Invariant theory for coincidental complex reflection groups

Anne V. Shepler; Eric Sommers; Victor Reiner

REVIEW 2 major objections 5 minor 40 references

Invariant theory for coincidental complex reflection groups

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single corrected Hilbert-series formula holds exactly for the coincidental complex reflection groups, and yields the cluster-combinatorics q-analogues along the way.

desk verdict A solid correction of Molchanov's conjecture with rigorous proofs for the infinite families and rank 2, but the four exceptional groups rest on an undocumented Mathematica check, so the main theorem is not fully verified as written. read the letter →

arxiv 1908.02663 v2 pith:2BR2URIH submitted 2019-08-07 math.CO math.RT

Victor Reiner , Anne V. Shepler , Eric Sommers This is my paper · ORCID

classification math.COmath.RT MSC 13A5005Axx20F55

keywords reflectiongroupsinvarianttheoryHilbertseriescoincidentalexponentsandcoexponentsq-Narayananumbersq-Kirkmanclustercomplexes

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 proves that one explicit product formula gives the Hilbert series of invariant mixed derivation–differential forms for exactly the coincidental complex reflection groups: the irreducible well-generated reflection groups whose exponents form an arithmetic progression. The formula corrects an earlier speculation offered for real reflection groups: the corrected version holds for all coincidental groups, including the infinite monomial family $G(d,1,n)$ and every non-real Shephard group, and fails for every other irreducible reflection group. Along the way the paper supplies Hilbert-series data for all irreducible complex reflection groups and derives product formulas for $q$-Catalan, $q$-Kirkman, and $q$-Narayana numbers, together with a $q$-analogue of the standard $h$-vector-to-$f$-vector transformation for finite type cluster and Cambrian complexes. A sympathetic reader should care because the result ties invariant-theoretic numerology to the combinatorics of noncrossing partitions and cluster fans through one formula.

What carries the argument

The load-bearing object is the triply graded Hilbert series of the invariant space $(S(V^*)\otimes \wedge V^*\otimes \wedge V)^W$, examined degree by degree in polynomial, dual-exterior, and exterior variables. The formula is organized around the exponents $e_i$ and coexponents $e^*_i$; for coincidental $W$ the coexponents form the arithmetic progression $1,1+a,\ldots,1+(n-1)a$, so the elementary symmetric function $\sigma_r(q^{e^*_1},\ldots,q^{e^*_n})$ simplifies to $q^{r+a\binom{r}{2}}\left[\begin{smallmatrix}n\\r\end{smallmatrix}\right]_{q^a}$. The proof runs through an explicit conjectured basis built from invariant derivations acting on exterior products of basic derivations; invariance and the correct degree sum reduce freeness to checking that certain determinant expressions are nonzero.

What would settle it

Recompute the Hilbert series for $G_{32}$ (or $H_3$) by direct Molien averaging over the group for small degrees, for instance all total degrees up to 12, and compare coefficient-by-coefficient with Theorem 1.1; any mismatch would refute the formula. A second check is to evaluate the determinant of the conjectured basis elements in Conjecture 4.1$'$ at a randomly chosen point in $\mathbb{C}^n$ and see whether it is zero.

Watch

Extended reading notes

Core claim

The central assertion, Theorem 1.1, is that for any coincidental complex reflection group $W$ acting on $V = \mathbb{C}^n$, $$\operatorname{Hilb}\left((S(V^*)\otimes \wedge V^*\otimes \wedge V)^W,q,t,s\right)=\sum_{r=0}^n s^r \sigma_r($q^{{e^*_1}}$,\ldots,$q^{{e^*_n}}$)\frac{\prod_{i=1}^r(1+$q^{{-e^*_i}}$t)\prod_{i=1}^{n-r}(1+$q^{{e_i}}$t)}{\prod_{i=1}^n(1-$q^{{d_i}}$)},$$ where the coefficient of $q^i t^k s^r$ is the dimension of $(S^i(V^*)\otimes \wedge^k V^*\otimes \wedge^r V)^W$. The proof treats the Weyl groups of type $A$ and the family $G(d,1,n)$ by reduction to known symmetric-function Hilbert-series formulas, all rank-two coincidental groups by determinant nonvanishing checks, and the four exceptional groups $H_3$, $G_{25}$, $G_{26}$, and $G_{32}$ by a computational verification that a conjectured explicit basis is linearly independent. The paper also shows that the formula holds for an irreducible reflection group if and only if the group is coincidental, and it tabulates the actual Hilbert series for the non-coincidental exceptional groups.

Load-bearing premise

The load-bearing premise is that the omitted Mathematica computation verifying Conjecture 4.1 for the four exceptional groups $H_3$, $G_{25}$, $G_{26}$, and $G_{32}$ is correct; no code, output, or determinant expressions are shown, and if that computation is wrong the main formula fails for those groups.

