From dual canonical bases to positroidal subdivisions

Ayush Kumar Tewari; Jian-Rong Li

REVIEW 3 major objections 4 minor 31 references

From dual canonical bases to positroidal subdivisions

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper proves that every dual-canonical-basis tableau of the Grassmannian cluster algebra induces a positroidal subdivision of the hypersimplex $\Delta(k,n)$, and that non-frozen prime tableaux give precisely the coarsest subdivisions…

desk verdict A sound dictionary between dual canonical tableaux and positroidal subdivisions, but the Gr(2,n) bijection in the abstract is ahead of the proof as written. read the letter →

arxiv 2506.19443 v2 pith:KZEQXJTV submitted 2025-06-24 math.CO math.QA

Jian-Rong Li , Ayush Kumar Tewari This is my paper · ORCID

classification math.COmath.QA MSC 52B1152B4005E1014M1513F60

keywords GrassmannianclusteralgebradualcanonicalbasishypersimplexpositroidalsubdivisionsemistandardYoungtableautropicalsplitmatroidpolytope

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

Every rectangular semistandard Young tableau with $k$ rows and entries in $[n]$ labels a distinguished element of the dual canonical basis of the coordinate ring of the Grassmannian $\operatorname{Gr}(k,n)$. The paper proves that the Speyer–Williams tropical map assigns to each such tableau a weight that induces a positroidal subdivision of the hypersimplex $\Delta(k,n)$, the $0$\textendash$1$ polytope whose vertices are the $k$-subsets of $[n]$. For $k=2$, it proves that non-frozen prime tableaux are exactly the one-column tableaux attached to the split, or coarsest, positroidal subdivisions of $\Delta(2,n)$. The paper also conjectures that the number of split positroidal subdivisions of $\Delta(k,n)$ is $\frac{k-1}{2} n(n-k-1)$ and verifies this by computation for a range of small $k,n$. If the main theorem is right, every dual canonical basis element acquires a canonical polyhedral incarnation, tying representation theory to discrete geometry.

What carries the argument

The load-bearing object is the Speyer–Williams map $F_{k,n}\colon \mathbb{R}^{(k-1)(n-k)} \to \operatorname{Span}_{\mathbb{R}}\{e_J : J\in \binom{[n]}{k}\}/L$, evaluated on $v_T=\sum_{i,j} c_{i,j} e_{i,j}$, where $c_{i,j}$ counts how many times the fundamental tableau $T_{i,j}$ (a one-column tableau with entries $[j,j+k]\setminus\{i+j\}$) appears in a factorization of $T$. Its tropicalization of Plücker coordinates sends tableaux to weight vectors. The argument's second machine is the positive Dressian: a weight vector induces a positroidal subdivision exactly when it lies on a cone of the positive Dressian, so the equality between the positive tropical Grassmannian and the positive Dressian converts tableau weights into subdivisions. For the rank-two theorem, the split decomposition theorem of polytopes reduces any subdivision to a refinement of compatible splits, and one-column prime tableaux encode the internal edges of the phylogenetic trees that index those splits.

What would settle it

Compute $F_{k,n}(v_{T_1\cup T_2})$ for any two weakly separated one-column tableaux not covered by Example 5.4, using the same Plücker-coordinate formulas; if the weight is not the sum of the individual weights, the additivity behind Theorem 5.2 is false. A direct check of the rank-two bijection is also possible: list all non-frozen prime tableaux in $\operatorname{SSYT}(2,n)$ and all split positroidal subdivisions of $\Delta(2,n)$ for $n=6$; any split not indexed by a pair $\{i,j\}$, or any pair whose weight is not split, would falsify the correspondence.

