BRANCHED CRYSTALS AND CATEGORY O

VYJAYANTHI CHARI, DIJANA JAKELIĆ, AND ADRIANO A. MOURA

Introduction

The theory of crystal bases introduced by Kashiwara in [6] to study the category of integrable representations of quantized Kac–Moody Lie algebras has been a major development in the combinatorial approach to representation theory. In particular Kashiwara defined the tensor product of crystal bases and showed that it corresponded to the tensor product of representations. Later, in [7] he defined the abstract notion of a crystal, the tensor product of crystals and showed that the tensor product was commutative and associative.

In this paper, using some ideas from [4] and [9], we give a definition of branched crystals adapted to the study of the Bernstein–Gelfand–Gelfand category O of the quantized enveloping algebra of 𝔰𝔩_{2} which coincides with the usual definition of crystals for the integrable modules. In [6], Kashiwara essentially defined a crystal basis for the Verma modules. However, it is not hard to see that his tensor product rule, even for the case of 𝔰𝔩_{2}, does not give the decomposition (as a direct sum of indecomposable modules) of the corresponding tensor product of modules. Also, the restriction of the basis to a particular color does not reflect the decomposition of the module as a direct sum of indecomposable modules for the corresponding 𝔰𝔩_{2}. We define a notion of the tensor product of branched crystals which extends Kashiwara’s definition and the definition in [9] and prove in a combinatorial manner that the tensor product decomposes in the same way as the corresponding representations. Using this, we are then able to prove that the tensor product is both associative and commutative. It is not hard to generalize the definition of branched crystals to the higher rank case although the connection with O is probably much harder to establish. However, the results of Section 1 make it plausible that this is essentially the only possible theory of crystals for O which would satisfy the requirement that the crystal corresponding to a representation V, when restricted to a particular color, is the crystal of V regarded as a module for U_{q}(𝔰𝔩_{2}).

The paper is organized as follows. In Section 1 we define the notion of a (one–colored) branched crystal, its indecomposable components and classify the indecomposable branched crystals. In Section 2 we define the tensor product of two branched crystals and show that the result is also a branched crystal. Finally, we identify the indecomposable components (with multiplicities) in the tensor product of indecomposable branched crystals. In Section 3 we make a connection with the representation theory of U_{q}(sl_{2}). Using the results of Section 1 we see that given a module V in O, we can associate to it in a purely formal but natural and unique way a branched crystal B(V) so that direct sums are preserved. Further, the results of Section 2 make it clear that this association preserves tensor products, i.e., B(V ⊗ W) ≅ B(V) ⊗ B(W) for all V, W ∈ O.

Acknowledgements: We thank J. Greenstein for bringing [9] to our attention and for useful discussions. D.J. thanks the Max-Planck-Institut für Mathematik in Bonn for hospitality.

1. Branched Crystals

In this section we introduce the notion of a (one colored) branched crystal. These combinatorial objects are analogous to the notion of normal crystals defined by Kashiwara in [7].

1.1.

Definition. A branched crystal is a nonempty set B together with maps ẽ, f̃ : B ⊔ {0} → B ⊔ {0}, wt : B → Z, ε, φ : B → Z satisfying the following axioms:

(i) ẽ0 = f̃0 = 0 and if ẽb ∈ B (resp.f̃b ∈ B), then wt(ẽb) = wt(b) + 2 (resp. wt(f̃b) = wt(b) − 2)), (ii) if f̃b ∈ B, then ẽf̃b = b, (iii) ẽb ≠ ẽb′ if b, b′ ∈ B are such that ẽb, ẽb′ ∈ B but f̃ẽb ≠ b and f̃ẽb′ ≠ b′, (iv) ε(b) = min{s ∈ Z^{+} : f̃ẽ^{s+1}b ≠ ẽ^{s}b},

(v) φ(b) = wt(b) + ε(b). If there exists l > 0 such that f̃^{l−1}b ≠ 0 and f̃^{l}b = 0, then l = φ(b) + 1. Further, if b ∈ B is such that ẽb ∈ B but f̃ẽb ≠ b, then φ(ẽb) = 0.

An element b ∈ B is called a branch point if ẽb ∈ B but f̃ẽb ≠ b and B^{br} denotes the set of branch points in B. Note that f̃b ∉ B^{br} for b ∈ B. A subcrystal of B is a subset B′ of B such that B′ together with the restrictions of the maps wt, ẽ, f̃, ε, φ is a branched crystal. Given b ∈ B, let B_{b} be the subcrystal

B_{b} = {f̃^{r}ẽ^{s}b : r, s ∈ Z^{+}}.

We say that B_{b} is cyclic on b. If ẽb = 0, then B_{b} has no branch points.

Given branched crystals (B, wt, ẽ, f̃, ε, φ) and (B′, wt, ẽ, f̃, ε, φ) a map Φ : B ⊔ {0} → B′ ⊔ {0} is called a morphism of crystals if

(i) Φ(0) = 0, (ii) if b ∈ B and Φ(b) ∈ B′, then wt(Φ(b)) = wt(b) and ε(Φ(b)) = ε(b), (iii) if b ∈ B and Φ(b), Φ(f̃b) ∈ B′ (resp. Φ(ẽb) ∈ B′), then Φ(f̃b) = f̃Φ(b) (resp. Φ(ẽb) = ẽΦ(b)).

A morphism is strict if it satisfies

(iii’) Φ(f̃b) = f̃Φ(b) and Φ(ẽb) = ẽΦ(b) for all b ∈ B.

An injective morphism is called an embedding and an isomorphism is a bijective morphism.

1.2. Given b, b′ ∈ B, say that b ∼ b′ iff B_{b} ∩ B_{b′} ≠ ∅. It follows easily from Definition 1.1 (ii) that if ẽb = ẽb′ = 0 then b ∼ b′ iff b = b′.

Proposition. Let b, b′ ∈ B. Then b ∼ b′ iff there exist l, l′ ∈ Z^{+} such that ẽ^{l}b = ẽ^{l′}b′ ∈ B. Further, ∼ is an equivalence relation on B.

Proof. The first statement is obvious from Defintion 1.1(ii). To prove the second it is only neccesary to check that ∼ is transitive. Choose nonnegative integers l_{i}, 1 ≤ i ≤ 4, such that

ẽ^{l_{1}}b_{1} = ẽ^{l_{2}}b_{2}, ẽ^{l_{3}}b_{2} = ẽ^{l_{4}}b_{3}.

If l_{2} ≥ l_{3}, then

ẽ^{l_{4}+l_{2}−l_{3}}b_{3} = ẽ^{l_{2}}b_{2} = ẽ^{l_{1}}b_{1},

and hence B_{b_{1}} ∩ B_{b_{3}} ≠ ∅. The case l_{2} < l_{3} is similar and the lemma is proved. □

An equivalence class of B with respect to ∼ is a subcrystal and is called an indecomposable component of B. A crystal B is indecomposable if b ∼ b′ for all b, b′ ∈ B. Thus, an indecomposable component of B is a maximal indecomposable subcrystal of B.

Corollary.

(i) An indecomposable component of a branched crystal contains at most one element b such that ẽb = 0.

(ii) Suppose that B_{i}, i ∈ I, are a family of indecomposable subcrystals of a branched crystal B such that B = ∪_{i∈I}B_{i} and B_{i} ∩ B_{j} = ∅ if i, j ∈ I, i ≠ j. Then B_{i}, i ∈ I, are the indecomposable components of B.

1.3.

Proposition. Let B be a branched crystal and assume that B^{br} ≠ ∅. Let b ∈ B^{br}, then

(i) wt(b) ≤ −2, (ii) ẽ^{l}b is not a branch point for all 0 < l ≤ −wt(b) − 1 and ẽ^{−wt(b)−1}b ≠ 0, (iii) ẽ^{−wt(b)}b = 0, (iv) f̃^{l}b ≠ 0 ∀ l ≥ 0.

