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Square root crystals and the square root of $B(\infty)$

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Pith's one-line read Set-valued tableaux build a square root of the infinite crystal B(∞).

desk verdict A genuine square-root B(∞) with solid character work, but the direct-limit proof is too compressed and a sign typo needs fixing. read the letter →

arxiv 2608.11009 v1 pith:2V5GWG75 submitted 2026-08-11 math.RT math.CO

Eric Marberg , Travis Scrimshaw This is my paper · ORCID
classification math.RTmath.CO MSC 05E0517B3705A1914M15
keywords squarerootcrystalN-rootbaseB(∞)directlimitset-valuedtableauxGrothendieckpolynomialDemazurefiltrationBZLword
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets up a general monoidal category of N-root crystals, where instead of the usual single-step raising and lowering operators one works with nth roots of those operators, and then specializes to square root gl_n-crystals. In that setting it constructs an infinite crystal, SetTab_n(∞), by taking the direct limit of the finite set-valued-tableau crystals T_{−λ}⊗SetTab_n(λ) along insertion maps. The authors prove that SetTab_n(∞) is a square root analog of the direct limit crystal B(∞): it has a unique highest weight element, a Demazure filtration indexed by the symmetric group, several explicit models (marginally large tableaux, a Lusztig/PBW-type vector parameterization, and an embedding into a looped path crystal built from the BZL reduced word), and a closed product character. For a reader, the point is that the K-theoretic, set-valued-tableau side of type A representation theory now has an infinite object that organizes its characters the way B(∞) organizes classical characters, and that collapses back to B(∞) under a simple squaring-and-forgetting functor.

What carries the argument

The machinery is the N-root crystal formalism specialized to N_i=2. A square root gl_n-crystal is a set with operators e_i, f_i and weights moving by half-simple roots: $α_i^{{(0)}}$=−e_0+e_i and $α_i^{{(1)}}$=e_0−e_{i+1}, so two applications of e_i or f_i move by the ordinary simple root α_i=e_i−e_{i+1}, while the extra e_0 coordinate records the β-defect. The construction of SetTab_n(∞) uses the tensor product rule for these crystals and a directed system of insertions insert_μ:SetTab_n(λ)→SetTab_n(λ+μ). The other main ingredient is the looped path crystal $P^{{1/2}}$_j, an infinite string whose graph contains 2-cycles, used to form $P^{{1/2}}$_{BZL} and to give the direct-limit embedding; the vector crystals $Vec^{{1/2}}$_{j,n}, whose coordinates are independent half-integers, make the product character transparent.

What would settle it

Enumerate the elements of SetTab_n(∞) of low weight using the marginally large set-valued tableau model for a small n (say n=3 or 4) and compare each coefficient with the product ∏_{1≤i<j≤n}(1+βx_j)/(1−$x_jx_i^{{−1}}$); any mismatch among finitely many coefficients, or any failure of the square-root axioms on a finite subgraph, would disprove the direct limit construction and its character formula.

Watch

Extended reading notes

Core claim

The central claim is that a square root analog of the direct limit crystal B(∞) exists and is computable. Starting from the N-root crystal category for gl_n with all simple roots split into halves, the paper forms the direct limit SetTab_n(∞) of the crystals T_{−λ}⊗SetTab_n(λ), where SetTab_n(λ) is the square-root crystal of semistandard set-valued tableaux of shape λ. It shows this limit is upper regular, connected, lower generated by its unique highest weight element u_∞, and carries a Demazure filtration; it is isomorphic to the tensor product $Vec^{{1/2}}$_{n−1,n}⊗···⊗$Vec^{{1/2}}$_{1,n} and embeds into the looped path crystal $P^{{1/2}}$_{BZL}. As a direct corollary the character is ∏_{1≤i<j≤n}(1+βx_j)/(1−$x_jx_i^{{−1}}$), which factors as ch(B(∞))∏_{i=1}^n(1+βx_i)^{i−1}, and the functor F^+ that squares operators and keeps only defect-zero elements sends SetTab_n(∞) isomorphically to B(∞). The model is rigid: it works for the BZL reduced word but fails for others, because the looped path crystals satisfy only restricted braid relations and the simple roots are split asymmetrically.

