Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\ell>0$ or a quantum group at an $\ell$-th root of unity. We define a tensor ideal of singular $\mathbf{G}$-modules in the category $\mathrm{Rep}(\mathbf{G})$ of finite-dimensional $\mathbf{G}$-modules and study the associated quotient category $\mathrm{\underline{Re}p}(\mathbf{G})$, called the regular quotient. Our main results are a 'linkage principle' and a 'translation principle' for tensor products: Let $\mathrm{\underline{Re}p}_0(\mathbf{G})$ be the essential image in $\mathrm{\underline{Re}p}(\mathbf{G})$ of the principal block of $\mathrm{Rep}(\mathbf{G})$. We first show that $\mathrm{\underline{Re}p}_0(\mathbf{G})$ is closed under tensor products in $\mathrm{\underline{Re}p}(\mathbf{G})$. Then we prove that the monoidal structure of $\mathrm{\underline{Re}p}(\mathbf{G})$ is governed to a large extent by the monoidal structure of $\mathrm{\underline{Re}p}_0(\mathbf{G})$. These results can be combined to give an external tensor product decomposition $\mathrm{\underline{Re}p}(\mathbf{G}) \cong \mathrm{Ver}(\mathbf{G}) \boxtimes \mathrm{\underline{Re}p}_0(\mathbf{G})$, where $\mathrm{Ver}(\mathbf{G})$ denotes the Verlinde category of $\mathbf{G}$.


Introduction
In the representation theory of groups and of Hopf algebras, it is often helpful to decompose categories of representations as a direct sum of blocks.One can then hope to obtain stronger or more fine-grained results by considering one block of the category at a time.However, this strategy is generally not well suited for understanding the monoidal structure of the category, because a tensor product of two representations, each belonging to a given block, may have indecomposable direct summands in many different blocks.The main results of this article provide a way of partially overcoming this obstacle, for categories of representations of simple algebraic groups (over fields of positive characteristic) and quantum groups (at roots of unity).More precisely, we use minimal tilting complexes (see [Gru22a]) to define a tensor ideal of singular modules in the representation categories.When considering representations modulo this tensor ideal, it turns out that the principal block is closed under tensor products and that the monoidal structure of the entire category is governed to a large extent by the resulting monoidal structure on the principal block.We refer to these results as a linkage principle and a translation principle for tensor products, in analogy with the classical results describing the block decomposition of the categories in question (due to H.H. Andersen [And80] and J.C. Jantzen [Jan74]).In the following, we briefly recall linkage and translation before discussing our results in more detail.
The categories of (finite-dimensional) representations of simple algebraic groups and of quantum groups have many structural properties in common, which often makes it possible to treat the two cases simultaneously.We refer to the representation theory of simple algebraic groups as the modular case and to the representation theory of quantum groups as the quantum case, and we fix the following notational conventions:

The modular case
Here G is a simply connected simple linear algebraic group over an algebraically closed field of characteristic ℓ > 0. We write Rep(G) for the category of finite-dimensional rational G-modules.
The quantum case Here G = U ζ (g) is the specialization at a complex ℓ-th root of unity ζ of Lusztig's divided powers version of the quantum group corresponding to a complex simple Lie algebra g.We write Rep(G) for the category of finite-dimensional G-modules of type 1.
In either of the two cases, G comes equipped with a simple root system Φ and a weight lattice X.For this introduction (and for most of this article), we suppose that ℓ ≥ h, the Coxeter number of Φ.In the quantum case, we further assume that ℓ is odd (and not divisible by 3 if Φ is of type G 2 ).From now on, we use the term G-module to refer to the objects of Rep(G); in particular, all G-modules that we consider are implicitly assumed to be finite-dimensional.The category Rep(G) is a highest weight category with weight poset X + ⊆ X the set of dominant weights with respect to a fixed positive system Φ + ⊆ Φ, and we denote by the simple G-module, the standard module, the costandard module and the indecomposable tilting G-module of highest weight λ ∈ X + .Let W fin and W aff = ZΦ⋊W fin be the finite Weyl group and the affine Weyl group of Φ, respectively, and denote the natural embedding ZΦ → W aff by γ → t γ .We consider the ℓ-dilated dot action of W aff on X, defined by t γ w • λ = w(λ + ρ) − ρ + ℓγ for γ ∈ ZΦ, w ∈ W fin and λ ∈ X, where ρ = 1 2 α∈Φ + α.Finally, we write C fund for the fundamental alcove in X R = X ⊗ Z R (see Subsection 1.5).For a weight λ ∈ C fund ∩ X, the linkage class Rep λ (G) of λ is the full subcategory of Rep(G) whose objects are the G-modules all of whose composition factors have highest weight in X + ∩ W aff • λ.The linkage principle asserts that Rep(G) admits a decomposition The linkage class Rep 0 (G) is called the principal block of G. 1 Furthermore, for λ, µ ∈ C fund ∩ X, the linkage classes of λ and µ are related via a translation functor which is an equivalence of categories with quasi-inverse T λ µ .Our linkage principle and translation principle for tensor products build on the notion of singular G-modules, which we define below, using minimal tilting complexes and negligible tilting modules.As explained in [Gru22a], associated to every G-module M , there is a unique (up to isomorphism) bounded minimal complex C min (M ) of tilting G-modules, called the minimal tilting complex of M , such that A tilting G-module is called negligible if it has no direct summands T (λ) with λ ∈ C fund ∩ X.It is well-known that the negligible tilting modules form a tensor ideal in the category of tilting G-modules (see [GM94,AP95]).Now we are ready to give the key definition: Definition.A G-module M is called singular if all terms of C min (M ) are negligible.Otherwise, we say that M is regular. 2 1 Not all of the linkage classes are blocks (in the usual sense that they can not be decomposed any further), but those corresponding to weights in C fund ∩ X are.A precise description of the blocks of Rep(G) can be found in Section II.7.2 of [Jan03].
2 Our terminology is justified by the fact that, for λ ∈ X + , the simple G-module L(λ) is regular if and only if its highest weight λ is ℓ-regular, i.e. if λ ∈ W aff • λ ′ for some λ ′ ∈ C fund ∩ X (see Lemma 3.3 below).
By results of [Gru22b], the singular G-modules form a tensor ideal in Rep(G).For every G-module M , we can write M ∼ = M sing ⊕ M reg , where M sing is the direct sum of all singular indecomposable direct summands of M and where M reg is the direct sum of all regular indecomposable direct summands of M (for a fixed Krull-Schmidt decomposition).By the Krull-Schmidt theorem, the G-modules M sing and M reg are well-defined up to isomorphism, but there may be no canonical choice of direct sum decomposition M ∼ = M sing ⊕ M reg .We call M sing and M reg the singular part and the regular part of M , respectively.
We can now state our linkage principle for tensor products, which asserts that the monoidal structure of Rep(G) is compatible with the decomposition into linkage classes, when we consider Rep(G) modulo the tensor ideal of singular G-modules.Suppose for the rest of the introduction that ℓ ≥ h, the Coxeter number of G (so that C fund ∩ X is non-empty).
Theorem A. Let λ ∈ C fund ∩ X and let M and N be G-modules in the linkage classes of 0 and λ, respectively.Then (M ⊗ N ) reg belongs to the linkage class of λ.
