and cups into morphisms U i and D i respectively (this is possible since a i = 1), and then applied (3-11). Taking all of these together shows that Next we check that the terms B i v are idempotents. Recall that B i v for i = j is defined as a linear combination of the following two morphisms where we also write w = p (i) . For i = j, we use the same definition for X, but set Y = 0. To compute the various products of these elements, we will use the following partial trace rule for E v−w from (4-10), which holds by assumption PT(v): Since we know from (3-11) that the loop L i v−2w is annihilated by postcomposing with another down morphism D i E v−2w , we also obtain from (4-11) that v−w Thus we calculate The latter holds since (L i v−2w ) 2 = 0, which follows from (3-11). (Observe that the above relations also hold in the special case i = j where Y = 0.) Now we verify that B i v is an idempotent: where we have used XY = YX and fq(a i −1) gq(a i −1) = f q (a i ).
4C. Categorified fusion rules for eves. For this section, we fix a, b ∈ N with 1 a, b < p∨ and k, t ∈ N 0 . We have seen in Proposition 4.11 that the decomposition of a tensor product of simple tilting modules T(ap (k) − 1) ⊗ T(bp (t) − 1) into indecomposable tilting modules is multiplicity-free. In fact, its decomposition is Weyl-multiplicity-free, meaning that no Weyl factor appears twice. This implies that the problem of computing categorified fusion rules for eves is well-posed.

Lemma 4.22 There exists a unique decomposition of E ap
where v ranges over the set specified by (4-5).
Proof. The existence of an expansion into idempotents as illustrated follows from the fact that the tilting modules T(v − 1) form a complete collection of indecomposable objects (up to isomorphism) in Tilt k,q and the associated primitive idempotents are the mixed JW projectors E v−1 . The summands that occur are determined by (4-5) since E v−1 represents T(v − 1). The orthogonality and uniqueness follow from the fact that T(ap (k) − 1) ⊗ T(bp (t) − 1) is Weyl-multiplicity-free since T(ap (k) − 1) ∼ = ∆(ap (k) − 1), and T(bp (t) − 1) ∼ = ∆(bp (t) − 1).

