Pith. sign in

REVIEW 3 major objections 5 minor 34 references

The paper proves the Grothendieck ring of shifted-Yangian category O is isomorphic to the Cox ring of a new pro-variety, the open bi-infinite Bott-Samelson variety, and hence to a cluster algebra.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 08:37 UTC pith:25E3XXHS

load-bearing objection Strong, serious paper that settles several open conjectures; the main theorem is stated unconditionally but depends on an explicit unproved associativity conjecture, and the paper must delimit where that assumption enters before it is ready. the 3 major comments →

arxiv 2607.04480 v2 pith:25E3XXHS submitted 2026-07-05 math.RT math.AGmath.QA

Category mathcal{O} for truncated shifted Yangians and the bi-infinite Bott-Samelson variety

Joel Kamnitzer , Antoine Labelle , Alexis Leroux-Lapierre , Th\'eo Pinet , Alex Weekes This is my paper
classification math.RT math.AGmath.QA MSC 17B3714M1513F60
keywords shifted Yangianscategory OGrothendieck ringCox ringBott-Samelson varietiescluster algebrasextended QQ-systemmonomial crystals
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to describe the Grothendieck ring of the category O for shifted Yangians — the ring that records how representations of these algebras, which deform the universal enveloping algebra of the current Lie algebra, decompose under tensor product. Its central claim is that the complexified ring of the integral finite-length part of this category is isomorphic to the Cox ring of a new pro-variety, the open bi-infinite Bott-Samelson variety, built from bi-infinite arrays of points in partial flag varieties with incidence conditions; the isomorphism is equivariant for the Langlands dual group. If correct, this single statement identifies three seemingly different objects: a representation ring, a geometric coordinate ring, and a cluster algebra generated by the formal solutions of the extended QQ-system. Along the way the paper proves the conjectured description of the ring as that cluster algebra, shows that the ℓ-characters of the distinguished chamber modules satisfy the QQ-system relations, and identifies the spectrum of the ring with a Coxeter-independent version of the scheme of bands. A reader should care because the result converts a complicated representation-theoretic structure into an explicitly presented ring — a quotient of a polynomial ring by four families of relations — and settles two standing conjectures.

Core claim

On the paper's own terms, the discovery is its Theorem 11: the morphism Ω from the Cox ring R of the open bi-infinite Bott-Samelson pro-variety Z^o_∞ to the complexified Grothendieck ring K_C(O^Z_sh) of the integral category O for shifted Yangians is an isomorphism of C-algebras intertwining the natural actions of the Langlands dual group G^∨, and it restricts to isomorphisms between spaces of sections of line bundles on the compactification Z_∞ and the subspaces K_C(O^λ_sh(R)) attached to product monomial crystals. The map is assembled by matching the fundamental subspaces K_C(O^{ϖ_i}_sh(a)) with line-bundle sections, then proving that each of the four families of explicit relations present

What carries the argument

The load-bearing object is the bi-infinite Bott-Samelson pro-variety Z_∞ together with its open cell Z^o_∞. Z_∞ parametrizes collections (x_{i,a}) indexed by (i,a) ∈ I×2Z with each x_{i,a} a point of the partial flag variety G^∨/P_i and with an incidence condition linking x_{i,a} and x_{j,a+1} whenever the Dynkin nodes i and j are adjacent; the Cox ring R — the ring formed by all spaces of sections of line bundles on Z^o_∞ — is explicitly a quotient of a polynomial ring by four families of relations. The central mechanism is the morphism Ω: R → K_C(O^Z_sh), built in three steps: match the fundamental line-bundle sections with the fundamental Grothendieck-ring subspaces K_C(O^{ϖ_i}_sh(a)) via

Load-bearing premise

The monoidal conclusions rest on the unproved Conjecture 5.14 — that V1⊗(V2⊗V3) and (V1⊗V2)⊗V3 are isomorphic for all objects of O_sh, even though the shifted coproducts are known not to be co-associative — while the ring-level isomorphism leans on a quoted equivalence between the rational and trigonometric shifted frameworks whose proof is imported rather than given here.