Watch

Extended reading notes

Core claim

Formally, the central assertion is Theorem 4.1: for every tableau $T\in \operatorname{SSYT}(k,[n])$, the vector $F_{k,n}(v_T)$ obtained by evaluating the Speyer–Williams map on the tableau's fundamental-tableau multiplicities lies in the positive Dressian, hence its lower hull is a positroidal subdivision of $\Delta(k,n)$. The proof routes through the equality of the positive tropical Grassmannian and the positive Dressian and through the classification of cones of the positive Dressian. In rank two, Theorem 5.2 identifies the non-frozen prime tableaux $\{i,j\}$ with the split positroidal subdivisions, those with exactly two maximal cells; the paper's stated correspondence is that these tableaux are precisely the coarsest subdivisions of $\Delta(2,n)$. The paper further presents a conjectural enumeration and computational evidence for the split subdivision count in higher rank.

Load-bearing premise

The rank-two correspondence assumes that the Speyer–Williams weight is additive under tableau unions, $F(v_{T_1\cup T_2}) = F(v_{T_1}) + F(v_{T_2})$; the paper states this as Conjecture 5.3 and does not prove it, and the bijective direction through phylogenetic trees is asserted rather than proved.

Editorial extensions

If this is right

Every element of the dual canonical basis of $\mathbb{C}[\operatorname{Gr}(k,n)]$ comes with a canonical positroidal subdivision of $\Delta(k,n)$, so representation-theoretic data encoded by tableaux can be read polyhedrally.
For $\operatorname{Gr}(2,n)$, the coarsest subdivisions of $\Delta(2,n)$ are indexed by pairs $\{i,j\}$ with $i<j$, the same data as non-frozen prime tableaux; this gives a uniform description of the split subdivisions in rank two.
If Conjecture 5.8 holds, the number of split positroidal subdivisions of $\Delta(k,n)$ is $\frac{k-1}{2} n(n-k-1)$, a count already verified computationally for the small cases listed in the paper.
If Conjecture 4.2 holds, every coarsest subdivision coming from a tableau without frozen factors must come from a prime tableau, giving a representation-theoretic obstruction to coarseness.

Reading between the lines

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

A natural test is whether the additivity identity of Conjecture 5.3 can be upgraded to a tableau-calculus rule: if it holds, the subdivision of a multi-column tableau is the common refinement of the splits of its weakly separated one-column factors, making the subdivision visibly computable from the tableau.
The exceptional prime tableaux in $\operatorname{Gr}(3,8)$ that are not coarsest align with the difference between the positive tropical Grassmannian and the cluster complex; one could test whether, in general, non-coarsest prime tableaux are in bijection with cluster variables of degree greater than one.
The conjectural split count has the flavor of a simple closed form; a bijective proof might come from encoding a split by a pair $(i,j)$ together with a cyclic-gap datum, extending the rank-two phylogenetic-tree picture to higher $k$.
The framework suggests a dictionary between dual canonical basis factorizations and common refinements of subdivisions: prime tableaux should correspond to indecomposable subdivisions, so the tableau poset and the subdivision refinement poset may be compared directly.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this paper gives every rectangular dual-canonical-basis tableau a positroidal subdivision of the hypersimplex via the Speyer-Williams map, and it does it honestly—Theorem 4.1 is a corollary of known results (Speyer–Williams, positive Dressian equals positive tropical Grassmannian, ray classification), not a hidden-fitting argument. The new part is the dictionary itself and the conjectural enumeration of split subdivisions, and that is worth taking seriously.

What is good. The main construction is simple and credible: each tableau gives a weight in the positive Dressian, so each gives a positroidal subdivision. The paper then specializes to Gr(2,n). The structural lemma that non-frozen prime tableaux are two-element one-column tableaux is clean, and it does establish a genuine bijection between those tableaux and split subdivisions—if the additivity step holds. They also provide code and computational evidence for the higher-rank split conjecture for nine parameter pairs, which is more reproducible than many papers in this area. The conjectural formula (k-1)n(n-k-1)/2 is concrete and testable.