Remark 4.23
In general, is not Weyl-multiplicity-free or not even multiplicity-free, then such decompositions are no longer unique. In Section 4B, we will encounter this issue when giving a complete description of the case T(v − 1) ⊗ T(1).
The problem of explicitly computing the idempotents in the decompositions from Lemma 4.22 in full generality is beyond the scope of this paper. Instead, we will compute the idempotents projecting onto summands T(v − 1) of generation at most one. In particular, we completely determine the decomposition in the complex quantum group case.
We will consider morphisms of the form To describe the idempotent of E ap (k) −1 ⊗ E bp (t) −1 that factors through E v−1 we will consider the ansatz with scalars x Lemma 4.24 Let T(v − 1) and T(v − 1) be summands of T(ap (k) − 1) ⊗ T(bp (t) − 1) as specified by (4)(5).
with uniquely determined scalars d ap (k) ,bp (t) (v,S) ∈ k, and the sum running over all down-admissible Proof. This is a direct consequence of the fact that loops form a basis for the endomorphism algebra of E v−1 and that T(ap (k) − 1) ⊗ T(bp (t) − 1) is Weyl-multiplicity-free.
To warm up, we will compute the digon expansion (4-16) in the case when v is of generation zero, i.e. when the direct summand T(v − 1) of T(ap (k) − 1) ⊗ T(bp (t) − 1) is simple. In this case, the endomorphisms of E v−1 are just its scalar multiples, and the single relevant scalar d ap (k) ,bp (t) v,∅ can be determined using the values of theta webs from [KL94, Chapter 6].
Proof. We consider the analogous digon morphism in k(v). (This is possible since all of the involved projectors are simple JW projectors). Abusing notation, we find that for some scalar c ∈ k(v). Now taking the trace on both sides gives . It follows that this scalar specializes to a well-defined scalar d ap (k) ,bp (t) v,∅ in k and the claim follows. Note that we have no assumptions on gen v in Proposition 4.26.
Proof. Throughout the proof we will use the quantities x, m, and n, defined as above. First, we shall argue thatd ap (k) ,bp (t) v,∅ descends to k since the denominator m+x does not vanish upon specializing from v to q. Let us expand (allowing zeros on the left). As both ap (k) = (m + x) + 1 and bp (t) = (n + x) + 1 are eves, the quantum Lucas' theorem Proposition 2.8 implies since all digits of (m + x) and (n + x) are greater than or equal to the corresponding digit of x.
Next we assume that T(v −1) is a summand of T(ap (k) −1)⊗T(bp (t) −1). To show d ap (k) ,bp (t) v,∅ = 0, first assume k > t and T(v − 1) takes the form as a typical summand in (4-5). The relevant p -adic expansions in d ap (k) ,bp (t) v,∅ are given by The digits of x are constrained by x t b−1 2 and x j p∨ −1 2 for 0 j t − 1. Consequently we also have where we again want to use the quantum Lucas' theorem. To this end, observe that all digits below the kth one behave as in the k > t case. So let us focus on the digit of m + n + x + 1 which involves x k . For (4-6), this digit is a + b − x k − 1, which is bigger or equal to x k because 2x k a + b − 1. For (4-7), this digit for the x-sum therein is |a − b| + 3x k − 1 for x k at least 1, which is clearly bigger or equal to x k . For the y-sum, this digit is a + b − p∨ − x k − 1, which is bigger or equal to x k by the allowed range for x k which gives 2x k a + b − p∨ − 1.
For the final part of this proof, we assume that T(v − 1) is not a summand of T(ap (k) − 1) ⊗ T(bp (t) − 1). To see that the scalar vanishes in this case, we again use the quantum Lucas' theorem. In particular, the relevant calculations of v + x and v stay the same, but now at least one of the digits of x leaves the specified range and the corresponding (quantum) binomial in the factorization for v+x x q vanishes.
We now return to the task of computing digon expansions (4-16). Suppose that gen v = 1 with corresponding minimal down-admissible stretch S, which reflects down along c = dp (i) . Then we define another scalar in k(v) bỹ Theorem 4.27 Retaining notation as above and for where d ap (k) ,bp (t) v,S is obtained fromd ap (k) ,bp (t) v,S by specializing v to q.
The proof appears below.
For the proof of Theorem 4.27 we need the following lemma about simple JW projectors. We were not able to find it in the literature, but it is probably known. If c = 1, then the top part is what we want, while the bottom can be simplified using induction. For c > 1, we expand bothẽ a+b−1 once more. In doing so, we obtain a diagram of the same type (up to partial trace, which we can remove up to a scalar using (2-4)), but with smaller projectors. This process is repeated until we obtain our desired result.
Proof of Theorem 4.27. We work in the generic setting of TL k(v),v and specialize to TL k,q at the end. Note that we can expand the semisimple p JW projector e v−1 via Lemma 3.15 as follows where we point out that gen v−c = 0 and thus, e v−c−1 =ẽ v−c−1 . The two summands of e v−1 are orthogonal idempotents. Hence, expanding the projectors at the top and bottom we get The first summand is covered by classical recoupling theory and, using [KL94, Section 9.10], it evaluates tod Here we have rewritten the scalar in [KL94, Section 9.10] by collecting terms into quantum binomials, giving us the expression in (4-17), and then we applied the definition of e v−1 backwards.
The second summand in (4-20) can be evaluated using Lemma 4.29 twice to give Collecting terms and using Lemma 4.30, proves the claimed formula. Finally, note that the left-hand side of (4-19) has non-negative p -adic valuation, and the morphisms e v−1 andL v S descend to the basis elements E v−1 and L S v−1 of the endomorphism algebra of E v−1 . Hence, the scalars in (4-17) and (4-18) can also be specialized to (k, q).
with scalars For gen v = 0, the second term in (4-21) is dropped.
Proof. For ease of notation, let We now proceed to calculate the compositions A 2 , AB and B 2 . Theorem 4.27 and Proposition 4.26 imply To compute the other compositions, we work in the generic setting of TL k(v),v . Applying Lemma 4.29 to each of the sandwiched projectors, we obtain and thus applying Theorem 4.27, gives us By expansion of the projectors E v−1 and E v[S]−1 in the following diagram where all other summands in the expansion are annihilated since there are no common Weyl factors. Further, by applying Lemmas 4.29 and 4.25, which is symmetric, i.e. AB = BA.
It is now straightforward to verify that x