Load-bearing premise

Everything rests on a prior theorem about the finite crystals of set-valued tableaux: each one is a single connected piece generated from one highest tableau and has a well-behaved filtration by Bruhat order; if even one shape fails that property, the insertion maps defining the infinite limit could stop being honest inclusions and the product character formula would not follow.

Editorial extensions

If this is right

  • The character of SetTab_n(∞) is the closed product ∏_{1≤i<j≤n}(1+βx_j)/(1−x_jx_i^{−1}), which factors as ch(B(∞))∏_{i=1}^n(1+βx_i)^{i−1}; setting β=0 recovers the character of the classical B(∞).
  • SetTab_n(∞) has a unique highest weight element, is upper regular, and has a Demazure filtration indexed by the symmetric group, extending the Demazure/Lascoux structure of the finite crystals SetTab_n(λ) to the infinite limit.
  • The square root object is described explicitly by marginally large set-valued tableaux, by the tensor product Vec^{1/2}_{n−1,n}⊗···⊗Vec^{1/2}_{1,n} (a Lusztig/PBW-type parameterization), by an embedding into P^{1/2}_{BZL}, and by string data; these models are isomorphic as square root crystals.
  • Tensoring SetTab_n(∞) with the one-element crystal R_λ and taking the connected component of r_λ⊗u_∞ recovers SetTab_n(λ), so the infinite object packages the whole family of finite set-valued tableau crystals the way B(∞) packages B(λ).
  • The direct-limit embedding works only for the BZL word (and its restricted commutation class), not for arbitrary reduced words, so the square root theory is genuinely more rigid than the classical theory that works for every reduced expression of the longest element.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable next step is to check the paper's conjectural product formula for the Demazure subcrystals SetTab_n(∞)_w (Conjecture 4.9); if it holds, it would yield explicit finite identities among the polynomial refinements of Schubert classes indexed by permutations.
  • One could run the same direct-limit recipe for the parabolic directed sets P^+_J mentioned in Remark 3.40, producing parabolic versions of the square root B(∞); comparing their characters with the product formula would test how much of the classical parabolic B(∞) story survives at the square root level.
  • A natural next step—hinted at in the paper's final section—is to build the dual square root category using the opposite splitting of the simple roots, which may connect to different root-system combinatorics and to a different family of Grothendieck-type symmetric functions.
  • One can ask whether the β-defect behaves as a grading: since F^+ is monoidal on the defect subcategory but not on all square root crystals, SetTab_n(∞) may be viewed as a generating object for a filtered structure whose associated graded is B(∞), and it is worth testing whether the same pattern appears for every finite SetTab_n(λ).
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces a category of abstract N-root crystals, which generalizes Kashiwara crystals by splitting each simple root into N_i parts and allowing half-integer statistics when N_i=2. It then specializes to square-root gl_n-crystals (N_i=2), reviews Yu's crystals SetTab_n(λ) on semistandard set-valued tableaux, and constructs a direct limit SetTab_n(∞) from weight-shifted versions T_{−λ}⊗SetTab_n(λ). The main claims are that SetTab_n(∞) is a square-root analog of B(∞); that it has a Demazure filtration, a Lusztig-type vector parameterization as Vec^{1/2}_{n−1,n}⊗⋯⊗Vec^{1/2}_{1,n}, an explicit product character, a looped-path/polyhedral realization B^{1/2}(∞) inside P^{1/2}_{BZL}, and a string parameterization; and that F^+(SetTab_n(∞)) recovers B(∞). The paper also documents several non-classical phenomena such as non-braid behavior and asymmetry under root-system dualities.