Soft spots. The rank-two proof has a load-bearing gap that the stress-test note catches exactly. In Theorem 5.2, after decomposing the weight of a subdivision as a sum of split weights, the proof claims that surjectivity of the Speyer-Williams map gives tableaux T_i whose union is T and whose individual images are the splits. Surjectivity does not give that. The needed identity F(v_{union T_i}) = sum_i F(v_{T_i}) is exactly Conjecture 5.3, stated later and never proved. Without it, the contradiction for a one-column prime tableau does not go through. The converse direction, identifying split subdivisions with those one-column tableaux via phylogenetic trees, is also asserted informally rather than derived from a stated theorem. These are likely repairable—the examples all work, and the additivity conjecture might follow from known weak-separation facts—but as written the abstract's precise "correspond" claim is stronger than the proof supports. That should be fixed before publication.

The authors lean on several of their own prior results, including the tableau-basis correspondence. That is normal here and not a red flag; the results are established and the code is available. The paper is not circular in any fitting sense; there are no fitted parameters, and the claims are not tuned to examples.

Who this is for: anyone working on tropical Grassmannians, matroid subdivisions, and cluster algebras. The dictionary is worth having. My own verdict: send it to a serious referee. It needs a revision, but the core construction is sound and the main gap is identifiable and plausibly fixable. If the authors add the missing additivity lemma or clearly demote the Gr(2,n) claim to a conjecture, the paper is publishable in a good combinatorics venue.

Referee Report

3 major / 4 minor

Summary. The paper connects dual canonical basis elements of Grassmannian cluster algebras, indexed by rectangular semistandard Young tableaux, to positroidal subdivisions of the hypersimplex. The main theorem (Theorem 4.1) asserts that for every T in SSYT(k,[n]), the Speyer--Williams weight F_{k,n}(v_T) lies in the positive Dressian and hence induces a positroidal subdivision. For k=2, Theorem 5.2 claims that non-frozen prime tableaux induce split (coarsest) subdivisions, and the abstract states a precise correspondence with coarsest subdivisions. The paper also states Conjectures 4.2 and 5.8 about the k>2 case and supplies SageMath and polymake data.

Significance. The potential significance is substantial: if Theorem 4.1 holds, every dual canonical basis element acquires a canonical polyhedral subdivision, giving a new bridge between the representation-theoretic basis of Grassmannian cluster algebras and tropical geometry. The paper's main positive contribution is Theorem 4.1, which is a concise corollary of Speyer--Williams' parametrization together with the equality of the positive tropical Grassmannian and the positive Dressian. The computational evidence and the public code repository are concrete strengths. However, the advertised rank-two correspondence is not established as written: the proof of Theorem 5.2 relies on the unproved additivity Conjecture 5.3, and the converse direction is only asserted informally. The core idea is promising and likely repairable, but the abstract's precise 'correspond precisely' claim currently outruns the proof.

major comments (3)

[§5.1, proof of Theorem 5.2] The proof invokes the split decomposition theorem to write wΣ = w1 + ... + wj and then asserts that surjectivity of F_{k,n} gives tableaux T1,...,Tj with T = ∪ Ti and with F(v_{T_i}) inducing S_i. Surjectivity alone does not yield such a tableau decomposition of T; it only gives existence of some preimage for each weight. The needed identity F(v_{∪ Ti}) = Σ F(v_{Ti}) is exactly the content of Conjecture 5.3, which is stated later in the paper and is not proved. Without this identity, the contradiction that a one-column prime tableau cannot decompose is not obtained. This is a load-bearing gap in the forward direction of the rank-two theorem.
[§5.1, paragraph after Example 5.5] The converse of the claimed correspondence is only asserted informally. The text says that split positroidal subdivisions correspond to phylogenetic trees with exactly one internal edge and that non-frozen prime tableaux provide a canonical indexing, and it then lists the cells of the subdivision. No theorem or proof is supplied that every split positroidal subdivision of Δ(2,n) arises from the image of a non-frozen prime tableau under F_{2,n}. Since the abstract claims these tableaux 'correspond precisely' to the coarsest subdivisions, this direction must be stated as a theorem and proved.
[§5.1, Conjecture 5.3] Even if Conjecture 5.3 were proved, it is not immediately applicable in the proof of Theorem 5.2 as written: the conjecture assumes a union of pairwise weakly separated one-column tableaux, but the proof's split decomposition yields weights w_i whose preimages are not shown to be pairwise weakly separated. Thus the additivity identity used in the proof is conditional on an additional structural assertion that is not stated or proved.