Proof. If b is a branch point then ẽb ∈ B and by Definition 1.1(i),(iv),(v) we have

0 ≤ ε(ẽb) = −wt(ẽb) = −wt(b) − 2

proving (i). If wt(b) = −2, then wt(ẽb) = 0 = ε(ẽb). The first equality together with (i) implies that ẽb ∉ B^{br} and the second equality implies that ẽ^{2}b = 0 by Definition 1.1(iv). Suppose now that wt(b) < −2. It follows by induction on l that ε(ẽ^{l}b) = ε(ẽb)−(l−1) = −wt(b)−1−l > 0 for 0 < l < −wt(b)−1. Hence ẽ^{l}b ∉ B^{br} and ẽ^{l+1}b ≠ 0. Further, ε(ẽ^{−wt(b)−1}b) = 0 and wt(ẽ^{−wt(b)−1}b) = −wt(b) − 2 > 0. This proves (ii). It also follows that ẽ^{−wt(b)}b = 0. If l ∈ Z^{+} is minimal such that f̃^{l}b = 0 then by Definition 1.1 (v) we have l = φ(b) + 1 = wt(b) + 1. But this is impossible since by (i) the right hand side is negative. □

Corollary.

(i) An indecomposable branched crystal has at most one branch point. (ii) Every indecomposable crystal has one element b such that ẽb = 0. (iii) If φ(b) ≥ 0, then ẽ^{l}b ∉ B^{br} for all l ≥ 0.

Proof. Let B be an indecomposable branched crystal and assume that b_{1}, b_{2} ∈ B^{br}, b_{1} ≠ b_{2}. It follows from Proposition 1.3(iii) and Corollary 1.2 that

ẽ^{−wt(b_{1})−1}b_{1} = ẽ^{−wt(b_{2})−1}b_{2}

and hence wt(b_{1}) = wt(b_{2}). Applying f̃ and using Proposition 1.3(ii) we get ẽb_{1} = ẽb_{2}. But this is impossible by Definition 1.1(iii). This proves (i). If B^{br} ≠ ∅, (ii) follows from Proposition1.3(iii). If B^{br} = ∅, then ε(b) = ε(ẽb) + 1 > 0 for all b ∈ B such that ẽb ∈ B. Hence ẽ^{ε(b)+1}b = 0 for all b ∈ B. To prove (iii), note that φ(b) < 0 for all b ∈ B^{br}. Therefore, it suffices to show that if φ(b) ≥ 0 then φ(ẽ^{l}b) ≥ 0 for all l ≥ 0 such that ẽ^{l}b ∈ B. This is easily done inductively. □

1.4. The main result of this section is:

Theorem. Let B be an indecomposable branched crystal. Then B is isomorphic to a branched crystal in one of the four infinite families defined below. Let r, s ∈ Z, r ≥ 0.

B(V (r)) = {b_{j} : 0 ≤ j ≤ r},

f̃b_{j} = b_{j+1},  ẽb_{j} = b_{j−1},  wt(b_{j}) = r − 2j,  ε(b_{j}) = j,  φ(b_{j}) = r − j.

B(M(s)) = {b_{j} : j ≥ 0},

f̃b_{j} = b_{j+1},  ẽb_{j} = b_{j−1},  wt(b_{j}) = s − 2j,  ε(b_{j}) = j,  φ(b_{j}) = s − j

B(T (r)) = {b_{(j)} : j ≥ 0} ⊔ {b_{j} : j ≥ 0}, where ẽb_{0} = b_{(r)}, and

f̃b_{(j)} = b_{(j+1)},  ẽb_{(j)} = b_{(j−1)},  wt(b_{(j)}) = r − 2j,  ε(b_{(j)}) = j,  φ(b_{(j)}) = r − j,
f̃b_{j} = b_{j+1},  ẽb_{j+1} = b_{j},  wt(b_{j}) = −r − 2 − 2j,  ε(b_{j}) = j,  φ(b_{j}) = −r − 2 − j.

B(M(r)^{σ}) = {b_{(j)} : 0 ≤ j ≤ r} ⊔ {b_{j} : j ≥ 0}, where ẽb_{0} = b_{(r)}, and

f̃b_{(j)} = b_{(j+1)},  ẽb_{(j)} = b_{(j−1)},  wt(b_{(j)}) = r − 2j,  ε(b_{(j)}) = j,  φ(b_{(j)}) = r − j.
f̃b_{j} = b_{j+1},  ẽb_{j+1} = b_{j},  wt(b_{j}) = −r − 2 − 2j,  ε(b_{j}) = j,  φ(b_{j}) = −r − 2 − j.

Proof. It is not hard to verify that the elements of the families given in the theorem satisfy the conditions for branched crystals. Now let B be an indecomposable branched crystal. Suppose first that B^{br} = ∅ and choose b ∈ B such that ẽb = 0. Then, B_{b} = {f̃^{r}b : r ∈ Z^{+}} is a subcrystal of B. Since ε(f̃^{r}b) = r for all r ∈ Z^{+} such that f̃^{r}b ∈ B, it follows that B_{b} is isomorphic to B(M(wt(b))) if f̃^{r}b ≠ 0 for all r ∈ Z^{+}, and to B(V (wt(b))) if f̃^{wt(b)+1}b = 0. It remains to show that B = B_{b}. But this is clear, for given b′ ∈ B there exists s ∈ Z^{+} with ẽ^{s}b′ = 0 or equivalently ẽ^{s−1}b′ = b and hence f̃^{s−1}b = b′. If B^{br} = {b}, set ẽ^{−wt(b)−1}b = b̄ and let b′ ∈ B\{b}. If ẽ^{s}b′ = b for some s ∈ Z^{+}, then b′ = f̃^{s}b ∈ B_{b}. Otherwise ẽ^{s}b′ ≠ b for all s ∈ Z^{+} and ẽ^{k}b′ = b̄ for some k ∈ Z^{+}. Hence b′ = f̃^{k}b̄ ∈ B_{b}. It is not hard to see now that B_{b} is isomorphic to B(T (−wt(b) − 2)) or to B(M(−wt(b) − 2)^{σ}). □

We conclude this section with some additional results which are needed later.

1.5. Assume that {B_{j}}_{j∈J} is a family of branched crystals. There exists an obvious structure of a branched crystal on the disjoint union ⊔_{j∈J}B_{j} which we denote by ⊕_{j∈J}B_{j}. Namely the maps wt, ẽ, f̃, ε, φ on ⊕_{j∈J}B_{j} are defined by requiring their restriction to B_{j} to be the corresponding map on B_{j}. The canonical inclusion ι_{j} : B_{j} → ⊕_{j∈J}B_{j} is a strict embedding of branched crystals. This proves,

Proposition. Let B be a branched crystal and let B_{j}, j ∈ J be the indecomposable components of B. Then, B = ⊕_{j∈J}B_{j}. Furthermore, if Φ : B → ⊕_{l∈L}B′_{l} is an isomorphism of branched crystals with B′_{l} indecomposable, then there exists a bijective map ϕ : J → L such that Φ|_{B_{j}} : B_{j} → B′_{ϕ(j)}, j ∈ J is an isomorphism of branched crystals.

1.6. Given a branched crystal B, set

B^{hw} = {b ∈ B\(∪_{b′∈B^{br}}B_{b′}) : ẽb = 0}

and

B^{br,σ} = {b ∈ B^{br} : f̃ẽb = 0}.

The following is easily seen as a consequence of Theorem 1.4.

Proposition. Let B be a branched crystal. The indecomposable components of B are B_{b}, where

b ∈ B^{hw} ∪ B^{br}.

Moreover we have isomorphisms of branched crystals,

B_{b} ≅ B(M(wt(b))), if b ∈ B^{hw}, f̃B_{b} ⊂ B,
B_{b} ≅ B(V (wt(b))), if b ∈ B^{hw}, f̃^{wt(b)+1}b = 0,
B_{b} ≅ B(T (−wt(b) − 2)), if b ∈ B^{br}\B^{br,σ},
B_{b} ≅ B(M(−wt(b) − 2)^{σ}), if b ∈ B^{br,σ}.

