Generalized negligible morphisms and their tensor ideals

We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $\mathcal{C}$ over a local ring $R$. If the maximal ideal of $R$ is generated by a single element, we show that any thick ideal of $\mathcal{C}$ admits an explicitely given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

1. Introduction 1.1. Negligible morphisms. We are interested in the structure of tensor ideals [Co18] [Ba05] [BKN17] [CH17] in a rigid spherical monoidal category C over a field k. Roughly speaking we have two notions of tensor ideals: • tensor ideals, submodules I(X, Y ) ⊂ Hom C (X, Y ) (for any X, Y ∈ C) that are closed under composition and tensor products with morphisms; • thick (tensor) ideals, subsets of ob(C) which are closed under tensor products with arbitrary objects of C and also closed under retracts. Arguably the most important tensor ideal is the tensor ideal of negligible morphisms By [AK02] it is the largest proper tensor ideal of C, and the only tensor ideal such that the categorial quotient C/N can be semisimple. Its associated thick ideal N = {X ∈ C | X ∼ = 0 ∈ C/N } consists of direct sums of indecomposable objects X whose categorial dimension dim(X) = 0 vanishes (the negligible objects).
1.2. Generalized negligible morphisms. It is the aim of this article to present a generalization of the notion of negligible morphism which will lead to a measure -the nullity -for the negligibility of an object X ∈ C. While N and dim(X) can always be defined under our assumptions on C, the definition of the generalized negligible tensor ideals N I and the nullity requires more restrictive conditions. In order to define these tensor ideals, we consider a deformation or lift of C to a monoidal category C R over a local ring R as follows. Let C R denote a monoidal rigid spherical category whose Hom spaces are free R-modules satisfying End(1) = R. For any ideal I of R we put the tensor ideal I-negligible morphisms in C R . An object X ∈ C R is called I-negligible if T r X (a) ∈ I for all a ∈ End(X). For a fixed ideal I we can also say that f : X → Y is k-negligible with respect to I if T r(f • g) is in I k . We will only use this in the special situation where I = m, the maximal ideal of R. We then obtain the k-negligible morphisms N k := N m k which form a decreasing chain of tensor ideals in C R and likewise for the N k .
In order to define these for the category C over k we suppose now that we have a surjective and full tensor functor C R → C with k = R/m. A special case of this is if C is the mod m evaluation of C R : here the objects of C are the same as those of C R with Hom C (X, Y ) = Hom C R (X, Y )/mHom C R (X, Y ) ∼ = Hom C R (X, Y ) ⊗ R R/m.
When passing from C R to C, the images of the N I and N I define tensor ideals and thick ideals in C respectively (possibly zero if we are not in the special situation of the mod m evaluation) which we denote again by N I and N I (or N k and N k ). Note that N 1 and N 1 are mapped to N and N in C. We call an indecomposable object of C k-negligible if X ∈ N k . Its nullity is the smallest k such that X ∈ N k . 1.3. Examples. Given a monoidal category C the question is whether it admits a lift to a monoidal category C R . Our main examples are the following: Theorem 1.1. The following categories can be obtained as mod m evaluations: (1) The category of (quantum) tilting modules T ilt(U q (g), Q(q)), where g is a semisimple complex Lie algebra and q a nontrivial primitive ℓ-th root of unity with ℓ odd, ℓ > h and not divible by 3 if g contains g 2 , is the mod m evaluation of T ilt(U v (g), R) where R is the completion of Q [v] (v−q) , the polynomial ring localized at (v − q), i.e. all rational functions over Q which are evaluable at v = q.
(2) The category of (modular) tilting modules T ilt(G, k), where G is a semisimple simply connected algebraic group over a perfect field k of characteristic p > 0, is the mod m evaluation of T ilt(G, W (k)) where W (k) is the ring of Witt vectors of k. The proof of theorem 1.1 1) and 2) is explained in sections 6 -7. The field Q(q) can of course be replaced by C. Part 3) can be generalized to incorporate the q-versions Rep(U q (gl t )) and Rep(U q (o t )) [HW]. In this case however we deal with two parameter versions, e.g. the local ring is the completion of C[r, ξ] r−ξ n−1 ,ξ−q (using BMW notation) for the q-version of O t . Using 3) it is possible to define k-negligible ideals also for certain categories of representations of supergroups. In this case the nullity is related to the atypicality.
1.4. Modified dimensions and link invariants. In recent years there has been a lot of interest in the construction of modified traces and dimension functions [GKPM11] [GKPM13] [GPMV13] [GPM18]. One of the main motivations for the introduction of these modified trace and dimension functions is the construction of knot invariants since the invariant (in the sense of Reshetikhin-Turaev) of an indecomposable object X with dim(X) = 0 vanishes. It is very difficult to show that a given ideal has a nontrivial modified trace function. In many cases the only known thick ideal to admit such a nontrivial modified trace is the ideal of projective objects in C, the smallest nontrivial thick ideal. Moreover, in most cases these modified trace functions are not explicitely given.
The situation changes if there is a surjective and full tensor functor C R → C. In this case we can often renormalize the usual trace in C R and consider its image in C to get a modified trace function.
Theorem 1.2. Let R be a local domain (which is not a field) whose maximal ideal (p) is generated by the element p. Let C R be a rigid spherical monoidal category whose Hom spaces are free R-modules. Let I be a thick ideal all of whose objects are k-negligible (with respect to (p)), such as e.g. the ideal N k of all k-negligible objects. For X ∈ I and a ∈ End(X) (1) T r The proof of this theorem is essentially trivial. The whole difficulty lies in the construction of an appropriate lift of C to an analogous category over a local ring R. Since any proper thick ideal is contained in the ideal of negligible objects, we obtain Corollary 1.3. Under the assumptions of the theorem, every thick ideal in C admits a modified trace function.
Since in each case in theorem 1.1 the maximal ideal is generated by a single element, we obtain Corollary 1.4. Each thick ideal of T ilt(U q (g), Q(q)), T ilt(G, k) and the Deligne categories Rep(S t ), Rep(GL t ) and Rep(O t ), t ∈ C, admits a nontrivial modified trace function.
The most interesting example for this is the case of T ilt(U q (g), C). The classical way of Reshetikhin-Turaev to define link invariants colored by objects of T ilt(U q (g), C) yields a trivial invariant L unless the objects are all in the fundamental alcove. Due to our lifting theorem, we can directly define a link invariant in the sense of Reshetikhin-Turaev over the local ring R, the completion of C[v] (v−q) . This invariant can be normalized by 1 p k like the dimension function and yields an R/(p) valued invariant for T ilt(U q (g), C).
Theorem 1.5. (see theorem 4.1) Assume that the components of the link have been colored with the objects X 1 , . . . , X m ∈ C R . Let k be the nullity of ..,Xm) (L) |v=q , which is valid for its evaluation on any m-component link L.
1.5. Thick ideals for tilting modules. What does the nullity capture in the tilting module case?
• In the modular case, the maximal ideal m of the complete discrete valuation ring W (k) is generated by p. If a tilting module over W (k) is in N k , then in particular rank(T (λ)) ∈ (p) k . There are however tilting modules over W (k) satisfying p k |rank W (k) T (λ) while • In the quantum case the maximal ideal is generated by (v − q). The dimension is an element in the subring Q[v] (v−q) ; and the dimension over Q(q) is obtained by evaluating this rational function at v = q.
As for the modular case, T (λ) ∈ N k implies that the multiplicity of (v − q) k in the numerator is at least k; and conversely if T (λ) is irreducible.
In the modular and quantum case the thick ideals are sums of thick ideals attached to a right p-cell or a right cell in the affine Weyl group W + p (or W + ℓ ) (see section 8.1) by results of [AHR17] [O97]. The combinatorics of these cells is however very difficult and not fully understood, especially in the modular case [Je17]. We construct thick ideals I(F ) associated to minimal facets F and compute their nullities in Proposition 8.8. This suggests a description of tensor ideals as a collection of positive cones associated to certain facets.
For the quantum type A n−1 every thick ideal is a sum of thick ideals attached to Young diagrams λ of size n (which parametrize the two-sided cells of the affine Weyl group). We attach a standard facet F 0 (λ) to every such Young diagram and prove: Theorem 1.6. (see theorem 9.11 for details) The thick ideal I(λ) = I(F 0 (λ)) generated by the tilting modules T (ν) for which ν + ρ ∈ F 0 (λ) coincides with the thick ideal constructed by Ostrik for the cell in the dominant Weyl chamber corresponding to the two-sided cell labeled by the Young diagram λ T . In particular, the nullity of any generating module T (ν) of that ideal is equal to the value of Lusztig's a-function of that cell.