Significance. If the direct-limit construction is properly established, this is a substantial contribution: it provides a genuinely new analog of B(∞) in a K-theoretic square-root crystal setting, with several explicit and independently checkable models (vector coordinates, looped path crystals, marginally large tableaux) and a strikingly simple character formula that factorizes through ch(B(∞)). The general N-root crystal framework is also of independent interest and appears well motivated by existing work on Grothendieck positivity. The paper is constructive and does not assume its target, and the character formula is derived from an explicit vector model rather than from the direct limit alone. However, the central direct-limit theorem is not proved as written: the proof of Theorem 3.31 asserts a commutation statement that is false, and no alternative complete proof is supplied. Because nearly every subsequent result depends on Theorem 3.31, the current version cannot be accepted without repair.

major comments (3)
  1. [§3.4, Theorem 3.31] The proof of Theorem 3.31 is not valid as written. The sentence 'By the j-shifted version of Lemma 3.30, the crystal operators f_i commute with inserting a {j} into row j, yielding the claim' asserts an elementwise commutation that can fail. For example, take n=3, λ=(2,1), T∈SetTab_3(λ) with row reading {3}⊗{1}⊗{2} (top row 1 2, bottom row 3), and μ=(1,1); then T'=insert_μ(T) has row reading {2}⊗{3}⊗{1}⊗{1}⊗{2}. A direct tensor-product computation gives f_2(T)=0 while f_2(T')≠0, so the insertion map does not commute with f_2 elementwise. This does not by itself disprove Theorem 3.31, because Definition 2.12 only requires preservation of f_i when both images are nonzero; however, it disproves the stated proof. Lemma 2.16, which is the stated criterion, requires commutativity with the e_i, and Lemma 3.30 only covers the one-row case. The paper supplies no separate argument that e_i commutes with insert_μ for multi-row λ and μ; Remark 3.32 merely sketches a possible signature-rule proof. Since Theorem 3.33, Corollary 3.35, Theorem 4.7, Corollary 4.8, and Theorem 4.18 all rest on the existence of this directed system of embeddings, this is the central load-bearing gap. Please supply a complete proof of the e_i commutation (or otherwise prove that each Ψ_{λ,μ} is a crystal morphism), and do not rely on the false stronger commutation claim.
  2. [§4.1, Eq. (4.5)] Equation (4.5) is inconsistent with Corollary 4.8 and with Eq. (1.1). Corollary 4.8 gives ch(SetTab_n(∞)) = ∏_{1≤i<j≤n}(1−x_j x_i^{−1})^{−1} · ∏_{i=1}^n (1+βx_i)^{i−1}, while Eq. (4.5) prints a factor (1−βx_i)^{i−1}. For n=2 the correct factor is 1+βx_2; a minus sign would give the wrong character. The sign should be corrected to (1+βx_i)^{i−1} in Eq. (4.5), matching the abstract and Eq. (1.1).
  3. [§3.4, Theorem 3.33 and Corollary 3.35] These results are stated as immediate consequences of Theorem 3.31 plus Yu's Theorem 3.15. If the direct-limit embeddings are repaired, the arguments appear plausible, but the dependence should be made explicit: Theorem 3.15 supplies connectedness, upper regularity, lower generation, and the Demazure filtration of each SetTab_n(λ), and Theorem 3.31 is what transfers these properties to the limit. In particular, the sentence 'Since the crystals are upper regular, the crystal operator formulas just given simplify to ...' relies on the embeddings existing, so these formulas are only valid once Theorem 3.31 is proved.
minor comments (5)
  1. [§3.3, Lemma 3.29 and §3.4, Proposition 3.39] Both results are introduced with 'We omit the proof'. Lemma 3.29 is used for the set-theoretic injectivity of the insertion maps, and Proposition 3.39 is used to justify the marginally large tableau model and Figure 3. The omitted arguments are described as straightforward, but since these statements are part of the main constructions, a proof or a precise reference should be included.
  2. [§4.2, proof of Theorem 4.18] The proof refers to 'the isomorphism in Theorem 4.4', but the relevant statement is Theorem 4.7. Please correct the cross-reference.
  3. [§3.1] The text 'symmetric fucntions' appears to be a typo for 'symmetric functions'.
  4. [§4.2, Definition 4.21] The displayed BZL word in Definition 4.21 is abbreviated '(n−1,n−2,n−1,...,1,2,...,n−1)', which is not the full BZL word from (4.1). Please write the full word or explain the ellipsis convention.
  5. [§5.1, Example 5.1] In the displayed equations, it would help to identify explicitly which elements of SetTab_3(∞) are denoted b_{12} and b_{21}; the current inline mention of Figure 6 is terse.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the direct-limit construction and character formula are derived from explicit external inputs, not from the target claims.