minor comments (4)

[§5.1, proof of Theorem 5.2] There is a notation mismatch: the split subdivisions are indexed by j, while the tableau decomposition is written as T = ∪_{i=1}^m T_i; the proof should use a single index consistently.
[§5.1, proof of Theorem 5.2] The statement that 'F_{k,n} is a bijection' should specify the domain and codomain; F is a piecewise-linear bijection from the positive tropical Grassmannian fan to the corresponding subfan of the secondary fan, and this precision matters for the subsequent surjectivity argument.
[§3.2, definition of SSYT] The phrase 'The empty tableau is denoted by 1' is confusing because 1 is also used as an entry of tableaux; consider denoting the identity tableau by ∅ or by a separate symbol.
[§5.2, Conjecture 5.8] The enumeration formula (k-1)/2 · n(n-k-1) is stated as a conjecture for all k≥2, but for k=2 it coincides with the number of split subdivisions explicitly verified in Section 5.1; a short comment clarifying the relationship would help the reader calibrate the strength of the conjecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main derivation applies external Speyer–Williams and positive Dressian results, and the rank-two proof gap is a missing-support issue, not a circular reduction.

full rationale

The paper’s central claim, Theorem 4.1, is a direct application of established external results: Speyer and Williams’ parametrization of the positive tropical Grassmannian [27], the equality of the positive tropical Grassmannian with the positive Dressian [28, 1], and the classification of cones of the positive Dressian. The vector v_T is constructed from the fundamental-tableau factorization of T and is then fed into the Speyer–Williams map; no parameter is fitted to the target subdivisions and later renamed a prediction. There is no self-definitional loop, no ansatz smuggled through a citation, and no known result merely renamed in new coordinates. The paper does rely on prior work co-authored by the authors, notably [4] for the tableau indexing of dual canonical basis elements and [5] for the one-column classification of non-frozen prime tableaux in rank two, but these are prior structural theorems with independent content rather than citations whose only function is to force the present conclusions. The proof of Theorem 5.2 does contain a genuine missing-support issue: it passes from a sum of split weights to a tableau decomposition T = ∪ T_i by appealing to surjectivity/bijectivity of F_{k,n}, which would require an additivity property F(v_{∪ T_i}) = Σ F(v_{T_i}) that is not proved and is closely related to Conjecture 5.3. This is a correctness gap in the written derivation, not a circularity: Theorem 5.2 is not equivalent by construction to Conjecture 5.3, and no input is merely renamed as the output. For the reasons above, the derivation chain does not reduce to its own inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no fitted constants and no new algebraic entities beyond the tableaux and the Speyer-Williams map already in the literature. The main unproved input is the additivity of the map under tableau union, stated as Conjecture 5.3 but used in the proof of Theorem 5.2. The remaining axioms are standard theorems or domain assumptions from the cited literature, several of which are co-authored by Li and Tewari.

assumptions (7)