For the relation between N k and the values of the a-function in all types see remark 9.12. For type A we can also give the already alluded geometric description of tensor ideals via positive cones associated to certain facets, see Theorem 9.9 for details. This follows fairly easily from earlier work of [Shi], where we have benefitted from its description in [Co08]. In particular, we obtain an explicit description of the N k . Moreover, this approach also suggests a description of the ideal structure for the modular case which is done in section 9.7. 1.6. Structure of the article. In section 2 we introduce basic properties of the generalized negligible ideals. Modified trace functions are studied in section 3 and modified link invariants in section 4. Sections 6 -7 deal with the case of tilting modules. In section 9 we give a description of the thick ideals in quantum type A. We end the article with some open questions in section 10. A second article [HW] will treat the case of Deligne categories. A third article will deal with open questions about the thick ideals and the N k for modular and quantum tilting modules.
2. k-negligible morphisms and their tensor ideals 2.1. Preliminaries. In the following let R be a local ring with maximal ideal m. We assume C R (or sometimes simply C) to be a monoidal rigid spherical category whose Hom spaces are free R-modules and such that End(1) = R (see e.g. [EGNO15, Section 4.7] for details). For simplicity we additionally assume that C R is braided. Otherwise we would have to distinguish between left thick ideals and right thick ideals and between partial and modified traces for the left and right versions. However, the followings notions make sense without the added braided if one is willing to either work with left or right versions of these.
Recall that under these assumptions there exist, for each object X in C R , canonical morphisms i X : 1 → X ⊗ X * ,d X : X ⊗ X * → 1 via which we can define the trace T r X on End(X) by T r X (a) =d X (a ⊗ id X * )i X , for all a ∈ End(X); here i X and d X * : X * * ⊗ X * → 1 are the morphisms in the definition of rigidity for the objects X and X * , andd X = d X * • (s X ⊗ 1). The isomorphisms s X : X → X * * from the spherical structure are normalized such that dim(X) = dim(X * ) for all objects X in C, where dim(X) = T r(id X ). For elements a ∈ End(X ⊗ Y ), we can also define the partial trace or conditional expectation E X : The name partial trace is justified by the equation • for all X, Y, Z, W ∈ C and g ∈ Hom(X, Y ) and h ∈ Hom(Z, W ) and likewise from the right. A collection of objects I in a monoidal category C is called a thick ideal of C if the following conditions are satisfied: (i) X ⊗ Y ∈ I whenever X ∈ C and Y ∈ I.
(ii) If X ∈ C, Y ∈ I and there exist α : X → Y , β : Y → X such that β • α = id X , then X ∈ I. To any tensor ideal I we can associate the thick ideal I given by One of the major reasons to study the tensor ideals and thick ideals in C is due to the fact that the morphisms that are sent to zero under a monoidal functor C → C ′ to another monoidal category C ′ form a tensor ideal; and the objects of C that are sent to zero form a thick ideal.
2.3. Generalized negligible morphisms. Let R be a local ring and C R as in section 2.1.
Definition 2.1. a) Let I ⊂ R be an ideal. We call a morphism f : X → Y Inegligible if T r X (g • f ) ∈ I and T r Y (f • g) ∈ I for all morphisms g : Y → X. An object X is called I-negligible if T r X (a) ∈ I for all a ∈ End(X).
b) If f is I-negligible with respect to I = m k , we simply say that f is k-negligible. An object is k-negligible if it is I-negligible for I = m k .
Lemma 2.2. The I-negligible morphisms form a tensor ideal N I in the category C R . The I-negligible objects form a thick ideal N I in C R .
Proof. It is easy to see that N I is an ideal using T r(f • g) = T r(g • f ) for composable morphisms f : . Now let f ∈ N I (X, Y ) and g ∈ Hom(W, Z) arbitrary for X, Y, W, Z in C R . Let h ∈ Hom(Y ⊗ Z, X ⊗ W ). Then Let now X be an I-negligible object, and Y any object in C. If a ∈ End(X ⊗ Y ), we have T r X⊗Y (a) = T r X (E X (a)) ∈ I, as E X (a) ∈ End(X) and X is I-negligible. Hence X ⊗ Y is an object in N I as well.
Remark 2.3. The definition is similar to the the one of the Jantzen filtration on morphisms defined by the form T r(f • g).
2.4. The mod m evaluation. We are primarily interested in categories whose Hom spaces are vector spaces over a field. Recall that if M is a free R-module of rank r, we obtain a well-defined vector space M/mM over k = R/m of dimension r. We call the mod m evaluation (or reduction modulo m) of C R the category C over k whose objects are in 1-1 correspondence with the ones of C R , and where In the following, the notations Hom, End etc will refer to the evaluation category C. The corresponding spaces for C R will be denoted by Hom R , End R etc. We call C R a lift of C.
2.5. Examples. We give some examples of the lifting of a monoidal category C over k to a monoidal category over a local ring.
2.5.1. Fusion categories. Let k be any field, R a local ring with R/m ∼ = k. If C is a split fusion category over k, a lifting of C in the sense of [EGNO15,9.16] is a split fusion categoryC over R such that C is the mod m evaluation ofC.
Theorem 2.4. [EGNO15, Theorem 9.16.1] If the global dimension of C is non-zero, C admits a lifting to R and this lifting is unique up to equivalence.
Of particular interest is the situation where k is a perfect field of prime characteristic p. Then the ring of Witt vectors W (k) is a discrete valuation ring with maximal ideal generated by p and W (k)/pW (k) ∼ = k. If C is a non-degenerate (symmetric/braided) fusion category category over k, then it admits a (symmetric/braided) lifting to W (k) by [ENO05, Theorem 9.3, Corollary 9.4]. Since we are interested here in the construction of tensor ideals, the semisimple case is not relevant to us. 2.5.2. Algebraic groups. While we can define the mod m evaluation for any monoidal category over the local ring R, it is often not the correct category one is interested in. Consider the case of an algebraic group G over the local ring R. Then extension of scalars of Rep(G, R) defines a monoidal functor The image of Rep(G, R) under this functor is the mod m evaluation, but it is not the category Rep(G ⊗ R/m, R/m) (unless we are in the semisimple case). Indeed the canonical functor where G R/m is the algebraic group over k obtained by extension of scalars, is in general not bijective [J03, 10.14].
2.5.3. Deligne categories. For every field k Deligne [Del07] defined symmetric monoidal categories Rep(S t ), Rep(GL t ) and Rep(O t ), t ∈ k, which interpolate the representation categories of the symmetric group, the general linear group and the orthogonal group. Each of this categories is constructed in the following way: One defines a skeletal subcategory corresponding to the tensor powers of the permutation representation V of S n , the standard representation V of O(n) or the tensor product V ⊗ V ∨ of the standard representation V of GL(n) and its dual. The object corresponding to such a tensor power V ⊗r is denoted r in the S n and O(n)-case and (r, s) in the GL(n)-case. The endomorphism algebras of these objects are by definition (1) End Rep(St) (r) = kP r (t), the partition algebra for the parameter t.
To get the full category, we take the additive karoubian envelope of the skeletal subcategory. The categories Rep(S t ), Rep(GL t ) and Rep(O t ) admit a lift to the completion of the local ring of evaluable rational functions R = k[T ] (T −t) [HW]. Indeed the construction described above makes sense over R as well. This can be generalized to inlcude the q-deformations of Rep(O t ) and Rep(GL t ).
2.5.4. Tilting modules. Let T ilt denote the monoidal category of modular/quantized tilting modules. Then T ilt admits a lift to the category of tilting modules over the ring of Witt vectors W (k) (where k is a perfect field of characteristic p > 0) respectively a lift to the category of tilting modules over the completion of Q[v] v−q . For details we refer the reader to sections 5, 6 and 7.
Lemma 2.5. The tensor ideals N I of C R define tensor ideals in the mod m evaluation C. The thick ideals N I define thick ideals in the mod m evaluation. The tensor ideal N 1 corresponds to the ideal of negligible morphisms in C and the thick ideal N 1 corresponds to the indecomposable objects of categorial dimension 0.
In particular we obtain a chain of tensor ideals and likewise a chain of thick ideals The number k can often be seen as a measure for the vanishing of dim(X). If X ∈ N k with k minimal, we say that X has nullity k. For the explicit meaning of this nullity we refer to the examples that appear later in the article.
Question 2.6. The following question was raised by Kevin Coulembier and Victor Ostrik: Can one find a local ring R such that the N J , where J runs over the ideals of R, is a complete list of thick ideals in C? While we do not know the answer, the existence of a lifting to a local ring seems delicate in the non-semisimple case.