What would settle it

Produce three simple objects in O_sh whose double tensor product depends on parenthesization (a concrete failure of Conjecture 5.14), which would collapse the monoidal categorification claims; or, for a small Lie algebra such as sl_3, compute the classes in an explicit truncated category and check that the four families of relations presenting the Cox ring — in particular the extended QQ-system identity [L_{wϖ_i,a}][L_{wsiϖ_i,a+2}] − [L_{wsiϖ_i,a}][L_{wϖ_i,a+2}] = ∏_{j∼i}[L_{wϖ_j,a+1}] among chamber-module classes — hold in K0(O^Z_sh); the first failure would refute the isomorphism of Theorem

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The ring K_C(O^Z_sh) is now given by an explicit presentation — a quotient of a polynomial ring by four families of relations — so classes of representations in the category can in principle be computed and compared by generators and relations.
  • Two standing conjectures follow: the Grothendieck ring is isomorphic to the cluster algebra generated by the Q-variables of the extended QQ-system, and the ℓ-character of every chamber module L_{wϖ_i,a} equals the corresponding Q-variable.
  • Spec K_C(O^Z_sh) is a Coxeter-independent version of the scheme of bands: every choice of Coxeter element c yields an isomorphism of this spectrum with the scheme B(G^∨,c) of bi-infinite sequences (g_s) in G^∨ with g_s g_{s+1}^{-1} in the double Bruhat cell, so the scheme of bands no longer depends on an orientation.
  • The Grothendieck ring carries a G^∨-action with G^∨-equivariant multiplication, and each subspace K_C(O^λ_sh(R)) is a G^∨-module isomorphic, by a Borel-Weil-type statement, to the space of sections of the corresponding line bundle on the compact pro-variety Z_∞.
  • Shifted coproducts descend to truncated shifted Yangians, giving multiplication maps V(λ1,R1)⊗V(λ2,R2)→V(λ1+λ2,R1∪R2); these quantize the multiplication maps of generalized affine Grassmannian slices and make the truncated categories a directed system under inclusion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the missing associativity (Conjecture 5.14) is proved, the ingredients assembled here — generic R-matrices, real and prime chamber modules, and the Corollary 5.15 criterion — look designed to complete the proof that O^Z_sh is a monoidal categorification of the cluster algebra; the paper stops short of that step.
  • Because the spectrum is presented independently of any Coxeter element, the collection of height-function projections Spec R → N_−\G should glue into a universal object over the space of all orientations; a testable consequence is that the scheme of bands itself can be recovered as the total space of that atlas.
  • The natural grading on parity KLRW algebras suggests a t-deformation of the Grothendieck ring; if it is compatible with the Poisson structure on Bott-Samelson varieties, it would quantize the scheme of bands and match existing quantum deformations of the cluster algebra — an identity that can be checked on generating series.
  • The authors note that all constructions work over any field of characteristic zero; a concrete next experiment is to test the truncation-category comparisons in positive characteristic, where the KLRW equivalence and the characteristic-cycle arguments may fail independently.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a representation-theoretic and geometric framework for the Grothendieck ring of the integral category O of shifted Yangians associated to a simply-laced Lie algebra. It constructs a new pro-variety, the open bi-infinite Bott–Samelson variety Z^o_infinity, and its Cox ring R, and proves the central theorem that the map Ω : R → K_C(O^Z_sh) is a G^vee-equivariant isomorphism of C-algebras. From this isomorphism and prior work of Geiss–Hernandez–Leclerc and Francone–Leclerc, the paper derives Conjecture A (the Grothendieck ring is the GHL cluster algebra), Conjecture B (the ℓ-character of each chamber module is the corresponding Q-variable), the extended QQ-system, a G^vee-action on the Grothendieck ring, truncated shifted coproducts, and a generalization of Hernandez–Leclerc duality. The paper is very substantial and contains many auxiliary results on monomial crystals, KLRW algebras, GT-characters, and Bott–Samelson varieties.

Significance. If the central isomorphism is unconditional as stated, the paper resolves several conjectures in the representation theory of shifted Yangians and connects them to a Coxeter-independent geometric object. The construction of Z^o_infinity and the Cox-ring presentation of Spec K_C(O^Z_sh) is a significant contribution in its own right, as is the proof that chamber modules solve the extended QQ-system. The paper is careful to cite prior results and distinguishes proved statements from conjectural inputs. However, the scope of the results is currently somewhat unclear because a central structural assumption — Conjecture 5.14 on associativity of the tensor product in O_sh — is explicitly assumed in the Interlude but the main theorems are stated unconditionally.