Note that for G-modules M and N in Rep 0 (G), Theorem A implies that (M ⊗ N ) reg also belongs to Rep 0 (G).The next result is our translation principle for tensor products.
The translation functors T λ 0 and T µ 0 are equivalences for λ, µ ∈ C fund ∩ X, so Theorem B implies that the monoidal structure of Rep(G) modulo singular G-modules is completely determined by the monoidal structure of Rep 0 (G) modulo singular G-modules.Therefore, much like the classical linkage principle and translation principle, the linkage principle and translation principle for tensor products can be used to reduce questions about the structure of tensor products of arbitrary G-modules (modulo singular direct summands) to questions about the structure of tensor products of G-modules in the principal block (see also Remark 3.9 below).We point out that the coefficients c ν λ,µ in Theorem B are the structure constants of the Verlinde category Ver(G) (i.e. the quotient of the category of tilting G-modules by the tensor ideal of negligible tilting modules) and that they can be computed as an alternating sum of dimensions of weight spaces of Weyl modules (see Subsection 1.6).

Now let us write
Here, the box product denotes the additive closure of the category whose objects are pairs consisting of a tilting G-module and a G-module with regular part in Rep 0 (G), and whose Hom-spaces are tensor products of Hom-spaces in Ver(G) and Rep 0 (G).See Section 6 for more details.
To conclude the introduction, we indicate how the paper is organized.In Section 1, we set up our notation and summarize relevant results about representations of algebraic groups and quantum groups.In Section 2, we recall results about minimal tilting complexes from [Gru22a,Gru22b] and we investigate in detail the minimal tilting complexes of Weyl modules and simple G-modules (see Propositions 2.4 and 2.5).In Section 3, we define the tensor ideal of singular G-modules and study the associated quotient category.We first prove more functorial versions of Theorems A and B in Lemma 3.4 and Theorem 3.7 and then prove the versions of these theorems that are stated above in Lemma 3.12 and Theorem 3.14.In Section 4, we discuss a notion of strong regularity, which is more well-behaved with respect to tensor products than the notion of regularity defined above.This allows us to generalize results of A. Parker [Par03] about the good filtration dimension of Weyl modules and simple G-modules to iterated tensor products these modules in Theorem 4.5.As an application of our results that may be of independent interest, we use singular G-modules in Section 5 to prove that a composition of translation functors between linkage classes corresponding to weights in C fund ∩ X is naturally isomorphic to a translation functor.Finally, in Section 6, we prove Theorem C and discuss some related external tensor product decompositions, and in Section 7 we give some examples of regular parts of tensor products for G of type A 2 .
1. Preliminaries 1.1.Roots and weights.Let Φ be a simple root system in a euclidean space X R with scalar product (− , −).For α ∈ Φ, we denote by α ∨ = 2α (α,α) the coroot of α.The weight lattice of Φ is and the Weyl group of Φ is the (finite) subgroup The index of the root lattice ZΦ in the weight lattice X is finite, and the quotient X/ZΦ is called the fundamental group of Φ.Now fix a positive system Φ + ⊆ Φ corresponding to a base Π of Φ, and let be the set of dominant weights with respect to Φ + .We consider the partial order on X that is defined by λ ≥ µ if and only if λ − µ is a non-negative integer linear combination of positive roots.Furthermore, we write αh and α h for the highest root and the highest short root in Φ + , respectively, with the convention that αh = α h (and that all roots are short) if Φ is simply laced.We let ρ = 1 2 α∈Φ + α be the half-sum of all positive roots and write h = (ρ, α ∨ h ) + 1 for the Coxeter number of Φ.The Weyl group W fin is a Coxeter group with simple reflections S fin = {s α | α ∈ Π}, and we write w 0 ∈ W fin for its longest element.1.2.Algebraic groups and quantum groups.The root system Φ is at the heart of the structure of two kinds of Lie theoretic objects whose finite-dimensional simple modules are canonically indexed by X + : simple algebraic groups (over a field of positive characteristic) and quantum groups (at a root of unity).The representation theory of quantum groups parallels that of algebraic groups to a large extent, so we will mostly treat the two cases simultaneously.When a distinction becomes necessary, we refer to the representation theory of the algebraic group as the modular case and to the representation theory of the quantum group as the quantum case.
The modular case.We follow the notational conventions from Section II.1 in [Jan03].Let G Z be a split simply-connected simple algebraic group scheme over Z with split maximal torus T Z , such that the root system of G Z with respect to T Z is isomorphic to Φ.The positive system Φ + ⊆ Φ determines a Borel subgroup B + Z , and the negative roots −Φ + determine a Borel subgroup B Z .We fix an algebraically closed field k of characteristic ℓ > 0 and denote by G = G k the simply-connected simple algebraic group scheme over k corresponding to G Z , with maximal torus T = T k and Borel subgroup B = B k .
The quantum case.We follow the conventions of Sections II.H.1-II.H.6 in [Jan03].Let g be the complex simple Lie algebra with root system Φ, let U q (g) be its quantized enveloping algebra and let U Z q (g) be Lusztig's integral form of U q (g), a Z[q, q −1 ]-subalgebra of U q (g) generated by divided powers.Now let ζ ∈ C be a primitive ℓ-th root of unity, where ℓ ∈ Z ≥0 is odd and ℓ = 3 if Φ is of type G 2 .Then there is a unique ring homomorphism Z[q, q −1 ] → C with q → ζ, and we define q (g) as the extension of scalars of U Z q (g) via this homomorphism.In order to be able to treat the quantum case and the modular case simultaneously, we write k = C and G = U ζ (g) in the quantum case.
1.3.Representation categories.In the modular case, we write Rep(G) for the category of finitedimensional G-modules (in the sense of Section I.2.3 in [Jan03]), and in the quantum case we write Rep(G) for the category of finite-dimensional G-modules of type 1 (see Section II.H.10 in [Jan03]).In either case, we will from now on refer to the objects of Rep(G) as G-modules.For two G-modules M and N , we write Hom G (M, N ) for the space of homomorphisms from M to N in Rep(G) and Ext i G (M, N ) for the Ext-groups.If N is indecomposable then we denote by [M : N ] ⊕ the multiplicity of N in a Krull-Schmidt decomposition of M .
The category Rep(G) is a highest weight category with weight poset (X + , ≤) and we write for the simple G-module, the costandard module, the standard module and the indecomposable tilting G-module of highest weight λ ∈ X + , respectively.The costandard modules are also called induced modules and the standard modules are called Weyl modules.A G-module is said to have a good filtration (or a Weyl filtration) if it has a filtration with successive quotients isomorphic to induced modules (or Weyl modules), and a tilting G-module is (by definition) a G-module that admits both a good filtration and a Weyl filtration.Every tilting G-module decomposes as a direct sum of indecomposable tilting G-modules T (λ) with λ ∈ X + .We write Tilt(G) for the full subcategory of tilting G-modules in Rep(G).
The category Rep(G) also has a rigid monoidal structure and a braiding.In the modular case, this follows from Section I.2.7 in [Jan03], and the braiding is induced by the canonical symmetric braiding on the category of k-vector spaces.In the quantum case, Rep(G) is rigid monoidal because G is a Hopf algebra, and the braiding is constructed in [Lus10,Chapter 32].For all λ ∈ X + , we have ∆(λ) * ∼ = ∇ − w 0 (λ) , and a tensor product of two induced modules has a good filtration by results of S. Donkin [Don85], O. Mathieu [Mat90] and J. Paradowski [Par94].This also implies that Tilt(G) is closed under tensor products in Rep(G).