2.7.
Compatibilities and k-semisimplicity. The following lemma follows immediately from the definition.
Lemma 2.7. Let C be the mod-m evaluation of C R over the local ring R.
Since N k ⊂ N k+1 we obtain a chain of full and surjective tensor functors If C is a tensor category so that C/N is semisimple, this loosely suggests to interpret C/N k as k-semisimple. This can be made more precise by considering the trace. Recall [Del07, Proposition 5.7] that a tensor category is semisimple if and only if the trace pairing Hom(X, Y ) × Hom(Y, X) → k is non-degenerate, i.e. T r(f g) = 0 for all g : Y → X implies f = 0. For a morphism f in C/N k the deviation for the failure of the non-degeneracy condition can be seen as an element in I ≤k−1 by considering the lift of f in C R /N k .
2.8. A generalization. If a monoidal functor is full and surjective (on objects), the images of the tensor ideals and thick ideals N k are again tensor ideals and thick ideals. Therefore we can more generally define k-negligible morphisms and objects provided we have a full and surjective monoidal functor C R → C.  [BCK17]. An ideal J in a supercategory is an ideal as in an ordinary category with the extra assumption that J(X, Y ) is a graded subgroup of Hom(X, Y ) for all X, Y ∈ C. The notion of a tensor ideal is otherwise unchanged. Thick ideals can be defined as for monoidal categories.

Modified traces and dimensions
We recall the definition of a modified trace function. The existence of such trace functions on thick ideals is in general difficult and often only known on the thick ideal of projective objects. We show that these exist for our rigid spherical category provided it admits a lift to a local ring whose maximal ideal is principal.
3.1. The concept of a modified trace. Let C R be rigid spherical monoidal over a local ring R. As in Section 2.1 we assume that C R is braided in order to identify left and right duals. We follow [GKPM11] [GKPM13] [GPMV13] [ GPM18] in the definition of a trace function. Recall that for any objects X, Y ∈ C and any endomorphism f ∈ End C (X ⊗ Y ) we have the left trace t L (f ) ∈ End C (X) and the right trace t R (f ) ∈ End C (Y ) defined as follows Definition 3.1. If I is a thick ideal in C then a trace on I is a family of linear functions where V runs over all objects of I and such that the following two conditions hold.
(1) If X ∈ I and Y ∈ C then for any f ∈ End C (X ⊗ Y ) we have (2) If X, Y ∈ I then for any morphisms f : Given such a trace on a thick ideal I, {t V } V ∈I , define the modified dimension function on objects of I as the modified trace of the identity morphism: 3.2. Existence of modified traces. As was shown in [GKPM13], modified trace functions exist on the thick ideal of projective objects in a number of examples. Beyond that, little seems to be known for general thick ideals.
Let R be a local domain (which is not a field) whose maximal ideal (p) is generated by the element p. Let I be a thick ideal all of whose objects are k-negligible (with respect to (p)), such as e.g. the ideal N k of all k-negligible objects. Then we define the modified trace T r (k) . We list some elementary properties of these modified traces.
Lemma 3.2. Let X, Y be objects in I, and let Z be an object in C. Then we have (a) T r (k) Proof. These properties hold for the ordinary trace. By our assumptions the renormalizations T r (k) are also well-defined for the elements to which they are applied in our statements. As the renormalization factor is the same on both sides of the equation, the statements are also true for T r (k) .
Taking the images of these modified traces defines modified trace functions T r Corollary 3.3. Suppose that R is not a field and that m = (p) is principal. Every thick ideal in a mod m evaluation carries a nontrivial modified trace functions.
Corollary 3.4. Every thick ideal in the following categories admits a nontrivial modified trace.
(1) The Deligne categories (2) Let T ilt(U q (g), Q(q)) denote the category of tilting modules in the category of finite dimensional modules of Lusztig's quantum group where g is a semisimple Lie algebra and q a primitive ℓ-th root of unity where ℓ > h and ℓ is not divisible by 3 if g contains g 2 . (3) Let T ilt(G, k) denote the category of tilting modules in the category of finite dimensional representations of G, where G is a semisimple and simply connected algebraic group over a perfect field k of characteristic p > 0.
Proof. Each category is obtained as a mod m reduction of a monoidal category over a discrete valuation ring (see theorem 7.3 and [HW]).
Remark 3.5. The results of this section generalize to the situation of section 2.8 where we have a full and surjective monoidal functor C R → C for R a local ring with principal maximal ideal.

Modified link invariants
We define link invariants for objects in C R . These can be normalized according to the nullity of the objects and yield nontrivial link invariants for objects in C even if their categorial dimension is zero.
Let C be a ribbon category. Then one obtains for each labeling of components of a link by objects of C an invariant of that link, see e.g. Turaev's book [Tu]. For our purposes it will be enough to do this via braids as follows: By Markov's theorem, any link L with m components can be obtained as the closure of a braid β whose image in the canonical quotient map into S n would be a permutation with m cycles. Choose objects X i , 1 ≤ i ≤ m in C. We label the strands of β which corresponds to the i-th cycle by the object X i . After conjugating by a suitable braid, if necessary, we can assume that the first c 1 strands are labeled by X 1 , the next c 2 strands by X 2 etc, where c i is the number of strands labeled by X i . Using the braiding morphisms in C we obtain a linear map The link invariant L (X 1 , ...,Xm) (L) is then defined by We now assume that our ribbon category C is defined over a local ring R whose maximal ideal (p) is generated by an element p in R, as in the previous subsection.
Then we can similarly also define the link invariant L (X 1 , ...,Xm),(k) over the category C as follows: Let Then we have Theorem 4.1. (a) If the object X ⊗m is k-negligible, then we obtain a new link invariant L (X 1 , ...,Xm),(k) defined by which is well-defined. In particular, we obtain a well-defined invariant with values in R/(p).
Proof. The value of L (X 1 , ...,Xm),(k) (L) is just a renormalization of the value of L (X 1 , ...,Xm) (L) which does not depend on the particular presentations of L via a braid β. Hence it is a link invariant. Note that all constructions of L can be performed over the subring Remark 4.2. 1. Our modified trace depends on the choice of the generator p. If we choose a different generator p ′ , it is of the form p ′ = ap for an invertible element in R. Then the modified dimensions with respect to p and p ′ differ by the same element a k for all objects in I.
2. Observe that if X is a simple object in N k , then the dimension of X ⊗2 would be in I 2k . Nevertheless, X ⊗2 usually is not in N 2k . We would expect that the nullity of the object X ⊗m would just be the maximum of the nullities of the objects X i .
3. It would be interesting to define modified traces over local rings whose maximal ideals need more than one generator. 4. We expect that these constructions can also be applied to the theory of logarithmic Hopf link invariants as in [CG17, Section 3.1.3].