Applications
We now derive some consequences for the monoidal structure of Tilt k,q . These results are generalizations to mixed characteristic of well-known results.
5A. Tensor ideals, cells and Verlinde quotients. A (two-sided) ⊗-ideal I in a monoidal category is a collection of morphisms that is closed under composition and tensoring with arbitrary morphisms. Recall also that a thick ⊗-ideal J in a monoidal category is a ⊗-ideal that is generated by the identity morphisms on a set of objects. In general, not every ⊗-ideal is thick, so it is remarkable that these notations coincide for Tilt k,q , as we shall now see. Note that in part (d) we use again that e v−1 are secretly defined over Q(v) to take the p -adic valuation.
In the following theorem, we will use { − } ⊕ as a notation for the additive closure of a given set of objects, meaning the full subcategory additively (via taking direct sums) generated by the specified objects.
Theorem 5.1 Let J v−1 be the thick ⊗-ideal in Tilt k,q that is monoidally generated by T(v − 1), i.e. the thick ideal corresponding to set of objects containing the direct summands of T(v − 1) ⊗ T(w − 1) for any w ∈ N.
(c) If p = 2 and = 2, then In particular, by Theorem 5.1 the ⊗-ideals are the strongly connected components of the fusion graph of T(1), cf. Figure 3.
Proof. For part (a), we will use the fact that T(1) monoidally generates Tilt k,q , see e.g. Proposition 4.11. We argue inductively that the summands of T(p (k) − 1) ⊗ T(1) ⊗d belong to {T(w − 1) | w p (k) } ⊕ . Indeed, for d = 0 there is nothing to show, and the case with d > 0 then follows inductively from Proposition 4.7. It is also clear from Proposition 4.7 that every element of {T(w − 1) | w v} ⊕ will appear for some d big enough.
Next we prove part (b). Given some ⊗-ideal I = 0, take any non-zero g ∈ I ∩ Hom Tilt k,q T(w − 1), T(x − 1) , which exists for suitable w, x ∈ N. By tensor-hom adjunction, there is an associated "bent" morphism g b ∈ I ∩ Hom Tilt k,q 1, T(x − 1) ⊗ T(w − 1) , which is again non-zero. Considering the direct summands of T(w − 1) ⊗ T(x − 1), we deduce that there exists a non-zero f ∈ I ∩ Hom Tilt k,q 1, T(v − 1) and we assume that v ∈ N has been chosen to be minimal with this property. Thus, T(v − 1) has 1 = ∆(0) as a Weyl factor, which implies v = 2p (k) − 1 for some k by Proposition 3.3. Composing f with the inclusion T(2p (k) − 2) → T(p (k) − 1) ⊗ T(p (k) − 1), we obtain a non-zero morphism in I ∩ Hom Tilt k,q 1, T(p (k) − 1) ⊗ T(p (k) − 1) . Reversing the bending trick produces a non-zero endomorphism of T(p (k) − 1) that is in I. As T(p (k) − 1) is simple, we conclude id T(p (k) −1) ∈ I, so I contains at least J p (k) −1 . By minimality of k, we get I = J p (k) −1 .
The final statement is a consequence of Proposition 3.23, using Proposition 2.8.
A cell in Tilt k,q is defined as an equivalence class of indecomposable objects with respect to the preorder T(v − 1) J T(w − 1) :⇔ ∃x ∈ N : T(w − 1) is a direct summand of T(v − 1) ⊗ T(x − 1).