1.4.Affine Weyl group and alcove geometry.The affine Weyl group W aff and the extended affine Weyl group W ext of G are defined by The canonical embedding of X into W ext is denoted by λ → t λ .Since the action of W fin on X/ZΦ is trivial, W aff is a normal subgroup of W ext and W ext /W aff ∼ = X/ZΦ.Furthermore, W aff is a Coxeter group with simple reflections S = {s α | α ∈ Π} ⊔ {s 0 }, where s 0 = t α h s α h , and we write ℓ : W aff → Z ≥0 for the length function with respect to S.
We consider the ℓ-dilated dot action of W ext on X R given by for λ ∈ X, w ∈ W fin and x ∈ X R .For β ∈ Φ + and m ∈ Z, the fixed points of the affine reflection t mβ s β form an affine hyperplane and we call any connected component of X R \ β,m H β,m an alcove.A weight λ ∈ X is called ℓ-singular if λ ∈ H β,m for some β ∈ Φ + and m ∈ Z and ℓ-regular if λ ∈ C for some alcove C ⊆ X R .The group W aff acts simply transitively on the set of alcoves, and for every alcove C ⊆ X R , the closure C is a fundamental domain for the ℓ-dilated dot action of W aff on X R and C ∩ X is a fundamental domain for the ℓ-dilated dot action of W aff on X.The fundamental alcove is Let us write Ω = Stab Wext (C fund ) for the stabilizer of C fund in W ext .As C fund is a fundamental domain for the action of W aff on X R , we have W ext = W aff ⋊ Ω and We write x → ω x for the canonical epimorphism W ext → Ω with kernel W aff .
As W aff acts simply transitively on the set of alcoves, there is for every element • C fund , and we extend the length function on It is well-known that every (left or right) W fin -coset in W ext has a unique element of minimal length, and that we have 1.5.Linkage and translation.For λ ∈ C fund ∩ X, the linkage class Rep λ (G) of λ is the full subcategory of Rep(G) whose objects are the G-modules all of whose composition factors are of the form where pr λ M denotes the unique largest submodule of M that belongs to Rep λ (G); see Sections II.7.1-3 in [Jan03].This gives rise to a decomposition with projection functors pr λ : Rep(G) → Rep λ (G).The linkage class Rep 0 (G) containing the trivial G-module L(0) ∼ = k is called the principal block of G, and we call the extended principal block.
For λ, µ ∈ C fund ∩ X, let ν ∈ X + be the unique dominant weight in the W fin -orbit of µ − λ and define the translation functor from Rep λ (G) to Rep µ (G) via As observed in Sections II.7.6-7 in [Jan03], the simple module L(ν) in the definition of T µ λ can be replaced by any G-module of highest weight ν, such as ∇(ν), ∆(ν) or T (ν), without changing T µ λ (up to a natural isomorphism).If λ and µ are ℓ-regular then T µ λ is an equivalence with aff , by Propositions II.7.9, II.7.11 and II.7.15 in [Jan03].
1.6.Negligible tilting modules.Suppose from now on that ℓ ≥ h, the Coxeter number of G, so that By results of H.H. Andersen and J. Pardowski [AP95], the set N of negligible tilting modules forms a thick tensor ideal in Tilt(G), that is, N is closed under isomorphism, direct sums, retracts, and tensor products with arbitrary tilting G-modules.The quotient category 3 Ver(G) = Tilt(G)/N is a semisimple monoidal category called the Verlinde category or the semisimplification of Rep(G), cf.[EO18].The tilting modules T (λ) with λ ∈ C fund ∩X form a set of representatives for the isomorphism classes of indecomposable objects in Ver(G), and we write for the structure constants of Ver(G).By Proposition II.E.12 in [Jan03], we also have For later use, we need to establish two elementary properties of these structure constants.We first prove that they are invariant under the action of Ω = Stab Wext (C fund ) on C fund , in the following sense: In particular, we have Proof.As conjugation by ω is an automorphism of W aff , we have Writing ω = t γ w with γ ∈ X and w ∈ W fin , it is straightforward to see that and therefore dim ∆(λ Lemma 1.2.Let λ, µ ∈ C fund ∩ X and denote by ν be the unique dominant weight in the W fin -orbit of w 0 λ + µ.Then ν ∈ C fund ∩ X and c ν λ,µ = 0. Proof.As −w 0 ν is the unique dominant weight in the W fin -orbit of −w 0 λ − µ, we have and as the negligible tilting modules form a thick tensor ideal in Tilt(G), it follows that T (ν ′ ) is non-negligible and ν ′ ∈ C fund ∩ X.Furthermore, the existence of a non-zero homomorphism from the trivial G-module T (0) to the tensor product 3 By the quotient category Tilt(G)/N , we mean the category whose objects are the tilting G-modules, and where the Hom-space for two tilting G-modules T and T ′ is the quotient of HomG(T, T ′ ) by the subspace of homomorphisms that factor through a negligible tilting module.

Minimal tilting complexes
In this section, we recall the theory of minimal tilting complexes explained in [Gru22a].We write C b Tilt(G) for the category of bounded complexes of tilting G-modules, K b Tilt(G) for the bounded homotopy category of Tilt(G) and D b Rep(G) for the bounded derived category of Rep(G).Since Rep(G) is a highest weight category, the canonical functor is an equivalence of triangulated categories.Thus, for every bounded complex X of G-modules, there is a unique homotopy class of bounded complexes C of tilting G-modules such that C ∼ = X in the derived category.Furthermore, as Tilt(G) is a Krull-Schmidt category, every homotopy class of bounded complexes of tilting G-modules contains a unique minimal complex (up to isomorphism of complexes).The minimal tilting complex C min (X) of X is the unique (up to isomorphism) bounded minimal complex of tilting G-modules that is isomorphic to X in the derived category.The minimal tilting complex C min (M ) of a G-module M is defined as C min (0 → M → 0), with M in degree zero.By construction, C min (M ) is the unique bounded minimal complex of tilting G-modules with We next recall some elementary properties of minimal tilting complexes from [Gru22a] and [Gru22b].
(1) If M is a tilting module then C min (M ) = M , viewed as a one-term complex with M in degree 0.
(2) We have Proof.Parts (1)-(3) are Remark 2.10 and Lemma 2.12 in [Gru22a], and part (4) is Lemma 1.3 in [Gru22b].Let us introduce an additional piece of notation: For G-modules M and N , we write M ⊕ ⊆ N if there exists a split monomorphism from M into N .The next result is Lemma 2.17 in [Gru22a].
