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Highest weight components in a tensor product crystal

Statement

Assume the axiom of choice (The Axiom of Choice), inherited through Uniqueness: crystal bases of integrable modules decompose into highest weight crystals. Let $M_{1}, M_{2}$ be integrable chosen $P$ -graded type-one quantum weight modules with finite-dimensional label-weight spaces and weight sets contained in finite unions of cones, let $(L_{1}, B_{1}), (L_{2}, B_{2})$ be crystal bases of $M_{1}, M_{2}$ , and let $B_{1} \otimes B_{2}$ be the crystal of $M_{1} \otimes M_{2}$ (The crystal basis of a tensor product and the two-case operator rule, Tensor products of integrable weight modules and of their crystal lattices).

(a) Highest vertices. For $b = b_{1} \otimes b_{2} \in B_{1} \otimes B_{2}$ the following are equivalent: (i) $b$ is a highest weight element, i.e. ${\tilde{e}}_{i} b = 0$ for every $i \in I$ ; (ii) $ε_{i} (b_{1}) = 0$ and $ε_{i} (b_{2}) \leq ⟨ h_{i}, w t (b_{1}) ⟩$ for every $i \in I$ ; (iii) $ε_{i} (b_{1} \otimes b_{2}) = 0$ for every $i \in I$ . In this case $w t (b) = w t (b_{1}) + w t (b_{2})$ is dominant.

(b) Generated components. For every highest vertex $b \in B_{1} \otimes B_{2}$ of weight $λ$ , the lowering-generated subcrystal $F (b)$ is a connected component of $B_{1} \otimes B_{2}$ , is isomorphic as a crystal to the canonical $B (λ) = B (λ)$ of The canonical crystal $B (λ)$ is a highest weight crystal of highest weight $λ$ , and $λ \in P^{+}$ .

(c) Decomposition. $B_{1} \otimes B_{2}$ is the disjoint union of the subcrystals $F (b)$ over the highest weight elements $b \in B_{1} \otimes B_{2}$ , each of which is a highest weight crystal in the sense of Highest weight elements, lowering-generated sets and highest weight crystals; the multiplicity of $B (λ)$ as a component equals the multiplicity of $V (λ)$ in the tensor product module $M_{1} \otimes M_{2}$ .

No choice principle beyond the inherited one is used.

Facts & Assumptions

Given: Integrable chosen $P$ -graded type-one weight modules $M_{1}, M_{2}$ with finite-dimensional label-weight spaces and cone-bounded weight sets (Quantum weight modules, weights in $P$ , and integrability), crystal bases $(L_{1}, B_{1}), (L_{2}, B_{2})$ and the tensor product crystal $B_{1} \otimes B_{2}$ of The crystal basis of a tensor product and the two-case operator rule.

[F1]

The tensor $ε / φ$ formulas (The weight, $ε_{i}$ and $φ_{i}$ formulas for the tensor product crystal): $ε_{i} (b_{1} \otimes b_{2}) = \max (ε_{i} (b_{1}), ε_{i} (b_{2}) - ⟨ h_{i}, w t (b_{1}) ⟩)$ , with $ε_{i}, φ_{i}$ finite and nonnegative on every vertex.

[F2]

Crystal graph conventions (The crystal graph, $ε_{i}$ , $φ_{i}$ and the weight function): ${\tilde{e}}_{i} b = 0$ if and only if $ε_{i} (b) = 0$ , and $φ_{i} (b) = ε_{i} (b) + ⟨ h_{i}, w t (b) ⟩$ for all $b$ .

[F3]

The tensor rule theorem (The crystal basis of a tensor product and the two-case operator rule) and the integrability lemma (Tensor products of integrable weight modules and of their crystal lattices): $(L_{1} \otimes L_{2}, B_{1} \otimes B_{2})$ is a crystal basis of $M_{1} \otimes M_{2}$ , and $M_{1} \otimes M_{2}$ is integrable with finite-dimensional label-weight spaces and cone-bounded weight set.

