SYMMETRIC TENSOR CATEGORIES IN CHARACTERISTIC 2

DAVE BENSON AND PAVEL ETINGOF

Abstract. We construct and study a nested sequence of finite symmetric tensor categories Vec = C_{0} ⊂ C_{1} ⊂ ⋯ ⊂ C_{n} ⊂ ⋯ over a field of characteristic 2 such that C_{2n} are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius–Perron dimension. This generalizes the category C_{1} described by Venkatesh [Ven] and the category C_{2} defined by Ostrik. The Grothendieck rings of the categories C_{2n} and C_{2n+1} are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2^{n+2}-th root of unity, O_{n} = ℤ[2 cos(π/2^{n+1})].

1. Introduction

Let C be a symmetric (abelian) tensor category over an algebraically closed field k ([EtGNO], Definitions 4.1.1, 8.1.12). Assume that C is finite, i.e., equivalent to the category of representations of a finite dimensional algebra. When char(k) = 0, P. Deligne proved in [De2] that C admits a (necessarily unique) fiber functor to the category sVec_{k} of supervector spaces (see also [EtGNO], Theorem 9.11.5). In other words, C may be realized as the category of representations of a finite supergroup with an element of order ≤ 2 acting by parity.

For semisimple C, this result was generalized by V. Ostrik in [O2] to characteristic p > 0. Namely, he showed that C admits a fiber functor to the category Ver_{p}(k), the semisimplification of the category Rep_{k}ℤ/p of representations of the group ℤ/p over k. For instance, if p = 2 then Ver_{p}(k) = Vec_{k} is the category of vector spaces, so Ostrik’s result says that C = RepG, where G is a finite group scheme over k. In other words, any such C admits a (unique) fiber functor C → Vec_{k}.

One may wonder if this is true without the semisimplicity assumption. This turns out not to be the case. Namely, let C_{1} be the tensor category of representations of the Hopf algebra k[d]/d^{2} with the commutativity isomorphism given by

s(v ⊗ w) = w ⊗ v + dw ⊗ dv.

Then C_{1} does not admit a fiber functor to Vec_{k} ([Ven], 1.5).

So one may ask if any category C as above has a fiber functor to C_{1}. This is also false, as was recently shown by V. Ostrik. Namely, V. Ostrik constructed a symmetric tensor category C_{2} equivalent to C_{1} ⊕ Vec_{k} as an abelian category, which does not admit a fiber functor to C_{1} ([O3]). So one may ask if any C as above has a fiber functor to C_{2} until the next counterexample is found, and so on.

The goal of this paper is to construct and study an infinite ascending chain of finite symmetric tensor categories

Vec_{k} = C_{0} ⊂ C_{1} ⊂ C_{2} ⊂ C_{3} ⊂ ⋯

in characteristic 2 such that C_{2n} is incompressible, i.e., does not admit a tensor functor to a finite tensor category of smaller Frobenius–Perron dimension (even a non-symmetric one).

This shows that there is no finite symmetric tensor category D over k such that any finite symmetric tensor category C over k admits a fiber functor to D.

This motivates the following question, which currently remains open.

Question 1.1. Consider the infinite category C_{∞} = ⋃_{n≥0} C_{n}. Does any finite symmetric tensor category over k admit a fiber functor to C_{∞}?

The organization of the paper is as follows. In Section 2 we state the main theorem, giving a construction of the categories C_{n}. In Section 3 we prove the main theorem, and show that C_{2n} categorifies the ring O_{n} = ℤ[2 cos(π/2^{n+1})] (i.e., this ring is the Grothendieck ring of C_{2n}). In Section 4 we study further properties of the categories C_{n}, in particular show that they are incompressible, compute their Cartan matrices and prove their universal property. We also give an application of the categories C_{n} to modular representation theory of finite groups. In Section 5 we compute the structure of C_{n} for small n.

Acknowledgements. This material is based on work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. The work of the second author was also partially supported by the NSF grant DMS-1502244. The second author is very grateful to V. Ostrik for sharing his construction of the category C_{2} and for numerous useful discussions, without which this paper would not have appeared. The authors also thank A. Davydov, R. Guralnick, D. Nikshych, V. Serganova, P. Tiep and G. Williamson for useful discussions.

2. The main theorem

Throughout the paper, we will use the basics on tensor categories from [EtGNO].

Let k be an algebraically closed field of characteristic p > 0. Consider the symmetric rigid monoidal category Tilt_{p} of tilting modules over SL(2, k) (see [J], Appendix E). Its indecomposable objects are T_{m}, m ≥ 0 (the tilting module with highest weight m). It is known that this category contains a descending chain of tensor ideals I_{1} ⊃ I_{2} ⊃ ⋯, where I_{n} is spanned by the objects T_{m}, m ≥ p^{n} − 1. Let T_{n,p} = Tilt_{p}/I_{n}. If n = 1, the category T_{n,p} is semisimple abelian and equivalent to the Verlinde category Ver_{p} = Ver_{p}(k), but for n ≥ 2 it is in general only Karoubian and not abelian.

Now assume that p = 2. Recall that in any symmetric tensor category C over k and X ∈ C, the Frobenius twist X^{(1)} is the cohomology (i.e., the kernel modulo the image) of the operator

d = 1 + s: X ⊗ X → X ⊗ X,

where s is the commutativity isomorphism (note that d^{2} = 0, so the cohomology is well defined). Let K be the algebraic closure of the fraction field of the ring of Witt vectors W (k). Let Ver_{2^{n+1}}(K) denote the category of tilting modules over the quantum group SL_{q}(2, K) at a 2^{n+2}th primitive root of unity q modulo negligible objects (see [EtGNO], Subsection 8.18.2), and let Ver_{2^{n+1}}^{+}(K) be its even part, spanned by modules with even highest weight.

Our main result is the following theorem.

Theorem 2.1. Let char(k) = 2. There exists an ascending chain of finite symmetric tensor categories over k, Vec_{k} = C_{0} ⊂ C_{1} ⊂ C_{2} ⊂ ⋯ (with fully faithful symmetric tensor embeddings) having the following properties.

(i) For each n ≥ 0, C_{2n} contains T_{n+1,2} as a generating full rigid monoidal Karoubian subcategory.

(ii) There are no symmetric tensor functors C_{i+1} → C_{i} and no tensor functors C_{2i+2} → C_{2i+1}, i ≥ 0.

(iii) There exists a (non-symmetric) tensor functor F_{i}: C_{2i+1} → C_{2i}, i ≥ 0. Namely, for i ≥ 1, the category C_{2i} is equipped with a distinguished self-dual simple object X_{i}, and C_{2i+1} is the category of finite dimensional modules over the commutative cocommutative Hopf algebra A_{i} = ∧X_{i} with a nontrivial triangular structure defining the symmetric braiding of C_{2i+1}. For i = 0, the construction is the same except that X_{0} = 0 and A_{0} = ∧𝟙 with a nontrivial triangular structure. In particular, C_{2i+1} has the same simple objects as C_{2i}.

(iv) One has

FPdim(C_{2n}) = FPdim Ver_{2^{n+1}}(K) = \frac{2^{n}}{sin^{2}(π/2^{n+1})},

FPdim(C_{2n−1}) = FPdim Ver_{2^{n+1}}^{+}(K) = \frac{2^{n−1}}{sin^{2}(π/2^{n+1})}.

(v) All simple objects of C_{n} are self-dual.

(vi) One has

FPdim(X_{n}) = 2 cos(π/2^{n+1}) = \sqrt{2 + \sqrt{2 + ⋯ + \sqrt{2 + \sqrt{2}}}} (n roots),

and

dim(X_{n}) = 0.

(vii) One has A_{n} = 𝟙 ⊕ X_{n} ⊕ 𝟙 as an object of C_{2n}, with multiplication X_{n} ⊗ X_{n} → 𝟙 being the evaluation morphism of X_{n} composed with the (arbitrarily normalized) identification X_{n} ≅ X_{n}^{∗} in the first component. In particular,

FPdim(A_{n}) = 2 + 2 cos(π/2^{n+1}) = 4 cos^{2}(π/2^{n+2}).

(viii) As an abelian category, C_{2n+1} is indecomposable; moreover, all simple objects of this category occur in the projective cover of 𝟙.

(ix) We have C_{2n+2} = C_{2n+1} ⊕ M_{n}, and M_{n} is equivalent to C_{2n} as a C_{2n+1}-bimodule category, where the bimodule structure is defined by the functor F_{n} and the symmetric braiding of C_{2n+1}. Under this equivalence, X_{n+1} corresponds to 𝟙 ∈ C_{2n}. In particular, C_{2n} has 2^{n} simple objects. Namely, the simple objects of C_{2n} are X_{S} := ⨂_{i∈S}X_{i}, where S ⊂ {1, …, n}. Thus,

C_{2n} = C_{2n−1} ⊕ C_{2n−3} ⊕ ⋯ ⊕ C_{1} ⊕ C_{0}

as abelian categories, i.e., it has n blocks.

(x) The projective cover of a simple object Y ∈ C_{2n+1} in C_{2n+1} (and C_{2n+2}) has the form A_{n} ⊗ P_{Y}, where P_{Y} is the projective cover of Y in C_{2n}.

(xi) One has X_{n+1}^{(1)} = X_{n}.

Theorem 2.1 is proved in the next section.

The categories C_{n} for n ≤ 2 were known before. Namely, the category C_{1} is the category of modules over the Hopf algebra k[d]/d^{2}, with braiding defined by

c = s(1 ⊗ 1 + d ⊗ d);

it is a reduction modulo 2 of the category of supervector spaces sVec_{K} = Ver_{4}^{+}(K) and appears in [Ven]. The category C_{2} is the category T_{2,2}, which happens to be abelian; this fact was observed by V. Ostrik. It has two simple objects 𝟙 and X = X_{1} and one more indecomposable P – the projective cover of 𝟙, which is a nontrivial extension of 𝟙 by 𝟙, and one has X ⊗ X = P, so FPdim(X) = \sqrt{2}. This category is a reduction modulo 2 of the Ising category Ver_{4}(K).