major comments (3)
  1. [Interlude; Theorems 11.2/11.8; §5.2–5.4] The paper assumes Conjecture 5.14 in the Interlude ('we work under the assumption that Conjecture 5.14 holds'), yet Theorem 11 is stated as an unconditional isomorphism Ω : R → K_C(O^Z_sh). The proof of Ω uses the multiplicative structure on K_C(O^Z_sh): Theorem 2 constructs the multiplication maps (11) using generic simplicity of tensor products (Theorem 5.23), whose proof in §5.2 uses associativity isomorphisms (54)–(55), and Theorem 8.13 uses tensor products of chamber modules and Corollary 8.3. The text does not delimit which of these steps requires the full Conjecture 5.14, as opposed to only the special associators from [Zha24] or the K0-ring associativity obtained via the injective ℓ-character map (Theorem 4.13). If any step in the dependency chain of Theorems 11.2/11.8 uses the assumed monoidal structure, then the central isomorphism is conditional on an unproved conjecture. This
  2. [§1.1, footnoted 'interchangeable' framework] The paper freely passes between the rational (shifted Yangian) and trigonometric (quantum affine) frameworks, citing [VV25], [DK25], [GT16], [HZ25] and [Kam+19b]. This bridge is load-bearing because Conjectures A and B, the cluster algebra A of [GHL24], and Francone–Leclerc’s scheme of bands [FL25] are formulated in the trigonometric setting, while Theorem 11 is proved in the shifted Yangian setting. The compatibility is asserted informally rather than stated as a precise theorem. Please state exactly which categories, rings, ℓ-characters, simple-class correspondences, truncations, and G^vee-actions are identified by this chain, and verify that the isomorphism (4) and the identification Spec R ≅ B(G^vee,c) concern the same object after transfer. Without this, the deductions of Conjectures A and B from Theorem 11 are not fully documented.
  3. [Theorem 2 and Remark 5.36] Theorem 2 is stated under the non-vanishing hypotheses that V(λ1)_μ1 and V(λ2)_μ2 are non-zero, and Remark 5.36 notes the hypotheses can be dropped only in type A. Since Theorem 2 is used to define the multiplication map (11) and hence the ring structure on K_C(O^Z_sh), the paper should clarify whether the hypotheses are automatically satisfied in all cases needed for Proposition 7.8 and Theorem 7.9. If not, the proof of the multiplication map, and therefore of Theorem 11, is incomplete for arbitrary parameters R1, R2. This is a local but important point that should be addressed explicitly.
minor comments (5)
  1. [Remark 8.8] There is an unresolved reference 'Corollary ??' in Remark 8.8; this should be fixed before publication.
  2. [Notation in §1.13 and §2] The notation for C^λ and the variables R_i,s is occasionally ambiguous; for instance, 'extracts the u^s-coefficient' should be stated with the sign convention used in (35).
  3. [Example 5.43] The displayed matrices in Example 5.43 are difficult to read and appear to have a typographical issue in the h(u) matrix. Please reformat.
  4. [Corollary 5.20 and Interlude] The statement in the Interlude that O_sh^(0) is monoidal 'by Corollary 5.20' is misleading: Corollary 5.20 is a statement about Grothendieck groups, not about associativity of the categorical tensor product. This should be rephrased to avoid confusion with the assumed Conjecture 5.14.
  5. [Theorem 8.16 proof] The proof of Theorem 8.16 says the identity τ(Σ_{wϖ_i,a}) = Σ_{wϖ_i,a+2} follows directly from the definition of the Σ's in [FH24]. This is plausible but nontrivial; a short derivation or precise reference would help.

Circularity Check

0 steps flagged

No significant circularity: the main isomorphism is assembled from independent prior theorems and geometric computations; the explicit assumption of Conjecture 5.14 is a genuine limitation but not a circularity.

full rationale

The claimed derivation chain K_C(O^Z_sh) ≅ R ≅ coord(bands) ≅ A is not circular in construction. The map Ω is built by identifying the fundamental G^∨-modules K_C(O^{ϖ_i}_sh(a)) with V(ϖ_i) and then extending to the geometric Cox ring R, whose presentation is obtained independently from the geometry of the bi-infinite Bott–Samelson variety (Corollary 10.20). The four families of defining relations of R are verified inside K_0(O^Z_sh) using height-function subcategories (Theorem 8.6), the extended QQ-system for chamber modules (Theorem 8.13), and G∨-equivariance; injectivity is obtained either through the Geiss–Hernandez–Leclerc isomorphism (4) or through the Borel–Weil-type Theorem 4. None of these steps defines R in terms of K_0 or fits a parameter to the predicted quantity. The self-citations, notably [Kam+19b] for KLRW-truncation equivalences and crystal characterizations, [Fin+18] for shifted coproducts, and [Kam+19a] for product monomial crystals, are load-bearing but are prior published results whose stated assumptions do not include the target theorems; they therefore constitute independent evidence rather than a circular self-support chain. The main genuine weakness is the Interlude's explicit working assumption of Conjecture 5.14 (co-associativity of the shifted tensor product), which makes the monoidal structure on O^Z_sh conditional. Some intermediate arguments, such as Theorem 5.6 and generic simplicity, rely on special associator isomorphisms from [Zha24] rather than the full conjecture, and the paper does not fully delimit which uses of the tensor product require the full conjecture. This is an unresolved hypothesis and a potential correctness gap, but it is not a circularity: no theorem is being deduced from an equivalent form of itself, and the ring-level statement Theorem 11 can in principle stand independently of the categorical associativity conjecture.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 3 invented entities