Tilting modules in the modular case
We recall some statements about tilting modules over a field k and a complete discrete valuation ring R.
5.1. Weights. Let k be a field of characteristic p > 0 and G a semisimple, simply connected algebraic group over k. We denote by Rep(G) the category of finite dimensional representations. We fix a maximal torus T and a Borel subgroup B. We denote by R the set of roots and by R + the set of positive roots. The dominant integral weights are

Induced modules and Weyl modules. The induction functor from
where k λ stands for k regarded as a B-module via λ ∈ X(T ). All these H i (λ) are finite dimensional over k [J03, I.5.12.c]. The following properties are well-known: (1) For λ ∈ X(T ) + we have H 0 (λ) = 0.
(3) V has a Weyl filtration iff V * has a good filtration.
(4) The tensor product of two modules with a good filtration has again a good filtration.
5.3. Tilting modules over k. A finite dimensional G-module is called a tilting module if it has a Weyl filtration and a good filtration. We denote by T ilt(G, k) the full subcategory of tilting modules where k is a field of characteristic p > 0. The direct sum and the tensor product of two tilting modules is tilting. If V 1 , V 2 are tilting modules, then Ext 1 Proposition 5.1. [J03, Lemma E.6] For each λ ∈ X + there exists a unique indecomposable tilting module T (λ) such that the weight space T (λ) λ is free of rank 1 over k and such that T (λ) µ = 0 implies µ ≤ λ. Every tilting module can be written in a unique way as a direct sum of these T (λ).

5.4.
Tilting modules over discrete valuation rings. The entire theory of tilting modules can be developed over more general rings. We assume now that R is a complete discrete valuation ring.
For a split reductive group G over Z we denote by G R the group over R obtained by extension of scalars to R (but we may omit the subscript if there is no risk of confusion). As for R = k, one defines The analogue of Kempf's vanishing theorem permits to develop the theory of good filtrations and Weyl filtrations also over R.
Note that if a G-module M has a good filtration, then M is free over R (since all H 0 (µ) are so) and for each R-algebra In particular we can define tilting modules for G R . Any tilting module for G R is free of finite rank over R.
Lemma 5.3. [J03, Lemma E.19, Proposition E.22] Suppose that R is a complete discrete valuation ring. For each λ ∈ X + there exists a unique indecomposable tilting module T R (λ) such that the weight space T R (λ) λ is free of rank 1 over R and such that T R (µ) (µ) = 0 implies µ ≤ λ. Every tilting module can be written in a unique way as a direct sum of these T R (λ).
Remark 5.4. The existence of the T (λ) with the correct properties works in greater generality (for instance if R is a Dedekind ring which is a principal ideal domain, see [J03, Section E]). While any tilting module for such R can be decomposed into a direct sum of indecomposable tilting modules, their decomposition will in general not be unique.
From this one concludes that End G R/m (T R (λ)) ⊗ R R/m) is a local ring and that therefore T R (λ) ⊗ R/m is indecomposable.

Tilting modules for quantum groups
We review some results about quantized tilting modules which are needed to show that the category of tilting modules can be obtained as a mod m evaluation.
6.1. Lusztig's integral form. We denote by A = Z[v, v −1 ] the Laurent polynomial ring in an indeterminate v and by U = U A Lusztig's integral form (with divided powers) of the Drinfeld quantized enveloping algebra U v over Q(v) [L93]. Any commutative ring R with 1 and a fixed invertible element v can be regarded as a commutative A-algebra via the homomorphism φ : A → R such that φ(v n ) = v n for all n ∈ Z. For any A-algebra R we denote U R = U A ⊗ A R. We apply this for R = Q(q) and specialize the generic parameter v to a primitive ℓ-th root of unity q with ℓ odd, and assume that ℓ is not divisible by 3 if g ∼ = g 2 . We also assume ℓ > h, the Coxeter number of g. Then we denote by Rep(U q (g)) the category of finite dimensional representations of U q (g) of type 1 as in [APW91].
6.2. Quantized tilting modules over Q(q). In Rep(U q (g)) we have analogs of Weyl and induced modules (where the latter can be defined by means of the triangular decomposition of U q (g)) which we denote again by V (λ) and H 0 (λ), λ ∈ X + . These satisfy exactly the same Ext-vanishing conditions as in the modular case. In particular Ext i (V (λ), H 0 (λ)) = 0 for i > 0, λ ∈ X + (see [An92] and the references therein). This uses the quantum variant of Kempf's vanishing theorem. We can then define modules with a good filtration and a Weyl filtration and define tilting modules as in the modular case. The main statement (proposition 5.1) about indecomposable tilting modules still holds and we denote by T (λ) the indecomposable tilting module attached to λ ∈ X + . 6.3. Quantized tilting modules over R. As in the modular case, the entire theory of tilting modules can be developed over a (complete) discrete valuation ring. The crucial ingredient here is that Kempf's vanishing theorem holds over the ground ring A as well. This follows from work of Ryom-Hansen [RH03] and Kaneda [Ka00] [ . For an A-algebra R we denote by H 0 R (λ) and V R (λ) the corresponding induced module and Weyl module. For any base change A → R we have ∀λ ∈ X + and likewise for V R (λ). Again by Kempf's vanishing theorem over A and standard facts about H i we get the vanishing of ) for all λ, µ ∈ X + and i > 0 (see also ). The remaining arguments are the same as in the modular case. In particular for each λ ∈ X + there is an indecomposable tilting module T R (λ), λ ∈ X + , such that its λ-weight space is free of rank 1 [Ka98-2, Theorem 7.6] and every tilting module decomposes uniquely into a direct sum of the T R (λ).
Since the λ-weight space is free of rank 1, the arguments of Jantzen [J03, E.20] go through and we obtain Corollary 6.1. Let R be a complete discrete valuation ring with residue field R/m = Q(q) (or C) of characteristic 0. Then

k-negligible ideals for tilting modules
The remaining necessary properties to define the k-negligible ideals for either quantized or modular tilting modules follow from Lusztig's theory of canonical bases [L93, Part 4]. 7.1. Weights and Weyl groups. We will use the following notation (similar to [PW93]) in the following: • R an irreducible root system in an euclidian space E, R + a fixed set of positive roots, φ the simple roots. • h the Coxeter number of R, ρ the half sum of the positive roots.
Recall that the affine Weyl group acts on X. The fundamental domain for this action is the closed alcove C ℓ . The affine Weyl group W ℓ can be identified with the set of alcoves in X by matching w ∈ W ℓ with w · C ℓ . The alcove corresponding to w is denoted by C w . We denote The simple objects inC are parametrized by X + , i.e. every simple object is isomorphic to (L(λ), B(λ), Ψ λ ) for some unique λ ∈ X + . Therefore any nontrivial based module admits a filtration with subquotients isomorphic to (L(λ), B(λ), Ψ λ ) for some λ ∈ X + .
Lemma 7.1. [Ka98, Corollary 1.9] Let M, M ′ be two based modules. Then M A ⊗ A M ′ A admits a filtration of U A -modules with each subquotient isomorphic to some L A (λ), λ ∈ X + .

We can now base change from
. Then the lemma immediately implies that the tensor product of two modules with a Weyl filtration (or good filtration) has a again a Weyl filtration (good filtration).
Let G k as in section 5 denote a semisimple simply connected algebraic group over a field k of characteristic p. Then (as in [Ka98, 1.10]) we can view Z as an A-algebra and obtain a ring isomorphism for the Chevalley Z-form G of G k . The Dist(G)-modules that are free of finite rank over Z are G-modules such that L A (λ) ⊗ A Z is the Weyl module for G of highest weight λ, λ ∈ X + . Therefore lemma 7.1 implies in the modular case as well that the tensor product of two modules with a Weyl filtration (or good filtration) has a again a Weyl filtration (good filtration).
Corollary 7.2. The tensor product of two quantized tilting modules over R and the tensor product of two modular tilting modules over the Witt ring W (k) is a tilting module.
7.3. The mod m evaluation. We want to realize now T ilt(U q (g), Q(q)) and T ilt(G, k) as mod m evaluations. In the modular case (by section 5) we should take a complete discrete valuation ring of characteristic zero with residue field k of characteristic p. In order to measure the p-divisibility, its maximal ideal should be generated by p. By [Se04, II.Theorem 3] for every perfect field k of characteristic p, there exists a unique complete discrete valuation ring which is absolutely unramified [i.e., p is a uniformizing element] and has k as its residue field: namely W (k), the ring of Witt vectors. The results of section 5, 6 and 7.2 imply As both rings are discrete valuation rings, all the N I are of the form N k for k = 1, 2, . . .. (1) We have (2) dim V (λ) = 0 if and only if λ is ℓ-regular..
(3) If w ∈ W ℓ satisfies w · λ ∈ X + , then for the length function l on W .
The dimension formula can be rewritten as a product where [ ] denotes the q number. We deduce from this formula the following corollary.