Corollary 2.2. Let O_{n} be the ring of integers in the field ℚ(2 cos(π/2^{n+1})) = ℚ(ζ_{n} + ζ_{n}^{−1}), where ζ_{n} = exp(πi/2^{n+1}); that is, O_{n} = ℤ[2 cos(π/2^{n+1})] ([W], p.16, Proposition 2.16). Then the map FPdim defines an isomorphism of the Grothendieck ring of C_{2n} onto O_{n}. Under this isomorphism, the simple objects X_{S} of C_{2n} map to the numbers d_{S} := ∏_{j∈S} d_{j}, where d_{j} := ζ_{n}^{2^{n−j}} + ζ_{n}^{−2^{n−j}}.

Proof. It is well known that [ℚ(ζ_{n} + ζ_{n}^{−1}) : ℚ] = 2^{n}. Also, FPdim(X_{S}) = d_{S}. It is clear that any number of the form ζ_{n}^{r} + ζ_{n}^{−r}, r = 1, …, 2^{n} − 1 is a linear combination of d_{S} (by taking the binary expansion of r). This implies that FPdim: Gr(C_{2n}) → O_{n} is an isomorphism mapping X_{S} to d_{S}. □

Corollary 2.3. Let S ⊂ [1, n] and i ∈ [1, n]. Let k_{S}(i) be the largest integer ≤ i which does not belong to S. Then in Gr(C_{2n}) we have

X_{i}X_{S} = X_{k_{S}(i)∪S\[k_{S}(i)+1,i]} + 2 ∑_{k=k_{S}(i)+1}^{i} X_{S\[k,i]}

where we agree that for any S, X_{0∪S} = 0.

Proof. This follows immediately from Corollary 2.2. □

Corollary 2.3 gives the following recursion to determine the structure constants N_{ST}^{U} of the multiplication of Gr(C_{2n}), such that X_{S}X_{T} = ∑_{U} N_{ST}^{U} X_{U} (where S, T, U ∈ [1, n]). Namely, given S, T, U ⊂ [1, n − 1], we have

N_{ST}^{n∪U} = N_{n∪S,T}^{U} = N_{S,n∪T}^{U} = N_{n∪S,n∪T}^{n∪U} = 0,
N_{n∪S,T}^{n∪U} = N_{S,n∪T}^{n∪U} = N_{ST}^{U} ,

and

N_{n∪S,n∪T}^{U} = N_{ST}^{[1+max U,n−1]∪U\max U} + 2 ∑_{k=max U}^{n−1} N_{ST}^{[k+1,n−1]∪U}, U ≠ ∅;

N_{n∪S,n∪T}^{∅} = 2 ∑_{k=0}^{n−1} N_{ST}^{[k+1,n−1]}.

It is easy to see that in fact this sum can contain at most one nonzero term. Namely, we have the following corollary.

Corollary 2.4. Let K be the largest integer ≤ n − 1 contained in none or both of the sets S, T. Then if U ∉ ∅ then

N_{n∪S,n∪T}^{U} = \begin{cases}N_{ST}^{[1+max U,n−1]∪U\max U}, K = max U \\ 2N_{ST}^{[K+1,n−1]∪U}, K > max U \\ 2N_{ST}^{[max U+1,n−1]∪U}, K < max U \end{cases}

and

N_{n∪S,n∪T}^{∅} = 2N_{ST}^{[K+1,n−1]}.

In particular, all nonzero numbers N_{ST}^{U} are powers of 2.

3. Proof of the main theorem

We will construct the categories C_{n} inductively starting from C_{0} = Vec_{k}. Assume they have been constructed until C_{2n}, with the claimed properties (except (i)), and let us construct C_{2n+1}, C_{2n+2}. Then we will separately prove (i).

3.1. Construction of C_{2n+1} and proof of (iii),(x), and the second part of (iv). Consider the algebra A_{n} = 𝟙 ⊕ X_{n} ⊕ 𝟙 as defined in the theorem. This is a commutative and cocommutative Hopf algebra (as we are in characteristic 2). Hence C_{2n+1} := A_{n}-mod is a finite tensor category. Moreover, since A_{n} is a local algebra, the simple objects of C_{2n+1} are the same as those of C_{2n}, and the projective cover of a simple object Y ∈ C_{2n} in C_{2n+1} is the free A_{n}-module A_{n} ⊗ P_{Y}, where P_{Y} is the projective cover of Y in C_{2n}. This implies that

FPdim(C_{2n+1}) = FPdim(C_{2n}) FPdim(A_{n}) =

= \frac{2^{n}}{sin^{2}(π/2^{n+1})} · 4 cos^{2}(π/2^{n+2}) = \frac{2^{n}}{sin^{2}(π/2^{n+2})}.

Now let us put a symmetric structure on C_{2n+1}. We will define this structure by the formula

c := s ∘ R,

where R ∈ Hom(𝟙, A_{n} ⊗ A_{n}) is a triangular structure on A_{n} (the universal R-matrix); then R defines a functorial isomorphism R_{Y Z}: Y ⊗ Z → Y ⊗ Z for any Y, Z ∈ C_{2n+1}. Namely, let τ: 𝟙 → X_{n} ⊗ X_{n} be the coevaluation map, and define R by the formula

R = 1 ⊗ 1 + τ + a ⊗ a,

where a: 𝟙 → A_{n} is the tautological inclusion in the top degree. It is easy to see that under suitable normalizations of τ and a we have

(1)  Δ(a) = a ⊗ 1 + 1 ⊗ a + τ, τ_{13} · τ_{23} = τ ⊗ a,

which implies that R satisfies the hexagon axioms (Δ⊗1)(R) = R_{13}R_{23}, (1⊗Δ)(R) = R_{13}R_{12}. Also, (1) implies that

τ^{2} = dim(X_{n})a ⊗ a = 0,

so R_{21}R = 1 ⊗ 1, hence R is a triangular structure, which defines a symmetric braiding on C_{2n+1}.

Thus, we have established parts (iii),(x), and the second part of (iv).

3.2. Proof of (ii) for C_{2n+1}. Let us show that there are no symmetric tensor functors C_{2n+1} → C_{2n}. Observe that the Frobenius–Perron dimensions of the simple objects X_{S} of C_{2n} are linearly independent over ℚ (in fact, they form a ℚ-basis of ℚ(cos(π/2^{n+1}))). This implies that any tensor functor F: C_{2n+1} → C_{2n} must map X_{S} to X_{S} for each S. In particular, F (X_{1,...,n}) = X_{1,...,n}, which is projective, which implies that F is a surjective tensor functor, in fact already so when restricted to C_{2n} (see [EtGNO], Subsection 6.3). Therefore, by [EtGNO], Proposition 6.3.4, F |_{C_{2n}} : C_{2n} → C_{2n} is an equivalence, and we may assume (by post-composing with its inverse) that F |_{C_{2n}} : C_{2n} → C_{2n} is the identity.

Now let F : C_{2n+1} → C_{2n} be a symmetric tensor functor. Let G = \underline{Aut}_{⊗}(F) be the group scheme in C_{2n} corresponding to F, and K = π_{1}(C_{2n}) ⊂ G be the fundamental group of C_{2n} ([De1], Section 8). Since the functor F is a split surjection, the inclusion K ↪ G is a split injection. Thus, G ≅ K ⋉ N, where N is a closed normal group subscheme in G. Let H = O(N)^{∗} be the group algebra of N (a cocommutative Hopf algebra in C_{2n}). Then C_{2n+1} is the category of H-modules in C_{2n}, and F is the forgetful functor (forgetting the structure of an H-module).

The Frobenius-Perron dimension of H equals FPdim(C_{2n+1})/ FPdim(C_{2n}) = 2+FPdim(X_{n}). Hence the augmentation ideal I of H is isomorphic to X_{n} ⊕ 𝟙 (since there are no nontrivial extensions between 𝟙 and X_{n} in C_{2n}), i.e., H = 𝟙 ⊕ X_{n} ⊕ 𝟙. Moreover, it is clear that I acts trivially in every simple H-module, hence it is nilpotent. This easily implies that H ≅ A_{n} as an algebra (namely, the product X_{n} ⊗ X_{n} → 𝟙 ⊂ I is nonzero since H is a Hopf algebra). Thus, we may assume that F is isomorphic to F_{n} as an additive functor. In other words, we may assume that F is obtained from F_{n} by a Drinfeld twist J of the Hopf algebra A_{n}. The twist J must have the form

J = 1 ⊗ 1 + λτ + μa ⊗ a,

where λ, μ ∈ k. Since the functor F is symmetric, we must have (J^{21})^{−1}RJ = 1 ⊗ 1, which implies τ = 0, a contradiction. This proves (ii) for C_{2n+1}.

3.3. Proof of (viii). Let us show that every simple object of C_{2n+1} occurs in the projective cover P_{n}(𝟙) of 𝟙 in C_{2n+1}; in particular, this category is indecomposable. Indeed, for n = 0 this is clear, so we can consider n ≥ 1. We know that every simple object of C_{2n−1} occurs in P_{n−1}(𝟙), and P_{n}(𝟙) = ∧X_{n} ⊗ P_{n−1}(𝟙). Since ∧X_{n} = 𝟙 ⊕ X_{n} ⊕ 𝟙, we conclude that P_{n}(𝟙) involves all simple objects of C_{2n}, hence of C_{2n+1}. This proves (viii).

3.4. Construction of C_{2n+2}. Now let us construct the symmetric tensor category C_{2n+2}. For this, we can use [DaR], Theorem 2.8, for S = C_{2n}, H = A_{n}, σ = 1 + a, β = 1, g = 1, ω = 1, Γ: H^{∗} → H defined by R = (σ ⊗ σ)Δ(σ)^{−1}, λ(a) = 1, λ(1) = 0. Namely, Theorem 2.8 of [DaR] constructs a braided category, and in our case it is easy to see that the braiding is symmetric (since σ^{2} = 1).