The central isomorphism rests on a large body of external theorems (HZ24, Kam+19b, BFN19/Fin+18, Web17, VV11/Web15, Kas03, Gib21, GHL24, FL25, FH25, VV25) whose assumptions do not contain the target conjectures; plus one internally assumed conjecture (5.14) used for categorical monoidality, and the cited rational<->trigonometric bridge. There are no numerical free parameters or fitted constants. The new geometric objects are constructed explicitly (not postulated), each with independent checkable characterizations such as explicit relations and character formulas.

axioms (7)
  • domain assumption Conjecture 5.14: associativity of tensor products, V1⊗(V2⊗V3) ≅ (V1⊗V2)⊗V3, assumed from the Interlude onward.
    Stated in Section 5.2; the Interlude says 'we work under the assumption that Conjecture 5.14 holds.' Needed for O^Z_sh to be a monoidal category in the categorical sense, for Corollary 5.15, Theorem 13, and the monoidal-categorification framing. Ring-level results appear to avoid it via injective l-characters (Theorem 4.13).
  • domain assumption Equivalence between rational (shifted Yangian) and trigonometric (shifted quantum affine) frameworks, with compatibility of l-characters, simplicity, and Grothendieck rings.
    Invoked in the Section 1.1 Remark: 'we freely pass between the trigonometric and rational frameworks,' citing [VV25, Cor 1.2.1], [HZ25, Thm 5.4], [Kam+19b, Cor 5.22]. This bridge imports the cluster-algebra isomorphism (4) and the Francone-Leclerc band-scheme result (5) into the Yangian setting.
  • standard math Nakajima's monomial crystal B is a normal g^vee-crystal with the stated epsilon/phi data [Kas03, Prop 3.1, Thm 4.3].
    Foundational for Section 2 (product monomial crystals, Theorem 2.12, Theorem 2.15) and for Theorem 4.19 identifying l-weights of fundamental modules with crystal elements.
  • domain assumption KLR/KLRW categorification theorems: cyclotomic KLR algebras categorify V(lambda), KLRW algebras categorify tensor products, and dual canonical bases match simple classes [Web17; VV11; Web15; KK12].
    Used throughout Sections 6-8 and for Theorem 9 (8.6), Corollary 10, and the dual-canonical-basis results. These are prior published theorems with stated assumptions not containing the paper's conjectures.
  • domain assumption Hernandez-Zhang l-character machinery: injective ring morphism K0(O_sh) -> E^ell, classification of simples by highest l-weights, and the combinatorial simplicity criterion [HZ24, Thm 3.12, Thm 3.14, Cor 5.10], extended to E8 via [Neg25, Thm 1.9].
    The backbone of Section 4-5 and Theorem 5.23. For E8 the paper asserts a generalization based on [Neg25, Thm 1.9] rather than proving it; this is flagged in the text ('generalizable to this special type because of...').
  • domain assumption Faithfulness and GK-dimension results for truncated categories from [Kam+24, Prop 9.21, Lemma 4.11, Lemma 4.12] and the quantized Coulomb branch description [BFN19; Wee19].
    Used in Theorem 5.33 to prove faithfulness of generic simples (all parameters in distinct integrality classes), which is the key step in the proof of Theorem 2 (truncated coproducts).
  • domain assumption Gibson's Demazure character formula [Gib21] and Bott-Samelson theory, used to compute sections of line bundles on Z_infinity.
    Ingredient for Theorem 4 (Theorem 10.26), H^0(Z_infinity, L_{lambda,R}) ≅ V(lambda,R), and for the (2)=>(1) direction of Theorem 2.15.
invented entities (3)
  • Z_infinity, the bi-infinite Bott-Samelson pro-variety independent evidence
    purpose: Geometric model whose points are arrays (x_{i,a}) in I×2Z-graded flag varieties with incidence conditions; sections of its line bundles L_{lambda,R} reproduce the G^vee-modules V(lambda,R) = K_C(O^lambda_sh(R)).
    Fully pinned down by incidence conditions and projections to flag varieties for each height function; the character computation via Gibson's formula and the isomorphism to V(lambda,R) provide checkable handles independent of the main theorem.
  • Z^o_infinity and its Cox ring R independent evidence
    purpose: The open cell (x_{i,a} ≠ x_{i,a+2}) whose Cox ring is the target of the central isomorphism Omega : R -> K_C(O^Z_sh).
    R has an explicit presentation as a quotient of a polynomial ring by four families of relations (Corollary 10.20, Theorem 5), and Spec R is shown to be the universal principal torus bundle over Z^o_infinity and isomorphic to the band scheme — all checkable constructions.
  • The torus A with character lattice P = B/Gamma, and its action on K_C(O^Z_sh) independent evidence
    purpose: Torus whose weight decomposition matches the block decomposition of O^Z_sh indexed by P = B/Gamma; used in the gluing of the G^vee-action and the Cox-ring grading.
    Points are explicit arrays (t_{i,a}) satisfying t_{i,a}t_{i,a+2} = prod_{j~i} t_{j,a+1}; the weight spaces are the blocks of Theorem 4.22, an independent structural result.

