REVIEW 2 major objections 3 minor 5 references
On a Schur-positive function
T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash
Pith's one-line read The symmetric function built by averaging over e-th roots of unity is Schur positive.
desk verdict Resolves the MS24 conjecture for all e and gives a clean SYT interpretation; the Gessel–Viennot proof has a correctable endpoint error, but the Aitken-based proof is solid. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing objects are the skew shape ξ(λ)=ν/μ, which stretches λ by the factor e and adds a staircase-like border, and the ring homomorphism φ^{(e)} on symmetric functions defined by φ^{(e)}(h_n)=1/(en)!. The positivity proofs hinge on two determinantal identities: an expression for a_λ as a determinant of reciprocal factorials, derived from the standard determinantal formula for Schur functions, and the classical determinant formula that counts standard Young tableaux of a skew shape as a determinant of reciprocal factorials. Matching these determinants gives the tableau interpretation. The third proof runs through the Grothendieck group of finite-dimensional modules for the quiver H
What would settle it
Take e=2 and n=3, and compute a_{(2,1)} two ways: from the determinant in Corollary 2.2 (entries 1/(2(λ_i+j-i))!), and from the number of standard Young tableaux of shape ξ(2,1) as defined in Section 4. If the two numbers disagree, the central tableau interpretation is false. The same check can be repeated for any small e and λ by hand or by a few lines of code.
Extended reading notes
Core claim
The central claim is that for every positive integer e and every n, the symmetric function W^{(e)}_n is Schur positive: its expansion in Schur functions has only positive integer coefficients. The paper proves the stronger statement that the coefficient a_λ counts standard Young tableaux of the skew shape ξ(λ)=ν/μ, where ν_i = eλ_i + (ℓ-i)(e-1) and μ_i = (ℓ-i)(e-1). A determinantal formula for a_λ follows from the standard determinantal expansion of Schur functions, and positivity is shown three ways: through non-intersecting lattice paths, through a classical determinant formula for skew tableau counts, and by identifying a_λ with the dimension of a simple module for a quiver Hecke algebra
Load-bearing premise
The load-bearing premise is that the classical determinant formula for skew tableau counts, applied to the specific pair of partitions ν and μ, produces exactly the determinant formula derived for a_λ; the paper cites this formula rather than proving it or verifying the substitution in detail.
Editorial extensions
If this is right
- For every partition λ of n, the coefficient a_λ is the explicit positive integer #SYT(ξ(λ)), so the Schur expansion has a direct combinatorial description.
- The identity Σ_{λ⊢n} #SYT(λ) #SYT(ξ(λ)) = (en)!/(e!)^n follows, giving an e-parameter analogue of a classical permutation-counting identity.
- W^{(e)}_n is the Frobenius characteristic of an explicit symmetric-group module, so its Schur coefficients are genuine representation dimensions.
- Each simple quiver Hecke module L(λ) has dimension #SYT(ξ(λ)) and, in characteristic zero, is isomorphic to a skew-shape module with the same Young diagram.
- The e=1 case reduces the combinatorial identity to the classical correspondence for permutations, suggesting a bijective 'e-version' of that correspondence.
Reading between the lines
- The skew shape ξ(λ) suggests that a_λ may admit a hook-length-style product formula; evaluating the skew hook-length formula for ξ(λ) might simplify to a product over the cells of λ, a direct check the paper does not carry out.
- The argument uses the root-system structure of affine type A, so the same averaging construction over other finite cyclic groups, or in other affine types, may yield Schur-positive families by identical methods.
- Since L(λ) is shown to be a skew-shape module, W^{(e)}_n might be the Frobenius characteristic of a module built by classical symmetric-group constructions, not necessarily requiring quiver Hecke algebras for its definition.
- A q-analogue of the determinant formula would produce a one-parameter deformation of W^{(e)}_n that is Schur positive at generic q and reduces to the present function at q=1; checking whether such a deformation exists is a natural next step.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines symmetric functions W_n^{(e)} from average powers of sums of e-th roots of unity, and proves that they are Schur positive. It gives a determinantal formula for the Schur coefficients a_λ, then offers three proofs of positivity: a Gessel–Viennot nonintersecting lattice path argument, a proof via Aitken's determinantal formula for standard Young tableaux of skew shapes, and a representation-theoretic proof using KLR algebras. The second proof yields the explicit combinatorial interpretation a_λ = #SYT(ξ(λ)) for a skew shape ξ(λ) constructed from λ.
Significance. The main result is a clean Schur-positivity theorem with a concrete SYT interpretation, which is a genuine contribution. The second proof is elegant and essentially self-contained modulo Aitken's classical determinant formula, and it also produces the nontrivial identity in Corollary 4.2. The connection to KLR algebras is a suggestive additional perspective. These strengths make the paper worth publishing, provided the flaws in the other two proofs are addressed.
major comments (2)
- [§3 (First proof)] The Gessel–Viennot path interpretation is misstated. For an entry binom(N,K) with N=e(λ_i+ℓ−i) and K=e(ℓ−j), the number of east/north paths from (0,−N) to (K,K) is binom(N+2K,K), not binom(N,K). Thus the determinant is not equal to the number of path families described. Concretely, for e=2, n=2, λ=(1,1), the asserted formula gives 13 instead of the true a_λ=5. The proof can be repaired by changing the endpoints to (e(ℓ−j),−e(ℓ−j)), for which the path count is binom(N,K). As written, this first proof is invalid.