full rationale

No load-bearing circular step was found. SetTab_n(∞) is defined as the direct limit (3.13) of finite crystals T_{-λ} ⊗ SetTab_n(λ), whose crystal structure is imported from Yu's independent Theorem 3.15, and the direct-system maps are given explicitly by Ψ_{λ,μ}(t_{-λ} ⊗ T) = t_{-λ-μ} ⊗ insert_μ(T) in (3.12); the morphism property is then proved in Theorem 3.31 rather than assumed. The row-vector model Vec^{1/2}_{j,n} is built from an explicit parameterization of one-row tableaux and its character is computed independently as ∏(1+βx_i)/(1-x_i x_j^{-1}) in Proposition 4.5; Theorem 4.7 and Corollary 4.8 derive the character of SetTab_n(∞) from that product rather than presupposing it. The identification F^+(SetTab_n(∞)) ≅ B(∞) is benchmarked against Cliff's and Hong–Lee's external descriptions of B(∞), and no parameter is fitted to a prediction. Self-citations to [MT25, MTY26] occur, notably Theorem 3.20, but they are used for auxiliary character structure (e.g., Example 5.7) and not to prove the central product formula or Demazure filtration. The proof of Theorem 3.31 is only one sentence, and Lemma 3.29 and Proposition 3.39 omit proofs; those are rigor/correctness gaps, not circularity, because the assertions are not equivalent to the paper's own inputs by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The central claim rests on the new N-root axioms, the explicit split-root data (3.1), and Yu's theorem that finite set-valued tableau crystals are connected and Demazure-filtered. There are no numerical parameters fitted to data; the structural choices N=2 and the split roots are ad hoc to the square root program.

free parameters (2)
  • Root-splitting data α_i^(0) = −e_0 + e_i, α_i^(1) = e_0 − e_{i+1} (Eq. 3.1)
    The square root category and every subsequent construction depend on this specific choice of how simple roots are split. It is not derived; changing it gives different theories, as noted in Section 5.3 and Problem 5.10.
  • N_i = 2 for all i = 2
    The square root specialization fixes the root multiplicity to 2. Proposition 2.10 shows N_i > 2 is trivial, so this is the nontrivial choice for the theory.
assumptions (4)
  • ad hoc to paper N-root crystal axioms in Definition 2.2, including statistics valued in (1/N_i)Z and the integrality of ⟨·,·⟩.
    The paper's framework is a new generalization; the central claims are proved inside these axioms, so they are postulates rather than standard background.
  • domain assumption The standard square root crystal S_n of Definition 3.2 is a regular square root crystal.
    The entire polynomial subcategory is generated from this crystal; its regularity is checked directly in the paper.
  • domain assumption Yu's theorem [Yu23, Thm. 5.1]: SetTab_n(λ) is a connected polynomial square root crystal with Demazure filtration and character L_{wλ}.
    This imported prior result is used to define the direct system and lift Demazure filtrations to SetTab_n(∞). It is not re-proved here.
  • standard math Standard root system, weight lattice, and tensor product rule from [BS17, Kas93].
    Background from Lie theory and crystal base theory that is assumed throughout.
invented entities (2)
  • Abstract N-root crystal
    purpose: Generalize Kashiwara crystals by taking N_i-th roots of the raising and lowering operators.
    Definition 2.2 introduces a new framework. It has no external falsifiable handle, but its properties are proved internally from the axioms.
  • Looped path crystal P_j^{1/2}
    purpose: Square root analog of the elementary string crystal E_j, used to build the polyhedral realization of SetTab_n(∞).
    Defined in Definition 4.11 as a new combinatorial object. It is validated by the internal theorem that SetTab_n(∞) embeds into a tensor product of such crystals.