The tensor category C_{2n+2} may also be constructed as a ℤ/2-extension of C_{2n+1} using the methods of [EtNO] extended to the non-semisimple case in [DaN1]. Namely, we define C_{2n+2} = C_{2n+1} ⊕ M_{n}, where M_{n} = C_{2n} as a C_{2n+1}-module category (which is exact since the functor F_{n} is surjective), and make it a C_{2n+1}-bimodule category by using the symmetric braiding on C_{2n+1}. To define the tensor structure on C_{2n+2}, we first need to show that the bimodule category M_{n} is invertible, and moreover defines an involution in the Brauer-Picard group BrPic(C_{2n+1}). To this end, note that the R-matrix of A_{n} is nondegenerate, i.e., defines a Hopf algebra isomorphism Γ: A_{n}^{∗} → A_{n}, and this isomorphism is symmetric since R_{21} = R. Hence, analogously to [EtNO], Proposition 9.3, M_{n} is an involution in BrPic(C_{2n+1}), as desired.

Now, according to [EtNO], the first obstruction to lifting this structure to a structure of a tensor category on C_{2n} lies in H^{3}(ℤ/2, I), where I is the group of invertible objects of the Drinfeld center of C_{2n+1}. This Drinfeld center is the category of pairs (Z, ϕ), where Z ∈ C_{2n+1} and ϕ: Z⊗? →?⊗Z is a functorial isomorphism satisfying consistency conditions. If (Z, ϕ) is invertible then Z is invertible, hence Z = 𝟙. Thus I = Aut_{⊗}(Id), the category of tensor automorphisms of the identity functor of C_{2n+1}. But any such tensor automorphism has finite order d and defines a ℤ/d-grading on C_{2n+1}. Since C_{2n+1} is indecomposable, this grading must be trivial and d = 1. Hence I = 1 and the first obstruction vanishes. Also H^{2}(ℤ/2, I) vanishes, hence there is a unique way to pass to the next step of the extension process.

The second obstruction now lies in H^{4}(ℤ/2, k^{×}), which vanishes. So the extension exists, and moreover it is unique since H^{3}(ℤ/2, k^{×}) also vanishes (as we are in characteristic 2).

Moreover, by using a version of the extension theory of [EtNO] for braided and symmetric categories ([DaN2]) the category C_{2n+1} can be endowed with a braided structure, which in fact turns out to be symmetric. Namely, following [DaN2], to a finite braided tensor category C over k one can attach a simply connected classifying space B_{2}C with three nontrivial homotopy groups, π_{2}, π_{3} and π_{4}. Then if G is an abelian group then G-extensions of C as a braided category correspond to homotopy classes of maps ϕ : K(G, 2) → B_{2}C, where K(G, 2) is the Eilenberg-Mac Lane space attached to G. Such maps can also be classified using classical obstruction theory. Namely, at the first step we have to fix a homomorphism ρ : G → π_{2}. Then we get the first obstruction O_{4}(ρ) ∈ H^{4}(K(G, 2), π_{3}). If it vanishes, then we go to the second step and get to make a choice b in a torsor over H^{3}(K(G, 2), π_{3}), and we get the second obstruction O_{5}(ρ, b) ∈ H^{5}(K(G, 2), π_{4}). If it vanishes, then we go to the third step and get to choose an element γ in a torsor over H^{4}(K(G, 2), π_{4}), and it determines the map ϕ uniquely up to homotopy.

In our situation, C = C_{2n+1}, G = ℤ/2, and we have already chosen the map ρ given by the module category M_{n}. Also, π_{3} is the group of invertible objects in the symmetric center of C_{2n+1}, i.e., π_{3} = 1. Thus, O_{4} automatically vanishes and we have a unique choice for b. Finally, π_{4} = k^{×}, so since H_{4}(K(G, 2)) and H_{5}(K(G, 2)) are 2-groups (see e.g. [EiM], Theorem 22.1), we get H^{4}(K(G, 2), π_{4}) = H^{5}(K(G, 2), π_{4}) = 0 (as char(k) = 2, so k^{×} is a uniquely 2-divisible group). This implies that the map ϕ (and hence the corresponding braided extension) is uniquely determined by ρ.

Furthermore, by using the version of the same theory for symmetric categories (with H^{j} replaced by H^{j+1}, π_{i} with π_{i+1}, B_{2} with B_{3} and K(G, 2) replaced by K(G, 3)), also described in [DaN2], one can show that the obtained braided extension is in fact symmetric.

3.5. Proof of (ii),(iv),(v),(vi) and (ix). Now we have a symmetric tensor category C_{2n+2}. By construction, FPdim(C_{2n+2}) = 2 FPdim(C_{2n+1}), which yields (iv). Also, C_{2n+2} contains a simple object X_{n+1} ∈ M_{n} corresponding to 𝟙 ∈ C_{2n+1}, and the simple objects of C_{2n+2} are of the form X_{S} and X_{S} ⊗ X_{n+1}, where S ⊂ {1, …, n}. This establishes (ix).

Now, since FPdim(C_{2n+1}) = FPdim(M_{n}) ([EtNO], Theorem 3.5.2)^{1}, we have

FPdim(X_{n+1})^{2} = \frac{FPdim(C_{2n+1})}{FPdim(C_{2n})} = 4 cos^{2}(π/2^{n+2}),

hence FPdim(X_{n+1}) = 2 cos(π/2^{n+2}). Also for Frobenius–Perron dimension reasons, in the Grothendieck group Gr(C_{2n+2}), we have

X_{n+1}^{2} = 2 · 𝟙 + X_{n},

1Theorem 3.5.2 of [EtGNO] applies to semisimple categories, but the proof extends in a straightforward way to the non-semisimple case.

hence dim(X_{n+1}) = 0, establishing (vi). Also X_{n+1} is self-dual again for dimension reasons, proving (v). Finally, once again for dimension reasons, there are no tensor functors C_{2n+2} → C_{2n+1}, establishing (ii).

3.6. Proof of (vii) and (xi). We will need the following lemmas.

Lemma 3.1. Let C be a symmetric tensor category over a field of characteristic 2, and X ∈ C. Then X ⊗ X has a 3-step filtration whose successive quotients are ∧^{2}X, X^{(1)}, ∧^{2}X.

Proof. The filtration is given by F_{1} = Im(1 + s), F_{2} = Ker(1 + s), F_{3} = X ⊗ X. □

Lemma 3.2. Let C be a symmetric tensor category over a field of any characteristic, and X ≠ 0 be an object of C such that ∧^{2}X = 0 (or, equivalently, S^{2}X ≅ X ⊗ X). Then X is invertible.

Proof. Let us first show that X is simple. Assume the contrary, i.e., that we have a 2-step filtration on X with successive quotients Y, Z ≠ 0. Then grS^{2}X is a quotient of

S^{2}grX = S^{2}(Y ⊕ Z) = S^{2}Y ⊕ (Y ⊗ Z) ⊕ S^{2}Z.

This is a contradiction with S^{2}X ≅ X ⊗ X, since

gr(X ⊗ X) = Y ⊗ Y ⊕ 2(Y ⊗ Z) ⊕ Z ⊗ Z,

i.e., has strictly larger length than grS^{2}X.

Now note that s|_{X⊗X} = 1, hence s|_{X⊗X^{∗},X⊗X^{∗}} = 1. Thus by the above argument X ⊗ X^{∗} is simple, hence X ⊗ X^{∗} = 𝟙 and X is invertible. □

Now we are ready to prove (vii) and (xi). Namely, let us compute ∧^{2}X_{n+1} and ∧^{3}X_{n+1}. Since X_{n+1} is self-dual, by Lemma 3.1 we have a 3-step filtration on X_{n+1} ⊗ X_{n+1} with successive quotients ∧^{2}X_{n+1}, X_{n+1}^{(1)}, ∧^{2}X_{n+1}. Since X_{n+1} is not invertible, by Lemma 3.2 this implies that ∧^{2}X_{n+1} = 𝟙 and X_{n+1}^{(1)} = X_{n}, proving (xi).

Also note that the cyclic permutation c is not the identity on X_{n+1}^{⊗3} (as by Lemma 3.2, s|_{X_{n+1}⊗X_{n+1}} ≠ 1). This implies that 1 + c + c^{2} is not invertible (as c^{3} − 1 = 0). Thus Ker(1 + c + c^{2}) = V ⊗ k^{2}, where k^{2} is the 2-dimensional irreducible S_{3}-module and V a nonzero object (namely, V = S_{(2,1)}X_{n+1}, the Schur functor attached to the partition (2, 1), which is well defined in characteristic 2). But X_{n+1}^{⊗3} = 2X_{n+1} ⊕ X_{n} ⊗ X_{n+1}, hence V = X_{n+1}. But we also have X_{n+1} = ∧^{2}X_{n+1} ⊗ X_{n+1} = ∧^{3}X_{n+1} ⊕ V. Hence ∧^{3}X_{n+1} = 0, proving (vii).

3.7. The invariants in tensor powers of X_{n}. Let us compute the dimension of the space of invariants in X_{n}^{⊗r}. It is clear that if r is odd then this space is zero, so it suffices to compute the invariants in X_{n}^{⊗2m}. Let d_{mn} = dim Hom(𝟙, X_{n}^{⊗2m}).

Proposition 3.3. Let f_{n}(z) = ∑_{m≥0} d_{mn}z^{m}. Then

(2)  f_{n}(z) = \frac{(t + t^{−1})(t^{2^{n}−1} − t^{−2^{n}+1})}{t^{2^{n}} − t^{−2^{n}}},

where z = (t + t^{−1})^{−2}.

Proof. Since X_{0} = 0, we have f_{0}(z) = 1.