1.7.

Lemma. Let B be a branched crystal and let b, b′ ∈ B. Then B_{b} ∩ B_{b′} ≠ ∅ iff B_{b′} ⊂ B_{b} or vice versa.

Proof. Suppose B_{b} ∩ B_{b′} ≠ ∅. Let a = ẽ^{ε(b)}b and a′ = ẽ^{ε(b′)}b′. Then B_{b} = B_{a} and B_{b′} = B_{a′}. We have three cases:

(i) a, a′ ∈ B^{br}, (ii) ẽa = ẽa′ = 0, (iii) a ∈ B^{br} and ẽa′ = 0 or vice versa.

It follows from Corollary 1.3(i) (resp. Corollary 1.2) that a = a′ in case (i) (resp. case (ii)). Similarly in case (iii) we have ẽ^{−wt(a)−1}a = a′, hence B_{a′} ⊂ B_{a}. □

2. A tensor product rule for branched crystals

2.1. We now define an analogue for branched crystals of Kashiwara’s tensor product rule for crystals. Extending [9], we also define ψ : B → Z as follows:

ψ(b) = max{ε(b), ε(b) − φ(b) − 1}.

Definition. Given two branched crystals B, B′, let B ⊗ B′ be the set B × B′ equipped with the maps ẽ, f̃: (B ⊔ {0}) × (B′ ⊔ {0}) → (B × B′) ⊔ {0} and wt : B × B′ → Z defined below.

wt(b ⊗ b′) = wt(b) + wt(b′)

f̃(b ⊗ b′) = f̃b ⊗ b′, if b, b′ satisfy F1,

= b ⊗ f̃b′, if b, b′ satisfy F1′,

= f̃^{φ(b)+1}b ⊗ ẽ^{φ(b)}b′, if b, b′ satisfy F2,

ẽ(b ⊗ b′) = ẽb ⊗ b′, if b, b′ satisfy E1,

= b ⊗ ẽb′, if b, b′ satisfy E1′,

= ẽ^{φ(b′)+1}b ⊗ f̃^{φ(b′)}b′, if b, b′ satisfy E2,

= f̃^{−wt(b)−wt(b′)−2}b ⊗ ẽ^{−wt(b)−wt(b′)−1}b′, if b, b′ satisfy E2′, = 0 otherwise,

where we say that b, b′ satisfy

(i) the condition F1 if φ(b) < 0 or ψ(b′) < φ(b), (ii) the condition F1′ if f̃b′ ≠ 0 and ψ(b′) ≥ φ(b) ≥ 0, (iii) the condition F2 if f̃b′ = 0 and ψ(b′) ≥ φ(b) ≥ 0, (iv) the condition E1 if either wt(b) < φ(b) < −1 or ψ(b′) ≤ φ(b),

(v) the condition E1′ if ψ(b′) > φ(b) ≥ 0, (vi) the condition E2 if either φ(b) = −1, f̃^{φ(b′)+1}b′ = 0, and 0 ≤ φ(b′) ≤ ε(b)−1 or b ∈ B^{br}, ∄a′ ∈ B^{′br}

such that b′ = ẽ^{l}a′ for some l ≥ 0, and 0 ≤ φ(b′) ≤ −wt(b) − 2, (vii) the condition E2′ if either φ(b) = −1, ẽb ∈ B, and −wt(b) − 1 ≤ φ(b′) ≤ −wt(b) − 2 + ε(b′), or

φ(b) < 0, ε(b) = 0, and −wt(b) − 1 ≤ φ(b′) ≤ −wt(b) − 2 + ε(b′).

Here we understand 0 ⊗ b′ = b ⊗ 0 = 0 and define ẽ0 = f̃0 = 0. Finally, set

ε(b ⊗ b′) = ∞, if f̃ẽ^{s+1}(b ⊗ b′) = ẽ^{s}(b ⊗ b′) for all s ≥ 0,
= min{s ∈ Z^{+} : f̃ẽ^{s+1}(b ⊗ b′) ≠ ẽ^{s}(b ⊗ b′)}, otherwise

and

φ(b ⊗ b′) = wt(b ⊗ b′) + ε(b ⊗ b′).

2.2. The next few subsections are devoted to the proof the following theorem.

Theorem. Let B, B′ be branched crystals and assume that B_{i}, i ∈ I and B′_{j}, j ∈ J are the indecomposable components of B and B′, respectively. Then B⊗B′ is a branched crystal and B⊗B′ = ⊕_{i,j}B_{i}⊗B′_{j}.

Assuming that B ⊗ B′ is a branched crystal the second statement is proved as follows. Let ẽ_{B,B′} : (B × B′) ⊔ {0} → (B × B′) ⊔ {0} and ẽ_{B_{i},B′_{j}} : (B_{i} × B′_{j}) ⊔ {0} → (B_{i} × B′_{j}) ⊔ {0} be defined as above. It is clear that for all (b, b′) ∈ B_{i} × B′_{j} we have ẽ_{B,B′}(b ⊗ b′) = ẽ_{B_{i},B′_{j}}(b ⊗ b′) ∈ (B_{i} × B′_{j}) ⊔ {0} (and similarly for f̃). It is now immediate that B ⊗ B′ = ⊕_{i,j}B_{i} ⊗ B′_{j}.

2.3. We proceed now with the proof of Theorem 2.2. It is clear that the condition (i) of Definition 1.1 is satisfied.

For (ii), let b ∈ B, b′ ∈ B′ be such that f̃(b ⊗ b′) ∈ B ⊗ B′. If b, b′ satisfy F1 (resp. F1′) then it is obvious that f̃b, b′ (resp. b, f̃b′) satisfy E1 (resp. E1′). If b, b′ satisfy F2, we have

φ(f̃^{φ(b)+1}b) = −1, φ(ẽ^{φ(b)}b′) = φ(b), ε(f̃^{φ(b)+1}b) = ε(b) + φ(b) + 1.

Here we used Corollary 1.3(iii). Hence f̃^{φ(b)+1}b, ẽ^{φ(b)}b′ satisfy E2 and ẽf̃(b ⊗ b′) = b ⊗ b′. This proves

Let us now check condition (iv). Let b ∈ B, b′ ∈ B′ and B̄, B̄′ be the indecomposable components they belong to respectively. It is evident that ẽ^{l}(b⊗b′) ∈ (B̄×B̄′)⊔{0}. We claim that ε(b⊗b′) ∈ Z^{+} and, therefore, condition (iv) of Definition 1.1 is satisfied. To prove the claim it suffices to check that there exists l ∈ Z^{+} such that ẽ^{l}(b⊗b′) = 0. If that was not the case we would have wt(ẽ^{l}(b⊗b′)) = wt(b⊗b′)+2l for all l ∈ Z^{+}. But this is impossible since the weight function is clearly bounded from above on B̄ ×B̄′.

2.4. To prove (iii) and (v), we begin with the following proposition which characterizes the branch points in B ⊗ B′.

Proposition. Suppose that B, B′ are branched crystals. Let b ∈ B, b ∈ B′. Then

(a) f̃(b ⊗ b′) = 0 iff b, b′ satisfy F2 and f̃^{φ(b)+1}b = 0.

(b) Let b, b′ be such that ẽ(b ⊗ b′) ≠ 0. Then f̃ẽ(b ⊗ b′) ≠ b ⊗ b′ iff one of the following happens: (i) b′ ∈ B^{′br} and b, b′ satisfy E1′, (ii) b ∈ B^{br} and b, b′ satisfy E2, (iii) b, b′ satisfy E2′, (iv) φ(b) = −wt(b′) − 2 ≥ 0 and φ(b′) < −1 (or equivalently, ψ(b′) = φ(b) + 1 and φ(b′) < −1).

