Growth rates of the number of indecomposable summands in tensor powers

In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.


Introduction and main results
A central, yet hard, problem in representation theory is the decomposition of tensor products of representations into indecomposable summands.Computations of these decomposition numbers are often major unsolved problems in representation theory.In this paper we take a different perspective and we are interested in asymptotic properties of the number of indecomposables in tensor products of representation rather than explicit decompositions.In contrast to the question of explicit decompositions, we obtain results in extensive generality.
To get started, let Γ be a finite group with a finite dimensional representation V over some field k.
Definition 1.1.We define b Γ,V n := #indecomposable summands in V ⊗n counted with multiplicities (sometimes simply denoted by b n when Γ and V are clear from the context), where 'indecomposable' means as Γ-representation.Let further n+m , so that β Γ,V is well-defined by (a version of) Fekete's Subadditive Lemma.There is no chance to compute b Γ,V n explicitly in this generality, but it turns out that the 'limit' β Γ,V can be understood.As a first step we note that we have the following lemma whose the proof is immediate.
Lemma 1.2.We have b Γ,V n ≤ (dim V ) n , and consequently A classical result of Bryant-Kovács [BK72, Theorem 1] shows that there exists some n for which V ⊗n contains a projective direct summand (projective over Γ if V is a faithful and projective over the appropriate factor group of Γ otherwise).As observed in [BS20, Theorem 6.3], a consequence of this is that the bound for β Γ,V in Lemma 1.2 is actually an equality, that is: Theorem 1.3.We have A Mathematica loglog plot of n √ b n (y-axis) for n ∈ {1, . . ., 1000} (x-axis) gives , and indeed the limit is two, as predicted by Theorem 1.3.Precisely, 1000 √ b 1000 ≈ 1.99265.However, the asymptotic growth rate of b n is different than 2 n .As we will see in Example 2B.4,we get (Here and throughout, we use f ∼ g for f is equal to g asymptotically, meaning the ratio of f and g converges to one.)We have 2/π ≈ 0.798 and Mathematica's log plot gives: , .
For a precise statement see Example 2B.4 below.3 A broader formulation of the ideas in Definition 1.1 and Theorem 1.3 is the following setup.
Notation 1.8.Let D be a k-linear Karoubian monoidal category that is Krull-Schmidt, with a k-linear faithful monoidal functor F : D → Vect K to the category Vect K of finite dimensional vector spaces over a field extension K of k.Note that, if K is a finite extension of k, then the existence of F implies that morphism spaces in D are finite dimensional, so D is automatically Krull-Schmidt.
For any object X ∈ D, we can define b X n similarly as in Definition 1.1 as the number of indecomposable direct summands in X ⊗n and we have Again, β D,X is well-defined by (a version of) Fekete's Subadditive Lemma.We prove the following result, which also generalizes most of the examples in Theorem 1.4.
(b) Assume that char(k) = 0.If D is symmetric and F can be lifted to a symmetric monoidal functor F ′ : D → SV ect K to the category of super vector spaces SV ect K , then Theorem 1.3 holds, i.e.: β D,X = dim F (X).
(c) Let k = C and D = Rep C SL 2 .For every m ∈ Z ≥2 , there exists a faithful monoidal functor F m : D → Vect C , which sends the vector SL 2 -representation X to C m .Hence, for m ≥ 3 we have Here and throughout, Rep k Γ denotes the category of finite dimensional (rational) Γ-representations over k.
Remark 1.10.(a) Note that the functors F m in Theorem 1.9.(c) are not symmetric for m ≥ 3, so Theorem 1.9.(a) does not apply.
(b) Faithfulness of F in Theorem 1.9.(a) is required to ensure the estimate b X n ≤ dim F (X) n and cannot be dropped, since it is easy to construct counterexamples with Deligne's categories of [Del07].Of course, even without faithfulness, the bound dim F (X) ≤ β D,X remains valid.
Acknowledgments.Want to thank Pavel Etingof for comments on a draft of this paper, in particular for Remark 6.2 which is Pavel's observation, Andrew Mathas for help with the literature on symmetric groups and Schur algebras, Volodymyr Mazorchuk for email exchanges about monoids, and Jonathan Gruber and Arun Ram for discussions about growth rates.We also thank the referee for very valuable comments and remarks.
We also thank the MFO workshop 2235 "Character Theory and Categorification" for bringing us together in Oberwolfach in August/September 2022 -this project started during this fantastic workshop.V.O. and D.T. were supported by their depressions.

