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arxiv: 2605.00013 · v1 · submitted 2026-04-07 · 🧮 math.RT

The Dual Canonical Basis in the Spin Representation via the Temperley-Lieb Algebra

Rachel Chen This is my paper

Pith reviewed 2026-05-10 19:28 UTC · model grok-4.3

classification 🧮 math.RT
keywords dual canonical basisspin representationTemperley-Lieb algebraHecke algebracanonical basisspherical modulediagrammatic construction
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The pith

The Temperley-Lieb algebra supplies explicit formulas for the full dual canonical basis of the spin representation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The spin representation, the n-fold tensor product of two-dimensional space, carries a dual canonical basis introduced by Lusztig that appears across algebra, geometry, and physics. Khovanov had shown that part of this basis arises from Temperley-Lieb diagrams, but the paper supplies a simpler construction that reaches the entire basis and writes closed-form expressions for every vector. As a direct result the canonical basis of the spherical module becomes computable, and the Hecke algebra is used to recover the duality between the two bases, yielding parallel statements for the dual modules. The work closes with an axiomatic characterization of the canonical basis stated purely in diagram language.

Core claim

The full dual canonical basis of (C^2)^⊗n is realized through the standard diagrammatic action of the Temperley-Lieb algebra. This action produces explicit formulas for each basis element and, as a byproduct, for the canonical basis of the spherical module. The Hecke algebra then reproves that the two bases are dual, giving corresponding results for the dual spaces of the spherical and aspherical modules, while an alternative definition of the canonical basis is stated in purely diagrammatic terms.

What carries the argument

The diagrammatic action of the Temperley-Lieb algebra on the spin representation, which encodes basis vectors through its generators, relations, and idempotents.

Load-bearing premise

The known relations of the Temperley-Lieb algebra together with its duality to the Hecke algebra let the diagrammatic action produce the complete dual canonical basis for every n without gaps or extra obstructions.

What would settle it

For n=3 compute the vectors given by the explicit formulas and check whether they match the standard dual canonical basis elements defined by Lusztig.

Figures

Figures reproduced from arXiv: 2605.00013 by Rachel Chen.