Proof. Part (a) is obvious. For part (b), we first prove that f̃ẽ(b ⊗ b′) ≠ b ⊗ b′ if (i)–(iv) happens. This is obvious in cases (i) and (ii). For (iii) it is also obvious unless −wt(b) − wt(b′) − 2 = 0. In this case we need to show that F1′ does not apply to b ⊗ ẽb′. This is true since φ(b) < 0. In case (iv) we see that E1′ applies to b, b′ and, hence, we need to show that F1′ does not apply to b ⊗ ẽb′. But this is clear since φ(b′) < −1 implies ψ(ẽb′) = ψ(b′) − 2. The converse is proved similarly. □

Corollary. Let b ∈ B, b′ ∈ B′ satisfy ẽ(b ⊗ b′) ≠ 0 and f̃ẽ(b ⊗ b′) ≠ b ⊗ b′. Then f̃^{l}(b ⊗ b′) ∈ B ⊗ B′ for all l ≥ 0.

Proof. Suppose that b, b′ satisfy condition (b)(i) of the proposition. Then, by Proposition 1.3 we see that f̃^{l}b′ ∈ B′ for all l ≥ 0. Since ψ(f^{l}b′) > ψ(b) it follows that b and f^{l}b′ satisfy F1′ for all l ≥ 0 and hence the corollary follows in this case. Suppose now that b, b′ satisfy E2. Then, f̃^{l}b ∈ B for all l > 0 and f̃^{l}b, b′ satisfy F1 and hence f̃^{l}(b ⊗ b′) = f̃^{l}b ⊗ b′ ∈ B. A similar computation proves the result if b, b′ satisfy condition (b)(iii). Finally, if we have φ(b) = −wt(b′) − 2 ≥ 0 and φ(b′) < −1, then by Definition 1.1(v) we have that f̃^{l}b′ ∈ B′ for all l ≥ 0 and since

ψ(b′) = −wt(b′) − 1 ≥ −wt(b′) − 2 = φ(b) ≥ 0,

it follows that b and f̃^{l}b′ always satisfy F1′ and the corollary is proved. □

2.5. We can now prove that condition (v) of Definition 1.1 is satisfied. Let b ⊗ b′ be such that ẽ(b ⊗ b′) ≠ 0 and f̃ẽ(b ⊗ b′) ≠ b ⊗ b′. Suppose b′ ∈ B^{′br} and b, b′ satisfy E1′. Then b, ẽ^{l}b′ satisfy E1′ for all 0 ≤ l ≤ −φ(b) − wt(b′) − 1 and F1′ for all 2 ≤ l ≤ −φ(b) − wt(b′) − 1. Therefore,

ẽ^{l}(b ⊗ b′) = b ⊗ ẽ^{l}b′ ≠ 0, for all 1 ≤ l ≤ −φ(b) − wt(b′) − 1
f̃ẽ^{l+1}(b ⊗ b′) = ẽ^{l}(b ⊗ b′) for all 1 ≤ l ≤ −φ(b) − wt(b′) − 2.

Next, using E1 and F1 we have

ẽ^{−φ(b)−wt(b′)−1+l}(b ⊗ b′) = ẽ^{l}b ⊗ ẽ^{−φ(b)−wt(b′)−1}b′ ≠ 0 for all 0 ≤ l ≤ ε(b),
f̃ẽ^{−φ(b)−wt(b′)−1+l}(b ⊗ b′) = ẽ^{−φ(b)−wt(b′)−2+l}(b ⊗ b′) for all 1 ≤ l ≤ ε(b),

and since ẽ^{ε(b)+1}b = 0,

ẽ^{−φ(b)−wt(b′)+ε(b)}(b ⊗ b′) = 0.

This proves that

ε(ẽ(b ⊗ b′)) = −wt(b) − wt(b′) − 2,

i.e that φ(ẽ(b ⊗ b′)) = 0. A similar computation takes care of the case b ∈ B^{br} and b, b′ satisfy E2. Now suppose b, b′ satisfy E2′, then

ẽ(b ⊗ b′) = f̃^{−wt(b′)−wt(b)−2}b ⊗ ẽ^{−wt(b′)−wt(b)−1}b′,

and using E1 and F1, it follows that

ẽ^{l+1}(b ⊗ b′) = ẽ^{l}f̃^{−wt(b′)−wt(b)−2}b ⊗ ẽ^{−wt(b′)−wt(b)−1}b′ for all 0 ≤ l ≤ −wt(b′) − wt(b) − 2,
f̃ẽ^{l+1}(b ⊗ b′) = ẽ^{l}(b ⊗ b′) for all 1 ≤ l ≤ −wt(b′) − wt(b) − 2.

Since none of the rules E1, E1′, E2, E2′ apply to b ⊗ ẽ^{−wt(b′)−wt(b)−1}b′ we conclude that ε(ẽ(b ⊗ b′)) = −wt(b′) − wt(b) − 2, i.e. φ(ẽ(b ⊗ b′)) = 0. Finally, consider the case φ(b) = −wt(b′) − 2 ≥ 0 and φ(b′) < −1. This time, we find that ẽ(b ⊗ b′) = b ⊗ ẽb′ and using E1 and F1 we have

ẽ^{l+1}(b ⊗ b′) = ẽ^{l}b ⊗ ẽb′ ≠ 0 for all 0 ≤ l ≤ ε(b),
f̃ẽ^{l+1}(b ⊗ b′) = ẽ^{l}(b ⊗ b′) for all 1 ≤ l ≤ ε(b),

and ẽ^{ε(b)+2}(b ⊗ b′) = 0.

Hence ε(ẽ(b ⊗ b′)) = ε(b). We must check that ε(b) = −wt(b) − wt(b′) − 2, but this follows from φ(b′) < −1 and ψ(b′) = φ(b) + 1.

To complete the proof of (v), we must show that if l is minimal such that f̃^{l}(b ⊗ b′) = 0 then l = φ(b ⊗ b′) + 1. It clearly suffices to consider the case l = 1. By Proposition 2.4 we know that b, b′ satisfy F2. Then using E1′ and F1′ we see that

ẽ^{l}(b ⊗ b′) = b ⊗ ẽ^{l}b′ ≠ 0, for all 0 ≤ l ≤ ε(b′) − φ(b),

and

f̃ẽ^{l}(b ⊗ b′) = ẽ^{l−1}(b ⊗ b′), for all 1 ≤ l ≤ ε(b′) − φ(b).

Then, using E1 and F1 we get

ẽ^{ε(b′)−φ(b)+l}(b ⊗ b′) = ẽ^{l}b ⊗ ẽ^{ε(b′)−φ(b)}b′ ≠ 0, for all 0 ≤ l ≤ ε(b),
f̃ẽ^{ε(b′)−φ(b)+l}(b ⊗ b′) = ẽ^{ε(b′)−φ(b)+l−1}(b ⊗ b′), for all 1 ≤ l ≤ ε(b),

and ẽ^{ε(b′)−φ(b)+ε(b)+1}(b ⊗ b′) = 0. Hence, ε(b ⊗ b′) = ε(b′) − φ(b) + ε(b) = −wt(b′) − wt(b), i.e., φ(b ⊗ b′) = 0.

2.6. Finally, we must prove that condition (iii) of Definition 1.1 holds:

Proposition. Suppose that b_{i} ∈ B, b′_{i} ∈ B′ are distinct elements such that ẽ(b_{i} ⊗ b′_{i}) ∈ B ⊗ B′ and f̃ẽ(b_{i} ⊗ b′_{i}) ≠ (b_{i} ⊗ b′_{i}) for i = 1, 2. Then

ẽ(b_{1} ⊗ b′_{1}) ≠ ẽ(b_{2} ⊗ b′_{2}).

Proof. Assume ẽ(b_{1} ⊗ b′_{1}) = ẽ(b_{2} ⊗ b′_{2}). We consider four cases depending on the various possibilities for the pairs b_{1}, b′_{1} given Proposition 2.4.