The general linear group and consequences
Let us start by defining some of our main players: Definition 2.1.We define an affine semigroup scheme (over k) to be a semigroup object in the category of affine k-schemes (the opposite of the category of commutative k-algebras).Equivalently, we can think of it as a representable functor from the category of k-algebras to the category of semigroups.Concretely, an affine semigroup scheme corresponds to a commutative bialgebra, potentially without counit.The special case of an affine monoid scheme corresponds precisely to a bialgebra with a counit, and the further special case of an affine group scheme corresponds to a Hopf algebra.
We have the same notions in the 'super' version, by replacing the category of commutative k-algebras with the category of graded commutative Z/2Z-graded k-algebras.
Proof.The case M = 1 is immediate.In Section 2C and Section 2D we prove the statement for M > 1. □ We also fix some notation: Notation 2.3.(a) Recall that k denotes an arbitrary field, and we thus have that its characteristic char(k) ∈ N. We always let p ∈ Z >1 ∪ {∞} denote the additive order of 1 ∈ k × .Thus, char(k) = p except that for char(k) = 0 we set p = ∞.When we state char(k) = p > 0 it thus unambiguously means positive characteristic.
(b) Throughout this section, we will set Γ M = SL M .Since the decomposition of V ⊗n M into indecomposable summands is identical for SL M and GL M , we can focus just on Γ M in Proposition 2.2.
Remark 2.4.The proof of Proposition 2.2 will crucially exploit the theory of tilting representations, see for example [Jan03, Part II.E] (note that the relevant section in [Jan03] works over an arbitrary field of characteristic char(k) = p > 0) or [AST18] and the appendix of its arXiv version for background.The point is that V M is tilting, and by abstract theory direct summands of tensor products of tilting representations are tilting.Thus, the summands of V ⊗n M are tilting.Before proving Proposition 2.2, we extract some of its consequences.
Proof of Theorem 1.9.(a).Let D and F be as in Theorem 1.9.(a), and let X be an object of D. We have β X ≤ dim F (X) by the analog of Lemma 1.2.We will show that b dim F (X) n ≤ b X n , which implies the claim by Proposition 2.2.
To this end, recall that p = char(k) ∈ N ∪ {∞}.Recall that simple representations of the symmetric group S n are labeled by p-regular partitions λ of n, see e.g.[Mat99,Theorem 3.43] for an even more general statement.Let D λ be the simple representation labeled by λ.By Schur-Weyl duality, see e.g.[Jan03, E.17], we have an epimorphism

Hence the number of indecomposable summands in V ⊗n
M equals the number of primitive idempotents of kS n , in a fixed decomposition of the identity, that are not sent to zero.Such a decomposition always consists of dim D λ idempotents corresponding to D λ , for each p-regular partition λ.Again by Schur-Weyl duality, this yields where the sum runs over all p-regular partitions of n with ≤ M rows.By assumption, we have algebra morphisms where the composite is the usual permutation action of S n .For any p-regular partition λ let e λ ∈ k[S n ] be a primitive idempotent such that k[S n ]e λ is the projective cover of D λ and let Π λ X be the direct summand e λ (X ⊗n ) of X ⊗n .Clearly Π λ X is not zero whenever Π λ F (X) := e λ F (X) ⊗n is not zero.Hence X ⊗n decomposes as a direct sum of Π λ X (where the latter need not be indecomposable) with multiplicities dim D λ .By the above, this shows that b X n is indeed bounded below by b dim F (X) n . □ Proof of Theorem 1.4.(a)-affine group schemes.For any affine group scheme (or any abstract group) Γ we have since V ⊗n , considered as a Γ-representation, is the restriction of the tensor power of the tautological GL(V )representation under the usual homomorphism Γ → GL(V ).Thus, in order to prove Theorem 1.3 we just need to combine the estimate b Γ,V n ≤ (dim V ) n from Lemma 1.2 with Proposition 2.2 and (2.5).□ Proof of Theorem 1.4.(a)-affine semigroup schemes.For any affine semigroup scheme (or any semigroup) Γ let D = Rep k Γ.We have a k-linear faithful monoidal functor F : D → Vect k sending X to its underlying k-vector space.Since this functor is symmetric, Theorem 1.9.(a) applies and we are done.□ There is also a proof for semigroups which does not rely on Schur-Weyl duality.Details will be given in Section 3 because Schur-Weyl duality fails in the super case in positive characteristic.
Finally, we conclude the list of consequences of Proposition 2.2 with the case of groupoids.Consider a groupoid (S : G) in the category of k-schemes, see [Del90, §1.6], with source and target morphisms G ⇒ S. If G → S × S is faithfully flat, then we say (S : G) is transitive.A representation of a groupoid (S : G) is a quasi-coherent sheaf on S with an action of G. Tensor products of representations are taken over O S .We denote by b (S:G),V n the analog of Definition 1.1.
Theorem 2.6.Consider a transitive groupoid (S : G) in k-schemes with a representation V on a locally free O S -module of finite rank.Then As observed in [Del90, §1], the monoidal category Rep k (S : G) of representations on locally free modules is actually abelian, and exact monoidal functors out of it are automatically faithful.For any field extension K of k for which S(K) ̸ = 0, taking stalks at the corresponding point of S therefore yields a faithful symmetric monoidal functor which sends V to a vector space of dimension rank(V ).We can thus apply Theorem 1.9(a).□ 2A.Finite groups.
Notation 2A.1.Recall the Bachmann-Landau notation, which we adjust as follows.
A function for all n 0 < n for some fixed n 0 ∈ N. Without the prime this is the classical Bachmann-Landau notation where Similarly, but only for either the lower or the upper bound, we write f ∈ Ω(g) and f ∈ O ′ (g), respectively.Finally, we write We also use f ∼ g, f is asymptotic to g, meaning lim n→∞ f (n)/g(n) = 1 (with g(n) ̸ = 0 for all n ≫ 1).
Proof.Without loss of generality, we assume that Γ acts faithfully on V .By [BK72, Theorem 1], there exists r ∈ N for which V ⊗r contains a projective direct summand.Using that projective representations form a tensor ideal, and the fact that projective indecomposables P satisfy dim P ≤ |Γ|, it then follows that where the ω i the fundamental weights.These are Γ M -representations defined integrally and these are simple for char(k) = 0, and we also have ∆(1, 0, . . ., 0) = V M .See [Jan03, Part II] for some background.
For char(k) = 0 the tensor product V ⊗n M decomposes into the simple summands for m 1 , . . ., m M −1 ∈ N M .Recall Weyl's character formula, see e.g.[FH91, Section 24], which shows that The following implies Proposition 2.2 for char(k) = 0: Proof.All the weights of the representation V ⊗n M are bounded by n in the sense that any coefficient of the expansion with respect to fundamental weights is less than n in absolute value, meaning that Thus, all simple summands of V ⊗n M have dimensions bounded by a polynomial in n, and a closer look at Weyl's character formula then implies that the polynomial is of degree M (M −1)