Let n, m ≥ 1. Recall that X_{n} ⊗X_{n} = A_{n−1}, the regular representation of the algebra A_{n−1} as an object of C_{2n−1}. Thus,

d_{mn} = dim Hom_{C_{2n−1}}(𝟙, A_{n−1}^{⊗m}) = dim Hom_{C_{2n−1}}(A_{n−1}, A_{n−1}^{⊗m−1}),

using that A_{n−1} ≅ A_{n−1}^{∗} as an A_{n−1}-module. Thus, by Frobenius reciprocity, we get

d_{mn} = dim Hom_{C_{2n−2}}(𝟙, A_{n−1}^{⊗m−1}) = dim Hom_{C_{2n−2}}(𝟙, (2 · 𝟙 ⊕ X_{n−1})^{⊗m−1}).

This implies that

d_{mn} = ∑_{s≥0} \binom{m − 1}{2s}2^{m−1−2s}d_{n−1,2s}.

In terms of generating functions, this recursion has the form

f_{n}(z) = 1 + zf_{n−1} (\frac{z^{2}}{(1 − 2z)^{2}} ) ,

and it determines f_{n}(z) for all n from the initial condition f_{0}(z) = 1.

It remains to observe that function (2) satisfies this recursion. □

3.8. Proof of (i). It remains to prove (i). For this we will use the following universal property of the tilting category.

Proposition 3.4. Let D be a Karoubian rigid symmetric monoidal category over a field k of any characteristic. Then k-linear symmetric monoidal functors F: Tilt(SL(2, k)) → D correspond to objects X ∈ D of dimension 2 such that ∧^{2}X ≅ 𝟙 and ∧^{3}X = 0, via F ↦ F (T_{1}).

Proof. Analogous to the proof of Theorem 2.4 of [O1]. □

Let Q_{n,p} be the polynomial (with integer coefficients) defined by the formula

Q_{n,p}(2 cos x): = \frac{sin(p^{n}x)}{sin x}.

We write Q_{n,p} = Q_{n,p}^{+} − Q_{n,p}^{−}, where Q_{n,p}^{+} is the sum of all the terms of Q_{n,p} with positive coefficients and Q_{n,p}^{−} is minus the sum of the terms with negative coefficients.

Corollary 3.5. Let D be a Karoubian rigid symmetric monoidal category over k. Then klinear symmetric monoidal functors F: T_{n,p} → D correspond to objects X ∈ D of dimension 2 such that ∧^{2}X ≅ 𝟙, ∧^{3}X = 0, and Q_{n,p}^{+}(X) ≅ Q_{n,p}^{−}(X), via F ↦ F (T_{1}).

Proof. The tensor ideal I_{n} is generated by the object T_{p^{n}−1}. This is the irreducible standard module St_{n} over SL(2, k) with highest weight p^{n} − 1 (the n-th Steinberg representation). Since characters of tilting modules are linearly independent, in the split Grothendieck ring of Tilt(SL(2, k)) we have T_{p^{n}−1} = Q_{n}(T_{1}) (since this identity holds at the level of characters). Hence T_{p^{n}−1} ⊕ Q_{n,p}^{+}(T_{1}) ≅ Q_{n,p}^{−}(T_{1}). This implies that if X = F (T_{1}) then Q_{n,p}^{−}(X) is always a direct summand in Q_{n,p}^{+}(X), and for a monoidal functor F : Tilt_{n,p} → D, the condition that F (Q_{n,p}^{+}(T_{1})) ≅ F (Q_{n,p}^{−}(T_{1})) (i.e., Q_{n,p}^{+}(X) ≅ Q_{n,p}^{−}(X)) is equivalent to the condition that F (T_{p^{n}−1}) = 0. Thus, the corollary follows from Proposition 3.4. □

Now let us return to the case char(k) = 2. To simplify notation, we write Q_{n} for Q_{n,2}. In C_{2n}, n ≥ 1, we have the object X_{n} such that dim X_{n} = 0 = 2 and ∧^{2}X_{n} = 𝟙, ∧^{3}X_{n} = 0. Hence by Proposition 3.4 we have a symmetric monoidal functor

F: Tilt(SL(2, k)) → C_{2n}.

Moreover, we have

F (T_{2^{n+1}−1}) = Q_{n+1}(X_{n})

in the Grothendieck ring of C_{2n}. But Q_{n+1}(X_{n}) = 0, since the eigenvalues of multiplication by X_{n} are Galois conjugates of 2 cos(π/2^{n+1}), which are exactly the roots of Q_{n+1}. This implies that the class of F (T_{2^{n+1}−1}) in the Grothendieck ring of C_{2n} is zero, thus F (T_{2^{n+1}−1}) is itself zero.

Thus, we obtain

Corollary 3.6. We have a symmetric monoidal functor F: Tilt_{n+1,2} → C_{2n} such that F (T_{1}) = X_{n}.

It remains to show that the functor F of Corollary 3.6 is fully faithful. To this end, we first prove the following lemma.

Lemma 3.7. Let C be a Karoubian monoidal category over a field k of characteristic p and F: T_{n+1,p} → C be an additive monoidal functor. Let St_{n} = T_{p^{n}−1} be the n-th Steinberg module. If F (St_{n}) ≠ 0 then F is faithful.

Proof. It suffices to show that if f: 1 → Z is a morphism in Tilt_{n+1,p} such that F (f) = 0 then f = 0. Assume the contrary, i.e., that f ≠ 0. Let I(f) be the tensor ideal generated by f. Consider the morphism f ⊗ 1: St_{n} → Z ⊗ St_{n}. We have Z ⊗ St_{n} = St_{n}^{⊕m} ⊕ M, where M is a direct sum of T_{i}, p^{n} ≤ i ≤ p^{n+1} − 1. Moreover, by the linkage principle ([J], Section 6) we have Hom(St_{n}, M) = 0. Hence there exists g: Z ⊗ St_{n} → St_{n} such that g ∘ (f ⊗ 1) = Id_{St_{n}}, i.e., Id_{St_{n}} ∈ I(f). Thus, since the functor F kills I(f), it kills Id_{St_{n}}, hence kills St_{n} itself, a contradiction. □

Proposition 3.8. The functor F is faithful.

Proof. By Lemma 3.7, it suffices to show that F (St_{n}) ≠ 0. But F (St_{n}) = Q_{n}(T_{1}), and any polynomial Q of degree less than 2^{n} such that Q(X_{n}) = 0 is identically zero (as X_{n} has 2^{n} distinct eigenvalues). This implies the statement. □

Proposition 3.9. The functor F is full.

Proof. It suffices to check that F is full on objects of the form T_{1}^{⊗r}. Thus, since F is faithful, it suffices to check that

dim Hom(𝟙, T_{1}^{⊗2m}) = d_{mn}.

Let D_{mn} := dim Hom(𝟙, T_{1}^{⊗2m}). Let T̃_{i} be the lift of T_{i} to the category Ver_{2^{n}}(K). Then, since Ext_{SL(2,k)}^{1}(T_{i}, T_{j}) = 0, we have dim Hom(T_{i}, T_{j}) = dim Hom(T̃_{i}, T̃_{j}). Hence

D_{mn} = dim Hom(𝟙, T̃_{1}^{⊗2m}).

But dim Hom(𝟙, T̃_{1}^{⊗2m}) just equals the number of paths on the Dynkin diagram of type A_{2^{n}−1} of length 2m beginning and ending at the left end. Hence, it follows from elementary combinatorics that

∑_{m≥0} D_{mn}z^{m} = \frac{(t + t^{−1})(t^{2^{n}−1} − t^{−2^{n}+1})}{t^{2^{n}} − t^{−2^{n}}},

where z = (t + t^{−1})^{−2}. Thus, the Proposition follows from Proposition 3.3. □

Propositions 3.8 and 3.9 show that F is fully faithful.

Finally, note that the object X_{n} generates C_{2n}. Indeed, some tensor power of X_{n} contains St_{n} = X_{{1,...,n}} as a subquotient. But St_{n} is projective, and any indecomposable projective is a direct summand in X_{n}^{⊗m} ⊗ St_{n} for some m, as desired.

This completes the proof of part (i) of Theorem 2.1. Thus, Theorem 2.1 is proved.

3.9. Remarks.

Remark 3.10. Observe that all indecomposable objects in C_{2} are direct summands of tensor powers of X_{1}, which implies that T_{2,2} = C_{2}. Hence T_{2,2} is an abelian category, as was shown by V. Ostrik.

Remark 3.11. We claim that any braiding on C_{2n} is necessarily symmetric. Indeed, let X := X_{n}. Since C_{2n} is tensor generated by X, it suffices to show that if c_{XX} : X⊗X → X⊗X is the braiding map then c_{XX}^{2} = Id. Since X ⊗ X is the regular A_{n−1}-module, we have End(X ⊗ X) = k[t]/(t^{2}). Thus c_{XX} = α + βt, where α, β ∈ k. Thus c_{XX}^{2} = α^{2} is a scalar. This means that c_{X⊗X,X}^{2} is also a scalar, namely α^{4}. But X ⊗ X contains 𝟙, hence α^{4} = 1. This implies that α = 1, hence c_{XX}^{2} = Id, as desired.

This gives another way to see that the braiding on C_{2n+2} constructed in the proof of Theorem 2.1 is symmetric.

Remark 3.12. Note that we had to use a nondegenerate R-matrix on A_{n}; otherwise (i.e., had we used the trivial one) the bimodule category M_{n} would not have been invertible. Moreover, it is easy to see that such a nondegenerate R-matrix is unique up to an isomorphism, i.e., there are no choices involved in the construction of C_{i}. Together with Remark 3.11, this implies that C_{2n} has a unique braiding up to equivalence, which is its symmetric structure.

Remark 3.13. Since the Frobenius twist is a monoidal functor, we have X_{S}^{(1)} = 0 if 1 ∈ S and X_{S}^{(1)} = X_{S−1} if 1 ∉ S, where S − 1 is the image of S under the map i ↦ i − 1.