Case 1. b_{1}, b′_{1} satisfy E1′ and b′_{1} ∈ B^{′br}. If b_{2}, b′_{2} also satisfy E1′ and b′_{2} ∈ B^{′br} then we get b_{1} = b_{2} and ẽb′_{1} = ẽb′_{2} which implies that b′_{1} = b′_{2}. If b_{2}, b′_{2} satisfy E2 with b_{2} ∈ B^{br}, then we get

b_{1} = ẽ^{φ(b′_{2})+1}b_{2}, ẽb′_{1} = f̃^{φ(b′_{2})}b′_{2},

i.e., b′_{2} = ẽ^{φ(b′_{2})+1}b′_{1}. But φ(b_{2}) = wt(b_{2}) ≤ −2 by Proposition 1.3, so we get a contradiction to the fact that b_{2}, b′_{2} satisfy E2. Next, suppose that b_{2}, b′_{2} satisfy E2′. This gives,

b_{1} = f̃^{−(wt(b_{2})+wt(b′_{2})+2)}b_{2},

which gives φ(b_{1}) = φ(b_{2}) + wt(b_{2}) + wt(b′_{2}) + 2 < 0, contradicting the fact that φ(b_{1}) ≥ 0. Finally, suppose that b_{2}, b′_{2} satisfy φ(b_{2}) = −wt(b_{2}) − 2 ≥ 0 and φ(b′_{2}) < −1. In particular, this means that b_{2}, b′_{2} satisfy E1′ and hence we get b_{1} = b_{2} and ẽb′_{1} = ẽb′_{2}. Since b′_{1} ∈ B^{′br} this implies that b′_{2} ∉ B^{′br} and so f̃ẽb′_{1} = b′_{2}, which gives

φ(b′_{2}) = wt(b′_{1}) + ε(f̃ẽb′_{1}) = wt(b′_{1}) + ε(ẽb′_{1}) + 1 = −1,

since ε(ẽb′_{1}) = −wt(b′_{1}) − 2 which contradicts φ(b′_{2}) < −1.

Case 2. b_{1}, b′_{1} satisfy E2 and b_{1} ∈ B^{br}. If b_{2} and b′_{2} also satisfy E2 and b_{2} ∈ B^{br}, assume without loss of generality that φ(b′_{1}) ≥ φ(b′_{2}). Since, ẽ^{φ(b′_{1})+1}b_{1} = ẽ^{φ(b′_{2})+1}b_{2}, we get by using Proposition 1.3 that ẽb_{1} = f̃^{φ(b′_{1})−φ(b′_{2})}ẽb_{2}. But this means that,

0 = φ(ẽb_{1}) = −φ(b′_{1}) + φ(b′_{2}),

which implies b_{1} = b_{2}. Since we also have f̃^{φ(b′_{1})−φ(b′_{2})}b′_{1} = b′_{2}, it follows that b′_{1} = b′_{2}.

Assume next that b_{2} and b′_{2} satisfy E2′. We get ẽ^{φ(b′_{1})+1}b_{1} = f̃^{−wt(b′_{2})−wt(b_{2})−2}b_{2} and hence b_{2} = ẽ^{φ(b′_{1})−wt(b′_{2})−wt(b_{2})−1}b_{1}. This implies that

φ(b_{2}) = φ(ẽb_{1}) + φ(b′_{1}) − wt(b′_{2}) − wt(b_{2}) − 2 ≥ 0,

which is again a contradiction. Finally, let us assume that φ(b_{2}) = −wt(b′_{2}) − 2 ≥ 0 and φ(b′_{2}) < −1, then ẽ(b_{2} ⊗ b′_{2}) = b_{2} ⊗ ẽb′_{2} and so we get ẽ^{φ(b′_{1})+1}b_{1} = b_{2}, which gives φ(b_{2}) = φ(b′_{1}) < 0 which is again a contradiction.

Case 3. b_{1} and b′_{1} satisfy E2′. If b_{2} and b′_{2} also satisfy E2′, then we get f̃^{−wt(b_{1})−wt(b′_{1})−2}b_{1} = f̃^{−wt(b_{2})−wt(b′_{2})−2}b_{2}. Since wt(b_{1}) + wt(b′_{1}) = wt(b_{2}) + wt(b′_{2})(= l), it follows immediately that b_{1} = b_{2}. Since ẽ^{l−1}b′_{1} = ẽ^{l−1}b′_{2} and φ(b′_{i}) ≥ 0, it follows from Corollary 1.3(iii) that b′_{1} = b′_{2}.

If b_{2} and b′_{2} satisfy φ(b_{2}) = −wt(b′_{2}) − 2 ≥ 0 and φ(b′_{2}) < −1, we get b_{2} = f̃^{−wt(b_{1})−wt(b′_{1})−2}b_{1}. This means that ẽ^{−wt(b_{1})−wt(b′_{1})−2}b_{2} = b_{1} and that ẽ^{l}b_{2} ∉ B^{br} for all 0 ≤ l ≤ −wt(b_{1}) − wt(b′_{1}) − 2 and hence φ(b_{1}) = φ(b_{2}) − wt(b_{1}) − wt(b′_{1}) − 2 ≥ 0 which is a contradiction.

Case 4 The remaining case is when b_{i} and b′_{i} satisfy φ(b_{i}) = −wt(b′_{i}) − 2 ≥ 0 and φ(b′_{i}) < −1, i = 1, 2. Again we get b_{1} = b_{2} and ẽb′_{1} = ẽb′_{2}. This means that the only time that it is not obvious that b′_{1} = b′_{2} is when b′_{1} ∈ B^{′br} and b′_{2} ∉ B^{′br}(or vice versa). But then we get b′_{2} = f̃ẽb′_{1} and so φ(b′_{2}) = −1 since φ(ẽb′_{1}) = 0. But this is again a contradiction. □

The proof of Theorem 2.2 is now complete.

2.7.

Proposition. Let B, B′ be branched crystals and let b ∈ B, b′ ∈ B′. Then ẽ(b ⊗ b′) = 0 iff one of the following holds:

(i) ε(b) = 0, b ∉ B^{br} (resp. ε(b′) = 0, b′ ∉ B^{′br}), and b, b′ satisfy E1 (resp. E1′),or (ii) b, b′ do not satisfy E1, E2, E1′, E2′.

Moreover, if (ii) holds, then either ε(b) = 0 or φ(b) = −1 Proof. Observe that if b, b′ satisfy E1 (resp. E1′) then ẽ(b ⊗ b′) = 0 iff ẽb = 0 (resp. ẽb′ = 0). It is also easy to see that if b, b′ satisfy none of the conditions E1, E1′, E2, E2′, then either ε(b) = 0 or φ(b) = −1. It remains to show that if b, b′ satisfy E2 or E2′, then ẽ(b ⊗ b′) ≠ 0. If b, b′ satisfy E2, observe that the upper bound for φ(b′) implies that ẽ^{φ(b′)+1}b ≠ 0 and we are done. If b, b′ satisfy E2′ then, since φ(b) < 0, we have f̃^{l}b ≠ 0 for all l ≥ 0. We need to show that ẽ^{−wt(b)−wt(b′)−1}b′ ≠ 0. It suffices to check that −wt(b) − wt(b′) − 1 ≤ ε(b′). But this is equivalent to φ(b′) ≥ −wt(b) − 1. □

2.8. Let B, B′ be indecomposable branched crystals and B = B ⊗ B′. We conclude this section by writing down the indecomposable components of B in a case by case fashion. By Proposition 1.6 it suffices to compute the sets B^{br}, B^{br,σ}, B^{hw} which is done by using Propositions 2.4 and 2.7 and Lemma 1.7. In the following we use the notation of Theorem 1.4. Also we understand p ∈ Z^{+} and use the convention B(M(−1)) = B(T (−1)) when convenient.

Case 1. Suppose that B = B(M(s)), B′ = B(V (r)).