2
. See Example 2B.2 for an example.Thus, from this and Lemma 1.2 we get The claim follows.□ Example 2B.4.We now strengthen Proposition 2B.3 for Γ 2 .For a Γ 2 -representation W , with weight spaces {W i ⊂ W |i ∈ Z}, its character is the Laurent polynomial with non-negative coefficients ch Let V be a representation of Γ 2 over k with char(k) = 0.Then, by the classification of simple SL 2 (k)representations, b G,V n equals the sum of dimensions of zero weight space and one weight space of V ⊗n .For example if V = V 2 is the vector representation, then ch n is the constant term or the coefficient of v of (v + v −1 ) n , depending on parity of n.Using the binomial theorem we see that By applying Stirling's formula we get that asymptotically which implies the claim in Example 1.7.Note that this is better than what we get from Proposition 2B.3 for M = 2.That is, the lower bound 2 n /n is what we get from Proposition 2B.3 and Mathematica's log plot gives: , .
More generally, let V be any nontrivial representation of Γ 2 .Then b Γ2,V n equals the sum of the constant term and the coefficient of v in (ch V ) n .The asymptotic of this number can be computed by using the central limit theorem.We get that where A = A(V ) ∈ R >0 is an easily computable constant depending on V .In the case when V is simple multinomials appear and one can use e.g. the results from [Ege14].The same approach applies in general.3 Remark 2B.5.In the semisimple case many related results are known, in particular for Lie algebras and Lie groups, see e.g.[PR20] for a recent publication.That paper studies the problem of finding the asymptotic of multiplicities of fixed simple representations instead of all simple representations.
A growth rate b Γ,V n ∈ Θ ′ (dim V ) n as in Proposition 2A.2 is very rarely the case, as the following result indicates: Proposition 2B.6.Recall that char(k) = 0.For an abstract group Γ with a finite dimensional representation V , the following are equivalent: Proof.We start by proving that (a) implies (b).Note that, with . By definition, an algebraic group is a torus over k if and only if its extension of scalars to k is a torus (a finite product of copies of the multiplicative group).Consequently, for this implication, we might as well assume that k is algebraically closed.
Replacing Γ by the Zariski closure of its image in GL(V ) does not change the numbers b Γ,V n , so we can assume that Γ is an algebraic group and V is a faithful Γ-representation.Now we argue by contradiction.We assume that (a) is satisfied.If (b) is not satisfied, then by [Mil17, Corollary 17.25] and the fact that every connected one-dimensional unipotent algebraic group is isomorphic to the additive group, it follows that Γ contains a copy of the additive group G a .We can restrict V to G a and get for some A ∈ R >0 .We can apply the Jacobson-Morozov theorem and find for the same A ∈ R >0 .This gives a contradiction with Example 2B.4,concluding the proof of this direction.Now we prove that (b) implies (a).For a finite field extension K of k, we can again consider a Γ- We can again replace Γ by the Zariski closure of its image.Since any torus splits after a finite field extension, we can thus assume that Γ = (G ×d m ) ⋊ H for d ∈ N and a finite group H.It is well-known that for such groups the (rational) representation theory is semisimple and the dimension of the simple representations are bounded by |H|.Conclusion (a) from this in the same way as in Proposition 2A.2.□ 2C.Proof of Proposition 2.2 -for M = 2.The case M = 2 is special since we have full access to the characters of tilting representations and these are the direct summands of V ⊗n 2 .As before, For example, for m = 52 and p = 2 we have m + 1 = (1, 1, 0, 1, 0, 1) so In particular, for v = 1 we get dim T (52) = 256. 3 This example implies: The following is a finer result than Proposition 2.2 itself, and thus, also implies Proposition 2.2.
Remark 2D.1.The bound as in Lemma 2C.2 is unavailable for M ≥ 3; the billiards conjecture in [LW18] and [Jen21] suggests that dimensions of tilting representations along the boundary grow exponentially already for Γ 3 = SL 3 (k).Our proof below "ignores" these tilting representations: we argue that we already have enough summands in the part where Donkin's tensor product formula applies up to a certain degree.
We will use the following to only consider the case when M is odd since this case has slightly nicer combinatorics: Lemma 2D.2.If Proposition 2.2 holds for M + 1, then it holds for M as well.
Using that Proposition 2.2 holds for M + 1 we get: We claim that thus lim n→∞ n b M n = M , as required.This can be seen as follows.Assume that for some fixed ϵ ∈ R >0 and all δ N ∈ R >0 there exists We plan to choose r = r(n) in such a way that the number of summands of V ⊗n M from ST r (Γ M ) is still about M n .Let us first estimate the number of occurrences of St r as a subquotient of a good filtration of V ⊗n M .This number depends only on the character of V ⊗n M and hence, is independent of the characteristic in the sense that the characters of both, V ⊗n M and St r ∼ = ∆ (p r − 1)ρ , are as in characteristic zero.In fundamental weight coordinates and SL M notation, we let ρ SL M = (1, . . ., 1) and there are choices involved how to lift this to GL M notation in standard coordinates.We will use Now the number of times that St r appears in V ⊗r M over SL M is at least the number of times it appears when we work over GL M , so we estimate the latter number.
In characteristic zero we can compute the involved characters via Schur-Weyl duality by applying the hook length formula to the partition where we from now on assume that n is divisible by M (which is sufficient to calculate the limit of n b M n ), that M is odd (which is justified by Lemma 2D.2) and that This is the partition (7, 4, 1) so that the row differences are p r − 1 = 3. 3 Now let us make the following concrete choice for r: Remark 2D.4.In fact, we could use r(n) = log p f (n) for every function f which grows slower than n.
The choice r(n) = ⌊log p ( √ n)⌋ is mostly for convenience as the formulas come out nicely.
The hook formula implies that, up to factors which will not contribute to the limit of the nth root, the number of times that St r(n) = ∆(λ) for λ as above appears in V ⊗n over GL M is approximately .
Hence, we get that We claim that We then can also see that the exponents converge to zero, and the claim follows.Consequently, using also ) in Bachmann-Landau notation, see Notation 2A.1.We get for A ∈ R >0 .This can be seen by using that for some B ∈ R >0 and g(n) ∈ Θ(n −1/2 ).Thus, since the limit n → ∞ of the nth root of the (marked in a blueish color) left-hand side is one, we see that nth root of this sequence converges to M and we get: Now let t n be the total dimension of summands of V ⊗n M which are in ST r(n) .Clearly, we have a(n) ≤ t n ≤ (dim V ) n and thus in conclusion Next, we estimate the dimensions of the indecomposable summands of V ⊗n M which are from ST r (Γ M ).We start with a general and well-known lemma.
Lemma 2D.5.We have dim T (a 1 , . . ., a M −1 ) Proof.Recall that i V M is a tilting Γ M -representation for all i ∈ {1, . . ., M − 1} (this follows since i V M is the Weyl representation ∆(0, . . ., 0, 1, 0, . . ., 0) for the ith fundamental weight ω i and this weight is minimal in the set of dominant integral weights).Now, essentially by their construction, the Γ M -representation Lemma 2D.6.Let D n denote the maximum of the dimensions of the indecomposable summands of Proof.Every such summand is of the form St r ⊗ T (r) where T is an indecomposable tilting representation and ( − ) (r) is the rth Frobenius twist, see [And18, Remark 2(1)].Hence, the highest weight of T should be bounded by n/p r(n) in the sense that sum of coefficients of the fundamental weights is bounded by this number (more restrictively even, if λ is the highest weight of T , then the weight (p r − 1)ρ + p r λ should appear in V ⊗n M ).If we let A denote the maximum A = max i { M i }, so that A = M (M −1)/2 , then by Lemma 2D.5, we know that if the relevant tilting module in ST r(n) (Γ M ) is to appear in V ⊗n M , then dim St r ⊗ T (r) ≤ p r A i ai ≤ p r A n/p r .