Editorial extensions

If this is right

The Hilbert series of invariant mixed forms is now known in closed product form for every coincidental complex reflection group, including all non-real Shephard groups.
Setting $t = -q^p$ yields product formulas for the $q$-Kirkman numbers, the graded multiplicities of $\wedge^r V$ in parking-space representations, for every coincidental group.
Setting $s=0$ and $t=-q^p$ expresses the $q$-Catalan number as a sum of $q$-Narayana numbers, recovering the earlier type $A$ and type $B/C$ product formulas.
Theorem 1.5 gives a $(q,t)$-analogue of the identity that converts an $h$-vector into an $f$-vector, and at $q=1$, $t=-q^{h+1}$ it recovers the face counts of the finite type cluster and Cambrian fans.
For every non-coincidental irreducible reflection group, the formula fails; the paper supplies the actual Hilbert series data for those groups.

Reading between the lines

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

The explicit conjectured basis, if it holds throughout the coincidental family, gives a uniform free-module structure that likely supports natural representation-theoretic bases for the $q$-Kirkman and $q$-Narayana spaces, not just their Hilbert series.
Because the $q$-analogues come from a single product formula, they may imply a cyclic sieving phenomenon for noncrossing partitions across all coincidental types, not only the classical Weyl cases, when $q$ is specialized to roots of unity.
The one computational step in the main theorem, the omitted Mathematica verification for the four exceptional groups, could be replaced by a human-checkable determinant computation; finding one would remove the only non-constructive part of the proof.
The exact dichotomy of coincidental versus non-coincidental suggests that the arithmetic-progression condition marks the boundary of any one-term product formula of this shape, so other reflection families may require multi-term but still uniform expressions.

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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

Dear colleague,

The headline: this is a real contribution, but the proof of the main theorem is not quite complete as written. Reiner–Shepler–Sommers correct Molchanov's 1992 conjecture, showing the proposed product formula fails in general and holds exactly for coincidental complex reflection groups. The corrected formula, with exponents and coexponents, gives a uniform Hilbert series for mixed invariant differential forms. That and the q-analogue transformation between f and h vectors, with consequences for q-Narayana and q-Kirkman polynomials, are new and worth having.

What they do well: the derivations for type A and G(d,1,n) via Kirillov–Pak and Koike are clean; the rank-2 case is proven by explicit determinant checks; the compiled data for non-coincidental groups is useful. The paper is clearly written and honest about what is and is not proven. The comparison to Molchanov's original hypothesis is careful.

The soft spot is exactly the one you would expect from Section 7: for the exceptional groups H3, G25, G26, and G32, the proof of Theorem 1.1 goes through Conjecture 4.1′, which they say they checked in Mathematica via Proposition 6.5, but they provide no code, no outputs, no determinant expressions. Since the conjecture implies the main theorem, this leaves the central claim unverified for four of the coincidental groups. I do not think this is a sign of a wrong result — the formula can be checked independently for these groups by a finite Molien sum, and the rest of the paper gives strong structural evidence — but it is a gap in the proof as written, and it should be fixed before the paper is accepted. A short appendix with the code or an independent verification would suffice.

There are a few minor typos (e.g., "exponent gap q" near the end of Section 10 should clearly be "gap a"), but nothing that affects the mathematics.

Who is this for: anyone working on reflection group invariants, cluster combinatorics, or q-Catalan/Narayana/Kirkman numbers. The paper deserves a serious referee. I would send it out, with the request that the computational gap be addressed.

Best,

Referee Report

2 major / 5 minor

Summary. The paper studies the triply graded Hilbert series Hilb((S(V*) ⊗ ∧V* ⊗ ∧V)^W, q, t, s) for finite complex reflection groups. It shows that Molchanov's proposed product formula is false in general and proves Theorem 1.1: for every coincidental complex reflection group W, this Hilbert series equals ∑_{r=0}^n s^r σ_r(q^{e*_1},...,q^{e*_n}) (∏_{i=1}^r (1+q^{-e*_i}t) ∏_{i=1}^{n-r} (1+q^{e_i}t)) / ∏_{i=1}^n (1-q^{d_i}). The proof treats type A and the family G(d,1,n) using Kirillov–Pak and Koike, treats all rank-2 duality groups by explicit determinant checks, and dispatches the exceptional groups H3, G25, G26, and G32 via a stated Mathematica verification of Conjecture 4.1′. The paper also computes analogous Hilbert series for all non-coincidental irreducible complex reflection groups, derives product formulas for q-Narayana and q-Kirkman polynomials, and proves a (q,t)-analogue of the standard h-vector-to-f-vector transformation.