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Pith. "Pith review of Square root crystals and the square root of $B(\infty)$." pith.science (2026). https://pith.science/paper/2V5GWG75

@misc{pith2026260811009,
  author       = {Pith},
  title        = {Pith review of: Square root crystals and the square root of $B(\infty)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2V5GWG75}},
  note         = {Machine review of arXiv:2608.11009}
}
abstract

We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of the first author, Tong, and Yu showed that regular square root $\mathfrak{gl}_n$-crystals can be a useful tool for proving Grothendieck positivity results. The objects studied here go beyond the regular case and allow us to construct a square root analog of the direct limit crystal $B(\infty)$. We give several descriptions of our square root of $B(\infty)$, using marginally large tableaux, the Lusztig or PBW parameterization, and the Nakashima--Zelevinsky polyhedral model. We show that this crystal has a simple character formula, exhibits a nontrivial Demazure filtration, and recovers Yu's semistandard set-valued tableau crystals after taking appropriate tensor products. We also investigate a number of differences between square root crystals and classical crystal constructions.

Figures

Figures reproduced from arXiv: 2608.11009 by the authors.

Figure 1
Figure 1. A connected component of S3 ⊗ S3 isomorphic to SetTab3(2Λ1). A semistandard Young tableau is a semistandard set-valued tableau T whose entries all have size 1; that is, with |Tij | = 1 for all (i, j) ∈ D. For each λ ∈ P + let SetTabn(λ) be the set of semistandard set-valued tableaux of shape Dλ. Denote uλ to be the unique set-valued tableau with all entries in row i filled with {i}; that is, with Tij = {i} for all (… view at source ↗
Figure 2
Figure 2. The crystal SetTab3(2Λ2). when λ = 2Λ2 and n = 3, as can be seen by comparing [PS22, [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. The p gl3 -crystal SetTab3(∞) up to depth at least 3 with the elements written as marginally large set-valued tableaux. Proposition 3.39. Each element of SetTabn(∞) may be expressed as [T] for a unique marginally large semistandard set-valued tableau T, and for this tableau, it holds that fiT ̸= 0 for all i ∈ I. For example, we have u∞ = [T], where T is the marginally large tableau of staircase shape λ = Λ1 + Λ2 + ·… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The p gln-crystal Vec1/2 n (2) for n = 3. φi(a) = ( ⌈ai⌉ − 1 2 if ai+1 ∈/ Z and ai > 0, ⌈ai⌉ otherwise, (4.2d) wt(a) = −ke0 + Xn i=1 ⌈ai⌉(ei + e0) = Xn i=1  ⌈ai⌉ei + (⌈ai⌉ − ⌊ai⌋)e0  . (4.2e) In terms of tableaux, the cases for the fi crystal operators are respective…
Figure 5
Figure 5. Figure 5: The p gl3 -crystal SetTab3(∞) up to depth at least 3 after applying the isomorphism (4.4) and writing (a) ⊗ (b, c) = (a, b, c) for simplicity. Compare [PITH_FULL_IMAGE:figures/full_fig_p036_5.png]
Figure 6
Figure 6. Figure 6: Crystal graph of B 1/2 (∞) for n = 3. In each boxed vertex, the word “ABC” stands for b = {−A 2 }2 ⊗ {− B 2 }1 ⊗ {−C 2 }2 = {−A 2 , − B 2 , − C 2 }BZL ∈ B1/2 (∞), the monomial on the second line is x wt(b) , and the last two lines show the respective values of ε1(b), ε…
Figure 7
Figure 7. Figure 7: Crystal graphs of SetTab4(Λ2) (left) and B 1/2 (Λ2) for n = 4 (right), where the word ABCDEF represents the element rΛ2 ⊗ {−A 2 , − B 2 , − C 2 , − D 2 , − E 2 , − F 2 }BZL. 4.3 String parameterization We now give our last description of SetTabn(∞) using the string dat…
Figure 8
Figure 8. Figure 8: For the string data (Σ1, Σ2, Σ3) with respect to the BZL word w0 = s2s1s2 (above) and dual BZL word w0 = s1s2s1 (below) for the p gl3 -crystal SetTab3(∞) up to depth at least 3. We write the string data as the word A1A2A3 with Ai = 2Σi . 5.1 Constructions using non-BZL…
Figure 9
Figure 9. Figure 9: For n = 4 and J = (2, 1, 3, 2, 1, 3), the crystal graph of B 1/2 J (Λ2) (left) and the connected component of rλ⊗ ⟨0, . . . , 0⟩BZL ∈ RΛ2 ⊗P1/2 BZL (right). The word ABCDEF represents the element rΛ2 ⊗ {−A 2 , − B 2 , − C 2 , − D 2 , − E 2 , − F 2 }J and rΛ2 ⊗ ⟨−A 2 , …