Now we can calculate
which concludes the proof.□ Now the total number of summands of V ⊗n M coming from ST r (Γ M ) is at least tn Dn .Hence, Hence, Proposition 2.2 follows for M odd.Then Lemma 2D.2 implies Proposition 2.2 for M even.

The general linear super group and consequences
In this section we will work in the category of super vector spaces over k (although we sometimes omit the word 'super'' to avoid too cumbersome phrasings).Since the latter reduces to the ordinary category of vector spaces in characteristic 2, we assume char(k) ̸ = 2.
Recall from Definition 2.1 that an affine group superscheme (in short: a supergroup) G over k is a representable functor from the category of commutative superalgebras (associative Z/2Z = { 0, 1}-graded algebras which are graded commutative) over k to the category of groups.For general background on the theory of supergroups we refer to, for example, [BK03a], [BK03], [Mas12] and [Mus12].
We refer to (M, N ) ∈ N ×2 as the 'super dimension' of the Z/2Z-graded vector space k M |N , and M + N as the 'dimension'.This should not lead to confusion as we will have no need for the 'categorical dimension' M − N , also something referred to as the (super) dimension.
Proposition 3.1.For a representation V of a supergroup G on a super vector space V of super dimension (M, N ), we have Here the numbers β G,V and b G,V n have the same meaning as before, i.e. they refer to the number of indecomposable summands in the G-representation V ⊗n .If we denote the representing commutative Hopf superalgebra for G by O(G), then a representation of G can either be interpreted as a Z/2Z-graded comodule for O(G), or equivalently as a homomorphism G → GL M |N of supergroups.By the latter interpretation, it is clearly sufficient to prove Proposition 3.1 for G = GL M |N and V = V M |N its vector representation on k M |N .
Before getting to the proof of Proposition 3.1, we derive some consequences.
Proof of Theorem 1.9.(b).If char(k) = 0, then We can therefore repeat the proof of Theorem 1.9.(a) from Section 2 to reduce to Proposition 3.1 for G = GL M |N .If one wants to write things out explicitly, the set of partitions is now those for which λ M +1 ≤ N .□ Remark 3.2.The proof of Theorem 1.9.(b) does not extend to positive characteristic.Indeed, in this case More concretely, it is observed in [CEKO22, §4], that for p = 3, M = 2 and N = 1, the number of indecomposable summands in V ⊗5 is 17, while the number of primitive idempotents in a decomposition of unity in kS 5 is only 16. Hence the action of the symmetric group on tensor powers of the vector representation of GL M |N is not sufficient to account for all indecomposable summands.
Because of this remark, we need an alternative proof for semigroups compared to the non-super case: Proof.As before, this follows from Lemma 3A.1 and the super analog of Lemma 1.2.□ 3B.Preparation for the proof: distributions and induction.By a 'subgroup' H < G of a supergroup we refer to a representable subgroup functor, or equivalently a closed subsuperscheme which is also closed under the group operation.
For the general linear supergroup GL M |N we consider the subgroups P + , P − < GL M |N .Here, for any superalgebra consists of all automorphisms which are expressed as (M + N )-block matrices in a way that the left down block of size N × M is zero.The subgroup P − corresponds similarly to a zero (M × N )-block.
For a supergroup G, we have the underlying affine group scheme G 0 , which can be defined as the restriction of the functor G to k-algebras (viewed as superalgebras contained in degree 0) or via the quotient of O(G) by the ideal generated by all odd elements.For G = GL M |N we have G 0 = P + ∩ P − = GL M × GL N .
For an affine group scheme G, one defines the distribution superalgebra as a subalgebra Dist G ⊂ O(G) * which is a cocommutative Hopf superalgebra, similarly to the classical case, see [BK03,§3].Explicit descriptions of these algebras for GL M |N and subgroups as P ± are also given loc.cit.
We also have the Lie superalgebra Lie G as a subspace of Dist G.For GL M |N this is the general linear Lie superalgebra gl M |N of square (M +N )-matrices with supercommutator.We have a vector space decomposition where g0 ⊕ g ± is the subalgebra corresponding to P ± < G.
For a supergroup G we denote by Rep k G the rigid monoidal category of finite dimensional (super) representations.In particular, for G an ordinary affine group scheme interpreted as a supergroup, this category is equivalent (as a k-linear additive category) to a direct sum of two copies of the classical representation category.
Proof.This can be proved by relying either on the theory of Harish-Chandra pairs from [Mas12] or the distribution algebras from [BK03].We choose the latter approach.