B_{b_{s+1}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s + 1, if s ≥ 0,
B_{b_{0}⊗b′_{r+s+1−p}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if s < 0,
B_{b_{0}⊗b′_{p}} ≅ B(M(r + s − 2p)), 0 ≤ p ≤ min{r, s},
B_{b_{0}⊗b′_{p}} ≅ B(M(r + s − 2p)), r + s + 2 ≤ p ≤ r.

Case 2. B = B(T (s)), B′ = B(V (r)).

B_{b_{(s+1)}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s + 1,
B_{b_{0}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if s < r,
B_{b_{0}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ r, if s ≥ r.

Case 3. B = B(M(s)^{σ}), B′ = B(V (r)).

B_{b_{0}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s + 1,
B_{b_{0}⊗b′_{r−p}} ≅ B(M(r + s − 2p)^{σ}), 0 ≤ p ≤ min{r, s}.

Case 4. B = B(V (r)), B′ = B(M(s)^{σ}).

B_{b_{r+s−2p}⊗b′_{p−s}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,
B_{b_{r−p}⊗b′_{0}} ≅ B(M(r + s − 2p)^{σ}), 0 ≤ p ≤ min{r, s},
B_{b_{0}⊗b′_{(r−s−1)/2}} ≅ B(M(−1)).

Case 5. B′ = B(M(r)), B = B(M(s)).

B_{b_{r+s−2p}⊗b′_{p+1}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,

B_{b_{r+1}⊗b′_{s−p}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s + 1, if r ≥ 0,

B_{b_{0}⊗b′_{r+s+1−p}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if r < 0,

B_{b_{0}⊗b′_{p}} ≅ B(M(r + s − 2p)), 0 ≤ p ≤ min{r, s},

B_{b_{0}⊗b′_{p}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2, if r < 0,

B_{b_{r+1}⊗b′_{p−r−1}} ≅ B(M(r + s − 2p)), p ≥ max{r, s} + 1, if r ≥ 0,

B_{b_{p}⊗b′_{0}} ≅ B(M(r + s − 2p)), r + s + 2 ≤ p ≤ r, if s < 0,

B_{b_{0}⊗b′_{(r+s+1)/2}} ≅ B(M(−1)), if r ≥ s.

Case 6. B = B(M(r)), B′ = B(M(s)^{σ}).

B_{b_{r−p}⊗b′_{0}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ s,

B_{b_{r+s−2p}⊗b′_{p−s}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,

B_{b_{r+1}⊗b′_{(s−p)}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s + 1, if r ≥ 0,

B_{b_{0}⊗b′_{(r+s+1−p)}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if r < 0,

B_{b_{r+1}⊗b′_{p−r−s−2}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2, if r ≥ 0,

B_{b_{0}⊗b′_{(p)}} ≅ B(M(r + s − 2p)), r + s + 2 ≤ p ≤ s, if r < 0,

B_{b_{0}⊗b′_{p−s−1}} ≅ B(M(r + s − 2p)), p ≥ s + 1, if r < 0,

B_{b_{0}⊗b′_{(r−s−1)/2}} ≅ M(−1), if r ≥ 0.

Case 7. B = B(M(s)^{σ}), B′ = B(M(r)).

B_{b_{0}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if s < r,

B_{b_{0}⊗b′_{r−p}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ r, if s ≥ r,

B_{b_{(r+s−2p)}⊗b′_{p+1}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s,

B_{b_{0}⊗b′_{p−s−1}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2, if r ≥ 0,

B_{b_{0}⊗b′_{p−s−1}} ≅ B(M(r + s − 2p)), p ≥ s + 1, if r < 0,

B_{b_{(p)}⊗b′_{0}} ≅ B(M(r + s − 2p)), s + r + 2 ≤ p ≤ s,

B_{b_{(0)}⊗b′_{(r+s+1)/2}} ≅ B(M(−1)), if s ≥ r.

Case 8. B = B(M(r)^{σ}), B′ = B(M(s)^{σ}).

B_{b_{(r+s−2p)}⊗b′_{p−s}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,

B_{b_{0}⊗b′_{(s−p)}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s + 1,

B_{b_{(r−p)}⊗b′_{0}} ≅ B(M(r + s − 2p)^{σ}), 0 ≤ p ≤ min{r, s},

B_{b_{0}⊗b′_{(p−r−1)}} ≅ B(M(r + s − 2p)), max{r, s} + 1 ≤ p ≤ r + s + 1,

B_{b_{0}⊗b′_{p−r−s−2}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2,

B_{b_{(0)}⊗b′_{(r−s−1)/2}} ≅ B(M(−1)).

Case 9. B = B(M(s)), B′ = B(T (r)).

B_{b_{s−p}⊗b′_{0}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ min{r, s},

B_{b_{r+s−2p}⊗b′_{p−r}} ≅ B_{b_{r+s−2p}⊗b′_{(p+1)}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s,

B_{b_{s+1}⊗b′_{(r−p)}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s + 1,

B_{b_{0}⊗b′_{(r+s+1−p)}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if s < 0,

B_{b_{s+1}⊗b′_{p−r−s−2}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2, if s ≥ 0,

B_{b_{s+1}⊗b′_{(p−s−1)}} ≅ B(M(r + s − 2p)), p ≥ max{r, s} + 1, if s ≥ 0,

B_{b_{0}⊗b′_{(p)}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2, if s < 0,

B_{b_{0}⊗b′_{(p−s−1)}} ≅ B(M(r + s − 2p)), p ≥ s + 1, if s < 0,

B_{b_{0}⊗b′_{(s−r−1)/2}} ≅ B_{b_{0}⊗b′_{((r+s+1)/2)}} ≅ B(M(−1)), if s ≥ r.

Case 10. B = B(T (r)), B′ = B(M(s)).

B_{b_{0}⊗b′_{s−p}} ≅ B(T (r + s − 2p)), 0 ≤ 2p ≤ r + s + 1, if s > r,

B_{b_{0}⊗b′_{s−p}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ s, if s ≤ r,

B_{b_{(r+1)}⊗b′_{s−p}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s + 1,

B_{b_{(r+s−2p)}⊗b′_{p+1}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,

B_{b_{0}⊗b′_{p−r−1}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2, if s ≥ 0,

B_{b_{0}⊗b′_{p−r−1}} ≅ B(M(r + s − 2p)), p ≥ r + 1, if s < 0,

B_{b_{(r+1)}⊗b′_{p−r−1}} ≅ B(M(r + s − 2p)), p ≥ max{r, s} + 1,

B_{b_{(p)}⊗b′_{0}} ≅ B(M(r + s − 2p)), s + r + 2 ≤ p ≤ r,

B_{b_{(0)}⊗b′_{(r+s+1)/2}} ≅ B(M(−1)), if s ≤ r.

Case 11. B = B(T (r)), B′ = B(T (s)).

B_{b_{(r+s−2p)}⊗b′_{(p+1)}} ≅ B_{b_{(r+s−2p)}⊗b′_{p−s}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,

B_{b_{(r−p)}⊗b′_{0}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ s,

B_{b_{(r+1)}⊗b′_{(s−p)}} ≅ B_{b_{0}⊗b′_{(s−p)}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s + 1,

B_{b_{(r+1)}⊗b′_{(p−r−1)}} ≅ B_{b_{0}⊗b′_{(p−r−1)}} ≅ B(M(r + s − 2p)), p ≥ s + 1,

B_{b_{(r+1)}⊗b′_{p−r−1}} ≅ B_{b_{0}⊗b′_{p−r−1}} ≅ (M(r + s − 2p)), p ≥ r + 1,

B_{b_{(0)}⊗b′_{((r+s+1)/2)}} ≅ B_{b_{(0)}⊗b′_{(r−s−1)/2}} ≅ B(M(−1)), if r ≥ s.

Case 12. B = B(T (r)), B′ = B(M(s)^{σ}).