Figure 1
Figure 1. Figure 1: Examples of some diagrams in the Temperley-Lieb algebra. One breakthrough in the study of representations of quantum groups is Lusztig’s discovery of the dual canonical basis, a basis satisfying certain upper triangularity and self-dual conditions in an arbitrary finite dimensional representation of a semisimple Lie algebra, in 1990 [20]. Lusztig’s dual 1 arXiv:2605.00013v1 [math.RT] 7 Apr 2026 [PITH_FULL… view at source ↗
Figure 2
Figure 2. Figure 2: The diagram e2 in Cℓ4(β) Lemma 2.3 ([7]). The set {e1, . . . , en−1} generates Cℓn(β); in other words, concatenation of the diagrams {e1, . . . , en−1} yield all possible n-diagrams. If we draw some diagrams, we can see that there are some relations between these generators. The following definition defines the Temperley-Lieb Algebra, the diagram algebra in terms of this basis instead of the diagrams. Defi… view at source ↗
Figure 3
Figure 3. Figure 3: An example of a (4, 1)-link state and its corresponding cup diagram. Remark. Note that the Temperley-Lieb algebra acts naturally on the vector space generated by all (n, p)-link states. This will be useful later in this paper. Lemma 3.3. There is a bijection between (n, p)-parenthesis diagrams, (n, p)-link states, and (n, p)- cup diagrams. 4. The Hecke Algebra and Kazhdan-Lusztig Basis 4.1. The Hecke Algeb… view at source ↗
Figure 4
Figure 4. Figure 4: The composition of a (3, 5)-diagram and a (5, 5)-diagram. Definition 5.2 (Temperley-Lieb Category). The Temperley-Lieb category, denoted TL, has objects {[n], n ∈ Z≥0}, where [n] is a set of n vertices. The morphisms Hom([m], [n]) is the vector space with a basis of diagrams consisting of (m, n)-diagrams. Define ϵ n i ∈ Hom([n], [n − 2]) to be the diagram connecting the ith and (i + 1)th vertices on the bo… view at source ↗
Figure 5
Figure 5. Figure 5: Diagrams for ϵ 4 2 and δ 4 2 , respectively, in Hom([4], [2]). Every diagram in the Temperley-Lieb category can be decomposed into certain generators, as described by the following theorem. Theorem 5.3 ([6]). The algebra L m,n∈Z≥0 Hom([m], [n]) is generated as an algebra by all ϵ n i and δ n i , where 1 ≤ i ≤ n − 1, with the following relations: (i) ϵ n i · δ n i = −q − q −1 , and (ii) (id ⊗ ϵ n i ) · (δ n… view at source ↗
Figure 6
Figure 6. Figure 6: The second relation in Theorem 5.3. obtaining a contractible loop. The second relation (see [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: , we write it in terms of the generators of TL by decomposing it as shown in the second and third diagrams. Explicitly, the first diagram decomposes into δ 4 1 δ 2 3 ϵ 2 1 ϵ 4 2 . • • • • • • • • • • • • • • • • • • • • • • • • • • • • [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Construction of the link state of v+ ♡ v+ ♡ v− ♡ v+ ♡ v−. We claim that this procedure gives a valid cup diagram of the top line. First of all, we claim that there are no unlinked vkm such that vki is linked to vkj and j < m < i. Since j < i, we have ki = + and kj = −. If km = +, vkm would have been linked with vkj ; if km = −, then vki would have been linked with vkm. Furthermore, there are no intersectin… view at source ↗
Figure 9
Figure 9. Figure 9: Construction of the next step for the diagram corresponding to v+ ♡ v+ ♡ v− ♡ v+ ♡ v−. After this step, let the number of quasi-simple links on the top line be p. On the bottom line, we draw the quasi-simple link connecting the (k − i)th vertex to the (k + 1 + i)th for all 0 ≤ i ≤ p − 1, and so on, drawing p quasi-simple links total. An example of this is shown in [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Computing the label for an element of the dual canonical basis given a diagram in IndTL5 TL2 ⊗ TL3 Ctriv. Proof. Given a label of the dual canonical basis vkn ♡ · · · ♡ vk1 , let D be the diagram con￾structed as above. By construction, the coefficient of vkn ⊗ · · · ⊗ vk1 of the action of D on v− ⊗ · · · ⊗ v− | {z } k ⊗ v+ ⊗ · · · ⊗ v+ | {z } n−k is equal to 1. Since diagrammatic basis elements map to dua… view at source ↗
Figure 11
Figure 11. Figure 11: Some diagrams in the basis of diagrams in IndTL4 TL2 ⊗ TL2 Ctriv. Explicitly computing the actions gives the following: A(v− ⊗ v− ⊗ v+ ⊗ v+) = v− ♡ v− ♡ v+ ♡ v+ B(v− ⊗ v− ⊗ v+ ⊗ v+) = v+ ♡ v− ♡ v− ♡ v+ C(v− ⊗ v− ⊗ v+ ⊗ v+) = v+ ♡ v+ ♡ v− ♡ v−. Note that we do not have enough information to confirm whether the action of diagrams explicitly corresponds with the element of the dual canonical basis for this e… view at source ↗
Figure 12
Figure 12. Figure 12: Example of the nested quasi-simple links described above. We have vkj = vkj+1 = · · · = vkj+c = v− and vki = vki−1 = · · · = vki−c = v+ from construction. In the action of the Temperley-Lieb algebra, δ will act on bj+1 and bi−1, bj+2 and bi−2, and so on, until bj+c and bi−c. In the decomposition of Theorem 8.1, in condition (iii), δ first appears for components vkj+c and vki−c , then vkj+c−1 and vki−c+1 ,… view at source ↗
Figure 13
Figure 13. Figure 13: An example of the embedding of TL3 into IndTL6 TL3 ⊗ TL3 Ctriv. Example 9.10. In the embedding described above, the procedure described in Theorem 9.9 is shown in [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