pith-pipeline@v1.3.0-alltime-deepseek · 76074 in / 23537 out tokens · 237874 ms · 2026-08-02T08:37:27.625857+00:00 · methodology

0 comments
read the original abstract

In this paper, we study the category $\mathcal{O}$ of representations of shifted Yangians associated to a simply-laced simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group $G^{\vee}$ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended $QQ$-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.

Figures

Figures reproduced from arXiv: 2607.04480 by Alexis Leroux-Lapierre, Alex Weekes, Antoine Labelle, Joel Kamnitzer, Th\'eo Pinet.

Figure 1
Figure 1. Figure 1: Part of a projectivized point of Z∞ for G∨ = SL4 (left) and the bi-infinite diagram parametrizing points of Z∞ for G∨ = SL6 (right). For every (i, a) ∈ I ×2Z, the pro-variety Z∞ carries a G∨-equivariant line bundle Oi,a(1) obtained by pulling back the antitautological line bundle O(1) on Gr(i). Thus, given any set of parameters R of size λ, we can define a G∨-equivariant line bundle Lλ,R by tensoring the l… view at source ↗
Figure 2
Figure 2. Figure 2: The heap H(2, 1, 3, 2, 1, 2, 3) = H(2, 3, 1, 2, 1, 2, 3). This heap is not alternating, as there is no bead on the 3rd rod between the highest beads of the 2nd rod (but adding a bead there would yield an alternating heap) [PITH_FULL_IMAGE:figures/full_fig_p086_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graph ΓH for H = H(2, 1, 3, 2, 1, 2, 3). Remark 9.10. For alternating heaps, ΓH is almost the same as the Hasse diagram of H. Specifically, the map h 7→ h+ gives an isomorphism between the Hasse diagram of H and the induced subgraph of ΓH obtained by deleting the minimal element p min i = (∅, Hi) of I(Hi) for every i. Moreover, p min i is connected to h+ if and only if i ∼ c(h) and there are no element… view at source ↗
Figure 4
Figure 4. Figure 4: The incidence diagram for Z∞ in type A4. Lemma 10.6. For every point (xi,a)i,a∈I×2Z of Z ◦ ∞ and every (i, b) ∈ I × op 2 Z, the wedge xi,b−1 ∧ xi,b+1 lies in the highest isotypic component V (2ϖi − αi) ∗ ⊆ V2 V (ϖi) ∗ , the tensor product N j∼i xj,b lies in the highest isotypic component V (2ϖi − αi) ∗ ⊆ N j∼i V (ϖj ) ∗ , and xi,b−1 ∧ xi,b+1 = N j∼i xj,b in P(V (2ϖi − αi) ∗ ). Proof. Pick two height functi… view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 7 linked inside Pith

  1. [19]

    Highest weights for truncated shifted Yangians and product monomial crystals

    [Kam+19a] Joel Kamnitzer, Peter Tingley, Ben Webster, Alex Weekes, and Oded Yacobi. “Highest weights for truncated shifted Yangians and product monomial crystals”. J. Comb. Algebra 3.3 (2019), pp. 237–303. [Kam+19b] Joel Kamnitzer, Peter Tingley, Ben Webster, Alex Weekes, and Oded Yacobi. “On categoryO for affine Grassmannian slices and categorified tenso...

    2019

  2. [22]

    Almost dominant generalized slices and convolution diagrams over them

    Graduate Studies in Mathematics. American Mathemat- ical Society, Providence, RI, 2000, pp. x+212. [KP21] Vasily Krylov and Ivan Perunov. “Almost dominant generalized slices and convolution diagrams over them”. Adv. Math. 392 (2021), Paper No. 108034,

    2000

  3. [24]

    Representations of shifted quantum affine algebras

    [Her23] David Hernandez. “Representations of shifted quantum affine algebras”. Int. Math. Res. Not. IMRN 13 (2023), pp. 11035–11126. [Her25] David Hernandez. “Symmetries of Grothendieck rings in representation theory”. arXiv preprint arXiv:2501.03024 (2025). [Her26] David Hernandez. “Representations and characters of quantum affine algebras at the crossro...