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Works this paper leans on

52 extracted references · 40 canonical work pages

  1. [1]

    Multiline queues with spectral parameters

    Erik Aas, Darij Grinberg, and Travis Scrimshaw. Multiline queues with spectral parameters. Comm. Math. Phys. , 374(3):1743--1786, 2020

    work page 2020

  2. [2]

    H. H. Andersen. Schubert varieties and D emazure's character formula. Invent. Math. , 79(3):611--618, 1985

    work page 1985

  3. [3]

    Crystal bases for the quantum superalgebra U_q( gl (m,n))

    Georgia Benkart, Seok-Jin Kang, and Masaki Kashiwara. Crystal bases for the quantum superalgebra U_q( gl (m,n)) . J. Amer. Math. Soc. , 13(2):295--331, 2000

    work page 2000

  4. [4]

    Lie groups and L ie algebras

    Nicolas Bourbaki. Lie groups and L ie algebras. C hapters 1--3 . Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. Translated from the French, Reprint of the 1989 English translation

    work page 1998

  5. [5]

    Crystal bases

    Daniel Bump and Anne Schilling. Crystal bases . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Representations and combinatorics

    2017

  6. [6]

    Colored five-vertex models and L ascoux polynomials and atoms

    Valentin Buciumas, Travis Scrimshaw, and Katherine Weber. Colored five-vertex models and L ascoux polynomials and atoms. J. Lond. Math. Soc. , 102(3):1047--1066, 2020

    work page 2020

  7. [7]

    A L ittlewood-- R ichardson rule for the K -theory of G rassmannians

    Anders Skovsted Buch. A L ittlewood-- R ichardson rule for the K -theory of G rassmannians. Acta Math. , 189(1):37--78, 2002

    work page 2002

  8. [8]

    String bases for quantum groups of type A_r

    Arkady Berenstein and Andrei Zelevinsky. String bases for quantum groups of type A_r . In I. M . G el'fand S eminar , volume 16, Part 1 of Adv. Soviet Math. , pages 51--89. Amer. Math. Soc., Providence, RI, 1993

    work page 1993

Show all 52 references
  1. [9]

    Tensor product multiplicities, canonical bases and totally positive varieties

    Arkady Berenstein and Andrei Zelevinsky. Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. , 143(1):77--128, 2001

    2001

  2. [10]

    Crystal bases and Y oung tableaux

    Gerald Cliff. Crystal bases and Y oung tableaux. J. Algebra , 202(1):10--35, 1998

    1998

  3. [11]

    Representation theory , volume 129 of Graduate Texts in Mathematics

    William Fulton and Joe Harris. Representation theory , volume 129 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1991. A first course, Readings in Mathematics

    1991

  4. [12]

    Kirillov

    Sergey Fomin and Anatol N. Kirillov. Grothendieck polynomials and the Y ang-- B axter equation. In Formal power series and algebraic combinatorics/ S \'eries formelles et combinatoire alg\'ebrique , pages 183--189. DIMACS, Piscataway, NJ, 1994

    1994

  5. [13]

    Quantum queer superalgebra and crystal bases

    Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim. Quantum queer superalgebra and crystal bases. Proc. Japan Acad. Ser. A Math. Sci. , 86(10):177--182, 2010

    2010

  6. [14]

    Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

    Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim. Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux. Trans. Amer. Math. Soc. , 366(1):457--489, 2014

    2014

  7. [15]

    Grojnowski and G

    I. Grojnowski and G. Lusztig. A comparison of bases of quantized enveloping algebras. In Linear algebraic groups and their representations ( L os A ngeles, CA , 1992) , volume 153 of Contemp. Math. , pages 11--19. Amer. Math. Soc., Providence, RI, 1993

    1992

  8. [16]

    Introduction to quantum groups and crystal bases , volume 42 of Graduate Studies in Mathematics

    Jin Hong and Seok-Jin Kang. Introduction to quantum groups and crystal bases , volume 42 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2002

    2002

  9. [17]

    Young tableaux and crystal B( ) for finite simple L ie algebras

    Jin Hong and Hyeonmi Lee. Young tableaux and crystal B( ) for finite simple L ie algebras. J. Algebra , 320(10):3680--3693, 2008