Now we proceed to study the minimal complexes of some specific G-modules.Let us assume from now on that ℓ ≥ h, the Coxeter number of G, and recall that we write x → ω x for the canonical epimorphism

Then
(1) T i = 0 for all i < 0 and all i > ℓ(x); (2 x , we have x ′ ∈ W + aff and ℓ(x ′ ) = ℓ(x).Hence, after replacing x by x ′ and λ by ω x • λ, we may (and shall) assume that x ∈ W + aff and ω x = e.We prove the claims by induction on ℓ(x).If ℓ(x) = 0 then x = e and ∆(λ) ∼ = T (λ), so ∆(λ) has minimal tilting complex 0 → T (λ) → 0 and all claims are satisfied.Now suppose that ℓ(x) > 0 and that the proposition holds for all y ∈ W + aff with ℓ(y) < ℓ(x).Then we can choose a simple reflection s ∈ S with xs ∈ W + aff and xs , s}, and consider the short exact sequence The short exact sequence (2.1) gives rise to a distinguished triangle By the induction hypothesis, we may assume that and that all weights ν ∈ X + with [B i : T (ν)] ⊕ = 0 for some i ∈ Z are of the form y • λ for some y ∈ W + aff with 0 ≤ i ≤ ℓ(xs) − ℓ(y).By Proposition 7.11 in [Jan03], we have for all i ∈ Z, and it follows that A i = 0 for i < 0 and i > ℓ(x) − 1.Further note that all tilting modules in Rep µ (G) are negligible because µ / ∈ C fund and that the translation functor T λ µ sends negligible tilting modules to negligible tilting modules, because negligible tilting modules form a thick tensor ideal in Tilt(G).It follows that the functor T λ µ • T µ λ sends all tilting modules to negligible tilting modules, so T λ µ T µ λ B i and A i are negligible for all i ∈ Z.We conclude that C i = A i ⊕ B i−1 = 0 for i < 0 and i > ℓ(x), that C i is negligible for all i = ℓ(x) and that As T i is a direct summand of C i for all i ∈ Z, this implies that T i = 0 for i < 0 and i > ℓ(x) and that T i is negligible for all i = ℓ(x).Furthermore, Lemma 2.3 yields is negligible and C ℓ(x)+1 = 0, and we conclude that T ℓ(x) ∼ = T (λ).Now suppose that ν ∈ X + such that aff with ℓ(z) ≤ ℓ(y) + 1, and it follows that (in either case) there is y It remains to show that T 0 ∼ = T (x • λ).By Section II.E.4 in [Jan03], there is a short exact sequence where M is a G-module with a Weyl filtration.By Corollary 2.16 in [Gru22a], we have C min (M ) i = 0 for i < 0, and using Lemma 2.3 as above, we obtain Proposition 2.5.Let x ∈ W + ext and λ ∈ C fund ∩ X, and write C min L(x • λ) as (2) Proof.As in the proof of Proposition 2.4, we can replace x by xω −1 x ∈ W + aff and λ by ω x •λ ∈ C fund ∩ X, so we will henceforth assume that x ∈ W + aff and ω x = e.By Sections II.2.12 and II.E.6 in [Jan03], there is a contravariant duality functor M → M τ on Rep(G) (denoted by M → τ M in [Jan03]) which fixes all simple G-modules and all tilting G-modules.(In the quantum case, this functor can be constructed using the involution ω from Lemma 4.6 in [Jan96].)Thus, the complex is a minimal tilting complex of L(x • λ), and by uniqueness, we have T i ∼ = T τ −i ∼ = T −i for all i.We prove the remaining claims by induction on ℓ(x).If ℓ(x) = 0 then x = e and L(λ) ∼ = T (λ), so L(λ) has minimal tilting complex 0 → T (λ) → 0 and all claims are satisfied.Now suppose that ℓ(x) > 0 and that the proposition holds for all y ∈ W + aff with ℓ(y) < ℓ(x).Consider the short exact sequence 0 Section II.2.14] and the minimal tilting complexes and observe that T i is a direct summand of C i := A i+1 ⊕ B i for all i ∈ Z, by Lemma 2.3.By the induction hypothesis and the linkage principle, we may assume that (1)-( 5) are satisfied for the minimal tilting complexes of all composition factors of rad G ∆(x • λ).Using Lemma 2.3 and induction on the length of a composition series of rad G ∆(x • λ), we see that every weight µ ∈ X + with [A i : T (µ)] ⊕ = 0 for some i ∈ Z is of the form y • λ, for some y ∈ W + aff with |i| ≤ ℓ(x) − ℓ(y) − 1.In particular, we have A i = 0 for all i ∈ Z with |i| ≥ ℓ(x).Now recall from Proposition 2.4 that B i is negligible for all i = ℓ(x), that B ℓ(x) ∼ = T (λ) and that every weight µ ∈ X + with [B i : T (µ)] ⊕ = 0 for some i ∈ Z is of the form y • λ, for some y ∈ W + aff with |i| ≤ ℓ(x) − ℓ(y).As C i = A i+1 ⊕ B i for all i ∈ Z, we conclude that every weight µ ∈ X + with [C i : T (µ)] ⊕ = 0 for some i ∈ Z is of the form y • λ, for some y ∈ W + aff with |i| ≤ ℓ(x) − ℓ(y).Furthermore, we have The claims (2), (3) and ( 5) are now immediate because T i is a direct summand of C i for all i ∈ Z.The first part of claim (4) follows from Lemma 2.3 because Analogously, we have Remark 2.6.In the quantum case, Propositions 2.4 and 2.5 can also be derived from Theorem 4.8 in [Gru22a], using combinatorial properties of Kazhdan-Lusztig polynomials.

The ideal of singular G-modules
For the rest of the article, we assume that ℓ ≥ h, the Coexeter number of G. Let I be a thick tensor ideal in Tilt(G).In [Gru22b, Lemma 2.3], it is shown that the set of G-modules is a thick tensor ideal with the 2/3-property: For any short exact sequence 0 → A → B → C → 0 of G-modules such that two of the G-modules A, B and C belong to I , the third also belongs to I .Furthermore, by Lemma 2.4 in [Gru22b], I is the smallest thick tensor ideal with the 2/3-property in Rep(G) that contains I. Recall from Subsection 1.6 that we write N for the thick tensor ideal of negligible tilting modules.Definition 3.1.We call N the ideal of singular G-modules and say that a G-module is regular if it does not belong to N .We refer to the quotient category Rep(G) := Rep(G)/ N as the regular quotient of Rep(G) and write q : Rep(G) → Rep(G) for the quotient functor.
The quotient category Rep(G) has the same objects as Rep(G), but for two G-modules M and N , the space of homomorphisms from M to N in Rep(G) is the quotient of Hom G (M, N ) by the space of homomorphisms that factor through a singular G-module.Since the set N of singular Gmodules is closed under retracts, a G-module M is regular if and only if q(M ) is non-zero in Rep(G).Furthermore, Rep(G) inherits the Krull-Schmidt property from Rep(G).
We first prove two results that justify our terminology.
Lemma 3.2.The ideal N of singular G-modules is the smallest thick tensor ideal in Rep(G) with the 2/3-property that contains all ℓ-singular linkage classes.