Remark 3.14. The category C_{2n} is filtered by full abelian subcategories C_{2n}^{r}, r ≥ 0, which consist of subquotients of direct sums of X_{n}^{⊗j}, j ≤ r. These categories are not closed under tensor product, but we have partial tensor products C_{2n}^{j} × C_{2n}^{r−j} → C_{2n}^{r} with associativity isomorphisms satisfying the pentagon relation. Moreover, the Frobenius twist functor X ↦ X^{(1)} is a monoidal functor C_{2n} → C_{2n−2} (as it sends X_{n} to X_{n−1}). Unfortunately, this functor is not a tensor functor, as it is not left or right exact. However, we expect that it is exact in any fixed degree m of the filtration and moreover is an equivalence preserving partial tensor products (i.e., a partial tensor functor) if n is sufficiently large compared to m. If so, we can define the limit C(∞) = lim_{n→∞} C_{2n} := ⋃_{m≥0} lim_{n→∞} C_{2n}^{m}, where C_{2n}^{m} is identified with C_{2n+2}^{m} for large n by means of the Frobenius twist functor. We expect that C(∞) ≅ RepSL(2, k). This is similar to the construction of the abelian envelope of the Deligne category Rep^{ab}(GL_{t}) out of representation categories of the supergroups GL(n + t|n) using the Duflo-Serganova homology functor DS in place of the Frobenius functor, [EHS].

4. Further properties of the categories C_{n}.

4.1. The matrix of multiplication by X_{n} on Gr(C_{2n}).

Proposition 4.1. Let B_{n} ∈ Mat_{2^{n}}(ℤ_{≥0}) be the matrix of multiplication by X_{n} on the Grothendieck ring Gr(C_{2n}) (in the basis of simple objects). Then the matrices B_{n} are computed recursively as follows: B_{0} = 0 and

B_{n+1} = (\begin{matrix}0 & 2 + B_{n} \\ 1 & 0 \end{matrix}) ,

where the blocks are of size 2^{n−1}. This matrix has distinct eigenvalues ζ_{n}^{2r+1} + ζ_{n}^{−2r−1}, r = 1, …, 2^{n}, where ζ_{n} = exp(πi/2^{n+1}).

Proof. We have X_{n}X_{S} = X_{S∪n} if n ∉ S and

X_{n}X_{S} = X_{n}^{2}X_{S\n} = (2 + X_{n−1})X_{S\n}.

This implies the first statement. Also B_{n} has an eigenvalue FPdim(X_{n}) = 2 cos(π/2^{n+1}), so the second statement follows from the Galois group action. □

4.2. The Cartan matrix of C_{n}. Let C_{n} be the Cartan matrix of C_{n}. Then by Theorem 2.1(ix),

C_{2n} = C_{2n−1} ⊕ C_{2n−3} ⊕ ⋯ ⊕ C_{1} ⊕ C_{0},

with C_{0} = 1, so it suffices to determine C_{n} for odd n.

Proposition 4.2. The matrices C_{2n+1} are determined from the recursion

C_{2n+1} = (\begin{matrix}2C_{2n−1} & C_{2n−1} \\ C_{2n−1} & 2C_{2n−2} \end{matrix})

with C_{1} = 2. In particular, C_{2n+1} is symmetric and all its nonzero entries are powers of 2.

Proof. First of all, C_{n} is symmetric for all n since the distinguished invertible object of C_{n} is 𝟙 (the only invertible object of C_{n}), hence all the projective objects of C_{n} (which are also injective) are self-dual, since so are all the simple objects (see [EtGNO], Subsection 6.4).

Let Y_{j} be the simple objects of C_{2n−2} (and C_{2n−1}). Then P_{2n−1}(Y_{j}) = P_{2n}(Y_{j}) and P_{2n+1}(Y_{j}) = A_{n} ⊗ P_{2n−1}(Y_{j}). Hence

Hom_{C_{2n+1}}(P_{2n+1}(Y_{j}), P_{2n+1}(Y_{r})) = Hom_{C_{2n+1}}(A_{n} ⊗ P_{2n−1}(Y_{j}), A_{n} ⊗ P_{2n−1}(Y_{r})) =
Hom_{C_{2n}}(P_{2n−1}(Y_{j}), A_{n} ⊗ P_{2n−1}(Y_{r})) =
Hom_{C_{2n}}(P_{2n−1}(Y_{j}), P_{2n−1}(Y_{r}))^{⊕2} ⊕ Hom_{C_{2n}}(P_{2n−1}(Y_{j}), X_{n} ⊗ P_{2n−1}(Y_{r})) =
Hom_{C_{2n}}(P_{2n−1}(Y_{j}), P_{2n−1}(Y_{r}))^{⊕2} =
Hom_{C_{2n−1}}(P_{2n−1}(Y_{j}), P_{2n−1}(Y_{r}))^{⊕2},

where we use that A_{n} = 𝟙 ⊕ X_{n} ⊕ 𝟙. This implies that the left upper block of C_{2n+1} is 2C_{2n−1}. Similarly,

Hom_{C_{2n+1}}(P_{2n+1}(X_{n} ⊗ Y_{j}), P_{2n+1}(Y_{r})) =
Hom_{C_{2n+1}}(A_{n} ⊗ P_{2n−1}(X_{n} ⊗ Y_{j}), A_{n} ⊗ P_{2n−1}(Y_{r})) =
Hom_{C_{2n}}(P_{2n−1}(X_{n} ⊗ Y_{j}), A_{n} ⊗ P_{2n−1}(Y_{r})) =
Hom_{C_{2n}}(X_{n} ⊗ P_{2n−2}(Y_{j}), A_{n} ⊗ P_{2n−1}(Y_{r})) =
Hom_{C_{2n}}(X_{n} ⊗ P_{2n−2}(Y_{j}), X_{n} ⊗ P_{2n−1}(Y_{r})) =

Hom_{C_{2n−2}}(P_{2n−2}(Y_{j}), P_{2n−1}(Y_{r})) =
Hom_{C_{2n−1}}(P_{2n−1}(Y_{j}), P_{2n−1}(Y_{r})),

which yields that the upper right and the lower left blocks of C_{2n+1} are both C_{2n−1}. Finally,

Hom_{C_{2n+1}}(P_{2n+1}(X_{n} ⊗ Y_{j}), P_{2n+1}(X_{n} ⊗ Y_{r})) =
Hom_{C_{2n+1}}(A_{n} ⊗ P_{2n−1}(X_{n} ⊗ Y_{j}), A_{n} ⊗ P_{2n−1}(X_{n} ⊗ Y_{r})) =
Hom_{C_{2n}}(P_{2n−1}(X_{n} ⊗ Y_{j}), A_{n} ⊗ P_{2n−1}(X_{n} ⊗ Y_{r})) =
Hom_{C_{2n}}(X_{n} ⊗ P_{2n−2}(Y_{j}), A_{n} ⊗ P_{2n−1}(X_{n} ⊗ Y_{r})) =
Hom_{C_{2n}}(X_{n} ⊗ P_{2n−2}(Y_{j}), P_{2n−1}(X_{n} ⊗ Y_{r}))^{⊕2} =
Hom_{C_{2n}}(X_{n} ⊗ P_{2n−2}(Y_{j}), X_{n} ⊗ P_{2n−2}(Y_{r}))^{⊕2} =
Hom_{C_{2n−2}}(P_{2n−2}(Y_{j}), P_{2n−2}(Y_{r}))^{⊕2},

which implies that the lower right block of C_{2n−1} is 2C_{2n−2}, as claimed. □

4.3. The incompressibility of C_{2n}.

Definition 4.3. (V. Ostrik) A tensor category C is called incompressible if any tensor functor F: C → D from C to a tensor category D is injective (i.e., a fully faithful embedding).

Theorem 4.4. The category C_{2n} is incompressible.

Proof. The proof is by induction in n. The base n = 0 is clear, so we only need to do the inductive step. Let F: C_{2n} → D be a tensor functor. We need to show that F is fully faithful. By replacing D with ImF, we may assume that D is finite and F is surjective. Also by the induction assumption we may assume that C_{2n−2} ⊂ D and F |_{C_{2n−2}} = Id.

Let V_{S} = F (X_{S}). By the induction assumption, V_{S} = X_{S} is simple if n ∉ S. Also it is clear that V_{n} is simple (as its FP dimension is < 2).

We claim that D does not contain nontrivial invertible objects. Indeed, if χ in an invertible object then χ must occur as a composition factor in V_{n}^{⊗r} for some r. This means that V_{n} ⊗ χ occurs in V_{n}^{⊗r+1}, which is impossible if r is odd for FP dimension reasons, since V_{n}^{⊗r+1} is a power of V_{n} ⊗ V_{n}, hence its composition factors are X_{S}, n ∉ S. Thus r is even, and χ = X_{S} for some S, hence χ = 𝟙.

Consider the tensor product V_{n} ⊗ V_{n} = F (A_{n−1}). As V_{n} is simple, we have

Hom(V_{n} ⊗ V_{n}, 𝟙) = Hom(𝟙, V_{n} ⊗ V_{n}) = k.

Also, we claim that

Hom(V_{n} ⊗ V_{n}, X_{n−1}) = Hom(X_{n−1}, V_{n} ⊗ V_{n}) = 0.

Indeed, otherwise V_{n} is a composition factor of X_{n−1} ⊗ V_{n}, so we have an object Y ∈ D with dimension

FPdim(Y) = FPdim(X_{n}) FPdim(X_{n−1}) − FPdim(X_{n}) =
= (ζ_{n} + ζ_{n}^{−1})(ζ_{n}^{2} + ζ_{n}^{−2}) − (ζ_{n} + ζ_{n}^{−1}) = ζ_{n}^{3} + ζ_{n}^{−3}.

But this is impossible, since this number has a larger Galois conjugate (namely, ζ_{n} + ζ_{n}^{−1}). Thus, we see that V_{n} ⊗ V_{n} is indecomposable with composition series 𝟙, X_{n−1}, 𝟙.

Now, we claim that Hom(V_{{1,...,n}}, V_{{1,...,n}}) = k. Indeed, we can write this space as

Hom(V_{n} ⊗ V_{n}, X_{{1,...,n−1}} ⊗ X_{{1,...,n−1}}) =
Hom(V_{n} ⊗ V_{n}, X_{{1,...,n−2}} ⊗ A_{n−2} ⊗ X_{{1,...,n−2}}).