    2008

  10. [18]

    Humphreys

    James E. Humphreys. Linear algebraic groups , volume No. 21 of Graduate Texts in Mathematics . Springer-Verlag, New York-Heidelberg, 1975

    1975

  11. [19]

    Humphreys

    James E. Humphreys. Reflection groups and C oxeter groups , volume 29 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1990

    1990

  12. [20]

    Humphreys

    James E. Humphreys. Representations of semisimple L ie algebras in the BGG category O , volume 94 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008

    2008

  13. [21]

    Quantum groups and their primitive ideals , volume 29 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]

    Anthony Joseph. Quantum groups and their primitive ideals , volume 29 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1995

    1995

  14. [22]

    Mirkovi\'c-- V ilonen cycles and polytopes

    Joel Kamnitzer. Mirkovi\'c-- V ilonen cycles and polytopes. Ann. of Math. (2) , 171(1):245--294, 2010

    2010

  15. [23]

    Crystalizing the q -analogue of universal enveloping algebras

    Masaki Kashiwara. Crystalizing the q -analogue of universal enveloping algebras. Comm. Math. Phys. , 133(2):249--260, 1990

    1990

  16. [24]

    Kashiwara

    M. Kashiwara. On crystal bases of the Q -analogue of universal enveloping algebras. Duke Math. J. , 63(2):465--516, 1991

    1991

  17. [25]

    The crystal base and L ittelmann's refined D emazure character formula

    Masaki Kashiwara. The crystal base and L ittelmann's refined D emazure character formula. Duke Math. J. , 71(3):839--858, 1993

    1993

  18. [26]

    On crystal bases

    Masaki Kashiwara. On crystal bases. In Representations of groups ( B anff, AB , 1994) , volume 16 of CMS Conf. Proc. , pages 155--197. Amer. Math. Soc., Providence, RI, 1995

    1994

  19. [27]

    Kirillov

    Anatol N. Kirillov. Notes on S chubert, G rothendieck and key polynomials. SIGMA Symmetry Integrability Geom. Methods Appl. , 12:Paper No. 034, 56, 2016

    2016

  20. [28]

    Crystal graphs for representations of the q -analogue of classical L ie algebras

    Masaki Kashiwara and Toshiki Nakashima. Crystal graphs for representations of the q -analogue of classical L ie algebras. J. Algebra , 165(2):295--345, 1994

    1994

  21. [29]

    Geometric construction of crystal bases

    Masaki Kashiwara and Yoshihisa Saito. Geometric construction of crystal bases. Duke Math. J. , 89(1):9--36, 1997

    1997

  22. [30]

    A. Lascoux. Transition on G rothendieck polynomials. In Physics and combinatorics, 2000 ( N agoya) , pages 164--179. World Sci. Publ., River Edge, NJ, 2001

    2000

  23. [31]

    Combinatorial aspects of the K -theory of G rassmannians

    Cristian Lenart. Combinatorial aspects of the K -theory of G rassmannians. Ann. Comb. , 4(1):67--82, 2000

    2000

  24. [32]

    On the combinatorics of crystal graphs

    Cristian Lenart. On the combinatorics of crystal graphs. I . L usztig's involution. Adv. Math. , 211(1):204--243, 2007

    2007

  25. [33]

    Crystal graphs and Y oung tableaux

    Peter Littelmann. Crystal graphs and Y oung tableaux. J. Algebra , 175(1):65--87, 1995

    1995

  26. [34]

    Cones, crystals, and patterns

    Peter Littelmann. Cones, crystals, and patterns. Transform. Groups , 3(2):145--179, 1998

    1998

  27. [35]

    Lakshmibai, C

    V. Lakshmibai, C. Musili, and C. S. Seshadri. Geometry of G/P . IV . S tandard monomial theory for classical types. Proc. Indian Acad. Sci. Sect. A Math. Sci. , 88(4):279--362, 1979

    1979

  28. [36]

    Structure de H opf de l'anneau de cohomologie et de l'anneau de G rothendieck d'une vari\'et\'e de drapeaux