Proof.Recall that a linkage class Rep µ (G) is called ℓ-singular if µ ∈ C fund \ C fund .For a G-module M in an ℓ-singular linkage class Rep µ (G), all terms of the minimal complex C min (M ) are negligible because they belong to Rep µ (G) by Lemma 2.2, so M ∈ N .Now let I be a thick tensor ideal with the 2/3-property that contains all ℓ-singular linkage classes.In order to show that I contains N , it suffices to verify that I contains N , since N is the smallest thick tensor ideal with the 2/3-property in Rep(G) that contains N .All indecomposable tilting modules of ℓ-singular highest weight belong to I by assumption, so now consider a negligible tilting module T (x • λ) of ℓ-regular highest weight, where λ ∈ C fund ∩ X and x ∈ W + aff with x = e.Let s ∈ S be a simple reflection with xs ∈ W + aff and xs • λ < x • λ.We can choose a weight µ ∈ C fund ∩ X with Stab W aff (µ) = {e, s}, and then by Section II.E.11 in [Jan03], we have , for ν the unique dominant weight in the W fin -orbit of λ − µ, we conclude that T (x • λ) belongs to I, as required.
Proof.Suppose first that λ is ℓ-regular and write λ = x • λ ′ for some x ∈ W + aff and λ ′ ∈ C fund ∩ X.By Propositions 2.4 and 2.5, the minimal tilting complexes of both ∆(λ) and L(λ) have the non-negligible tilting module T (λ ′ ) as their term in degree ℓ(x), and it follows that ∆(λ) / ∈ N and L(λ) / ∈ N .Conversely, if λ is ℓ-singular then the linkage class containing ∆(λ) and L(λ) is contained in N by Lemma 3.2, and it follows that ∆(λ) ∈ N and L(λ) ∈ N .
Our next goal is to prove two results that we consider as a 'linkage principle' and a 'translation principle' for tensor products.(See Remark 3.9 below for an explanation of this terminology.)The first one (Corollary 3.5) asserts that the principal block (and the extended principal block) are closed under tensor products in the regular quotient.The second one (Theorem 3.7) shows that the Krull-Schmidt decomposition of any tensor product in Rep(G) can be determined by looking at the Krull-Schmidt decomposition of (the projection to Rep 0 (G) of) a tensor product of G-modules in Rep 0 (G) and that the multiplicities of indecomposable direct summands are governed by the Verlinde category.Our main tool for proving these results will be the following lemma.
Proof.By the linkage principle, we have As a further consequence of Lemma 3.4, we prove that a translation functor with source in the extended principal block descends in the regular quotient to tensoring with a tilting module.
Corollary 3.6.Let λ ∈ C fund ∩ X and ω ∈ Ω.Then λ is the unique dominant weight in the W fin -orbit of ω • λ − ω • 0, and the canonical natural transformations give rise to an isomorphism of functors Proof.Writing ω = t γ w with γ ∈ X and w ∈ W fin , it is straightforward to see that ω • λ − ω • 0 = w(λ), so λ is indeed the unique dominant weight in the W fin -orbit of ω • λ − ω • 0. By Lemma 3.4, the component at a G-module N in Rep ω•0 (G) of either of the two natural transformations descends to an isomorphism in Rep(G), and the claim follows.
We are now ready to establish our 'translation principle' for tensor products.
Theorem 3.7.For λ, µ ∈ C fund ∩ X and ω, ω ′ ∈ Ω, there is a natural transformation of bifunctors Proof.We construct the natural transformation in several steps.

an isomorphism of functors upon passage to the regular quotient Rep(G).
(2) The braiding on Rep(G) gives rise to a natural isomorphism (3) The canonical projection to the linkage class of ωω ′ • 0 gives rise to a natural transformation which again descends to a natural isomorphism in Rep(G).(4) The tensor product T (λ) ⊗ T (µ) can be decomposed as a direct sum where N is a negligible tilting module.This decomposition gives rise to a natural isomorphism As N is negligible, the essential image of the bifunctor N ⊗ pr ωω ′ •0 (− ⊗ −) is contained in N , and it follows that q • N ⊗ pr ωω ′ •0 (− ⊗ −) = 0. Therefore, the projection onto the non-negligible part gives rise to a natural transformation which descends to an isomorphism of functors in Rep(G).(5) Again by Corollary 3.6, the canonical natural transformation induces an isomorphism of functors upon passage to the regular quotient.
All of the natural transformations in (1)-( 5) give rise to natural isomorphisms upon passage to the regular quotient Rep(G).Therefore, their composition is a natural transformation such that qΨ is a natural isomorphism.
Remark 3.8.The statement of Theorem 3.7 becomes more readable (but also slightly less general) if we set ω = ω ′ = e: For λ, µ ∈ C fund ∩ X, there is a natural transformation of bifunctors Ψ : , such that qΨ is an isomorphism of bifunctors.Taking the action of Ω into account complicates our notation here, but it will be useful in applications.
Remark 3.9.Let us briefly explain why we think of Corollary 3.5 and Theorem 3.7 as a 'linkage principle' and a 'translation principle' for tensor products.The usual linkage principle asserts that the category Rep(G) decomposes into linkage classes, and the usual translation principle establishes equivalences between the different ℓ-regular linkage classes.Thus, many questions about the structure of the category Rep(G) can be reduced to questions about the principal block Rep 0 (G).However, this strategy fails for two reasons when one tries to take the monoidal structure of Rep(G) into account.Firstly, the principal block is not closed under tensor products.In fact, the tensor product of two G-modules in Rep 0 (G) can have non-zero indecomposable direct summands in many different linkage classes, including ℓ-singular ones.Secondly, it is a priori not clear how structural information about tensor products of G-modules in the principal block can be used to deduce (precise) structural information about tensor products of G-modules in arbitrary ℓ-regular linkage classes.
The preceding results show that both of these obstacles can be partially resolved by passing to the regular quotient.Indeed, Corollary 3.5 tells us that the essential image Rep 0 (G) of the principal block in the regular quotient is closed under tensor products; hence, the decomposition of Rep(G) into linkage classes is, to some extent, compatible with the monoidal strucure of Rep(G).Furthermore, Theorem 3.7 enables us to describe (the regular parts of) tensor products of G-modules in arbitrary ℓregular linkage classes, once we know the structure of (the components in Rep 0 (G) of) tensor products of G-modules in Rep 0 (G).The reader should note, however, that all information about singular direct summands is lost in the process.
In the following, we present a second approach to the 'linkage principle' and the 'translation principle' for tensor products, which largely bypasses the quotient category Rep(G), but also loses the functoriality of Theorem 3.7.When studying tensor product of specific G-modules, rather than categorical properties of Rep(G), this second approach will turn out to be more convenient.Definition 3.10.For a G-module M , we write M ∼ = M sing ⊕ M reg , where for a fixed Krull-Schmidt decomposition of M , we define M sing to be the direct sum of the singular indecomposable direct summands of M and M reg to be the direct sum of the regular indecomposable direct summands of M .We call M sing the singular part of M and M reg the regular part of M .
Note that the decomposition M ∼ = M sing ⊕ M reg in the previous definition is neither canonical nor functorial.Nevertheless, the singular part and the regular part are uniquely determined up to isomorphism by the Krull-Schmidt decomposition of M .Lemma 3.11.For G-modules M and N , we have Proof.The first isomorphism is straightforward to see from the definition.The second one follows from the direct sum decomposition and the fact that singular G-modules form a thick tensor ideal.
The following lemma can be seen as another version of the 'linkage principle' for tensor products.
Lemma 3.12.Let λ ∈ C fund ∩ X and ω ∈ Ω, and let M and N be G-modules such that M reg belongs to Rep λ (G) and N reg belongs to Rep ω•0 (G).Then (M ⊗ N ) reg belongs to Rep ω•λ (G).