Proof. If T (λ) is simple, the nullity is determined by T r(id T (λ) ).
Example 7.6. The Steinberg module has nullity |R + |. Zeros occur in the dimension formula whenever (λ + ρ, β) ℓ is divisible by ℓ for β ∈ R + . For the weight of the Steinberg module this factor equals (ℓ · ρ, β) and hence each factor for β ∈ R + is divisible by ℓ. Since < St > is the smallest thick ideal of T ilt(U q (g), Q(q)), any indecomposable module in < St > has nullity |R + |.
Example 7.7. In the sl 2 -case the only nontrivial thick ideal is the ideal of negligible objects (the N 1 case). For sl 3 we have two nontrivial thick ideals (see [L] for pictures of the thick ideals in the rank 2 cases), namely N 1 and < St >= N 3 . (2) If w ∈ W p satisfies w · λ ∈ X + , then If T (λ) is irreducible, the nullity is already determined by T r(id T (λ) ), i.e. the p-divisibility. If M is a module of G, we denote by M [r] the G-module which coincides with M as a vector space on which the G-action is given by where the action on the right hand side is the original action of G on M , and F is the Frobenius map, see [J03] for more details.
Corollary 7.11. (a) If T (λ) = V (λ) is a simple tilting module, its nullity is equal to the number n p (λ) = α>0 k α , where the summation goes over all positive roots, and k α is the largest integer such that p kα |(λ + ρ, α).
Clearly T (λ) ∈ N k always implies p k | dim(T (λ)). See section 7.5.2 section 7.5.3 for examples that the converse doesn't hold. 7.5.1. The Steinberg modules in the modular case. Recall that we fixed a maximal torus T and a Borel subgroup T ⊂ B ⊂ G. We denote by F the Frobenius morphism. The notation G r T denotes the scheme (F r ) −1 (T ) and G r = ker(F r ). The where res : Rep(G) → Rep(G r T ) denotes the restriction functor and Rep(G r T ) is the stable category of Rep(G r T ). We denote the thick ideals that form the kernel of the functors ψ r by I r and the tensor ideals by I r . Clearly this gives an descending chain of thick ideals By [Co18, Proposition 5.2.3] and its proof the thick ideal I r is generated by St r . The key ingredient is the following lemma of Jantzen [J03, Lemma II.E.8]: Lemma 7.12. Let λ ∈ X(T ) + and r ∈ N >0 . Then T (λ) is projective as a G r T -module if and only if < λ, α ∨ >≥ p r − 1 for all simple roots α.
Since dim St r = p r|R + | and St r is simple, we obtain Corollary 7.13. The nullity of St r is r|R + |. In particular < St r >⊂ N r|R + | . 7.5.2. The modular cases A 1 . For SL(2) the I r are a complete list of thick ideals [Co18, Theorem 5.3.1]. A tilting module T (m) is in I r if and only if m ≥ p r − 1. In particular I 1 = N . For SL(2) we have dim(St r ) = p r . Therefore I r is the r-th negligible ideal, and every thick ideal is k-negligible for some k.
It follows from Jensen [Je17, Lemma 9.6] that for p > 2 where dim refers to the dimension of T (λ) as a vector space. In other words, N k measures the p-divisibility of the dimension of T (λ).
It is important to assume p > 2 here. Indeed the dimensions of the first tilting modules in the p = 2 case are Although dim T (2) = 4, it is not in N 2 . Over Z 2 we have T r(id T (2) ) = 4, but we can write T (2) ∼ = T (1) ⊗ T (1). Hence there is an endomorphism f of T (2) which permutes the two factors. It is easy to see that T r(f ) = 2, hence the trace is not always contained in (2) 2 and so T (2) / ∈ N 2 . 7.5.3. p-divisibility. The SL(2)-case seems to suggest that indeed we have T (λ) ∈ N k if and only if p k | dim T (λ) provided p > h (the Coxeter number). This is false. The following example was communicated to us by Thorge Jensen. For Sp(4) and p = 11 consider the following tilting modules (weights are expressed via fundamental weights) T (9, 20) 56870 By computations of Jensen these tilting modules all belong to the same pcell (see Section 8), but their p-valuation is not constant. All these modules belong to N 2 but the dimension of T (1, 20) is divisible by 11 3 . Therefore the nullity of a T (λ) is not simply given by the p-divisibility.

Combinatorics for tilting modules
By theorem 7.3 the k-negligible ideals are defined for modular and quantum tilting modules. Contrary to the case of Deligne categories, not every thick ideal is one of the N k . We would like to understand the negligible ideals N k for modular and quantum tilting modules. In both cases the thick ideals are governed by the intricated Kazhdan-Lusztig cell theory of the affine Weyl group which is largely not understood in the modular case. In the following we try to give a more direct geometric description of these cells. While the results in the current section are general, we focus on type A in section 9.6. 8.1. Classification of thick ideals. We first recall the classification of the thick ideals due to Ostrik and Achar-Hardesty-Riche. ) For λ, µ ∈ X + write λ ≤ q µ if there exists Q ∈ T ilt(U q (g), Q(q)) such that T (λ) is a summand of T (µ) ⊗ Q. If both λ ≤ q µ and µ ≤ q λ write λ ∼ q µ. The equivalence classes are called weight cells.
We remark that for any λ ∈ X + we have λ + ν ≤ q λ for all ν ∈ X + since T (λ + ν) is a summand of T (λ) ⊗ T (ν). The fundamental alcove C ℓ is the unique maximal weight cell in the ≤ q ordering. For a weight cell c we denote by T (≤ q c) the subcategory of objects in T ilt(U q (g), Q(q)) whose objects are direct sums of T (λ) with λ in a cell c ′ satisfying c ′ ≤ q c. Then T (≤ q c) is a thick ideal in T ilt(U q (g), Q(q)).
The division of X + into weight cells gives a division of W + ℓ . Recall that Lusztig and Xi have defined a partition of W + ℓ into right cells along with a right order ≤ R on the set of right cells.
The weight cells in X + (and therefore the thick ideals in T ilt(U q (g), Q(q))) correspond to the right cells in W + ℓ , i.e. for any right cell A ∈ W + ℓ the full subcategory T ilt(U q (g), Q(q)) ≤A ⊂ T ilt(U q (g), Q(q)) of direct sums of tilting modules T (λ) for By [O97, Remark 5.6] every thick ideal is a sum of ideals of the form T ilt(U q (g), Q(q)) ≤A for a right cell A so that this theorem yields the classification of thick ideals in T ilt(U q (g), Q(q)).

8.1.2.
Thick ideals in the modular case. The notion of a weight cell can be defined in complete analogy to the quantum case in section 8.1.1. We denote the modular analog of the preorder and the equivalence classes by ≤ T and ∼ T . An equivalence class is called a modular weight cell and C p is the largest modular weight cell. Contrary to the quantum case there are infinitely many modular weight cells (see section 7.5.1). Any modular weight cell c defines a thick ideal T (≤ T c).