In [BK03] it is proved that, for H denoting any of the supergroups in the lemma, Rep k H is equivalent to the category of integrable finite dimensional modules of Dist H. Here, 'integrable' essentially means weight module.Clearly, on the level of (Dist H)-representations, restriction has a left adjoint functor given by induction, for instance Dist G ⊗ Dist P − .By the explicit realization in [BK03,§4], it follows that Dist G ≃ Λg − ⊗ Dist P, respectively Dist P − ≃ Λg − ⊗ Dist G 0 as right (Dist P )-representations respectively right (Dist G 0 )-representations.It follows easily that induction sends integrable modules to integrable modules, providing the desired left adjoints in (a).Statements (b) and (c) then follow again from the above and the explicit forms of the distribution algebras in [BK03,§4].
Proof.As a direct consequence of adjunction and the definition of ind G H , we find for any n+j .Thus, assumption (i) and the super analog of Lemma 1.2 imply the claim.□ 3C.Proof of Proposition 3.1 -positive characteristic.Now we fix M, N > 0, G = GL M |N , P ± < G from Section 3B and thus G 0 = P 0 = GL M × GL N .Let V = k M |N be the vector representation of G. Since we will use comparison with characteristic zero, we do not yet make assumptions on char(k).
For partitions λ = (λ i ) 1≤i≤M , µ = (µ j ) 1≤j≤N of length at most M , N , we denote by L 0 (λ|µ) the corresponding simple polynomial G 0 -representation, which we also interpret as a P -representation in the usual way (for instance with trivial action of g + ).
Proof.By Lemma 3B.1(b), ind G P L 0 (λ|µ) is free as a Λg − -representation, so every P − -submodule contains u ⊗ L 0 (λ|µ) for u a non-zero element (unique up to constant) in the top degree of Λg − .Let v + be a highest weight vector of L 0 (λ|µ).It suffices to prove that the g + -submodule of ind G P L 0 (λ|µ) generated by u ⊗ v + contains 1 ⊗ v + , where we use ind G P L 0 (λ|µ) ≃ Λg − ⊗ L 0 (λ|µ).By choosing a conveniently ordered product of all root vectors in g − for u and a mirrored product of root vectors for g + for v, it follows that which concludes the proof.□ Now we fix a prime p and consider the partitions α, ν of lengths M , N − 1 given by Since α M = N is greater than the length of ν, the partition κ := αν t of length M + ν 1 makes sense.Concretely It follows that κ is a p-core.
Before coming to a crucial proposition, we need the following (well-known) lemma.
Lemma 3C.2.Let λ be a p-core partition, and T be a standard λ-tableaux.The primitive idempotent e T in QS r associated to T via Young symmetrizers (so that in particular (QS r )e T is isomorphic to the simple Specht module S λ ) belongs to Z (p) S r ⊂ QS r .
Proof.According to the theory of Young symmetrizers, see for example [Ful97, Section 7.2], we can clear denominators in e T and get a pseudo-idempotent ẽT ∈ ZS r .This pseudo-idempotent satisfies ẽT ẽT = n λ • ẽT (in other words e T = ẽT /n λ ), where n λ ∈ Z is the product of the hook lengths in λ by e.g.[Ful97, Section 7.4, Exercises 18 and 19].Moreover, the hook lengths of p-cores are never divisible by p, see e.g.[JK81, Statement 2.7.40].Hence, for a p-core we have n λ / ∈ pZ, and we are done.□ For the remainder of the section, we assume that char(k) = p > 2. By the above lemma, we can take an idempotent in e 0 ∈ Z (p) S r , with r = |κ|, which is a primitive idempotent corresponding to κ, when considered in QS r , and we denote by e ∈ F p S r ⊂ kS r its image modulo p.We denote by S κ V the corresponding direct summand e(V ⊗r ) in V ⊗r and we use the same notation S κ V for the summand e 0 (V ⊗r ) when working over Q. Proposition 3C.3.Working either over k ′ = Q or k ′ = k and with α, ν, κ as just introduced, we have Proof.By construction (and realizing V as the extension of scalars over a free Z (p) -module), the character of the left-hand side is identical for k and Q.The GL M × GL N -representation L 0 (α|ν) has the Weyl character, because ν is a p-core and α is just a 'shifted' p-core.Indeed, this follows from [JM97, Theorem 4.5], or from a direct application of tilting theory and Lemma 3C.2.It therefore follows from Lemma 3B.1(2) that also the character of ind G P L 0 (α|ν) is identical over k and Q.By Lemma 3C.1, the right-hand side is a simple representation.Indeed, we can calculate which is never zero in k or in Q.In particular, over Q, the right-hand side is the simple representation with highest weight α|ν.The isomorphism over Q now follows from [BR87, Section 4].Now working over k, by the combination of the previous two paragraphs, the two representations have the same character and the right-hand side is simple.They must thus be isomorphic.□ Now we can prove the main result.
Proof of Proposition 3.1.By the combination of Proposition 3C.3 and Lemma 3B.2 it suffices to prove Proposition 3.1 for the supergroup P acting on V .We prove the equivalent formulation in terms of P − .Proposition 3C.3 and Lemma 3B.1(c) imply that S κ V ≃ ind P − G0 L 0 (α|ν) as P − -representations.In particular, the result for (P − , V ) follows, via Lemma 3B.2 from the purely even case (G 0 , V ) in Section 2. □