domain assumption Dual canonical basis elements of C[Gr(k,n)] are in bijection with rectangular semistandard Young tableaux with k rows and entries in [n].
Invoked throughout Sections 3 to 5; taken from [4, Section 5], which includes author Li. The paper does not reprove it.
domain assumption Every point of R^{(k-1)(n-k)} maps via the Speyer-Williams map F_{k,n} to a point of the positive tropical Grassmannian, and F is a bijection onto it.
Used in Section 4 to assert that wt_T lies in Trop+Gr(k,n) and in Theorem 5.2 to invoke surjectivity; taken from [27].
domain assumption The positive tropical Grassmannian equals the positive Dressian, Trop+Gr(k,n) = Dr+(k,n).
Theorem 2.7, from [28] and [1]. This is the key bridge in the proof of Theorem 4.1.
domain assumption Points of the positive Dressian Dr+(k,n) correspond to positroidal subdivisions of Delta(k,n).
Theorem 2.6 and [1, Theorem 1.1], used to conclude Theorem 4.1.
standard math The weight of a common refinement of split subdivisions is the sum of the split weights, by the Split decomposition theorem.
Used in the proof of Theorem 5.2; cited as [18, Theorem 3.10].
ad hoc to paper The Speyer-Williams map is additive with respect to partitioning a tableau into weakly separated one-column tableaux, so wt(union S_i) = sum_i wt(S_i).
Unproved; this is exactly Conjecture 5.3. The proof of Theorem 5.2 relies on it to decompose T into a union of tableaux.
domain assumption Every non-frozen prime tableau in SSYT(2,[n]) has a single column.
Lemma 5.1, cited to [5]; used in the proof of Theorem 5.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of From dual canonical bases to positroidal subdivisions." pith.science (2026). https://pith.science/paper/KZEQXJTV

@misc{pith2026250619443,
  author       = {Pith},
  title        = {Pith review of: From dual canonical bases to positroidal subdivisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZEQXJTV}},
  note         = {Machine review of arXiv:2506.19443}
}

abstract

The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We establish that each such tableau induces a positroidal subdivision of the hypersimplex $\Delta(k,n)$ via a map introduced by Speyer and Williams. For $\text{Gr}(2,n)$, we prove that non-frozen prime tableaux correspond precisely to the coarsest positroidal subdivisions of $\Delta(2,n)$. Furthermore, we present computational evidence extending these results to $k>2$. In the process, we formulate a conjectural formula for the number of split positroidal subdivisions of $\Delta(k,n)$ for any $k \ge 2$ and explore the deep connections between the polyhedral combinatorics of $\Delta(k,n)$ and the dual canonical basis of $\mathbb{C}[\text{Gr}(k, n)]$.

Figures

Figures reproduced from arXiv: 2506.19443 by the authors.

**Figure 1.** Positive Tropical Grassmannian TropGr(2, 5) + with the five rays and maximal cones described as common refinements of rays along with the corresponding non-frozen prime tableaux corresponding to each of the cones. T = {ij} can be listed as paths in the following way: Σ = {(∪ i s=1{(s, t) ∶ t ∈ [i + 1, j]}) ∪ (∪ j s=i+1 {(s, t) ∶ t ∈ [s + 1, n]}) , (∪ i s=1{(s, t) ∶ t ∈ [s + 1, n]}) ∪ (∪ n s=i+1{(s, t) ∶ t ∈ [j + 1, … view at source ↗

**Figure 2.** A phylogenetic tree corresponding to a split positroidal subdivision Σ of ∆(2, n) where the leaves i and j are the elements in the non-frozen prime tableaux T = {ij} that corresponds to Σ [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Phylogenetic trees that correspond to the two split positroidal subdivisions of ∆(2, 4) where the maximal cells of the subdivision are indexed by the internal vertices v1 and v2 and the maximal cells can be described as paths passing through each internal vertex. i2 − i1 + i4 − i3 = k − 2, or of the form [1, i1] ∪ [i2, i3] ∪ [i4, n] for some 1 ≤ i1 < i2 − 1, i2 ≤ i3 < i4 − 1, i4 ≤ n. Conjecture 5.8. Let T ∈ SSYT(k,… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Phylogenetic trees that correspond to the five split positroidal subdivisions of ∆(2, 5) where the maximal cells of the subdivision are indexed by the internal vertices v1 and v2 and the maximal cells can be described as paths passing through each internal vertex [PI…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

[1]

Arkani-Hamed, T

N. Arkani-Hamed, T. Lam, and M. Spradlin , Positive configuration space, Communications in Mathematical Physics, 384 (2021), pp. 909–954

work page 2021
[2]