Proof.By Lemma 3.11 and the linkage principle, we have and it suffices to show that pr ν (M reg ⊗ N reg ) is singular for all ν ∈ C fund ∩ X with ν = ω • λ.This was already observed in the proof of Lemma 3.4, for arbitrary G-modules in the linkage classes Rep λ (G) and Rep ω•0 (G).
Next we give a reformulation of Corollary 3.6 in terms of regular parts of G-modules.
Corollary 3.13.Let λ ∈ C fund ∩ X and ω ∈ Ω, and let M be a G-module in Rep ω•0 (G).Then .12, we obtain The following result is a non-functorial version of the 'translation principle' for tensor products from Theorem 3.7.
Theorem 3.14.Let λ, µ ∈ C fund ∩X and let M and N be G-modules in the linkage classes Proof.By Lemma 3.11 and Corollary 3.13, we have and (M ⊗ N ) reg belongs to the linkage class Rep ωω ′ •0 (G) by Lemma 3.12.Again using Corollary 3.13, we obtain ) reg for all ν ∈ C fund ∩ X, and we conclude that For future applications, let us briefly explain how the action of Ω can be used to compare the regular parts of tensor products of G-modules with constituents belonging to different linkage classes in the extended principal block Rep Ω•0 (G).For ω ∈ Ω, consider the auto-equivalence Proof.We could deduce this as a special case of Theorem 3.14 where λ and µ belong to Ω • 0, but to avoid excessive indexing, we prefer to prove the claim directly.(The reader will note that the proof is also just a special case of the proof of Theorem 3.14.)By Lemma 3.11 and Corollary 3.13, we have where • 0) by Lemma 1.1.Again using Corollary 3.13, we obtain and the claim follows.

Strong regularity
For certain applications, it may be important to decide if the tensor product M ⊗ N of two Gmodules M and N is regular.In the quantum case, it is sufficient to assume that M and N are regular (see Remark 4.4 below), but we do not know if this is true in the modular case.To overcome this problem, we introduce the notion of strong regularity.For a non-zero G-module M , the good filtration dimension of M is defined as gfd(M ) = max d Ext d G (∆(λ), M ) = 0 for some λ ∈ X + .By Lemma 2.15 in [Gru22a], we also have  Remark 4.4.In the quantum case, we claim that the tensor product M ⊗ N of two regular Gmodules M and N is always regular, if ℓ > h.Observe that the claim is equivalent to the statement that singular G-modules form a prime ideal, i.e. that the tensor product M ⊗ N of two G-modules M and N is singular only if at least one of M and N is singular.By [Gru22b, Lemma 2.8] and its proof, we have where the intersection runs over the prime thick tensor ideals P in Rep(G) with the 2/3-property such that N ⊆ P. For any such tensor ideal P, we have N ⊆ P ∩ Tilt(G), and as N is maximal among the proper thick tensor ideals in Tilt(G) (by Lemma 1.2), it follows that N = P ∩ Tilt(G) and again by Lemma 2.8 in [Gru22b].Hence N is prime, as required.We do not know if the analogous statement is true in the modular case.
Lemma 4.3 allows us to prove the following generalization to tensor products of A. Parker's results from [Par03] about the good filtration dimension of Weyl modules and simple G-modules: Theorem 4.5.Let x 1 , . . ., x m , y 1 , . . ., y n ∈ W + ext and λ 1 , . . ., λ m , µ 1 , . . ., µ n ∈ C fund ∩ X.Then the tensor product is strongly regular and has good filtration dimension ℓ(x 1 ) + Proof.For 1 ≤ i ≤ m and 1 ≤ j ≤ n, the G-modules ∆(x i • λ i ) and L(y j • µ j ) are strongly regular by Remark 4.2, and their good filtration dimensions are given by gfd ∆(x i • λ i ) = ℓ(x i ) and gfd L(y j • λ j ) = ℓ(y j ).
The claim follows from Lemma 4.3, by induction on m + n.

An application to translation functors
In this section, we give a further application of the tensor ideal N of singular G-modules, which may be of independent interest.We prove that the composition of two translation functors between ℓ-regular linkage classes is naturally isomorphic to a translation functor.This statement should not be very surprising to experts in the field, but we are not aware of a proof in the literature.
Proposition 5.1.Let λ, µ, δ ∈ C fund ∩ X.Then there is an isomorphism of functors Proof.First suppose that δ = 0 and recall that T λ 0 = pr λ ∇(λ) ⊗ − , T µ 0 = pr µ ∇(µ) ⊗ − and T µ λ = pr µ ∇(ν) ⊗ − , where ν is the unique dominant weight in the W fin -orbit of µ − λ.Consider the functor Ψ := pr µ ∇(ν) ⊗ ∇(λ) ⊗ − , and note that the canonical embedding of functors T λ 0 =⇒ ∇(λ) ⊗ − gives rise to a natural transformation T µ λ • T λ 0 =⇒ Ψ.Furthermore, we have and the canonical projection ∇(ν) ⊗ ∇(λ) → ∇(µ) affords a natural transformation Ψ =⇒ T µ 0 .We claim that the composition of these natural transformations Let N be a complement of ∇(µ) ∼ = pr µ ∇(ν) ⊗ ∇(λ) in ∇(λ) ⊗ ∇(ν), and observe that pr µ N = 0.For a G-module M in Rep 0 (G), we have by the linkage principle and Lemma 3.2, and as (M ⊗ pr ν N ) reg belongs to Rep ν (G) by Lemma 3.12, we conclude that pr ν (M ⊗ N ) reg ∼ = (M ⊗ pr ν N ) reg for all ν ∈ C fund ∩ X.In particular, the functor pr µ (N ⊗ −) maps every G-module in Rep 0 (G) into the tensor ideal N of singular G-modules.As Ψ decomposes as the direct sum of the functors T µ 0 and pr µ (N ⊗ −), this implies that all components of the natural transformation Ψ =⇒ T µ 0 descend to isomorphisms in the regular quotient Rep(G).Similarly, the embedding of functors descends to a natural isomorphism in Rep(G) by Corollary 3.6, and it follows that the same is true for the natural transformation In particular, the component of ϑ at any simple G-module L(x•0) with x ∈ W + aff affords an isomorphism in Rep(G).Now L(x • 0) is non-zero in Rep(G) by Lemma 3.3, whence the endomorphism algebra of L(x • 0) in Rep(G) is also non-zero.Since the latter endomorphism algebra is a quotient of the endomorphism algebra of L(x • 0) in Rep(G), we conclude that the component of ϑ at L(x • 0) is non-zero; hence it affords an isomorphism between , by Schur's Lemma.Using the snake Lemma and induction on the length of a composition series, one easily deduces that the component of ϑ at every G-module in Rep 0 (G) is an isomorphism, so ϑ is a natural isomorphism, as claimed.Now since T λ 0 •T 0 λ is isomorphic to the identity functor on Rep λ (G), we further obtain isomorphisms of functors and such that the following diagram commutes, for all A, A ′ ∈ Ob(A) and B, B ′ ∈ Ob(B). (6.1) The vertical arrow on the left hand side is induced by the braiding in the category of k-vector spaces and the vertical arrows on the right hand side denote the composition of morphisms in C. By the above discussion, the additive k-linear category Ver(G) ⊠ Rep 0 (G) has a k-linear monoidal structure.For any tilting G-module T , the functor (T ⊗ −) : Rep 0 (G) → Rep(G) induces a functor because singular G-modules form a thick tensor ideal, and similarly, for any G-module M in Rep 0 (G), the functor (− ⊗ M ) : Tilt(G) → Rep(G) induces a functor because negligible tilting modules are singular.We have and the diagram (6.1) commutes because for all homomorphisms f : T → T ′ in Tilt(G) and g : M → M ′ in Rep 0 (G).Hence there exists a unique k-linear functor for all tilting G-modules T and all G-modules M in Rep 0 (G).Furthermore, the braiding on Rep(G) defines a natural isomorphism for tilting G-modules T and T ′ and G-modules M and M ′ in Rep 0 (G), and we have It is straightforward to check that these coherence maps endow F with the structure of a (strong) monoidal functor.Since Rep(G) is an additive k-linear category, the functor F extends uniquely to an additive k-linear functor and the coherence maps for F extend canonically to coherence maps for F , so that F becomes a monoidal functor.Remark 6.2.Let us write Ver ZΦ (G) for the full subcategory of Ver(G) whose objects are the direct sums of indecomposable tilting modules with highest weights in ZΦ, and Ver Ω (G) for the full subcategory of Ver(G) whose objects are isomorphic to direct sums of tilting modules of the form T (ω • 0) for certain ω ∈ Ω.Then Ver ZΦ (G) is closed under tensor products in Ver(G) because ZΦ is a subgroup of X and Ver Ω (G) is closed under tensor products in Ver(G) by Lemma 1.1.There is an equivalence of k-linear monoidal categories (6.2) Ver(G) ∼ = Ver ZΦ (G) ⊠ Ver Ω (G).