Since X_{n−1} does not occur in X_{{1,...,n−2}} ⊗ A_{n−2} ⊗ X_{{1,...,n−2}}, this space equals

Hom_{C_{2n−2}}(𝟙, X_{{1,...,n−2}} ⊗ A_{n−2} ⊗ X_{{1,...,n−2}}) =
Hom(X_{{1,...,n−1}}, X_{{1,...,n−1}}) = k,

as desired.

This means that V_{{1,...,n}} is indecomposable. Also, it is projective since X_{{1,...,n}} is projective, and F is surjective ([EtGNO], Theorem 6.1.16). Since D has no nontrivial invertible objects, the head and socle of V_{{1,...,n}} are isomorphic. Hence V_{{1,...,n}} is simple (as its endomorphism algebra is 1-dimensional). This implies that V_{S} is simple for all S (as it is a tensor factor of V_{{1,...,n}}). Thus, the simple objects of D are the objects V_{S}.

Now let P_{S} be the projective cover of X_{S} in C_{2n}, Q_{S} the projective cover of V_{S} in D. Let p_{S} = FPdim(P_{S}) and q_{S} = FPdim(Q_{S}). Let p = (p_{S}), q = (q_{S}). Then p, q are left Frobenius–Perron eigenvectors of the matrix of multiplication by V_{n}. Hence p = λq for some λ > 0. But we have shown that p_{{1,...,n}} = q_{{1,...,n}}, hence λ = 1. Thus F (P_{S}) = Q_{S} for all S, and F is an equivalence. □

4.4. Connection with modular representation theory of finite groups. The category T_{n,p} arises in the modular representation theory of the group SL(2, 𝔽_{q}), where q = p^{n}.

Namely, let k be an algebraically closed field of characteristic p, and pick an embedding 𝔽_{q} ↪ k. Let V = k^{2} be the 2-dimensional tautological representation of SL(2, 𝔽_{q}). Note that V is self-dual. Let D̃_{n,p} be the full subcategory of Rep_{k}SL(2, 𝔽_{q}) whose objects are finite direct sums of direct summands of tensor powers of V. It is clear that D̃_{n,p} is a rigid monoidal Karoubian category. Moreover, since all indecomposable objects T_{i} of Tilt_{p} occur as direct summands in tensor powers of the 2-dimensional module T_{1}, the natural restriction functor Res: Tilt_{p} → Rep_{k}SL(2, 𝔽_{q}) in fact lands in D̃_{n,p}, and the indecomposable objects of D̃_{n,p} are the direct summands in Res(T_{i}), i ≥ 0.

Recall that St_{n} denotes the n-th Steinberg module, St_{n} = T_{q−1}. The restriction of St_{n} to the group algebra of the p-Sylow subgroup of SL(2, 𝔽_{q}) is free (of rank 1), hence St_{n} generates the tensor ideal P of projective modules in Rep_{k}SL(2, 𝔽_{q}). Moreover, since

T_{1} ⊗ T_{i} = T_{i+1} ⊕ ⨁_{s<i} c_{is}T_{s},

this ideal contains Res(T_{i}) for i ≥ q − 1, i.e., Res(I_{n}) ⊂ P.

Let D_{n,p} := D̃_{n,p}/P be the stable category of D̃_{n,p}. It follows that Res descends to a symmetric monoidal functor F: T_{n,p} → D_{n,p}. It also follows that the indecomposable objects of D_{n,p} are the indecomposable (non-projective) direct summands of V^{⊗r}, r < q − 1 (we will see later that in fact there are no projective direct summands).

Moreover, since Res(St_{n−1}) is not projective (as its dimension p^{n−1} is less than the order q = p^{n} of a p-Sylow subgroup of SL(2, 𝔽_{q})), we have F (St_{n−1}) ≠ 0, hence by Lemma 3.7 F is faithful.

Proposition 4.5. The functor F is an equivalence of categories.

Proof. We will use the following lemma.

Lemma 4.6. If r, s < q then any homomorphism of SL(2, 𝔽_{q})-modules A: V^{⊗r} → V^{⊗s} is in fact a homomorphism of SL(2, k)-modules.

Proof. Let U be a maximal unipotent subgroup of SL(2, k). Then the condition that A commutes with u ∈ k ≅ U(k) is a system of polynomial equations with respect to u of degree max(r, s) < q. This system is satisfied for any u ∈ 𝔽_{q}. But any polynomial of degree < q which vanishes on 𝔽_{q} has to vanish on k. Thus these equations are satisfied for any u ∈ k, hence A commutes with U(k). But SL(2, k) is generated by its subgroups of the form U(k), which implies the statement. □

Corollary 4.7. If r < q then the module Res(T_{r}) is indecomposable.

Proof. Since T_{r} is a direct summand in T_{1}^{⊗r}, Lemma 4.6 implies that the functor Res induces an isomorphism End(T_{r}) ≅ End(Res(T_{r})). Hence End(Res(T_{r})) is a local algebra, which implies the statement. □

Corollary 4.7 implies that the functor F is essentially surjective, i.e., the indecomposable objects of D_{n,p} are V_{r} := Res(T_{r}), r < q − 1. Note that these modules are not projective, since F is faithful.

It remains to show that F is full. To this end, let f: V_{r} → V_{s} be a morphism in D_{n,p} (r, s < q − 1). Let f̃: V_{r} → V_{s} be a preimage of f in D̃_{n,p}. By Lemma 4.6, f̃ may be viewed as a morphism T_{r} → T_{s}. Let f̂ be its image in T_{n,p}. It is clear that F (f̂) = f, which implies the statement. Proposition 4.5 is proved. □

Proposition 4.5 and Theorem 2.1(i) imply

Corollary 4.8. The stable category D_{n+1,2} embeds as a full monoidal subcategory into the abelian symmetric tensor category C_{2n}.

Note that this is a rather striking property, as full monoidal subcategories of stable categories do not normally admit an abelian envelope, i.e., an additive monoidal embedding into an abelian monoidal category. For example, we have the following proposition.

Recall that an indecomposable finite dimensional representation E of a finite group G over k with char(k) dividing the order of G is endotrivial if E ⊗ E^{∗} = 𝟙 ⊕ P, where P is projective; in other words, E is invertible in the stable category of G.

Proposition 4.9. Let G ≠ 1 be a finite p-group and S_{k}(G) be the stable category of Rep_{k}G, where char(k) = p. Let E ≠ k be any endotrivial indecomposable representation of G. Then the full tensor subcategory E of S_{k}(G) generated by E does not admit an abelian envelope.

Proof. Let P be a free kG-module of finite rank, V a non-projective indecomposable finite dimensional G-module, and g: V → P, h: P → k be morphisms. We claim that h ∘ g = 0. Indeed, assume the contrary. We may assume that P has rank 1. Then Im(g) = P, hence V = P ⊕ V′, a contradiction.

Let f: E → k be a nonzero homomorphism (it exists since G is a p-group). It follows that f remains nonzero in the stable category. Since E is invertible and E ≠ k, this means that E does not have an abelian envelope. □

Example 4.10. For example, Proposition 4.9 applies when G = ℤ/p with p > 2 or G = (ℤ/2)^{2} and E = Ω(k).

Let B be a Borel subgroup of SL(2, k) (the subgroup of upper triangular matrices). Let B̃_{n,p} be the full subcategory of Rep_{k}B(𝔽_{q}) whose objects are finite direct sums of direct summands of V^{⊗r}. Let B_{n,p} be the stable category of B̃_{n,p}, i.e., its quotient by the tensor ideal of projective objects. Since B(𝔽_{q}) is the normalizer of a Sylow subgroup of SL(2, 𝔽_{q}), by Green’s correspondence, the restriction functor defines an equivalence D_{n,p} ≅ B_{n,p}. Moreover, we have the following proposition.

Let U = [B, B] and Ũ_{n,p} be the full subcategory of Rep_{k}U(𝔽_{q}) whose objects are finite direct sums of direct summands in tensor powers of V. Let U_{n,p} be the corresponding stable category. We have the symmetric monoidal restriction functor H: B_{n,p} → U_{n,p}.

Proposition 4.11. The functor H maps indecomposable objects to indecomposable ones. Hence it defines a bijection between isomorphism classes of objects of T_{n,p} = D_{n,p} = B_{n,p} and U_{n,p}. In particular, tensor powers of the U(𝔽_{q}) = (ℤ/p)^{n}-module V contain q − 1 distinct non-projective indecomposable direct summands (the restrictions of T_{r}, 0 ≤ r < q − 1).

Proof. It suffices to show that the restriction of T_{r} to U(𝔽_{q}) is indecomposable for r < q − 1. To this end, note first that the restriction of T_{r} to B is indecomposable, since it is already so for B(𝔽_{p^{N}}) for large N. Namely, we have shown that it is so modulo projectives, but for large N the restriction of T_{r} is too small to contain projective direct summands.

Now note that the restriction of any indecomposable finite dimensional rational B-module Y to U is indecomposable. Indeed, End_{B}(Y) = End_{U}(Y)^{B/U}, but B/U = 𝔾_{m} and acts trivially on the center of the semisimple algebra End_{U}(Y)/Rad(End_{U}(Y)), hence each primitive central idempotent of this algebra lifts to a B-invariant idempotent of End_{U}(Y), i.e., to an idempotent of End_{B}(Y). Hence End_{U}(Y) is local, as so is End_{B}(Y).