Bendle, J

D. Bendle, J. B ¨ohm, Y. Ren, and B. Schr ¨oter, Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians , Journal of symbolic computation, 120 (2024), p. 102224

work page 2024
[3]

Bossinger and J.-R

L. Bossinger and J.-R. Li, Cluster structures on spinor helicity and momentum twistor varieties , arXiv preprint arXiv:2408.14956, (2024)

arXiv 2024
[4]

Chang, B

W. Chang, B. Duan, C. Fraser, and J.-R. Li , Quantum affine algebras and Grassmannians , Math. Z., 296 (2020), pp. 1539–1583

work page 2020
[5]

Cheung, P.-P

M.-W. Cheung, P.-P. Dechant, Y.-H. He, E. Heyes, E. Hirst, and J.-R. Li , Clustering cluster algebras with clusters , Adv. Theor. Math. Phys., 27 (2023), pp. 797–828

work page 2023
[6]

De Loera, J

J. De Loera, J. Rambau, and F. Santos , Triangulations: structures for algorithms and appli- cations, vol. 25, Springer Science & Business Media, 2010

work page 2010
[7]

Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux

N. Early and J.-R. Li , Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux, arXiv preprint arXiv:2406.16879, (2024)

work page Pith review arXiv 2024
[8]

, Tropical geometry, quantum affine algebras, and scattering amplitudes , Journal of Physics A: Mathematical and Theoretical, 57 (2024), p. 495201

work page 2024

Show all 31 references

[9]

J. Edmonds, Submodular functions, matroids, and certain polyhedra , in Combinatorial Optimiza- tion—Eureka, You Shrink! Papers Dedicated to Jack Edmonds 5th International Workshop Aussois, France, March 5–9, 2001 Revised Papers, Springer, 2003, pp. 11–26

2001
[10]

Fulton, Young tableaux, vol

W. Fulton, Young tableaux, vol. 35 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry

1997
[11]

Gawrilow and M

E. Gawrilow and M. Joswig , Polymake: a framework for analyzing convex polytopes , in Poly- topes—combinatorics and computation, Springer, 2000, pp. 43–73

2000
[12]

C. Geiß, B. Leclerc, and J. Schr ¨oer, Cluster structures on quantum coordinate rings , Selecta Math. (N.S.), 19 (2013), pp. 337–397

2013
[13]

I. M. Gelfand, R. M. Goresky, R. D. MacPherson, and V. V. Serganova , Combinatorial geometries, convex polyhedra, and schubert cells , Advances in Mathematics, 63 (1987), pp. 301–316

1987
[14]

I. M. Gelfand, M. Kapranov, and A. Zelevinsky , Discriminants, Resultants, and Multidi- mensional Determinants, Springer Science & Business Media, 2009

2009
[15]

Hernandez and B

D. Hernandez and B. Leclerc , Cluster algebras and quantum affine algebras , Duke Math. J., 154 (2010), pp. 265–341

2010
[16]

Hernandez and B

D. Hernandez and B. Leclerc , Quantum grothendieck rings and derived hall algebras , Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2015 (2015), pp. 77–126

2015
[17]

Herrmann, A

S. Herrmann, A. Jensen, M. Joswig, and B. Sturmfels , How to draw tropical planes , the electronic journal of combinatorics, 16 (2009), p. R6. FROM DUAL CANONICAL BASES TO POSITROIDAL SUBDIVISIONS 15

2009
[18]

Herrmann and M

S. Herrmann and M. Joswig , Splitting polytopes, M¨ unster Journal of Mathematics, (2008)

2008
[19]

Kapranov, Chow quotients of Grassmannians i, im gelfand seminar, 29–110 , Adv

M. Kapranov, Chow quotients of Grassmannians i, im gelfand seminar, 29–110 , Adv. Soviet Math, 16 (1993)

1993
[20]

Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras , Duke Mathe- matical Journal, 63 (1991), pp

M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras , Duke Mathe- matical Journal, 63 (1991), pp. 465–516