Arguing as in the proof of Theorem 6.1, we further obtain equivalences of k-linear monoidal categories which can also be obtained from Theorem 6.1 and equation (6.2)

Examples
In this Section, let G be of type A 2 , with simple roots Π = {α 1 , α 2 } .Let ̟ 1 , ̟ 2 ∈ X + be the fundamental dominant weights with (̟ i , α ∨ j ) = δ ij , and let s 1 = s α 1 and s 2 = s α 2 be the simple reflections in W fin .The affine Weyl group W aff is generated by the simple reflections S = {s 0 , s 1 , s 2 }, where s 0 = t α h s α h and α h = α 1 + α 2 .For all x ∈ W + aff , we have L(x • 0) ⊗ L(0) ∼ = L(x • 0) ∼ = L(x • 0) ⊗ L(0) reg because L(x • 0) is regular by Lemma 3.3, and using Theorem 3.14, it follows that for λ, µ ∈ C fund ∩ X.So far, we have not used the fact that G is of type A 2 .Let us next discuss a less trivial example.A weight λ ∈ X + is called ℓ-restricted if λ = a̟ 1 + b̟ 2 with 0 ≤ a, b ≤ p − 1.The set of indecomposable direct summands of tensor products of simple G-modules with ℓ-restricted highest weight has been determined by C. Bowman, S. Doty and S. Martin in [BDM15].They show that every indecomposable direct summand of such a tensor product is either an indecomposable tilting module or a G-module of the form M (ν), for ν ∈ C fund ∩ X, whose radical series (and socle series) is displayed in the Alperin diagram below.5 (We replace simple modules L(x • ν) by their label x ∈ W + aff .)for all λ, µ ∈ C fund ∩ X.
Next observe that we have ℓ̟ 1 = t ̟ 1 • 0 = s 0 s 2 ω • 0 and ℓ̟ 2 = t ̟ 2 • 0 = s 0 s 1 ω • 0. If ℓ ≥ 5 then it is straightforward to see using the Frobenius twist functor (see Sections II.3.16 and II.H.7 in [Jan03]) that L(ℓ̟ 1 ) ⊗ L(ℓ̟ 2 ) ∼ = L(ℓα h ) ⊕ L(0), where ℓα h = t α h • 0 = s 0 s 1 s 2 s 1 • 0, and arguing as before, we see that Rep(G) for the quotient category of Rep(G) by the tensor ideal of singular Gmodules.Let Rep 0 (G) be the essential image of the principal block under the canonical quotient functor Rep(G) → Rep(G); so Rep 0 (G) is the full subcategory of Rep(G) whose objects are the G-modules M such that M reg belongs to Rep 0 (G).By Theorem A, Rep 0 (G) is closed under tensor products in Rep(G).The Theorems A and B can be combined to give the following external tensor product decomposition of Rep(G): Theorem C.There is an equivalence of k-linear monoidal categories

Lemma 2. 2 .
Let λ ∈ C fund ∩ X and let M be a G-module in Rep λ (G).Then all terms of C min (M ) belong to Rep λ (G).Proof.As M belongs to Rep λ (G) and the projection functor pr λ : Rep(G) → Rep λ (G) is exact, we have M ∼ = pr λ M ∼ = pr λ C min (M ) in D b Rep(G) .By part (3) of Lemma 2.1, C min (M ) admits a split monomorphism into pr λ C min (M ) in C b Tilt(G) , and the claim follows.

Lemma 3. 4 .
Let λ ∈ C fund ∩ X and ω ∈ Ω.For G-modules M and N in the linkage classes Rep λ (G) and Rep ω•0 (G), respectively, the canonical embedding and the lemma is equivalent to the statement that pr ν M ⊗ N ∼ = 0 in the regular quotient Rep(G), for all weights ν ∈ C fund ∩ X with ν = ω • λ.Observe that all terms of C min (M ) belong to Rep λ (G) and all terms of C min (N ) belong to Rep ω•0 (G) by Lemma 2.2.As C min pr ν (M ⊗ N ) admits a split monomorphism into the complex pr ν C min (M ) ⊗ C min (N ) by part (4) of Lemma 2.1, it suffices to prove that pr ν T (x • λ) ⊗ T (yω • 0) is negligible for all x, y ∈ W + aff and ν ∈ C fund ∩ X with ν = ω • λ.If x = e or y = e then T (x • λ) ⊗ T (yω • 0) is negligible, because the negligible tilting modules form a thick tensor ideal in Tilt(G).For x = y = e, we have T (λ) ⊗ T (ω • 0) ∼ = T (ω • λ) in the Verlinde category by Lemma 1.1, and it follows that pr ν T (λ) ⊗ T (ω • 0) is negligible for all ν ∈ C fund ∩ X with ν = ω • λ, as required.For λ ∈ C fund ∩ X, let us write Rep λ (G) for the essential image of the linkage class Rep λ (G) under the quotient functor q : Rep(G) → Rep(G), i.e. the full subcategory of Rep(G) whose objects are the G-modules that are isomorphic to a G-module in Rep λ (G), when considered as objects in Rep(G).We also write Rep Ω•0 (G) for the essential image of the extended principal block Rep Ω•0 (G) in Rep(G).As a consequence of Lemma 3.4, we obtain our 'linkage principle' for tensor products.Corollary 3.5.The subcategories Rep 0 (G) and Rep Ω•0 (G) are closed under tensor products.Proof.For ω, ω ′ ∈ Ω and G-modules M and N in the linkage classes of ω • 0 and ω ′ • 0, respectively, we have M ⊗ N ∼ = pr ωω ′ •0 (M ⊗ N ) in Rep(G) by Lemma 3.4, so M ⊗ N belongs to Rep ωω ′ •0 (G).The claim about Rep 0 (G) follows by setting ω = ω ′ = e.