- [§5 (Third proof)] This section is too terse to constitute a proof. The existence and character of the KLR module L are asserted without proof or reference, the isomorphism K0(C_n) ≅ Λ is quoted from [McN17] without stating the hypotheses needed, and the claim that [M] ↦ dim M/(en)! is a ring homomorphism requires justification (e.g., multiplicativity of dimensions under induction in these categories). The closing sentence about L(1^n) being one-dimensional does not by itself identify φ^{(e)}. If this is meant as a third proof, it needs to be expanded; otherwise it should be labeled as a representation-theoretic interpretation.
minor comments (3)
- [Equation (1.1)] The summation index 'λ⊢r' should be 'λ⊢n': the monomial symmetric functions m_λ have degree n, not r, and the multinomial coefficient is binom(en,eλ).
- [§3] Typo 'postivity' should be 'positivity'.
- [§5] Typo 'The the results' should be 'The results'.
Circularity Check
No substantive circularity: main Schur-positivity claim is proved by an explicit SYT count via Aitken's determinant formula; self-citations appear only in an optional KLR interpretation.
full rationale
The paper's main claim, Theorem 1.1, does not reduce to its own inputs. W_n^(e) is defined by a monomial expansion, and Theorem 2.1/Corollary 2.2 give a determinantal formula for a_lambda via Jacobi-Trudi. Section 4 then substitutes nu_i = e lambda_i + (ell-i)(e-1) and mu_j = (ell-j)(e-1) into Aitken's classical determinant formula (4.1). The exponent becomes e(lambda_i + j - i), exactly matching the determinant in (2.1), and |nu/mu| = en, so a_lambda = #SYT(xi(lambda)). This is an explicit, non-circular identification: Aitken's formula is an external theorem and the substitution is stated rather than fitted to the target. The third proof (Section 5) does rely substantially on [McN17], a prior paper by the same author, for the KLR categorification; the passage 'For reasons of space we punt the definition of KLR algebras to [McN17]' explicitly defers definitions, and the line 'it follows from the fact that L(1^n) is one-dimensional' is a terse omitted proof. These are rigor or support gaps, but not circularity: the cited machinery does not assume W_n^(e) is Schur positive, and this proof is optional because Section 4 already proves Theorem 1.1. [MS24] is cited only as the source of the conjecture, not as evidence for it. I also note a correctness concern in Section 3: the Gessel-Viennot endpoints are likely misstated, since paths from (0,-N) to (K,K) number binom(N+2K,K), not binom(N,K). However, an erroneous or compressed step is not a circular step; no quantity is fitted to a later prediction, and no load-bearing uniqueness or ansatz is imported from the author's own prior work. Thus the circularity score is low: 0 for the central proof, 2 reflecting the optional but nontrivial self-citation load in Section 5.
Assumptions & free parameters
assumptions (6)
- standard math The complete symmetric functions {h_λ} are dual to the monomial symmetric functions {m_λ}; Schur functions form an orthonormal basis for the Hall inner product.
- standard math Jacobi-Trudi identity: s_λ = det(h_{λ_i - i + j})
- standard math Gessel-Viennot theorem: a determinant of binomial coefficients equals the number of non-intersecting lattice paths when the entries are ordered.
- standard math Aitken's determinant formula (4.1): for a skew shape θ=α/β, #SYT(θ)/|θ|! = det(1/(α_i + j - i - β_j)!).
- domain assumption KLR algebra results from [McN17]: the category C_n has Grothendieck group isomorphic to Λ, the induction product corresponds to multiplication, and the simple modules L(λ) exist with the stated decomposition of L^{∘n}.
- standard math The map φ^{(e)} defined on h_n extends to a ring homomorphism because the h_n are algebraically independent generators of Λ.
Cite this review
Pith. "Pith review of On a Schur-positive function." pith.science (2026). https://pith.science/paper/XYREMQIL
@misc{pith2026260728682,
author = {Pith},
title = {Pith review of: On a Schur-positive function},
year = {2026},
howpublished = {\url{https://pith.science/paper/XYREMQIL}},
note = {Machine review of arXiv:2607.28682}
}
read the original abstract
We prove Schur-positivity for a family of symmetric functions.
Reference graph
Works this paper leans on
-
[1]
and Savage, Alistair , TITLE =
McNamara, Peter J. and Savage, Alistair , TITLE =. Forum Math. Sigma , FJOURNAL =. 2024 , PAGES =. doi:10.1017/fms.2024.102 , URL =. 2312.11766 , archivePrefix =
arXiv 2024
-
[2]
Gessel, Ira and Viennot, G\'erard , TITLE =. Adv. in Math. , FJOURNAL =. 1985 , NUMBER =. doi:10.1016/0001-8708(85)90121-5 , URL =
- [3]
- [4]
-
[5]
Aitken, A. C. , TITLE =. Proc. Roy. Soc. Edinburgh Sect. A , FJOURNAL =. 1943 , PAGES =
1943
Reviewed August 3, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.