Thus, T_{n}|_{U} is indecomposable. Now using the argument in the proof of Lemma 4.6, we conclude that the restriction map End_{U(k)}(T_{r}) → End_{U(𝔽_{q})}(T_{r}) is an isomorphism. Hence the algebra End_{U(𝔽_{q})}(T_{r}) is local, since so is End_{U(𝔽_{q})}(T_{r}), as desired. □

4.5. Ext^{1} between simple objects. Let D_{m} := dim Ext_{C_{m}}^{1}(X_{S}, X_{T}).

Proposition 4.12. The numbers D_{m}(S, T) are zeros and ones and are determined by the formulas

D_{2n+1}(S, T) = D_{2n}(S, T) = D_{2n−1}(S, T), n ∉ S, T;
D_{2n+1}(S, T) = D_{2n}(S, T) = D_{2n−2}(S \ n, T \ n), n ∈ S, T;
D_{2n+1}(S, T) = δ_{S\n,T} , n ∈ S, n ∉ T; D_{2n+1}(S, T) = δ_{S,T \n}, n ∈ T, n ∉ S;

and

D_{2n}(S, T) = 0

if n is contained in exactly one of the sets S, T, with initial conditions D_{1}(∅, ∅) = 1 and D_{0}(∅, ∅) = 0.

Proof. Since C_{2n} = C_{2n−1} ⊕C_{2n−2} ⊗X_{n}, we obtain the statement for D_{2n}(S, T). Also, we have

D_{2n+1}(S, T) = dim Ext_{A_{n}}^{1}(X_{S}, X_{T}) = D_{2n}(S, T) + dim Hom(X_{n} ⊗ X_{S}, X_{T}),

which implies the statement about D_{2n+1}(S, T). □

Corollary 4.13. Let S, T ⊂ ℤ_{>0} be finite subsets with max(S ∪ 0) = s and max(T ∪ 0) = t. Let n = max(s, t). Then D_{m}(S, T) are the same for all m ≥ 2n + 1. Moreover, if s = n > t then we have

D_{2n+1}(S, T) = δ_{S\n,T} , D_{2n}(S, T) = 0,

and if s < t = n then

D_{2n+1}(S, T) = δ_{S,T \n}, D_{2n}(S, T) = 0.

Further, if s = t = n > 0 then D_{2n+1}(S, T) = D_{2n}(S, T) = 0 unless S = T, and D_{2n+1}(S, S) = D_{n}(S, S) equals 0 if 1 ∈ S and 1 if 1 ∉ S. In particular, D_{m}(∅, ∅) = 1 if m > 0 and 0 if m = 0.

Proof. The corollary follows by straightforward application of Proposition 4.12. □

4.6. The universal property of C_{2n}.

Theorem 4.14. Let D be a symmetric tensor category over k. Then the following statements hold.

(i) Any faithful k-linear symmetric monoidal functor H : T_{n+1,2} → D factors through C_{2n} in a unique way: H = G ∘ F where F : T_{n+1,2} → C_{2n} is the inclusion of Theorem 2.1, and G: C_{2n} → D is a tensor functor. In other words, C_{2n} is the canonical abelian envelope of T_{n+1,2} in the sense of Deligne, [De3] (see also [EHS]).

(ii) Symmetric tensor functors G: C_{2n} → D correspond to objects X ∈ D of dimension 0 such that ∧^{2}X ≅ 𝟙, ∧^{3}X = 0, and Q_{n+1}^{+}(X) ≅ Q_{n+1}^{−}(X), but Q_{n}^{+}(X) ≇ Q_{n}^{−}(X), via F ↦ F (X_{n}).

Proof. (i) Recall that F is fully faithful by Theorem 2.1(i). Let us identify T_{n+1,2} with its image in C_{2n} under F. By Theorem 9.2.2 of [EHS], it suffices to show that^{2}

(1) any object Y of C_{2n} is the image of a morphism in T_{n+1,2} ⊂ C_{2n}, and

(2) for any epimorphism f : Y → Z in C_{2n}, there exists a nonzero T ∈ T_{n+1,2} ⊂ C_{2n} such that the epimorphism f ⊗ 1 : Y ⊗ T → Z ⊗ T splits.

To establish (1), let P_{Y} and I_{Y} be the projective cover and injective hull of Y, and let f : P_{Y} → I_{Y} be the composition of the natural morphisms P_{Y} → Y → I_{Y}. Then Y = Im(f), and f is a morphism in T_{n+1,2} (as P_{Y} , I_{Y} are both projective and thus contained in T_{n+1,2}).

To establish (2), it suffices to take T to be any nonzero projective object in C_{2n}. This proves (i).

(ii) By Corollary 3.5, symmetric monoidal functors H : T_{n+1,2} → D correspond to objects X ∈ D of dimension 0 such that ∧^{2}X ≅ 𝟙, ∧^{3}X = 0, and Q_{n+1}^{+}(X) ≅ Q_{n+1}^{−}(X). By Proposition 3.8, such H is faithful if and only if F (T_{2^{n}−1}) ≠ 0, i.e., Q_{n}^{+}(X) ≇ Q_{n}^{−}(X). Thus (i) implies (ii). □

5. The structure of the categories C_{n} for small n.

In this section we describe the structure of the categories C_{n} for n ≤ 6. Many of the results below were obtained by a computer calculation using MAGMA ([BCP]).

5.1. The structure of T_{m} for small m. We begin by describing the structure of the tilting modules T_{m} for SL(2, k), where k has characteristic two, for small values of m. Let L_{m} be the simple SL(2, k)-module with highest weight m; by Steinberg’s tensor product theorem this is a tensor product of Frobenius twists of the natural module V, indexed by the ones in the binary expansion of m. So for example we have L_{13} = V^{(3)} ⊗ V^{(2)} ⊗ V. The structure of these tilting modules is not hard to compute by hand. It may be found in the paper of Doty and Henke [DoH], and is reproduced here for convenience. We have T_{0} = L_{0} = k, the field with trivial action, and T_{1} = L_{1} = V, the natural module. Next, we have T_{2} = [0, 2, 0]. Here, we write [a, b, c, …] for a uniserial module with composition factors a, b, c, …, starting with the top composition factor. We have also abbreviated L_{m} to m, and we note that L_{2} = V^{(1)} is the Frobenius twist of V, L_{3} = V ⊗ V^{(1)}, and so on. Continuing this way, here are the structures of the first few tilting modules.

2Theorem 9.2.2 of [EHS] is stated for k = ℂ, but this assumption is not used in the proof, so the theorem applies to any algebraically closed field.

T_{0} = [0], T_{1} = [1], T_{2} = [0, 2, 0], T_{3} = [3], T_{4} = [2, 0, 4, 0, 2], T_{5} = [1, 5, 1], T_{6} =  0 2  4 0  6  0 4  2 0  ,

T_{7} = [7], T_{8} = [6, 4, 0, 8, 0, 4, 6], T_{9} = [5, 1, 9, 1, 5], T_{10} = [figure: 4; 6; 0; 4; 2; 8; 0; 10; 0; 8; 2; 4; 0; 6; 4] , T_{11} = [3, 11, 3],

T_{12} =[figure: 2; 0; 10; 4; 8; 2; 0; 12; 0; 2; 8; 4; 10; 0; 2] , T_{13} = [figure: 1; 5; 9; 1; 13; 1; 9; 5; 1] , T_{14} = [figure: 0; 2; 4; 8; 0; 6; 10; 0; 12; 0; 4; 8; 2; 14; 2; 8; 4; 0; 12; 0; 10; 6; 0; 8; 4; 2; 0] , T_{15} = [15].

Here the diagrams for the non-uniserial modules are in the sense of Alperin [A] or Benson and Carlson [BC].

The tensor product T_{n} ⊗ V is isomorphic to T_{n+1} plus a direct sum of T_{m} with m < n, with multiplicities given in the following table. This table extends in the obvious way for larger values of n.

1

2 1

1

2 1

1

2 2 1

1

2 1

1

2 2 1

1

2 1

1

2  2 2 1

1

2 1

1

2 2 1

1

2 1

1

2  2 2 1

1

2 1

1

2 2 1

1

2 1

1

2 2 2 2 1
...

5.2. The structure of C_{n} for small n. We now describe the structure of C_{n} with n small.

5.2.1. The categories C_{0} and C_{1}. We have C_{0} = Vec_{k}, with just one simple object 𝟙, and no self extensions.

Next, C_{1} again has just one simple object 𝟙, but its projective cover is an extension

P_{𝟙} = \begin{matrix}𝟙 \\ 𝟙 \end{matrix} . The cohomology ring is

Ext_{C_{1}}^{∗}(𝟙, 𝟙) = k[x]

where |x| = 1.

This is the only case where there is a forgetful tensor functor to vector spaces; but it is not a symmetric tensor functor. An object in C_{1} can be thought of as a vector space V together with a linear map d: V → V satisfying d^{2} = 0. The tensor product is as usual, with d: V ⊗ W → V ⊗ W given by d(v ⊗ w) = dv ⊗ w + v ⊗ dw. The commutativity isomorphism is given by s(v ⊗ w) = w ⊗ v + dw ⊗ dv. Thus for example a commutative algebra in C_{1} is a differential algebra satisfying ab − ba = da · db.

5.2.2. The category C_{2}. In the next case, C_{2} has two simple objects, 𝟙 = X_{∅} and V = X_{1}. The projective cover of 𝟙 is as in C_{1}, and V is a projective simple object. We have V ⊗V = P_{𝟙}. The cohomology ring is the same as for C_{1}. The Frobenius map acts on simple objects via 𝟙^{(1)} = 𝟙 and V^{(1)} = 0. The Frobenius–Perron dimensions are given by FPdim(𝟙) = 1 and FPdim(V) = \sqrt{2}.

5.3. The category C_{3}. The category C_{3} also has two simple objects, 𝟙 and V, with the same tensor products (except that P_{𝟙} is interpreted in C_{2}), the same Frobenius–Perron dimensions, and the same Frobenius map as in C_{2}, but the projective covers are more complicated. They are given by

P_{𝟙} = [figure: 𝟙; V; 𝟙; 𝟙; V; 𝟙] P_{V} = \begin{matrix}V \\ 𝟙 \\ 𝟙 \\ V \end{matrix} .