which descends to a natural isomorphism in Rep(G) by Lemma 3.4.Tensoring with T (λ) ⊗ T (µ) yields a natural transformation (4.1) gfd(M ) = max d C min (M ) d = 0 Definition 4.1.A non-zero G-module M with gfd(M ) = d is called strongly regular if C min (M ) d−1 is negligible and C min (M ) d is non-negligible.Remark 4.2.Observe that, for x ∈ W + ext and λ ∈ C fund ∩ X, the Weyl module ∆(x • λ) and the simple G-module L(x • λ) are both strongly regular of good filtration dimension ℓ(x).Indeed, by the description of the minimal tilting complexes of ∆(x • λ) and L(x • λ) in Propositions 2.4 and 2.5, the tilting modules C min ∆(x • λ) i and C min L(x • λ) i are negligible for i = ℓ(x) − 1, non-negligible for i = ℓ(x) and zero for i > ℓ(x).The fact that ∆(x • λ) and L(x • λ) have good filtration dimension ℓ(x) has already been observed by A. Parker in[Par03].Our interest in strongly regular G-modules is founded in the following result: Lemma 4.3.Let M and N be strongly regular G-modules.Then M ⊗ N is strongly regular and gfd(M ⊗ N ) = gfd(M ) + gfd(N ).

Proof.
Set d = gfd(M ) and d ′ = gfd(N ), so that C min (M ) i = 0 for i > d and C min (N ) i = 0 for i > d ′ by equation (4.1).Since M and N are strongly regular, there exist ν, ν ′ ∈ C fund ∩ X such thatC min (M ) d : T (ν) ⊕ = 0 and C min (N ) d ′ : T (ν ′ ) ⊕ = 0,and by Lemma 1.2, there exists δ ∈ C fund ∩ X with T (ν) ⊗ T (ν ′ ) : T (δ) ⊕ = 0. Thus T (δ) appears as a direct summand of the tensor product C min (M ) d ⊗ C min (N ) d ′ , which is the degree d + d ′ term of the tensor product complex C min (M ) ⊗ C min (N ).Furthermore, the degree d + d ′ − 1 termC min (M ) d−1 ⊗ C min (N ) d ′ ⊕ C min (M ) d ⊗ C min (N ) d ′ −1of the tensor product complex is negligible, and the terms in degree i > d + d ′ of the tensor product complex are zero.NowC min (M ⊗ N ) is the minimal complex of C min (M ) ⊗ C min (N ) by part(4) of Lemma 2.1, hence C min (M ⊗ N ) d+d ′ −1 is negligible and C min (M ⊗ N ) i = 0 for i > d + d ′ .As the terms of the tensor product complex C min (M ) ⊗ C min (N ) in degrees d + d ′ − 1 and d + d ′ + 1 are negligible or zero, respectively, Corollary 2.8 in[Gru22a] implies that0 = C min (M ) d ⊗ C min (N ) d ′ : T (δ) ⊕ = C min (M ⊗ N ) d+d ′ : T (δ) ⊕ .Finally, equation (4.1) yields gfd(M ⊗ N ) = d + d ′ , and it follows that M ⊗ N is strongly regular.

6.
Tensor product decomposition of Rep(G) In this section, we explain how the results from Section 3 give rise to an external tensor product decomposition Rep(G) ∼ = Ver(G) ⊠ Rep 0 (G) of the regular quotient.We first explain what we mean by the external tensor product of two additive k-linear categories.Let A and B be two additive k-linear categories.Following [Kel05, Section 1.4], we define the tensor product A⊗B as the k-linear category with objects Ob(A⊗B) = Ob(A)×Ob(B) and homomorphisms Hom A⊗B (A ⊠ B, A ′ ⊠ B ′ ) = Hom A (A, A ′ ) ⊗ Hom B (B, B ′ ) for A, A ′ ∈ Ob(A) and B, B ′ ∈ Ob(B), where we write A⊠B for the object of A⊗B corresponding to A and B. The composition of homomorphisms in A ⊗ B is induced by the composition in A and B in the obvious way.The category A ⊗ B need not be additive, so we define A ⊠ B to be its additive envelope. 4Thus A ⊠ B is an additive k-linear category.If A and B are endowed with k-linear monoidal structures then there is a canonical k-linear monoidal structure on A ⊗ B, and the latter extends uniquely to a k-linear monoidal structure on A ⊠ B. Now let C be another k-linear category.According to Section 1.5 in [Kel05], a k-linear functor F : A ⊗ B −→ C can equivalently be defined by two families of k-linear functors F A : B → C and G B : A → C, indexed by Ob(A) and Ob(B), respectively, such that Theorem 6.1.The functor F : Ver(G) ⊠ Rep 0 (G) −→ Rep(G), together with the canonical coherence maps, is an equivalence of k-linear monoidal categories.Proof.Recall that for λ ∈ C fund ∩ X, we write Rep λ (G) for the essential image of Rep λ (G) under the quotient functor q : Rep(G) → Rep(G).The projection functor pr λ : Rep(G) → Rep λ (G) induces a functor pr λ : Rep(G) → Rep λ (G) and the translation functor T λ 0 induces an equivalence between Rep 0 (G) and Rep λ (G) with quasi inverse induced by T 0 λ .We consider the functor G= λ∈C fund ∩X T (λ) ⊠ T 0 λ • pr λ (−) : Rep(G) −→ Ver(G) ⊠ Rep 0 (G)and claim that G is a quasi-inverse for F .Indeed, we have an isomorphism of functorsF • G = F • λ T (λ) ⊠ T 0 λ • pr λ (−) = λ T (λ) ⊗ T 0 λ • pr λ (−) ∼ = λ T λ 0 • T 0 λ • pr λ (−) ∼ = λ pr λ (−) ∼ = id Rep(G) ,where the first two equality signs follow from the definitions.The first isomorphism follows from Corollary 3.6 because T (λ) ⊗ − ∼ = T λ 0 as functors from Rep 0 (G) to Rep(G), the second isomorphism follows from the fact that T λ 0 is a quasi-inverse of T 0 λ , and the third isomorphism follows from the linkage principle and the fact that all G-modules in ℓ-singular linkage classes are singular (see Lemma 3.2).Conversely, for λ ∈ C fund ∩ X and M a G-module in Rep 0 (G), we haveG • F T (λ) ⊠ M = G T (λ) ⊗ M ∼ = G(T λ 0 M ) = T (λ) ⊠ T 0 λ T λ 0 M ∼ = T (λ) ⊠ M in Ver(G) ⊠ Rep 0 (G), again by Corollary 3.6.As Ver(G) is semisimple with simple objects T (λ), for λ ∈ C fund ∩ X, this gives rise to a natural isomorphism between G • F and the identity functor on Ver(G) ⊠ Rep 0 (G), whence G is a quasi inverse of F , as claimed.Thus F is an equivalence of categories, and even an equivalence of monoidal categories since F is monoidal (see Proposition 4.4.2 in [SR72]).
which is obtained from the short exact sequence in [Jan03, Proposition II.7.19] by taking duals.Furthermore, let us write