Ostrik's classification carries over to the modular case if we replace right cells by right p-cells (also called anti-spherical right p-cells).
Theorem 8.3. ([AHR17, Theorem 7.7, Corollary 7.8]) The modular weight cells in X + (and therefore the thick ideals in T ilt(G, k)) correspond to the right p-cells in W + p , i.e. for any right p-cell A ∈ W + p the full subcategory T ilt(G, k) ≤A ⊂ T ilt(G, k) of direct sums of tilting modules T (λ) for As in the quantum case, every thick ideal is a sum of ideals attached to right p-cells.
Example 8.4. By [An04] the set c r 1 = (p r − 1)ρ + X p r + p r C p is a weight cell which contains St r . We call this weight cell the r-th Steinberg cell. We have an equality of thick ideals < St r >= T (≤ c r 1 ). Remark 8.5. Since every thick ideal in the quantum and modular case is a sum of thick ideals attached to right cells, the nullity of a tilting module is constant on an alcove. Indeed, if T (λ) is a tilting module in a thick ideal I and T (λ) is contained in an alcove A, the entire alcove is contained in I since the ideal is a union of weight cells.
8.2. Affine Weyl groups. We review some basic combinatorics in connection with affine Weyl groups and tilting modules. See [J03], [Ja77] , [A03], [S1], [S2] and the literature quoted in these papers for more details. The use of facets for Kazhdan-Lusztig combinatorics already appeared before e.g. in [GW01].
Let X n be a finite root system, and let X (1) n be the root system of the corresponding untwisted affine Weyl group. It has a faithful representation in h * , where the generating reflections for X n act as usual, and the additional generator acts via the reflection in the hyperplane given by (θ, γ) = ℓ, where ℓ is a positive integer, θ is the long resp short root of greatest length if d|ℓ resp d ∤ ℓ; here d is the ratio of the square length of a long and a short root. We obtain a system of hyperplanes on h * from the orbits of the generating hyperplanes under the affine Weyl group. They can be described explicitly by H α,k = {x ∈ h * | (x, α) = kℓ}, α ∈ ∆ + , k ∈ Z, if d|ℓ. If d ∤ ℓ, we just replace the roots α by coroots in the definition above. The positive and negative sides of these hyperplanes are defined in the obvious way, replacing the equal sign in the definition by inequality signs. These hyperplanes make h * into a cell complex as follows: We call an intersection of k hyperplanes maximal if it has dimension n − k, and we denote by h * (n − k) the union of all maximal intersections of k hyperplanes. The set of j-cells then is given by all connected components of h * (j)\h * (j−1), with h * (−1) being the empty set. 8.3. Alcoves and facets. As usual, we call the n-cells alcoves, and lowerdimensional cells f acets. The (n−1)-cells which are in the closure of a given alcove A are called the walls of A. The fundamental alcove C ℓ is defined to be the unique alcove in the dominant Weyl chamber whose closure contains the origin 0. We say that a wall of C ℓ corresponds to the simple reflection s i if it is fixed by it. This defines a 1-1 correspondence between the walls of C ℓ and the simple reflections. We can now define a 1-1 correspondence between the alcoves and the elements w of the affine Weyl group as well as a labeling of the walls via simple reflections by induction on the length of w as follows: The element 1 corresponds to C ℓ . If the alcove A corresponds to the element w, and s i is a simple reflection such that ws i has greater length than w, then the alcove A ′ = As i corresponding to ws i is obtained by reflecting A in the wall labeled by s i . This reflection also defines the labeling of the walls of A ′ . Moreover, the action of the s i defines a right action of the affine Weyl group on the alcoves. The alcoves in the dominant Weyl chamber are in 1-1 correspondence with the shortest elements of the cosets of the finite Weyl group in the affine Weyl group. The Bruhat order then has the geometric interpretation that u < w is equivalent to the fact that whenever u(C ℓ ) and v(C ℓ ) lie on opposite sides of a hyperplane, u(C ℓ ) must be on the negative and w(C ℓ ) must be on the positive side of that hyperplane. We similarly define for two facets F 1 and F 2 that F 1 < F 2 if F 2 lies in or on the positive side of any hyperplane which contains F 1 .
For a given facet F , the stabiliser group W (F ) is the group generated by the reflections in the hyperplanes which contain F . We denote by C ℓ (F ) the unique alcove whose closure contains F , and which is on the positive side of every hyperplane which contains F . The set ∆(F ) denotes the positive roots corresponding to the hyperplanes which contain F and a wall of C ℓ (F ). By definition, C ℓ (F ) is on the positive side of each of these hyperplanes. We call the reflections corresponding to the roots in ∆(F ) the simple reflections of W (F ), and the roots in ∆ F the simple roots of W (F ). We also define the positive cone C + (F ) to be the region which is above all hyperplanes corresponding to the roots in ∆(F ). 8.4. Tilting modules and linkage. If the context is not specified, the statement holds for tilting modules of quantum groups at roots of unity as well as for tilting modules of algebraic groups in characteristic p. Let T (λ) be the unique indecomposable tilting module up to isomorphism whose highest weight is λ. We will use the well-known fact that if the Weyl module V (λ) is simple, it coincides with T (λ) and with the simple module L(λ) of highest weight λ.
Theorem 8.6. (Linkage Principle) The Weyl module V (µ) appears in a filtration of the tilting module T (λ) only if µ is in the orbit of λ under the affine Weyl group and µ ≤ λ in Bruhat order.
8.5. Minimal facets. The following lemma describes some tilting modules which are simple. We call a facet F minimal if it lies in the interior of the dominant Weyl chamber C and no other facet in its orbit which also lies in the interior of C can be smaller than it in Bruhat order. We then have the following easy lemma: Lemma 8.7. Let F be a minimal facet and let λ be an integral dominant weight such that λ + ρ ∈ F .
(a) In both the quantum group case and in the modular case, we have T (λ) = V (λ) and the nullity of T (λ) is equal to k(F ), the number of hyperplanes in which F lies.
(b) Consider the modular case in characteristic p. Let λ (r) = p r λ + (p r−1 − 1)ρ. Then T (λ (r) ) = V (λ (r) ) and the nullity of T (λ (r) ) is equal to r|R + | + k(F ). P roof. By definition of minimal facet and the linkage principle, there exists no dominant integral weight µ in the orbit of λ such that µ < λ in Bruhat order. Hence the only Weyl module which appears in the filtration of T (λ) is V (λ) itself. The statement about the nullity is a direct consequence of the dimension formula (for p > h in the modular case). The same argument also works for case (b), using Lemma 7.10. 8.6. Thick tensor ideals. Let I(F ) be the thick ideal generated by the tilting modules T (λ) with λ + ρ ∈ F . Recall that C + (F ) is the region consisting of all points x ∈ h * which are on the positive side of any hyperplane which contains F . Proposition 8.8. Let F be a minimal facet, and let λ and λ (r) be as in Lemma 8.7.
(a) In the quantum group case, the ideal I(F ) contains all modules T (ν) with ν + ρ in C + (F ). Any module in I(F ) has nullity ≥ k(F ).
(b) In the modular case, the ideal I(F (r) ) generated by all T (λ (r) ) with λ + ρ ∈ F contains all modules T (ν) with ν ∈ λ (r) + X + , where X + is the set of all dominant all dominant integral weights. Any module in I(F (r) ) has nullity at least r|R + | + k(F ).
Proof. Assume ν + ρ ∈ C + (F ). Then we can find a dominant integral weight λ such that λ + ρ ∈ F and ν − λ is a dominant weight. Hence the tensor product T (λ) ⊗ T (ν − λ) has highest weight ν. It follows that T ν ∈ I(F ), and hence has at least the nullity of F . Example 8.9. In the modular situation, the case λ = 0 corresponds to the Steinberg representations St r .
In order to get a more concrete description of these tensor ideals, it will be important to determine when two different facets generate the same tensor ideal.
Definition 8.10. Let F 1 and F 2 be minimal facets. We say that they are tensor equivalent if I(F 1 ) = I(F 2 ).
Lemma 8.11. Let I be the tensor ideal generated by the simple object T , and let S be an object in I with the same nullity as T . Then also S generates I. In particular, if F 1 and F 2 are tensor equivalent minimal facets, then k(F 1 ) = k(F 2 ).
Proof. By assumption, there exists an object W such that S ⊂ T ⊗ W . There exist maps ι : 1 → W ⊗ W * , d : W ⊗ W * → 1 such that d • ι = dim(W ). Moreover, by assumption, there exist morphisms ι S : S → T ⊗W and d S : T ⊗W → S such that d S •ι S = id S . Let a ∈ End(S) such that T r(a) has minimal nullity. We then define maps u : T → S ⊗ W * and v : S ⊗ W * → T by It follows that vu is an endomorphism of the simple object T , and hence vu = αid T for some scalar α. By functoriality of the trace operation, it follows from the definitions α dim(T ) = T r(vu) = T r(uv) = T r(a).
As T r(a) and dim(T ) have the same nullity, it follows that α is invertible. But then α −1 uv is an idempotent in End(S ⊗W ) whose image is isomorphic to T .
Remark 8.13. 1. In our cases, the nullity of the tensor ideal would correspond to the length of the longest element of the stabilizer of F . On the other hand, it is well-known that the longest element w 0 of a parabolic subgroup is in a cell for which the a-function is equal to the length of w 0 . For rank 2 and for type A n , these exhaust all two-sided cells.
2. All thick ideals are explicitly known in type A, see [Shi]. Each of them can be associated to a parabolic subgroup, and hence to a facet. However, there already seems to be a left cell for type D 4 whose value of the a-function, seven, would not be the length of the longest element of a parabolic subgroup of affine D 4 , see [DJ], [CCD].