The general linear quantum group and consequences
Let k still denote an arbitrary field.Further, fix q ∈ k * .We consider either of the following objects.
To (k, q) we associate the pair, often called the mixed characteristic of (k, q), by of the orders | − | of 1 and q 2 in the additive group underlying k.For example, the case p = ℓ ∈ N corresponds to a field k of positive characteristic and q = 1 and the case p = ℓ = ∞ corresponds to a field of characteristic zero with generic q.We also use the quantum numbers for a ∈ N and x ∈ k: where the second equality is only applicable for q 2 ̸ = 1.It then follows that ℓ is also the minimal value of n for which [n] = 0, or ∞ when no such value exists.
We call q ∈ k * generic if q is not a root of unity in k, e.g.q could be the formal variable in the field C(q) of rational complex functions.(a) For k = C(q) and generic q we have (p, ℓ) = (∞, ∞).For k = F 7 (q) and generic q we have (p, ℓ) = (7, ∞).These cases are both 'semisimple', in the terminology of Remark 4.2.
(d) For k = F 7 and q = 1 we have (p, ℓ) = (7, 7).In general, the representation category of Γ M is semisimple if and only if ℓ = ∞. 3 We consider Lusztig's divided power quantum group over k as in [Lus90] associated to a type A Cartan datum.We use Remark 4.2.Following Example 4.1, all of our discussions regarding Γ M split into four distinct cases: (a) For (p, ∞) the quantum group representations are semisimple for any p, and their combinatorics is the same as for G(C), for G = SL M or G = GL M .We call this the semisimple case.We discuss this case in Section 4A.
(b) The case (∞, ℓ) for ℓ < ∞ can be combinatorially identified with its special case k = C.We call this the complex quantum group case.We discuss this case in Section 4D.
(c) The strictly mixed case is p, ℓ < ∞ and p ̸ = ℓ.We discuss this case in Section 4C.
(d) The situation p = ℓ < ∞ prime is characteristic p.This reduces to the case in Section 2. This list is ordered in increasing order of difficulty, in the sense that the dimensions of the indecomposable summands are increasing, reading from top to bottom (and also more difficult to compute).Hence, using the same definitions as before, the convergence rate of n b Γ M ,V n is slower for the bottom cases compared to the top ones.
Let V M denote the quantum vector representation of Γ M .With the same notation as in the previous sections we have the analog of Proposition 2.2: Proposition 4.3.For any M ∈ Z >0 we have Proof.The case M = 1 is again immediate, and M > 1 is proven in Section 4B and Section 4C.□ Remark 4.4.As before, we make use of the fact that V ⊗n M is tilting, see e.g.[AST17, Proposition 2.3].In particular, V ⊗n M is a direct sum of indecomposable tilting representations.For (m 1 , . . ., m M −1 ) ∈ N M −1 we use the Weyl representations ∆(m 1 , . . ., m M −1 ) and the indecomposable tilting representations T (m 1 , . . ., m M −1 ) of highest weight (m 1 , . . ., m M −1 ).
We now consider any k-subalgebra Γ ⊂ U k q (gl M ).In this case V ⊗n is a Γ-representation by restriction, although the tensor product of representations does not need to exist in general.
Proof of Theorem 1.4 -quantum groups of type A and k-subalgebras.The only difference to the proof in Section 2 is that we use Proposition 4.3 instead of Proposition 2.2.□ Remark 4.5.As observed in the 1990s or even earlier, embeddings of Lie subalgebras g → gl M do not quantize properly.Consequently, contrary to Section 2 and Section 3, proving Theorem 1.4 for the general linear case does not imply it for other quantum groups.For this reason, quantum groups that are not of type A are outside of the scope of this paper.On the other hand, Theorem 1.4 does include coideal subalgebras, with the most prominent example being quantum symmetric pairs (also called ıquantum groups), which have been studied many people, see e.g.[NS95], [Let99] or [Kol14].
4A. Proof of Proposition 4.3 -semisimple case.In this case, the indecomposable tilting representation T q (m 1 , . . ., m M −1 ) is isomorphic to the Weyl representation ∆ q (m 1 , . . ., m M −1 ), and the latter has the quantum Weyl character, see e.g.[Saw06, Equation 2].We therefore get the same bound as for the group schemes, namely