[F4]

Uniqueness and decomposition (Uniqueness: crystal bases of integrable modules decompose into highest weight crystals, under the inherited AC): any crystal basis of an integrable module $M$ with finite-dimensional label-weight spaces and cone-bounded weight set decomposes, through an isomorphism $Φ : M \to ⨁_{j} V (λ_{j})$ , into the canonical components $B (λ_{j})$ ; each component has a unique highest weight element, of dominant weight $λ_{j}$ , equal to the lowering-generated subcrystal of that element; and $# {j : λ_{j} = λ} = [M : V (λ)]$ .

[F5]

The canonical component (The canonical crystal $B (λ)$ is a highest weight crystal of highest weight $λ$ ): $B (λ)$ is a highest weight crystal of highest weight $λ$ ; every of its vertices is an iterated lowering image of its highest element, and it has no other highest weight element.

[F6]

The definition of highest weight elements, $F (b)$ and highest weight crystals (Highest weight elements, lowering-generated sets and highest weight crystals).

[F7]

The axiom of choice (The Axiom of Choice), used only through [F4].

Proof

technique · direct

1.1F1F2given

The three highest-vertex conditions are equivalent. By [F1], ${\tilde{e}}_{i} (b_{1} \otimes b_{2}) \neq 0$ if and only if $ε_{i} (b_{1} \otimes b_{2}) = \max (ε_{i} (b_{1}), ε_{i} (b_{2}) - ⟨ h_{i}, w t (b_{1}) ⟩) > 0$ , which happens exactly when $ε_{i} (b_{1}) > 0$ or $ε_{i} (b_{2}) > ⟨ h_{i}, w t (b_{1}) ⟩$ ; hence (i), " ${\tilde{e}}_{i} b = 0$ for all $i$ ", is equivalent to (ii), and by [F2] also to (iii). If these hold, then for every $i$ one has $⟨ h_{i}, w t (b) ⟩ = φ_{i} (b) - ε_{i} (b) = φ_{i} (b) \geq 0$ because $ε_{i} (b) = 0$ and $φ_{i} (b)$ is a finite nonnegative integer by [F1]; hence $w t (b) = w t (b_{1}) + w t (b_{2})$ is dominant. This proves (a).

2.1step 1.1F3F4F5

Generated components are canonical. Let $b \in B_{1} \otimes B_{2}$ be a highest vertex of weight $λ$ . Apply [F4] to $M = M_{1} \otimes M_{2}$ with crystal basis $(L_{1} \otimes L_{2}, B_{1} \otimes B_{2})$ , whose hypotheses hold by [F3]: $B_{1} \otimes B_{2}$ decomposes into components $B (λ_{j})$ , and each component has a unique highest weight element $b_{λ_{j}}$ and equals $F (b_{λ_{j}})$ . Since $b$ is a highest vertex, it is the highest element of its component, so that component is $B (λ)$ with $λ = w t (b)$ ; consequently $F (b)$ is a connected component of $B_{1} \otimes B_{2}$ isomorphic to $B (λ)$ , and $λ \in P^{+}$ by (a) and [F4]. This proves (b).

3.1step 2.1F4F5F6F7∎

The decomposition and multiplicities. By [F4] the components of $B_{1} \otimes B_{2}$ are exactly the $B (λ_{j}) = Φ^{- 1} (B (λ_{j}))$ , indexed by the simple constituents $V (λ_{j})$ of $M_{1} \otimes M_{2}$ ; by [F5] each has a unique highest weight element, namely its standard highest element, and no other highest weight elements exist in the crystal. Hence the highest weight elements of $B_{1} \otimes B_{2}$ are exactly those elements, and the disjoint union of the subcrystals $F (b)$ over them is the whole crystal, each $F (b)$ being a highest weight crystal in the sense of [F6]. The multiplicity of $B (λ)$ among the components equals $# {j : λ_{j} = λ} = [M_{1} \otimes M_{2} : V (λ)]$ by the multiplicity clause of [F4]. The axiom of choice is used only through [F4], where it is already declared; no new nonconstructive selection occurs.

Depends on

Used by

Highest components in the A₂ tensor crystal B(λ₁)⊗ B(λ₁) Example

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Sami Assaf, Anne Dranowski and Nicolle Gonzalez, Extremal tensor products of Demazure crystals, arXiv:2210.10236 (standard reference, not scraped)
Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516 (standard reference, not scraped)