Quantum and modular tilting modules in type A
For U q (sl n ) two-sided cells of the affine Weyl group are parametrized by Young diagrams λ of size n. We show that the thick ideal I(F 0 (λ)) agrees with the thick ideal attached to the two-sided cell by work of Shi and Ostrik. This also connects the nullity with the values of Lusztig's a-function. We propose a geometric description of the thick ideals for U q (sl n ) and for SL(n). We assume throughout that ℓ and p are bigger than the Coxeter number h, which is equal to n in our case. 9.1. Description of I(F )s. We would like to get an elementary geometric description of the region in which the dominant integral weights λ lie for which T (λ) is in I(F ), for a given minimal facet. For rank 2 these can be found in [L]. It turns out that the regions corresponding to cells in the dominant Weyl chamber given by Lusztig coincide with the regions described in Proposition 8.8. We expect that the regions described in Proposition 8.8 would also describe cells beyond rank 2. We will illustrate this below for some cases.
The cells for the affine Weyl groups of type A have been determined by Shi in [Shi]. Using Ostrik's results, this implies that the thick ideals in type A n−1 are labeled by the partitions λ of the set {1, 2, ..., n}. As usual, we write partitions λ = [λ 1 , λ 2 , ... λ r ] where the λ i are integers satisfying λ 1 ≥ λ 2 ≥ ... λ r > 0. We identify them with Young diagrams, where λ i indicates the number of boxes in the i-th row. We denote weights of sl n by the projections of n-tuples of integers into the plane of R n given by vectors whose coordinate sum is equal to 0. We will usually only write the n-tuple of integers for simplicity of notation. We will also use the notation l for λ + ρ.
Definition 9.1. Let ℓ be a positive integer, ℓ > n, and let s be the number of columns of λ. Then we write the expression (i − 1)ℓ + x j−1 into the box of λ in the i-th row and j-th column, where we have the convention 0 = x 0 < x 1 < ... < x s−1 < ℓ. The standard facet F 0 (λ) consists of all n-tuples whose coordinates consist of all possible arrangements of numbers in the boxes of λ, written in descending order.
Definition 9.2. We call a facet F strongly minimal if F ′ < F for any F ′ with W (F ′ ) ∼ = W (F ). Here all facets are assumed to be in the interior of C.
Definition 9.3. Let F be a facet. We call α a root of the stabilizer W (F ) if there exists an integer n α such that (α, x) = n α ℓ for every x ∈ F . We call a root α positive if (α, x) > 0 for all x ∈ F . Finally, we call a collection R + F = {α i } of positive roots of W (F ) a set of simple roots of F if every positive root of W (F ) can be uniquely written as a linear combination of the α i s with nonnegative integer coefficients.
Remark 9.4. The simple roots of W (F ) allow us an easy description of the region C + (F ): If (α, y) = n α ℓ for all y ∈ F , then C + (F ) consists of all points x ∈ h * such that (α, x) ≥ n α ℓ for all simple α ∈ W (F ). It is tempting to speculate whether a description of the tensor ideal I(F ) can be given via the region where it would be enough to only consider strongly minimal facets F ′ on the right hand side. We study this question in more detail for type A in the following sections. 9.2. More facets. Let y be a point in the interior or lower closure of a given alcove. The following definition will be useful for Shi's algorithm for identifying cells which will be used later.
Definition 9.5. Let y be a point in the interior or lower closure of a given alcove. We call a subset {y i , i ∈ I} of the coordinates of y a y-chain if |y i − y j | ≥ ℓ for all i, j ∈ I. We call it an ℓ-strict (or just strict) y-chain if its elements are of the form y i − rℓ for 0 ≤ r < |I| − 1.
Observe that if y is a point in the standard facet F 0 (λ), the coordinates of y can be written as a disjoint union j Γ j of strict chains where |Γ j | = λ T j . Namely Γ j consists of all the coordinates of y which are congruent to x j−1 mod ℓ, where x 0 = 0.
There is a more general procedure to produce facets from any standard tableau of shape λ, i.e. for any filling of the boxes of λ such that the numbers increase along the rows and down the columns. Let i(r, c) be the number in the box in the r-th row and c-th column. Then we define the facet F t by the points y satisfying the equalities y i(r,c) − y i(r+1,c) = ℓ for any pair of boxes (r, c) and (r + 1, c) of our tableau. In general, these equations may describe a collection of facets. In this case, we pick the lowest one. It can be obtained by adding the additional inequalities y i − y i+1 < ℓ, 1 ≤ i < n. We now say that if F = F t for some standard tableau of shape λ. We now also define the region D(λ) by Remark 9.6. 1. Not every minimal facet is of the form F t for a standard tableau. E.g. for sl 3 the facet given by y 2 −y 3 = ℓ and y 1 −y 2 < ℓ is minimal, but can not be defined via a standard tableau t. However, we will see that any strongly minimal facet can be obtained from a standard tableau. 2. If 1 ≤ i ≤ n, we defineī = n+1−i. If a root α is given by α(y) = y i −y j , we define the rootᾱ byᾱ(y) = yj − yī. If F is a facet defined by α(y) = ℓ for certain roots α, we analogously define the facetF by the same equalities replacing each α byᾱ. We leave it to the reader to check that F 0 (λ) =F t , where t is the standard tableau obtained by filling the boxes of λ row by row.
9.3. Identifying minimal facets and their cells. Shi has given several algorithms how to identify the 2-sided cell to which a given alcove belongs. We review one of the here, and another one in the next section, following the presentation in [Co08], [Co10].
Given a point y in the interior of the dominant Weyl chamber, define a Young diagram µ = µ(y) as follows: µ 1 is the maximum of |Γ 1 |, for all possible y-chains Γ 1 . Assuming we know µ 1 up to µ i we then define µ i+1 by the condition where the Γ j s are mutually disjoint y-chains, and the maximum is taken over all possible collection of i + 1 disjoint y-chains.
Proposition 9.7. If y is a point on a facet F ∼ F 0 (λ) for the Young diagram λ, we have µ(y) = λ T . Moreover, F 0 (λ) is a minimal facet.
Proof. Let y be a point on the facet F . Let Γ i be the chain consisting of all the entries y i which are congruent to i − 1 mod ℓ; by construction it has λ T i elements. The claim would follow if we can show that for any other collection of mutually disjoint y-chainsΓ j we have Let I(m) be the number of indices k for which mℓ ≤ y k < (m + 1)ℓ. Then obviously the number of such indices in a disjoint union of i chains is less or equal to the minimum of i and I(m). Summing over all m from 0 to n shows that the largest possible number we can get is the number of boxes of λ in its first i columns. As by induction assumption µ j = λ T j for 1 ≤ j ≤ i, it follows that µ(y) = λ T .
To prove the second claim, let y be the point on F 0 (λ) with x j = j − 1. Observe that any pointỹ ∈ W.y ∪ C withỹ ≤ y needs to have the same L 1 norm, it needs to have the same number of coordinates congruent i mod ℓ as y for each i, and its coordinates need to be positive and strictly decreasing. By construction, the coordinates of y are the smallest possible numbers subject to these constraints. Hence ifỹ = y, we could find an i such that i j=1ỹ j > i j=1 y j , contradictingỹ ≤ y. 9.4. Description of D(λ). We now describe a second way how to determine the two-sided cell to which a given alcove in the dominant Weyl chamber belongs. It is also due to Shi. Given a point y in the dominant Weyl chamber, we construct a standard tableau t y as follows: We start with putting the number 1 into the top-left box. We then add the box with the number 2 on the right if y 1 − y 2 < ℓ, and we add it below the first box if y 1 − y 2 ≥ ℓ. Having placed boxes with the numbers 1 until i, we add the box containing i + 1 at the bottom of the left-most column such that y r − y i+1 ≥ ℓ, where r is the number in the lowest box of that column. Then one can show (see e.g. [Co08], Section 3.2 and 3.2): Proposition 9.8. If y is a dominant integral weight, the procedure above constructs a standard tableau t y . If λ is the associated Young diagram, and y is in the lower closure of the alcove A, then A is in the two-sided cell labeled by λ T .