2
. As before, an analog of Proposition 2B.3 follows from the above.This in turn implies Proposition 4.3 and Theorem 1.4.
We can use the same arguments as in the reductive group case above, with the following adaptations (the reader should compare Example 2C.1 and Example 4B.1 while reading the below): (a) Recall that p (i) = p i−1 ℓ for i > 0 and p (0) = 1.All appearances of p r should be replaced by p (r) .
(b) The results we need from [And18] hold, mutatis mutandis, for the quantum group as well, see [And18, Remark 2.2].That is, instead of the Frobenius twist one uses the Frobenius-Lusztig twist which, roughly speaking, acts as the Frobenius twist on digits a i for i > 0 and as its quantum analog on the zeroth digit, and the rest is the same.Taking all of the above together, Proposition 4.3 follows from the same arguments as for SL M , which proves Theorem 1.4.4D.Some extra observations for the complex quantum group case.For p = ∞ the combinatorics (in particular, the multiplicities of the decompositions) are the same as for the complex root of unity case.Here, we consider (p = ∞, ℓ < ∞), since the case (∞, ∞) is semisimple and has the same combinatorics as the complex group case.This case is therefore already addressed in Section 2B and Section 4A.
In this case the numbers (tilting:Weyl) are known to be given by parabolic Kazhdan-Lusztig polynomials, see [Soe97] and [Soe98].Even better, for Γ M = SL M the results of [Str97] imply that the parabolic Kazhdan-Lusztig polynomials are bounded.This in turn implies, again using the quantum Weyl character formula as in Section 4A, that the dimension of the tilting representation T (m 1 , . . ., m M −1 ) is a polynomial in m 1 , . . ., m M −1 of degree M (M −1) 2 , as before.□ Thus, Proposition 4D.1 implies that Proposition 4.3 holds for (∞, ℓ) and arbitrary M , but the result is even a bit stronger.We can even say a little more for M ∈ {2, 3}: Proof.(a).By Example 4B.1.(b).We will crucially use that the tilting characters (the Weyl multiplicities with the indecomposable tiling representations) are known explicitly by, for example, [Soe97], [Soe98] and [Str97].This explicit description of the characters is known as periodic patterns.
The quantum version of Weyl character formula gives dim ∆(m 1 , m 2 ) = (m 1 + 1)(m 2 + 1)(m 1 + m 2 + 2).We will use this as follows.The periodic pattern for U k q (sl 3 ) tilting representations are given by These patterns mean that e.

Counterexamples
In this section we will show that Theorem 1.3 does not extend arbitrarily, even over k = C.We will use analog notion as before.
Theorem 5.1.Let k = C.For every A ∈ R ≥0 there exists a cotriangular Hopf algebra Γ with a finite dimensional Γ-corepresentation V such that  Remark 5.2.There is also a quantum version of Theorem 5.1 where one replaces the trace condition by qtr(E T E −1 ) = −1 − q 2 ∈ k.

Questions
We list a few open questions regarding b Γ,V n and β Γ,V .Question 6.1.Let Γ 2 = SL 2 and let V 2 denote its vector representation.It is easy to observe that the number of summands in V ⊗n 2 in positive characteristic is bounded by the corresponding number in characteristic zero.In particular, Example 2B.4 and Lemma 2C.2 imply that for arbitrary p we have  s) , where f (s) = s − 1 + log 2 (1 + 2 s ).This can be proven using the character formula for T (m) in Example 2C.1.In particular, the average of 1/ dim T (m) is n f (−1) = n −2+log 2 (3/2) = n − log 2 (8/3) .Now, in characteristic zero the same type of argument would give that n −1 is proportional to E 1/ dim T (m) .By Example 1.7 this suggests to take the square root, which gives the correct result up to a factor, i.e. in characteristic zero we have For char(k) = 2 this then suggests to take δ = 1 2 log 2 (8/3) ≈ 0.708 which, empirically speaking, seems to be correct, see the Mathematica output above.

Proof.
It follows from Example 2C.1 that dim T (m) = 2 k a d p d , where k is the number of non-zero digits among the a d−1 , . . ., a 1 , a 0 in the p-adic extension of m + 1.This implies dim T (m) ≤ (m + 1) α : First, we have a d p d ≤ m + 1 so it remains to argue that 2 k ≤ (m + 1) log 2 2/ log 2 p .Note secondly that 2 k ≤ 2 d−1 and 2 d−1 = (m + 1) b for b = (d − 1) log 2 2/ log 2 (m + 1).However, m + 1 = a d p d + • • • + a 1 p + a 0 for a d ̸ = 0 which gives log 2 (m + 1) ≥ log 2 (p d−1 ) = (d − 1) log 2 p so that b ≤ log 2 2/ log 2 p.The result follows.□ Remark 2C.3.The number α = 1 + (log 2 p) −1 in Lemma 2C.2 converges to 1 for p → ∞, and p = ∞ is the semisimple case where dim T (m) = m + 1. Example 2C.4.For p = 2 (left) and p = 3 (right) we get , which are again Mathematica log plots.3 a n and the proof completes.□ Recall that the category of finite dimensional Γ M = SL M -representations, considered as an abelian category, has a direct summand ST r (Γ M ) = ST p r (Γ M ) consisting of representations which are linked with the Steinberg representation St r = St p r = T (p r − 1)ρ (note that these depend on p), see [And18, Section 3.5].These Steinberg representations are tilting and Weyl representations at the same time, and we will use this below.