    Pith/arXiv arXiv 2023

  4. [34]

    A functor for constructing R-matrices in the categoryO of Borel quan- tum loop algebras

    [Pin24] Th´ eo Pinet. “A functor for constructing R-matrices in the categoryO of Borel quan- tum loop algebras”. J. Lond. Math. Soc. (2) 109.1 (2024), Paper No. e12815,

    2024

  5. [36]

    q-opers, QQ-systems, and Bethe ansatz

    [Fre+24] Edward Frenkel, Peter Koroteev, Daniel S. Sage, and Anton M. Zeitlin. “ q-opers, QQ-systems, and Bethe ansatz”. J. Eur. Math. Soc. (JEMS) 26.1 (2024), pp. 355–

    2024

  6. [42]

    Weyl group symmetry of q-characters

    [FH25] Edward Frenkel and David Hernandez. “Weyl group symmetry of q-characters”. Se- lecta Math. (N.S.) 31.4 (2025), Paper No. 72,

    2025

  7. [45]

    Standard monomial theory for Bott-Samelson varieties

    [LLM02] Venkatramani Lakshmibai, Peter Littelmann, and Peter Magyar. “Standard monomial theory for Bott-Samelson varieties”. Compositio Math. 130.3 (2002), pp. 293–318. [LV11] Aaron D. Lauda and Monica Vazirani. “Crystals from categorified quantum groups”. Adv. Math. 228.2 (2011), pp. 803–861. [LT04] Niels Lauritzen and Jesper Funch Thomsen. “Line bundles...

    Pith/arXiv arXiv 2002

  8. [46]

    Inflations for representations of shifted quantum affine algebras

    [Pin25] Th´ eo Pinet. “Inflations for representations of shifted quantum affine algebras”. Adv. Math. 462 (2025), Paper No. 110093,

    2025

  9. [47]

    Hopf algebras and the quantum Yang-Baxter equation

    [Dri85] Vladimir G. Drinfeld. “Hopf algebras and the quantum Yang-Baxter equation”. Dokl. Akad. Nauk SSSR 283.5 (1985), pp. 1060–1064. [Dri87] Vladimir G. Drinfeld. “A new realization of Yangians and of quantum affine algebras”. Dokl. Akad. Nauk SSSR 296.1 (1987), pp. 13–17. [DK25] Ilya Dumanski and Vasily Krylov. “K-theoretic Hikita conjecture for quiver...

    arXiv 1985

  10. [51]

    On algorithms for solving vector space problems. II. Tame algebras

    [Pro07] Claudio Procesi. Lie groups. Universitext. An approach through invariants and rep- resentations. Springer, New York, 2007, pp. xxiv+596. [Rin80] Claus Michael Ringel. “On algorithms for solving vector space problems. II. Tame algebras”. Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979). Vol

    2007

  11. [52]

    Simple tensor products

    Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496. [Her10] David Hernandez. “Simple tensor products”. Invent. Math. 181.3 (2010), pp. 649–675. [Her19] David Hernandez. “Cyclicity and R-matrices”. Selecta Math. (N.S.) 25.2 (2019), Paper No. 19,

    1977

  12. [54]

    Heaps of pieces. I. Basic definitions and combinatorial lem- mas

    [Vie86] G´ erard Xavier Viennot. “Heaps of pieces. I. Basic definitions and combinatorial lem- mas”. Combinatoire ´ enum´ erative (Montreal, Que., 1985). Vol

    1985

  13. [68]

    Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case

    [GHL24] Christof Geiss, David Hernandez, and Bernard Leclerc. “Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case”. Proc. Lond. Math. Soc. (3) 129.3 (2024), Paper No. e12630,

    2024

  14. [69]

    Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras

    [KK12] Seok-Jin Kang and Masaki Kashiwara. “Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras”. Invent. Math. 190.3 (2012), pp. 699–742. [Kan+18] Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh. “Monoidal cate- gorification of cluster algebras”. J. Amer. Math. Soc. 31.2 (2018), pp. 349–426. [Kas03] Masaki Kashiwar...