Theorem 9.9. The indecomposable module T (ν) is in the ideal I(λ) generated by the facet F 0 (λ) if and only if ν + ρ is in the region D(λ) as defined in 3.
Proof. It follows from Proposition 9.7 that whenever ν + ρ ∈ F with F ∼ F 0 (λ), then T (ν) ∈ I(λ). Hence T (ν) ∈ I(λ) whenever ν + ρ ∈ D(λ) by Proposition 8.8. To prove the other inclusion, let T (ν) be an indecomposable tilting module with highest weight ν such that ν +ρ is in an alcove belonging to the two-sided cell labeled by λ T . Then we obtain a tableau t y of shape λ, where y = ν + ρ. Let i(r, c) be the number in the box in the r-th row and c-th column. Then by construction y i(r,c) − y i(r+1,c) ≥ ℓ. Define F as the set of points x such that x i(r,c) − x i(r+1,c) = ℓ. This is exactly the facet obtained from t y , see the discussion above 2. Then obviously y ∈ C + (F ). 9.5. Examples. In the following we list the strongly minimal facets F conjugate (as defined in 2) to F 0 (λ) for sl n , n ≤ 5, for each Young diagram λ with n boxes. We use the convention that 0 < x 1 < x 2 < ... < ℓ. The region C + (F ) is then given by all points y satisfying (y, α) ≥ ℓ for all roots α listed under simple roots. We start with sl 3 , where all strongly minimal facets are standard facets: In the following, we list all strongly minimal facets for sl 4 . Observe that we also have a strongly minimal facet which is not a standard facet for the diagram [2, 1 2 ].
It is well-known that it is sufficient (in type A) to compute the value of the a-function in the finite case W n = S n . By [EH17] a(λ T ) = r(λ) where r(λ) is the row number of λ, the sum over the row numbers of the boxes in the Young diagram where the row number of a box in the i-th row is i − 1 (note that we use λ T for the partition labeled by λ in [EH17]).
Theorem 9.11. The thick ideal I(λ) = I(F 0 (λ)) generated by the tilting modules T (ν) for which ν + ρ ∈ F 0 (λ) coincides with the thick ideal constructed by Ostrik for the cell in the dominant Weyl chamber corresponding to the two-sided cell labeled by the Young diagram λ T . In particular, the nullity of any generating module T (ν) of that ideal is equal to the a-function of that cell. Moreover, that thick ideal contains all tilting modules T (ν) for which ν + ρ ∈ D(λ), as defined below (2).
Proof. Let T (ν) be a tilting module for which y = ν + ρ ∈ F 0 (λ). As F 0 (λ) is a minimal facet by Proposition 9.7, T (λ) = V (λ). Hence its nullity can be determined from its dimension, see Corollary 7.5. By construction of F 0 (λ) y i is congruent y j mod ℓ if y i and y j are in the same column of λ. Hence the nullity is given by i , which is equal to the row number r(λ). This shows that the nullity of F 0 (λ) coincides with the a-function.
We have seen in Proposition 9.7 that if y ∈ F with F ∼ F 0 (λ) then µ(y) = λ T . Hence any module T (ν) with ν + ρ ∈ F is also in I(λ). The last statement now follows from Proposition 8.8.
Remark 9.12. It was shown by Ostrik for ℓ a large enough prime (see [O01] [O98]) that the dimension of any tilting module corresponding to a two-sided cell A is divisible by ℓ a(A) and for any cell there exists a tilting module such that its dimension is not divisible by ℓ a(A)+1 . Hence the relationship between nullity and a-function in Theorem 9.11 would hold whenever one can show that the nullity of such a module is given by its dimension. So it would seem reasonable to expect this to be true in general. But see the example in Section 7.5.2 where the nullity of a modular tilting module is not given by its dimension.
Remark 9.13. We were able to give a fairly explicit description of the tensor ideals thanks to the work of Shi [Shi]. In the formulation as D(F ), see Remark 9.4, our approach might also be useful to characterize tensor ideals for other Lie types. E.g. this formulation works for all rank 2 cases, but not for all ideals for type D 4 , see Remark 8.13.
Remark 9.14. The thick ideal N k is the sum of the I(λ) (λ partition of n) for which the nullity is ≥ k. 9.7. Regions for the modular case. We can use the results from the previous subsection also for the modular case; to conform with the usual notations, we replace ℓ by the prime p. In addition, we can use the "telescoping effect" in Lemma 8.7 (b) and Proposition 8.8 (b) to construct infinitely many thick ideals. We will also extend our conjectural description of these ideals and the related cells to this case. We use the same notation as in the previous subsection. Let F 0 (λ) be the standard minimal facet associated to the Young diagram λ, and let r be an integer, r ≥ 1. Then we construct the region D r (λ) by D 1 (λ) = D(λ) and D r (λ) = F ∼F 0 (λ) ( λ∈F λ (r) + X + ), r > 1 where λ (r) is as in Lemma 8.7 (and λ is interpreted as a weight), and the first union goes over all strongly minimal facets F which are equivalent to F 0 (λ). The following proposition again follows from Lemma 8.7 and Proposition 8.8.
Question 9.16. Is every thick ideal in type A a sum of ideals of the form D r (λ)?
An affirmative answer would give a geometric description of the right p-cells in the affine Weyl group.
Remark 9.17. 1. It is easy to see for r ≥ 1 that D r (∅) = D r−1 ([1 k ]). We will identify these regions in the following.
2. For type A 1 the thick ideals are generated by the Steinberg modules. It is easy to see that our regions do describe the thick ideals in this case. An explicit description of the tensor ideals has also been given for type A 2 , see section 9.7.1. One checks easily that this is compatible with our regions. 9.7.1. The modular A 2 -case. By [An04, Example 15] [Je17, Chapter 10] the thick ideals are given by the the Steinberg cells < St r >= T (≤ c r 1 ) and the tilting modules associated to the weight cell c r 2 = Y r \ (Y r+1 ∪ c r 1 ) . A beautiful picture illustrating the p = 5-case can be found in [An04] .

Questions
10.1. Extension to more general categories. It would be interesting to extend the definiton of k-negligible morphisms or ideals to other monoidal categories or supercategories.
10.2. q-Deligne categories at roots of unity. k-negligible ideals can be also defined for the q-versions of the Deligne categories in type ABCD. For generic q the classification of tensor ideals and thick ideals for these categories is the same as for the classical Deligne categories [Br17] [Co18], but is unknown for q a root of unity.
10.3. Modified traces. Currently we define modified traces only if the maximal ideal has one generator. It would be interesting to define modified traces if the maximal ideal is not principal. We also expect that our construction defines modified traces for other categories.
10.4. a-Function in the quantum and modular case. We expect an equality between the nullity and the a-function in the quantum case for ℓ large enough for all types (i.e. N k would correspond to the union of weight cells with a-value ≥ k for ℓ a large enough prime). There does not seem to be an accepted definition for the a-function in the modular case if the Kazhdan-Lusztig basis is replaced with the p-canonical basis. One might wonder if there is again a connection with the nullity.