The spin representation $(\mathbb C^2)^{\otimes n}$ has a dual canonical basis introduced by Lusztig that is important in many areas of algebra, geometry, and physics. Khovanov observed that a portion of the dual canonical basis can be viewed diagrammatically through the Temperley-Lieb algebra. We provide a simpler construction that we generalize to the entire dual canonical basis, and write explicit formulas to compute the dual canonical basis, and thus the canonical basis of the spherical module, as a byproduct. We reprove some of Khovanov's results using our new perspective. Furthermore, we use the Hecke algebra to reprove the fact that the canonical basis is indeed dual to the dual canonical basis, leading to similar results about the canonical basis in $\mathcal M^*$ and $\mathcal N^*$, the dual spaces to the spherical and aspherical modules, as a byproduct. Finally, we present an alternative axiomatic definition of the canonical basis using diagrams.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs the dual canonical basis of the spin representation (ℂ²)⊗n using a diagrammatic approach based on the Temperley-Lieb algebra. It generalizes Khovanov's partial construction to the full basis, provides explicit formulas for the basis elements (and thus the canonical basis of the spherical module as a byproduct), reproves some of Khovanov's results, uses the Hecke algebra to reprove duality between the canonical and dual canonical bases, and presents an alternative axiomatic definition of the canonical basis via diagrams. This yields analogous results for the dual spaces to the spherical and aspherical modules.

Significance. If the constructions and explicit formulas are correct, the work is significant for representation theory of quantum groups. It supplies a simpler, more explicit and computable diagrammatic route to the dual canonical basis, an object central to categorification, knot invariants, and applications in geometry and physics. The Temperley-Lieb perspective, Hecke-algebra reproof of duality, and alternative axiomatic definition could facilitate calculations and reveal new structural connections, building directly on standard relations and known duality results in the literature.

major comments (1)
  1. The generalization of the diagrammatic construction to the entire dual canonical basis (central to the abstract claim) relies on the standard Temperley-Lieb action extending without obstruction for arbitrary n. The manuscript should explicitly verify in the relevant construction section that the bar-involution invariance and triangularity properties hold for the newly constructed elements beyond Khovanov's range, as this is load-bearing for the explicit formulas and full-basis claim.
minor comments (1)
  1. The abstract and introduction would benefit from a brief comparison table or statement clarifying precisely which of Khovanov's results are reproved and which are new.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript's significance and for the constructive major comment. We address it point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The generalization of the diagrammatic construction to the entire dual canonical basis (central to the abstract claim) relies on the standard Temperley-Lieb action extending without obstruction for arbitrary n. The manuscript should explicitly verify in the relevant construction section that the bar-involution invariance and triangularity properties hold for the newly constructed elements beyond Khovanov's range, as this is load-bearing for the explicit formulas and full-basis claim.

    Authors: We agree that an explicit verification of these properties for arbitrary n strengthens the central claim. In the revised manuscript we will add a dedicated paragraph (immediately after the recursive definition of the diagrammatic elements in the construction section) that proves, by induction on n, that the newly defined basis vectors remain invariant under the bar involution and satisfy the required triangularity with respect to the standard basis. The inductive step relies only on the fact that the Temperley-Lieb generators act via the same diagrammatic rules for all n and that the relations are independent of n; the base cases recover Khovanov's range. This verification directly supports the explicit formulas and the full-basis statement without altering any existing arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs the dual canonical basis of the spin representation by generalizing Khovanov's diagrammatic Temperley-Lieb action to the full basis, providing explicit formulas and reproving duality via the Hecke algebra. All load-bearing steps rely on standard, externally established relations and duality results from the literature (Lusztig, Khovanov, and Hecke algebra theory), without any self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard algebraic properties of the Temperley-Lieb algebra and its action on the spin representation, which are drawn from prior literature rather than introduced here.

axioms (2)
  • domain assumption The Temperley-Lieb algebra acts on the spin representation via the standard diagrammatic generators satisfying the usual quadratic and braid relations.
    Invoked implicitly when extending Khovanov's partial construction to the full basis.
  • standard math The Hecke algebra provides a duality pairing between the canonical and dual canonical bases.
    Used to reprove the duality fact.

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We provide a simpler construction that we generalize to the entire dual canonical basis, and write explicit formulas to compute the dual canonical basis... via the Temperley-Lieb algebra... isomorphism L 0≤k≤n Ind TLn TLk ⊗ TL n−k Ctriv ≅ (C²)⊗n identifying diagrammatic basis with dual canonical basis

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matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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The paper appears to rely on the theorem as machinery.
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The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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