B_{b_{(r+s−2p)}⊗b′_{p−s}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,

B_{b_{(r−p)}⊗b′_{0}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ max{r, s},

B_{b_{0}⊗b′_{(s−p)}} ≅ B_{b_{(r+1)}⊗b′_{(s−p)}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s + 1,

B_{b_{0}⊗b′_{(p−r−1)}} ≅ B(M(r + s − 2p)), p ≥ r + 1, if r ≥ s,

B_{b_{0}⊗b′_{(p−r−1)}} ≅ B(M(r + s − 2p)), s + 1 ≤ p ≤ r + s + 1, if s > r,

B_{b_{0}⊗b′_{p−r−s−2}} ≅ B_{b_{(r+1)}⊗b′_{p−r−s−2}} ≅ B(M(r + s − 2p)), p ≥ r + s + 2,

B_{b_{(0)}⊗b′_{(r−s−1)/2}} ≅ B(M(−1)), if r ≥ s.

Case 13. B = B(M(s)^{σ}), B′ = B(T (r)).

B_{b_{(r+s−2p)}⊗b′_{(p+1)}} ≅ B_{b_{(r+s−2p)}⊗b′_{p−r}} ≅ B(T (r + s − 2p)), 2r + 2 ≤ 2p ≤ r + s,

B_{b_{(s−p)}⊗b′_{0}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ r,

B_{b_{0}⊗b′_{(r−p)}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s + 1,

B_{b_{0}⊗b′_{(p−s−1)}} ≅ B(M(r + s − 2p)), p ≥ r + 1,

B_{b_{0}⊗b′_{p−s−1}} ≅ (M(r + s − 2p)), p ≥ s + 1,

B_{b_{(0)}⊗b′_{((r+s+1)/2)}} ≅ B(M(−1)), if r ≤ s.

The cases B = B(V (r)) and B′ ∈ {B(V (s)), B(M(s)), B(T (s))} were done in [6], [9]. We give it here for completeness.

Case 14. B = B(V (r)), B′ = B(V (s)).

B_{b_{0}⊗b′_{p}} ≅ B(V (r + s − 2p)), 0 ≤ p ≤ min{r, s}.

Case 15. B = B(V (r)), B′ = B(M(s)).

B_{b_{r+s−2p}⊗b′_{p+1}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,
B_{b_{0}⊗b′_{p}} ≅ B(M(r + s − 2p)), 0 ≤ p ≤ min{r, s},
B_{b_{p}⊗b′_{0}} ≅ B(M(r + s − 2p)), r + s + 2 ≤ p ≤ r,
B_{b_{0}⊗b′_{(r+s+1)/2}} ≅ B(M(−1)), if r ≥ s.

Case 16. B = B(V (r)), B′ = B(T (s)).

B_{b_{r+s−2p}⊗b′_{(p+1)}} ≅ B_{b_{r+s−2p}⊗b′_{p−s}} ≅ B(T (r + s − 2p)), 2s + 2 ≤ 2p ≤ r + s,
B_{b_{r−p}⊗b′_{0}} ≅ B(T (r + s − 2p)), 0 ≤ p ≤ min{r, s},
B_{b_{0}⊗b′_{((r+s+1)/2)}} ≅ B_{b_{0}⊗b′_{(r−s−1)/2}} ≅ B(M(−1)), if r ≥ s.

3. The category O and branched crystals.

We recall the definition and properties of the category O of modules for U_{q}(sl_{2}) in 3.1–3.4. The relevant results can be found in [1], [2], [5], and [8].

3.1. Let C(q) be the field of rational functions in an indeterminate q and let U_{q}(𝔰𝔩_{2}) be the quantized enveloping algebra of 𝔰𝔩_{2} over C(q), i.e., the algebra generated by elements e, f, k^{±1} and relations

kk^{−1} = 1 = k^{−1}k, kek^{−1} = q^{2}e, kfk^{−1} = q^{−2}f,

ef − fe = \frac{k − k^{−1}}{q − q^{−1}} .

It is well–known that U_{q}(𝔰𝔩_{2}) is a Hopf algebra. Let σ : U_{q}(𝔰𝔩_{2}) → U_{q}(𝔰𝔩_{2}) be the involutive antiautomorphism obtained by extending the assignment

σ(e) = f, σ(f) = e, σ(k) = k^{−1}, σ(q) = q^{−1}.

Let Ω ∈ U_{q}(𝔰l_{2}) be the quantum Casimir element.

3.2. Recall that a module M of U_{q}(𝔰𝔩_{2}) is said to be a weight module of type 1 if we can write

M = ⊕_{r∈Z}M_{r}, M_{r} = {m ∈ M : km = q^{r}m}.

Let O be the category of type 1 U_{q}(𝔰𝔩_{2})–modules M such that dimM_{r} < ∞ for all r ∈ Z, and such that there exists an integer n depending on M such that M_{r} = 0 for all r ≥ n. It is obvious that O is an abelian category and that it is closed under taking tensor products. Given any M ∈ O let M^{σ} be the subspace of M^{∗} = Hom_{C(q)}(M, C(q)) consisting of elements m^{∗} such that m^{∗}(M_{r}) = 0 for all but finitely many r ∈ Z. The formula

(um^{∗})(m) = m^{∗}(σ(u)m), u ∈ U_{q}(𝔰𝔩_{2}), m^{∗} ∈ M^{∗}, m ∈ M,

defines an action of U_{q}(𝔰𝔩_{2}) on M^{σ} and it is easy to check that M^{σ} ∈ O. The module M^{σ} is called the restricted dual of M and we say M is self–dual if M ≅ M^{σ}. The assignment M ↦ M^{σ} is an exact contravariant functor on O and , if M, N ∈ O, then (M ⊗ N)^{σ} ≅ N^{σ} ⊗ M^{σ}. Given M ∈ O, and c ∈ C(q), let

M[c] = {m ∈ M : (Ω − c)^{n}m = 0, ∀ n >> 0}.

Clearly M[c] is a U_{q}(𝔰𝔩_{2})–submodule of M, M = ⊕_{c∈C(q)}M[c], and M[c] ≠ 0 implies c = (q^{r+1} + q^{−r−1})(q − q^{−1})^{−2} for some r ∈ Z.

3.3. Given r ∈ Z, r ≥ −1, let O_{r} be the abelian subcategory consisting of modules M ∈ O such that

M = M[(q^{r+1} + q^{−r−1})(q − q^{−1})^{−2}].

Since σ(Ω) = Ω it follows that M ∈ O_{r} iff M^{σ} ∈ O_{r}. Further,

O = ⊕_{r≥−1}O_{r}

and any object in O_{r} is isomorphic to a direct sum of indecomposable objects in O_{r}. The last statement is easily seen by an induction on dim(M_{r}) + dim(M_{−r−2}), r ≥ −1.

3.4. We now describe the indecomposable objects in O_{r}. Given s ∈ Z, let M(s) be the Verma module with highest weight s and highest weight vector m_{s}. Namely, M(s) is the U_{q}(𝔰𝔩_{2})–module generated by an element m_{s} with relations:

em_{s} = 0, km_{s} = q^{s}m_{s}.

Let V (s) be the unique irreducible quotient of M(s). If r ≥ 0, then

M(−r − 1) ≅ V (−r − 1) and dimV (r) = r + 1.

For r ∈ Z^{+}, let T (r) be the U_{q}(𝔰𝔩_{2})–module generated by an element t_{r} satisfying the relations:

e^{r+2}t_{r} = 0, kt_{r} = q^{−r−2}t_{r}, (Ω − (q^{r+1} + q^{−r−1})(q − q^{−1})^{−2})^{2}t_{r} = 0.

Note that dim(T (r))_{−r−2p} = 2 if p > 0 and dim T (r)_{r−2p} = 1 if 0 ≤ p ≤ r.

For r ∈ Z^{+}, the modules M(−r − 1), T (r), and V (r) are self dual. Let m_{r}^{∗} ∈ M(r)^{σ} be such that m_{r}^{∗}(m_{r}) = 1 and m_{r}^{∗}(M(r)_{s}) = 0 if s ≠ r. Then

U_{q}(𝔰𝔩_{2})m_{r}^{∗} ≅ V (r).