    2012

  15. [75]

    On a class of representations of the Yangian and moduli space of monopoles

    [Ger+05] Anton A. Gerasimov, Sergei M. Kharchev, Dmitry R. Lebedev, and Sergey V. Oblezin. “On a class of representations of the Yangian and moduli space of monopoles”.Comm. Math. Phys. 260.3 (2005), pp. 511–525. [Gib21] Joel Gibson. “A Demazure character formula for the product monomial crystal”. Algebr. Comb. 4.2 (2021), pp. 301–327. [GM25] Andrei Grigo...

    Pith/arXiv arXiv 2005

  16. [102]

    Generators for Coulomb branches of quiver gauge theories

    [Wee19] Alex Weekes. “Generators for Coulomb branches of quiver gauge theories”. arXiv preprint arXiv:1903.07734 (2019). [Zha20] Huafeng Zhang. “Yangians and Baxter’s relations”. Lett. Math. Phys. 110.8 (2020), pp. 2113–2141. [Zha24] Huafeng Zhang. “Theta series for quantum loop algebras and Yangians”. Comm. Math. Phys. 405.10 (2024), Paper No. 230,

    Pith/arXiv arXiv 1903

  17. [103]

    A diagrammatic approach to categorification of quantum groups I

    REFERENCES 137 [KL09] Mikhail Khovanov and Aaron D. Lauda. “A diagrammatic approach to categorification of quantum groups I”. Represent. Theory 13 (2009), pp. 309–347. [KL11] Mikhail Khovanov and Aaron D. Lauda. “A diagrammatic approach to categorification of quantum groups II”. Trans. Amer. Math. Soc. 363.5 (2011), pp. 2685–2700. [Kho+12] Mikhail Khovano...

    2009

  18. [107]

    Almost-dominant chamber modules and reverse plane partitions

    Math- ematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003, pp. xiv+576. [Kac90] Victor G. Kac. Infinite-dimensional Lie algebras. Third. Cambridge University Press, Cambridge, 1990, pp. xxii+400. [Kal+26] Artem Kalmykov, Joel Kamnitzer, Alexis Leroux-Lapierre, Th´ eo Pinet, and Alex Weekes. “Almost-dominant chamber modules...

    2003

  19. [110]

    Borel-Weil theorem for configuration varieties and Schur modules

    Progress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 1993, pp. xii+341. [Mag98a] Peter Magyar. “Borel-Weil theorem for configuration varieties and Schur modules”. Adv. Math. 134.2 (1998), pp. 328–366. [Mag98b] Peter Magyar. “Schubert polynomials and Bott-Samelson varieties”. Comment. Math. Helv. 73.4 (1998), pp. 603–636. [MT24] Andrew Mathas an...

    arXiv 1993

  20. [144]

    Baxter Q-operators and representations of Yangians

    Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2015, pp. viii+530. [Baz+11] Vladimir V. Bazhanov, Rouven Frassek, Tomasz /suppress Lukowski, Carlo Meneghelli, and Matthias Staudacher. “Baxter Q-operators and representations of Yangians”. Nuclear Phys. B 850.1 (2011), pp. 148–174. [B´ ed99] Robert B´ edard. “On commutatio...

    Pith/arXiv arXiv 2015

  21. [163]

    Homogeneous representations of Khovanov- Lauda algebras

    Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2005, pp. xiv+277. [KR10] Alexander Kleshchev and Arun Ram. “Homogeneous representations of Khovanov- Lauda algebras”. J. Eur. Math. Soc. (JEMS) 12.5 (2010), pp. 1293–1306. [KR11] Alexander Kleshchev and Arun Ram. “Representations of Khovanov-Lauda-Rouquier algebras and combinatorics ...

    2005

  22. [166]

    Shifted Yangians and finite W -algebras

    [BK06] Jonathan Brundan and Alexander Kleshchev. “Shifted Yangians and finite W -algebras”. Adv. Math. 200.1 (2006), pp. 136–195. [CW19] Sabin Cautis and Harold Williams. “Cluster theory of the coherent Satake category”. J. Amer. Math. Soc. 32.3 (2019), pp. 709–778. [Cha02] Vyjayanthi Chari. “Braid group actions and tensor products”. Int. Math. Res. Not. ...

    2006

  23. [248]

    Canonical bases in tensor products and graphical calculus for Uq(sl2)

    Contemp. Math. Amer. Math. Soc., Providence, RI, 1999, pp. 163–205. [FK97] Igor B. Frenkel and Mikhail G. Khovanov. “Canonical bases in tensor products and graphical calculus for Uq(sl2)”. Duke Math. J. 87.3 (1997), pp. 409–480. REFERENCES 135 [FWW25] Noah Friesen, Alex Weekes, and Curtis Wendlandt. “Braid group actions, Baxter polynomials, and affine qua...