.Proof.
Proof of Theorem 1.4.(a)-affine semigroup superschemes.A representation of an affine semigroup superscheme Γ corresponds to a semigroup homomorphism Γ → M at M |N , with M at M |N denoting the monoid superscheme of square (M + N )-matrices.In particular, the number of summands in V ⊗n over Γ is bounded from below by the number of summands over M at M |N .By considering O(M at M |N ) as a subcoalgebra of O(GL M |N ), we can identify the category of M at M |N -representations with the category of polynomial GL M |Nrepresentations, so the number of' direct summands in V ⊗n over M at M |N is the same as over GL M |N .In conclusion, the number of direct summands over Γ is bounded from below by the number of summands over GL M |N .The result thus follows from Proposition 3.1 for G = GL M |N .□ 3A.Proof of Proposition 3.1 -semisimple case.Assume that char(k) = 0. We get a stronger statement: Lemma 3A.1.The GL M |N -representation V ⊗n M |N is semisimple and the dimension of the simple representations occurring in V ⊗n M |N is bounded by a polynomial in n of degree M (M −1)+N (N −1) 2 That the tensor powers are semisimple is proved in [BR87, Theorem 5.14].The dimension of these simple representations is bounded by that of the Kac modules with same highest weight, see for instance [Mus12, §8.2].As induced modules, the dimension of the latter is given by a constant (depending on M N , see for instance Lemma 3B.1(b)) times the dimension of the simple (GL M × GL N )-representation with same highest weight, which is a weight appearing in V ⊗n M |N .The latter can be bounded by a polynomial in n of degree M (M −1)+N (N −1) 2 , as explained in Section 2B.□ Let b M,N n be the analog of b M n for GL M |N .Proposition 3A.2.We have

Lemma 3B. 1 .
Set G = GL M |N .(a) The forgetful functors res G P : Rep k G → Rep k P and res P − G0 : Rep k P − → Rep k G 0 have left adjoint functors ind G P and ind P − G0 .(b) As G 0 -representations and g − -representations, we have

Example 4. 1 .
Let us give a few examples:
g. the tilting representation with highest weight in a star pattern at the north east (the position of the highest weight is indicated by zero) has the twelve Weyl representations indicated by the circles in its Weyl filtration.All appearing highest weights of the Weyl representation are in the action orbit of the affine Weyl group on this alcove picture.Thus,dim ∆(m 1 , m 2 ) = (m 1 + 1)(m 2 + 1)(m 1 + m 2 + 2) implies that dim T (m 1 , m 2 ) ≤ 12(m 1 + 1)(m 2 + 1)(m 1 + m 2 + 2)for the star pattern.All other patterns have fewer Weyl factors and the claim follows.The same bound, as one easily checks, works for the periodic patterns along the boundary of the Weyl alcove as well.□ Similar to Section 4C we have certain summands that appear often enough to imply Theorem 1.3: Proposition 4D.3.Let M = 2 and ℓ = 3.Then there exist a family of summands of V ⊗n 2 whose number of appearance t n in V ⊗n 2 satisfies 2 n /n 5/2 ≤ t n .Proof.Via quantum Schur-Weyl duality, this is [KST22, Theorem 4E.2].□ Remark 4D.4.The restriction to ℓ = 3 in Proposition 4D.3 is used as [KST22] study the monoid version of the Temperley-Lieb calculus.Similar (and sharper) results can be obtained for any ℓ ∈ N by using Schur-Weyl duality and [Spe23, Propositions 9.4 and 9.5].

Proof.
The main player in this proof is the Temperley-Lieb categoryT L(−2) of Rumer-Teller-Weyl [RTW32] with circle parameter −2 ∈ C. It is the diagrammatic incarnation of D = Rep C SL 2 (C).Let m ≥ 2 and let X = C 2 be the vector representation of SL 2 (C).It follows from [Bic03, Theorem 1.1] that every matrix E ∈ GL m (C) with tr(E T E −1 ) = −2 ∈ Cgives a nonsymmetric fiber functorF E : D → Vect C , X → C m .Here we use the notion fiber functor as in e.g.[EGNO15, Definition 5.1.1].Since the number of indecomposable summands b D,X n does not depend on F E but only on Rep C SL 2 (C), the above implies that β D,X = 2 < dim X = m.

Finally, [ Bic03 ,
Theorem 1.1] and reconstruction theory as in e.g.[EGNO15, Theorem 5.4.1]provide a cotriangular Hopf algebra Γ E = H(F E ) for E ∈ GL m (C) as above such that, as monoidal categories,coRep C Γ E ∼ = Dand such that F E becomes the forgetful functor.It remains to argue that tr(E T E −1 ) = −2 admits a solution for every m ∈ N ≥2 .Indeed, we can take x a solution to the equationx 2 − x(m − 1) + 2 = 0 if m ̸ = 2 (which always has two solutions for m ∈ N ≥3 ), and x = 1 for m = 2. □ Theorem 5.1 implies Theorem 1.4.(b) and Theorem 1.9.(c).

n≤
B • 2 n /n 1/2for A, B ∈ R >0 and α = 1 + (log 2 p) −1 .So we ask: For fixed p, is there someδ = δ(p) ∈ R >0 for which b Γ2,V2 n ∈ Θ(2 n n −δ )?Note that in characteristic zero we have δ = 1/2 by Example 2B.4.Remark 6.2.The following observation was communicated to us by Pavel Etingof.For p = 2 the value δ ∈ R >0 in Question 6.1 appears to be δ = 1 2 log 2 (8/3), which is approximately 0.708.For example, , .areMathematica log plots for p = 2 with α as in Lemma 2C.2.The motivation for δ = 1 2 log 2 (8/3) is as follows.Consider the random variable dim T (m) s , where T (m) as before denotes an indecomposable tilting Γ 2 -representation with highest weight m in [n/2, n − 1], and consider the uniform distribution of m.Then the expectation value E dim T (m) s of this is, for n ≫ 0, proportional to n f ( >0 .Let F p r be the finite Galois field with p r elements.For k = F p r , since the number of indecomposable summands is bounded from above by the number of indecomposable summands for k = C, Example 1.7 implies that the growth rate of b Γ2,V2 For m 1 , . . ., m M −1 ∈ N M −1 we denote by ∆(m 1 , . . ., m M −1 ) the Weyl representations of Γ M = SL M of highest weight (m 1 , . . ., m M −1 ).This is in terms of the fundamental weights meaning that the highest weight is To approximate the above formula recall that, for all a ∈ Z ≥1 , we have