Corollary 5.2
The cells of Tilt k,q are of the form Thus, Tilt k,q has infinitely many finite cells if and only if p < ∞ and < ∞.
The quotients Tilt k,q p (k) = Tilt k,q /J p (k+1) −1 are k-linear, additive, idempotent closed, Krull-Schmidt, pivotal categories with finitely many indecomposable objects, namely T(v − 1) for v ∈ {1, ..., p (k+1) − 1}, and finite-dimensional hom-spaces (these properties are sometimes called fiat). for every object T ∈ I and every object X ∈ C and any f ∈ End C (T ⊗ X), we have Tr I T⊗X (f) = Tr I T Tr r X (f) . Below we will omit the subscripts if no confusion can arise.
Following ideas from e.g.
Theorem 5.7 Each ⊗-ideal in TL k,q admits a non-trivial mtrace Tr (k) which satisfies dim (k) where T(v − 1) ∈ J p (k) −1 \ J p (k+1) −1 and the product runs over all non-zero and non-leading digits of v. Every such mtrace descends to the corresponding categories Tilt k,q p (k) and Ver k,q p (k) .
Proof. By Theorem 5.1, we know that all ⊗-ideals are of the form J p (k) −1 , while Lemma 5.6 defines an mtrace on these. Moreover, (5-2) and Proposition 3.23 imply the claimed formula. The property that these descend to Tilt k,q p (k) and Ver k,q p (k) is clear.
5C. The Müger center is often trivial. We assume in this section that our parameter q has a square root. Recall that we have a braiding on TL k,q , using the Kauffman formula (2-1). Under these assumptions, recall that Tilt k,q is additive, idempotent closed, Krull-Schmidt, K-linear, braided and pivotal. Let C be a category having these properties, and let 1 and β denote the monoidal unit and the braiding of C.
Following [Müg03], we define the Müger center MZ(C) of C to be the full subcategory of C consisting of all objects X such that β X,Y β Y,X = id X,Y for any Y ∈ C.
Clearly 1 ⊕m ∈ MZ(C) for all m ∈ N, and we call MZ(C) trivial if 1 ⊕m are the only objects in MZ(C). In other words, MZ(C) Vect k , the latter being the category of finite-dimensional k-vector spaces.
Theorem 5.8 The Müger center MZ(Tilt k,q ) is non-trivial if and only if q = ±1.

Remark 5.9
In the case C is a ribbon, finite tensor category, having a trivial Müger center is equivalent to C being modular in the sense of Kerler-Lyubashenko -see [Shi19, Theorem 1.1]. In particular, Theorem 5.8 is a good indication that tilting modules in mixed characteristic may give rise to non-trivial link and 3-manifold invariants.
In order to prove Theorem 5.8, we need two auxiliary lemmas. To this end, recall that the endomorphism spaces in braided tensor categories have central elements given by "encircled identity morphisms", illustrated below. On simple objects these act as scalars, but T(v − 1) is usually not simple and so we obtain the following.
In contrast, the scalar resulting from an unlinked circle is −[2] v times the projector.

Proof.
For v ∈ Eve, in which case e v−1 =ẽ v−1 , it is well-known that (5-3) holds without lower order terms, see e.g. [KL94, Section 9.8]. We will use this throughout the proof. For the other cases, we calculate that The right-hand side in (5-5) is a linear combination of loopsL S v−1 . However, these are not well-defined over k, so we need to rewrite (5-5) in terms ofL S v−1 (the loops that specialize to L S v−1 ). We do not know the full change-of-basis matrix (see [TW19, Lemma 4.8] for generation 2), but we know that this matrix is unitriangular by an analog of [TW19, Lemma 3.17], which is proven verbatim as therein. In particular, the case a 0 = 0 in (5-3) follows, and also the coefficient of e v−1 is as claimed. To prove the case a 0 = 0 in (5-3), we expand v−1 This completes the proof.

Remark 5.11
In the previous proof, one may be tempted to argue that [2] q v[S] = [2] q v holds in k, and so one should be able to factor it out from the sum in (5-5). This is not allowed, since the individual summands are not well-defined in TL k,q . Indeed, this argument would predict that no lower order terms appear, not evenL {0} v−1 in (5-3), which is certainly wrong.
Proof of Theorem 5.8. For q = ±1, the Kauffman formula (2-1) shows that MZ(Tilt k,q ) = Tilt k,q , so let us focus on the case where q = ±1 and show that MZ(Tilt k,q ) is trivial.
To this end, it suffices to check that no indecomposable T(v−1) besides T(0) = 1 is Müger central.