    1999

  24. [271]

    Towards a math- ematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II

    [BFN18] Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. “Towards a math- ematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II”. Adv. Theor. Math. Phys. 22.5 (2018), pp. 1071–1147. [BFN19] Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. “Coulomb branches of 3dN = 4 quiver gauge theories and slices in...

    2018

  25. [298]

    Yangians and R-matrices

    [CP90] Vyjayanthi Chari and Andrew Pressley. “Yangians and R-matrices”. Enseign. Math. (2) 36.3-4 (1990), pp. 267–302. [CP91a] Vyjayanthi Chari and Andrew Pressley. “Fundamental representations of Yangians and singularities of R-matrices”. J. Reine Angew. Math. 417 (1991), pp. 87–128. [CP91b] Vyjayanthi Chari and Andrew Pressley. “Quantum affine algebras”...

    1990

  26. [304]

    Double Bruhat cells and total positivity

    [FZ99] Sergey Fomin and Andrei Zelevinsky. “Double Bruhat cells and total positivity”. J. Amer. Math. Soc. 12.2 (1999), pp. 335–380. [FZ02] Sergey Fomin and Andrei Zelevinsky. “Cluster algebras. I. Foundations”. J. Amer. Math. Soc. 15.2 (2002), pp. 497–529. [FL25] Luca Francone and Bernard Leclerc. “Cluster structures on schemes of bands”. arXiv preprint ...

    Pith/arXiv arXiv 1999

  27. [325]

    Monoidal categori- fication and quantum affine algebras II

    Contemp. Math. Amer. Math. Soc., Providence, RI, 2003, pp. 133–139. [Kas+24] Masaki Kashiwara, Myungho Kim, Se-jin Oh, and Euiyong Park. “Monoidal categori- fication and quantum affine algebras II”. Invent. Math. 236.2 (2024), pp. 837–924. [Kho97] Mikhail Khovanov. “Graphical calculus, canonical bases and Kazhdan-Lusztig the- ory”. Thesis (Ph.D.)–Yale Uni...

    2003

  28. [330]

    Progr. Math. Birkh¨ auser/Springer, Cham, 2019, pp. 133–

    2019

  29. [405]

    Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras

    [FM01] Edward Frenkel and Evgeny Mukhin. “Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras”.Comm. Math. Phys. 216.1 (2001), pp. 23–

    2001

  30. [506]

    Characters and blocks for finite-dimensional representations of quantum affine algebras

    Contemp. Math. Amer. Math. Soc., Providence, RI, 2010, pp. 49–81. [CM05] Vyjayanthi Chari and Adriano A. Moura. “Characters and blocks for finite-dimensional representations of quantum affine algebras”. Int. Math. Res. Not. 5 (2005), pp. 257–

    2010

  31. [831]

    2-Kac-Moody algebras

    Lecture Notes in Math. Springer, Berlin, 1980, pp. 137–287. [Rou08] Rapha¨ el Rouquier. “2-Kac-Moody algebras”. arXiv preprint arXiv:0812.5023 (2008). [SW24] Turner Silverthorne and Ben Webster. “Gelfand-Tsetlin modules: canonicity and cal- culations”. Algebr. Represent. Theory 27.2 (2024), pp. 1405–1455. [Soe86] Wolfgang Soergel. “ ´Equivalences de certa...

    Pith/arXiv arXiv 1980

  32. [874]

    Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians

    [KPW22] Joel Kamnitzer, Khoa Pham, and Alex Weekes. “Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians”. Adv. Math. 399 (2022), Paper No. 108281,

    2022

  33. [1234]

    Canonical bases and higher representation theory

    Lecture Notes in Math. Springer, Berlin, 1986, pp. 321–350. [Web15] Ben Webster. “Canonical bases and higher representation theory”. Compos. Math. 151.1 (2015), pp. 121–166. [Web17] Ben Webster. “Knot invariants and higher representation theory”. Mem. Amer. Math. Soc. 250.1191 (2017), pp. v+141. [WWY20] Ben Webster, Alex Weekes, and Oded Yacobi. “A quantu...

    1986

  34. [2025]

    Canonical bases and KLR-algebras

    [VV11] Michela Varagnolo and Eric Vasserot. “Canonical bases and KLR-algebras”. J. Reine Angew. Math. 659 (2011), pp. 67–100. [VV25] Michela Varagnolo and Eric Vasserot. “Representations of shifted affine quantum groups and Coulomb branches”. arXiv preprint arXiv:2503.06262 (2025). [Vaz99] Monica Joy Vazirani. Irreducible modules over the affine Hecke alg...

    arXiv 2011