The cohomology ring is

Ext_{C_{3}}^{∗}(𝟙, 𝟙) = k[x, y, z]/(y^{2} + xz)

where |x| = 1, |y| = 2, |z| = 3. The Poincaré series of this ring is given by

∑_{i} t^{i} dim Ext_{C_{3}}^{i}(𝟙, 𝟙) = \frac{1 + t^{2}}{(1 − t)(1 − t^{3})}.

One can also compute the Ext ring between V and itself. To this end, note that the minimal resolution of V is

⋯ → P_{V} → P_{𝟙} → P_{V} → P_{V} → P_{𝟙} → P_{V} → V → 0

and is periodic with period 3. Using this, one shows that

Ext_{C_{3}}^{∗}(V, V) = k[u, v]/(u^{2})

where |u| = 2, |v| = 3. The Poincaré series of this ring is

∑_{i} t^{i} dim Ext_{C_{3}}^{i}(V, V) = \frac{1 + t^{2}}{1 − t^{3}} .

Finally, we can compute Ext_{C_{3}}^{∗}(V, 𝟙) and Ext_{C_{3}}^{∗}(𝟙, V) as bimodules over Ext_{C_{3}}^{∗}(𝟙, 𝟙) and Ext_{C_{3}}^{∗}(V, V). Namely, using the above resolution, we see that Ext_{C_{3}}^{∗}(V, 𝟙) has a generator in degree one, and is annihilated by x and y in Ext_{C_{3}}^{∗}(𝟙, 𝟙) and by u in Ext_{C_{3}}^{∗}(V, V). The elements z and v both act as the periodicity generator in degree 3. For Ext_{C_{3}}^{∗}(𝟙, V) we have exactly the same structure. In both cases, the Poincaré series is t/(1 − t^{3}).

5.4. The category C_{4}. The category C_{4} has four simple objects, V_{0} = 𝟙 = X_{∅}, V_{1} = X_{1}, V_{2} = X_{2} and V_{3} = X_{12}. The projective covers of V_{0} and V_{1} are the same as the projective covers of 𝟙 and V in C_{3}, while the projective covers of V_{2} and V_{3} are the same as the projective covers of 𝟙 and V in C_{2}. Abbreviating V_{m} to m, the structures are as follows:

P_{0} = [figure: 0; 1; 0; 0; 1; 0] ,  P_{1} = [1, 0, 0, 1],  P_{2} = [2, 2],  P_{3} = [3].

Tensor products of simples are given by the following table:

\begin{matrix}V_{0} & V_{1} & V_{2} & V_{3} \\ V_{1} & [0, 0] & [3] & P_{2} \\ V_{2} & [3] & [0, 1, 0] & P_{1} \\ V_{3} & P_{2} & P_{1} & P_{0} \end{matrix}

The Frobenius map is given by

V_{0}^{(1)} = V_{0} V_{1}^{(1)} = V_{2},  V_{2}^{(1)} = 0,  V_{3}^{(1)} = 0,

and the Frobenius–Perron dimensions of the simples are given by

FPdim V_{0} = 1, FPdim V_{1} = \sqrt{2}, FPdim V_{2} = \sqrt{2 + \sqrt{2}}, FPdim V_{3} = \sqrt{4 + 2\sqrt{2}}.

The cohomology ring is the same as for C_{3}.

5.4.1. The category C_{5}. The category C_{5} also has four simple objects, V_{0}, V_{1}, V_{2} and V_{3}, with the same tensor products (but with the projectives interpreted in C_{4}), the same Frobenius– Perron dimensions and the same Frobenius map, but the projective covers are as follows.

P_{0} = [figure: 0; 1; 0; 2; 0; 1; 3; 2; 0; 0; 2; 3; 1; 0; 2; 0; 1; 0] , P_{1} = [figure: 1; 0; 3; 0; 2; 1; 1; 2; 0; 3; 0; 1] , P_{2} = [figure: 2; 2; 0; 0; 1; 1; 0; 0; 2; 2] , P_{3} = [3, 1, 0, 0, 1, 3].

The Poincaré series for the cohomology ring appears to be as follows.

∑ t^{i} dim Ext^{i}(𝟙, 𝟙) = \frac{1 + t^{2} + t^{3} + 2t^{4} + t^{5} + t^{6} + t^{8}}{(1 − t)(1 − t^{3})(1 − t^{7})}
= 1 + t + 2t^{2} + 4t^{3} + 6t^{4} + 8t^{5} + 11t^{6} + 14t^{7} + ⋯

(we have computed it up to degree 40).

The Cartan matrix, the dimension of Ext_{C_{5}}^{1}, and of Ext_{C_{5}}^{2} between simples is as follows.

\begin{aligned}& V_{0} & V_{1} & V_{2} & V_{3} \\ V_{0} & 8 & 4 & 4 & 2 \\ V_{1} & 4 & 4 & 2 & 2 \\ V_{2} & 4 & 2 & 4 & 0 \\ V_{3} & 2 & 2 & 0 & 2 \\ & & Cartan \end{aligned}  \begin{aligned}& V_{0} & V_{1} & V_{2} & V_{3} \\ V_{0} & 1 & 1 & 1 & 0 \\ V_{1} & 1 & 0 & 0 & 1 \\ V_{2} & 1 & 0 & 1 & 0 \\ V_{3} & 0 & 1 & 0 & 0 \\ Ext & {1 \atop C_{5}} \end{aligned}  \begin{aligned}& V_{0} & V_{1} & V_{2} & V_{3} \\ V_{0} & 2 & 1 & 1 & 0 \\ V_{1} & 1 & 1 & 0 & 0 \\ V_{2} & 1 & 0 & 2 & 1 \\ V_{3} & 0 & 0 & 1 & 1 \\ Ext & {2 \atop C_{5}} \end{aligned}

5.4.2. The category C_{6}. The category C_{6} has eight simple objects, V_{0} to V_{7}. The projective covers of V_{0} to V_{3} are as in C_{5}, while the projective covers of V_{4} to V_{7} are as in C_{4} but with all subscripts increased by four. Tensor products are given by the following table.

V_{0} V_{1}  V_{2}  V_{3}  V_{4}  V_{5} V_{6} V_{7}
V_{1} [0, 0]
V_{2} [3]  [0, 1, 0]
V_{3} [2, 2] [1, 0, 0, 1] W
V_{4} [5]  [6]  [7]  [0, 2, 0]
V_{5} [4, 4]  [7]  P_{6}  [1, 3, 1]  W′
V_{6} [7]  [4, 5, 4] P_{5} [2, 0, 1, 0, 2] P_{3} W′′
V_{7} P_{6}  P_{5}  P_{4}  P_{3}  P_{2} P_{1} P_{0}

where

W = [figure: 0; 1; 0; 0; 1; 0] , W′ = [figure: 0; 0; 2; 2; 0; 0] , W′′ = [figure: 0; 1; 2; 0; 3; 0; 2; 1; 0] .

The Frobenius–Perron dimensions and the Frobenius twists of the simple modules are as follows.

\begin{matrix}V & V_{0} & V_{1} & V_{2} & V_{3} & V_{4} & V_{5} & V_{6} & V_{7} \\ FPdim V & 1 & \sqrt{2} & \sqrt{2+\sqrt{2}} & \sqrt{4+2\sqrt{2}} & \sqrt{2+\sqrt{2 + \sqrt{2}}} & \sqrt{4+2\sqrt{2+\sqrt{2}}} & \sqrt{2+\sqrt{2}}.\sqrt{2+\sqrt{2+\sqrt{2}}} & \sqrt{4+2\sqrt{2}}.\sqrt{2+\sqrt{2+\sqrt{2}}} \\ V^{(1)} & V_{0} & V_{2} & V_{4} & V_{6} & 0 & 0 & 0 & 0 \end{matrix}

5.4.3. The category C_{7}. The Poincaré series for the cohomology ring appears to be as follows.

∑ t^{i} dim Ext^{i}(𝟙, 𝟙) = \frac{1+t^{2}+t^{3}+3t^{4}+4t^{5}+4t^{6}+3t^{7}+5t^{8}+4t^{9}+4t^{10}+4t^{11}+4t^{12}+4t^{13}+5t^{14}+3t^{15}+4t^{16}+4t^{17}+3t^{18}+t^{19}+t^{20}+t^{22}}{(1−t)(1−t^{3})(1−t^{7})(1−t^{15})}

(we computed it up to degree 26).

Remark 5.1. The cohomology computations were done as follows. First we computed the tilting modules T_{m} for m large enough. Then we computed the basic algebra of C_{2n} as End(⨁_{i=2^{n}−1}^{2^{n+1}−2}T_{i}). Then we used a standard MAGMA function to compute the dimensions of Ext^{j} between the simple modules over this basic algebra, for j up to a specified point.

Remark 5.2. In general, on the basis of these examples, we expect the following properties of the cohomology. Let R_{n} be the graded ring Ext^{•}(𝟙, 𝟙) in C_{2n−1} and C_{2n}, and let h_{n}(t) be its Hilbert series. Then we expect that:

1) R_{n} has a natural polynomial subalgebra k[x_{1}, …, x_{n}], where |x_{i}| = 2^{i} − 1, over which it is a free module of rank 2^{n(n−1)/2}. In particular, R_{n} is Cohen-Macaulay;

2) one has

h_{n}(t) = \frac{P_{n}(t)}{∏_{i=1}^{n}(1 − t^{2^{i}−1})},

where P_{n}(t) is a polynomial with nonnegative coefficients of degree 2^{n+1} − 2(n + 1); 3) the polynomial P_{n} is palindromic;

4) R_{n} is Gorenstein;

5) the homomorphism R_{n} → R_{n+1} induced by the inclusion C_{2n} → C_{2n+2} is an injection;

6) the direct limit lim_{n→∞} R_{n} has finite dimensional homogeneous subspaces; hence there exists a coefficientwise limit h_{∞}(t) := lim_{n→∞} h_{n}(t), which is the Hilbert series of Ext^{•}(𝟙, 𝟙) in C_{∞}.

It is an interesting question if R_{n} is an integral domain. If so, then by Stanley’s criterion ([St], Theorem 4.4), (1) and (3) imply (4).

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