The following proposition is proved in [8].

Proposition. Let r ∈ Z^{+}. The modules M(r), M(r)^{σ}, M(−r − 2), T (r), and V (r) are precisely the indecomposable modules in O_{r} while the only indecomposable module in O_{−1} is M(−1).

Corollary. Let r ∈ Z^{+} and V ∈ O_{r}. There exist unique integers n_{V} (T (r)), n_{V} (M(r)), etc, such that,

V ≅ T (r)^{⊕n_{V} (T (r))} ⊕M(r)^{⊕n_{V} (M(r))} ⊕M(−r −2)^{⊕n_{V} (M(−r−2))} ⊕V (r)^{⊕n_{V} (V (r))} ⊕(M(r)^{σ})^{⊕n_{V} ((M(r))^{σ})}.

3.5. It follows that the tensor product of any two objects of O can be written uniquely as a direct sum of indecomposable modules. We give these decompositions explicitly. It is enough to write down the formulas for modules V in the set

{V (r) ⊗ V (s), V (r) ⊗ M(s), V (r) ⊗ T (s), T (r) ⊗ M(s), T (r) ⊗ T (s), M(r) ⊗ M^{σ}(s), M(r) ⊗ M(s)}

since the other decompositions can be obtained by using σ. Our proof actually establishes that the tensor product in O is commutative, although this can be established as usual by using the R–matrix.

Case 1. If V = V (r)⊗V (s), r, s ≥ 0 then it is well–known that V (r)⊗V (s) ≅ ⨁_{p=0}^{min{r,s}} V (r +s−2p). Case 2. In all other cases, the element f acts freely on V and hence we can write,

(3.1)  V ≅ M_{<−1} ⊕ T ⊕ M_{≥0},

where

(3.2) M_{<−1} = ⨁_{s≥0} M(−s − 2)^{⊕n_{V} (M(−s−2))}, T = ⨁_{r≥−1} T (r)^{⊕n_{V} (T (r))}, M_{≥0} = ⨁_{r≥0} M(r)^{⊕n_{V} (M(r))},

where we set T (−1) = M(−1).

Case 2(a) Suppose that V = M(r) ⊗ M(s) and r, s ≥ 0. Then,

M_{≥0} = ⨁_{p=0}^{min{r,s}} M(r + s − 2p).

To see this, observe that (3.1) implies that

V^{σ} ≅ ⨁_{ℓ≥0}(M(ℓ)^{σ})^{⊕n_{V} (M(ℓ))} ⊕ T ⊕ M_{<−1}.

It is clear that the submodule

V_{tor}^{σ} = {v ∈ V^{σ} : f^{ℓ}v = 0, ℓ > 0}

of V is isomorphic to the submodule ⨁_{ℓ≥0} V (ℓ)^{⊕n_{V} (M(ℓ))} of ⨁_{ℓ≥0}(M(ℓ)^{σ})^{⊕n_{V} (M(ℓ))}. On the other hand, it is not hard to see that

V_{tor}^{σ} ≅ V (r) ⊗ V (s).

It follows immediately that

n_{V} (M(r + s − 2p)) = 1, 0 ≤ p ≤ min{r, s},

and n_{V} (M(ℓ)) = 0 otherwise. Since dim(M(r) ⊗ M(s))_{r+s−2p} = p + 1 for all p ≥ 0, a simple dimension counting now implies that

T = ⨁_{p=min\{r,s\}+1}^{[ \frac{r+s+1}{2} ]} T (r + s − 2p), and M_{<−1} = ⨁_{p=max\{r,s\}+1}^{∞} M(r + s − 2p).

Case 2(b) If V = V (r) ⊗ M(s) with r, s ≥ 0, then

M_{<−1}=0,
M_{≥0} = ⨁_{p=0}^{min{r,s}} M(r + s − 2p),

T = ⨁_{p=s+1}^{[ \frac{r+s+1}{2} ]} T (r + s − 2p), if r > s,
=0, if r ≤ s.

This can be proved similarly, see also [3].

Case 2(c). If V ∉ {M(r) ⊗ M(s), V (r) ⊗ M(s) : r, s ∈ Z^{+}}, then f acts freely on both V and V^{σ} and hence M_{≥0} = 0. This means that V_{p} ≅ T_{p} for all p ≥ −1, and so by comparing the dimension of the weight spaces on both sides of (3.1) we get

n_{V} (T (p)) = dim V_{p} − dim V_{p+2}, ∀ p ≥ −1,

and, for ℓ ≥ 0,

n_{V} (M(−ℓ − 2)) = dim V_{−ℓ−2} − ∑_{p=0}^{[ \frac{ℓ−1}{2} ]} n_{V} (M(−ℓ + 2p)) − 2 ∑_{p=0}^{[ \frac{ℓ}{2} ]} n_{V} (T (ℓ − 2p)) − ∑_{p>0} n_{V} (T (ℓ + 2p)).

Using these formulas, we write down the non–zero multiplicities in all cases. Below we assume p ∈ Z^{+}.

(i) V = V (r) ⊗ M(s) with s < 0.

n_{V} (T (r + s − 2p)) = 1, if 2p ≤ r + s + 1, n_{V} (M(r + s − 2p)) = 1, if r + s + 2 ≤ p ≤ r.

(ii) V = M(r) ⊗ M(s) with r < 0 or s < 0.

n_{V} (T (r + s − 2p)) = 1, if 2p ≤ r + s + 1, n_{V} (M(r + s − 2p)) = 1, if p ≥ r + s + 2.

(iii) V = V (r) ⊗ T (s).

n_{V} (T (r + s − 2p)) = 1, if p ≤ min{r, s}, n_{V} (T (r + s − 2p)) = 2, if 2s + 2 ≤ 2p ≤ r + s + 1.

(iv) V = M(r) ⊗ M(s)^{σ}.

n_{V} (T (r + s − 2p)) = 1, if 2p ≤ r + s + 1, n_{V} (M(r + s − 2p)) = 1, if p ≥ r + s + 2.

(v) V = T (r) ⊗ M(s).

n_{V} (T (r + s − 2p)) = 1, if 2p ≤ min{2r, r + s + 1}, n_{V} (T (r + s − 2p)) = 2, if 2r + 2 ≤ 2p ≤ r + s + 1,

and

(a) s ≥ 0:

n_{V} (M(r + s − 2p)) = 1, if max{r, s} + 1 ≤ p ≤ r + s + 1, n_{V} (M(r + s − 2p)) = 2, if p ≥ r + s + 2,

(b) s < 0:

n_{V} (M(r + s − 2p)) = 1, if r + s + 2 ≤ p ≤ r, n_{V} (M(r + s − 2p)) = 2, if p ≥ r + 1.

(vi) V = T (r) ⊗ T (s) ≅ T (r) ⊗ M(s) ⊕ T (r) ⊗ M(−s − 2).

3.6. Given V ∈ O, let B(V) be the branched crystal

B(V) = ⨁_{W} B(W)^{⊕n_{V} (W)},

where the sum is over the indecomposable objects W ∈ O. A straightforward comparison between sections 3.5 and 2.8 shows the following.

Theorem. For all U, V, W ∈ O we have:

(i) B(U) ⊗ B(V) ≅ B(U ⊗ V) = B(V ⊗ U) ≅ B(V) ⊗ B(U), (ii) B(U) ⊗ (B(V) ⊗ B(W)) ≅ B(U ⊗ (V ⊗ W)) = B((U ⊗ V) ⊗ W) ≅ (B(U) ⊗ B(V)) ⊗ B(W)).

In particular, the tensor product of branched crystals is commutative and associative.

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Department of Mathematics, University of California, Riverside, CA 92521

E-mail address: chari@math.ucr.edu, jakelic@math.ucr.edu, adrianoam@math.ucr.edu

Max-Planck-Institut für Mathematik, D-53111, Bonn, Germany

E-mail address: jakelic@mpim-bonn.mpg.de