1991
[21]

Li, Dual canonical bases for unipotent groups and base affine spaces , Journal of Algebra and its Applications, https://doi.org/10.1142/S0219498825503542, (2024)

J.-R. Li, Dual canonical bases for unipotent groups and base affine spaces , Journal of Algebra and its Applications, https://doi.org/10.1142/S0219498825503542, (2024)

2024 doi
[22]

Lusztig, Canonical bases arising from quantized enveloping algebras , Journal of the American Mathematical Society, 3 (1990), pp

G. Lusztig, Canonical bases arising from quantized enveloping algebras , Journal of the American Mathematical Society, 3 (1990), pp. 447–498

1990
[23]

Maclagan and B

D. Maclagan and B. Sturmfels , Introduction to tropical geometry, vol. 161, American Mathe- matical Society, 2021

2021
[24]

J. G. Oxley, Matroid theory, vol. 3, Oxford University Press, USA, 2006

2006
[25]

J. S. Scott, Grassmannians and cluster algebras, Proceedings of the London Mathematical Society, 92 (2006), pp. 345–380

2006
[26]

Speyer and B

D. Speyer and B. Sturmfels , The tropical Grassmannian., Advances in Geometry, 4 (2004)

2004
[27]

Speyer and L

D. Speyer and L. Williams , The tropical totally positive Grassmannian , Journal of Algebraic Combinatorics, 22 (2005), pp. 189–210

2005
[28]

, The positive Dressian equals the positive tropical Grassmannian, Transactions of the American Mathematical Society, Series B, 8 (2021), pp. 330–353

2021
[29]

D. E. Speyer, Tropical linear spaces, SIAM Journal on Discrete Mathematics, 22 (2008), pp. 1527– 1558

2008
[30]

https://www.sagemath.org

The Sage Developers , SageMath, the Sage Mathematics Software System (Version 10.4) , 2024. https://www.sagemath.org

2024
[31]

G. M. Ziegler, Lectures on polytopes, vol. 152, Springer Science & Business Media, 2012. Jian-Rong Li: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern Platz 1, 1090 Vienna, Austria Email address : lijr07@gmail.com Ayush Kumar Tewari: Johann Radon Institute for ...

2012

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Arkani-Hamed, T

N. Arkani-Hamed, T. Lam, and M. Spradlin , Positive configuration space, Communications in Mathematical Physics, 384 (2021), pp. 909–954

work page 2021

[2] [2]

Bendle, J

D. Bendle, J. B ¨ohm, Y. Ren, and B. Schr ¨oter, Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians , Journal of symbolic computation, 120 (2024), p. 102224

work page 2024

[3] [3]

Bossinger and J.-R

L. Bossinger and J.-R. Li, Cluster structures on spinor helicity and momentum twistor varieties , arXiv preprint arXiv:2408.14956, (2024)

arXiv 2024

[4] [4]

Chang, B

W. Chang, B. Duan, C. Fraser, and J.-R. Li , Quantum affine algebras and Grassmannians , Math. Z., 296 (2020), pp. 1539–1583

work page 2020

[5] [5]

Cheung, P.-P

M.-W. Cheung, P.-P. Dechant, Y.-H. He, E. Heyes, E. Hirst, and J.-R. Li , Clustering cluster algebras with clusters , Adv. Theor. Math. Phys., 27 (2023), pp. 797–828

work page 2023

[6] [6]

De Loera, J

J. De Loera, J. Rambau, and F. Santos , Triangulations: structures for algorithms and appli- cations, vol. 25, Springer Science & Business Media, 2010

work page 2010

[7] [7]

Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux

N. Early and J.-R. Li , Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux, arXiv preprint arXiv:2406.16879, (2024)

work page Pith review arXiv 2024

[8] [8]

, Tropical geometry, quantum affine algebras, and scattering amplitudes , Journal of Physics A: Mathematical and Theoretical, 57 (2024), p. 495201

work page 2024