    Alain Lascoux and Marcel-Paul Sch\"utzenberger. Structure de H opf de l'anneau de cohomologie et de l'anneau de G rothendieck d'une vari\'et\'e de drapeaux. C. R. Acad. Sci. Paris S\'er. I Math. , 295(11):629--633, 1982

    1982

  29. [37]

    Symmetry and flag manifolds

    Alain Lascoux and Marcel-Paul Sch\"utzenberger. Symmetry and flag manifolds. In Invariant theory ( M ontecatini, 1982) , volume 996 of Lecture Notes in Math. , pages 118--144. Springer, Berlin, 1983

    1982

  30. [38]

    Keys & standard bases

    Alain Lascoux and Marcel-Paul Sch\"utzenberger. Keys & standard bases. In Invariant theory and tableaux ( M inneapolis, MN , 1988) , volume 19 of IMA Vol. Math. Appl. , pages 125--144. Springer, New York, 1990

    1988

  31. [39]

    G. Lusztig. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. , 3(2):447--498, 1990

    1990

  32. [40]

    G. Lusztig. Canonical bases arising from quantized enveloping algebras. II . Number 102, pages 175--201. 1990. Common trends in mathematics and quantum field theories (Kyoto, 1990)

    1990

  33. [41]

    Introduction to quantum groups , volume 110 of Progress in Mathematics

    George Lusztig. Introduction to quantum groups , volume 110 of Progress in Mathematics . Birkh\"auser Boston, Inc., Boston, MA, 1993

    1993

  34. [42]

    Set-valued skyline fillings

    Cara Monical. Set-valued skyline fillings. Preprint, arXiv:1611.08777, 2016

    2016 arXiv

  35. [43]

    Crystal structures for symmetric G rothendieck polynomials

    Cara Monical, Oliver Pechenik, and Travis Scrimshaw. Crystal structures for symmetric G rothendieck polynomials. Transform. Groups , 26(3):1025--1075, 2021

    2021

  36. [44]

    Crystals for set-valued decomposition tableaux

    Eric Marberg and Kam Hung Tong. Crystals for set-valued decomposition tableaux. Algebr. Comb. , 8(4):857--896, 2025

    2025

  37. [45]

    Grothendieck positivity for normal square root crystals

    Erc Marberg, Kam Hung Tong, and Tianyi Yu. Grothendieck positivity for normal square root crystals. Adv. Math. , 501:111118, 2026

    2026

  38. [46]

    Polyhedral realizations of crystal bases for integrable highest weight modules

    Toshiki Nakashima. Polyhedral realizations of crystal bases for integrable highest weight modules. J. Algebra , 219(2):571--597, 1999

    1999

  39. [47]

    Polyhedral realizations of crystal bases for quantized K ac-- M oody algebras

    Toshiki Nakashima and Andrei Zelevinsky. Polyhedral realizations of crystal bases for quantized K ac-- M oody algebras. Adv. Math. , 131(1):253--278, 1997

    1997

  40. [48]

    K-theoretic crystals for set-valued tableaux of rectangular shapes

    Oliver Pechenik and Travis Scrimshaw. K-theoretic crystals for set-valued tableaux of rectangular shapes. Algebr. Comb. , 5(3):515--536, 2022

    2022

  41. [49]

    Combinatorial rules for three bases of polynomials

    Colleen Ross and Alexander Yong. Combinatorial rules for three bases of polynomials. S\'em. Lothar. Combin. , 74:Art. B74a, 11, [2015--2018]

    2015

  42. [50]

    Candidate for the crystal B(- ) for the queer L ie superalgebra

    Ben Salisbury and Travis Scrimshaw. Candidate for the crystal B(- ) for the queer L ie superalgebra. Kyoto J. Math. , 62(2):377--401, 2022

    2022

  43. [51]

    Grothendieck-to- L ascoux expansions

    Mark Shimozono and Tianyi Yu. Grothendieck-to- L ascoux expansions. Trans. Amer. Math. Soc. , 376(7):5181--5220, 2023

    2023

  44. [52]

    Set-valued tableaux rule for L ascoux polynomials

    Tianyi Yu. Set-valued tableaux rule for L ascoux polynomials. Comb. Theory , 3(1):Paper No. 13, 31, 2023

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.