Significance. If the main theorem is correct, it settles Molchanov's speculation in a clean way, unifies previous product formulas for cluster/Cambrian f-vectors, and gives explicit q-Narayana and q-Kirkman formulas for all coincidental groups. The paper's strengths are the detailed and checkable proofs for the infinite families and rank 2, the explicit Hilbert-series data for non-coincidental groups, and the crisp applications to known combinatorial objects. The main weakness is that the proof for the four exceptional groups rests on an undocumented computer check, which is the one place where the central claim is not independently auditable from the text.

major comments (2)

[Section 7, proof of Theorem 1.1, third bullet] The proof for H3, G25, G26, and G32 consists solely of the statement that Conjecture 4.1′ was checked in Mathematica via Proposition 6.5, using choices of {f_i} and {θ_i} from [30] and [26, App. B.3]. Since Proposition 4.4 shows that Conjecture 4.1′ implies Theorem 1.1, the main theorem for these four groups depends entirely on this unshown computation. No code, determinant expressions, or outputs are provided. This is a load-bearing gap, not a presentation issue: an error in the check would falsify Theorem 1.1 for a coincidental group. Please supply the verification artifacts (e.g., a script that computes the relevant determinants and prints their nonvanishing values) or replace this step by an independent check, such as directly comparing the finite Molien sum in equation (2.11) with the right-hand side of Theorem 1.1 for each of H3, G25, G26, and G32.
[Section 7, Proposition 6.5 and Section 11] The claim in Remark 7.1 that Theorem 1.1 holds if and only if W is coincidental is not fully demonstrated for the non-coincidental exceptional groups. While Remark 3.19 explicitly shows the failure for G(de,e,n) when e ≥ 2, the non-coincidental exceptional cases are supported only by the table in Section 11, whose entries are asserted without explanation of how they were computed or how they differ from the Theorem 1.1 expression. Since the 'if and only if' statement is advertised in the abstract, please either provide the computations behind the table or state explicitly, for each row, a coefficient or specialization that distinguishes the tabulated ν_r(W,q,t) from the formula predicted by Theorem 1.1.

minor comments (5)

[Section 10, paragraph after equation (10.2)] The phrase 'exponent gap q' should read 'exponent gap a'; q is already used as the grading variable.
[Section 8, first sentence] The phrase 'as notaion tha t' contains typos and should be 'as notation that'.
[Proposition 6.5] The word 'derviation' should be 'derivation' in the statement.
[Proposition 4.4 proof] The word 'appreviate' should be 'abbreviate'.
[Section 11] The table of ν_r(W,q,t) for non-coincidental exceptional groups would be much more useful if the manuscript stated the method used to obtain these polynomials, even briefly, or pointed to code or auxiliary files.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the Hilbert series formula is derived from external hook-content theorems and independent determinant checks, not from its own conclusion.

full rationale

The paper's central Theorem 1.1 is not assumed anywhere in its proof. For the infinite families, Section 3 computes the relevant Molien sums directly from Kirillov-Pak's and Koike's hook-content formulas for symmetric and monomial groups, then identifies the resulting expressions with the right-hand side of Theorem 1.1 using only the group's exponents and coexponents. For rank two groups, Section 8 proves the stronger Conjecture 4.1 by reducing, via Proposition 6.5, to checking nonzero determinants; this reduction uses the Gutkin-Opdam degree sum and Lemma 6.4, neither of which assumes the target Hilbert series. For the remaining exceptional groups H3, G25, G26, and G32, the proof cites a Mathematica verification of Conjecture 4.1 using Proposition 6.5 with explicit choices of basic invariants and derivations. This step is opaque because no code or output is included, but it is a computational independence check, not a circular use of the theorem. Proposition 4.4 only shows that Conjecture 4.1 implies Theorem 1.1, which is legitimate because the conjecture is then verified independently. The cited prior results [28], [32], and [33], though partly self-citations, are external theorems with stated assumptions and do not assume the present formula. The proof is therefore self-contained in the sense of not relying on its own conclusion; any concern about the undocumented exceptional-group computation is a reproducibility issue, not circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claims rest on standard theorems in invariant theory of reflection groups and on classification data; the only additional burden is the undocumented computer check for the four exceptional groups.

assumptions (8)

standard math Shephard-Todd-Chevalley theorem: S(V*)^W is a polynomial algebra with degrees d_i.
Used throughout Section 2 to define exponents and basic invariants.
standard math Solomon's theorem: (S⊗∧V*)^W is an exterior algebra over S^W on df_i.
Used in Section 2 to express the Hilbert series for invariant differential forms (eq. 2.6).
standard math Orlik-Solomon theorem: (S⊗∧V)^W is an exterior algebra over S^W on basic derivations.
Used in Section 2 to express the Hilbert series for invariant derivation forms (eq. 2.7).
standard math Gutkin-Opdam lemma: formula for the sum of U-exponents via hyperplane sums.
Used in Section 6 to predict degree sums and in Proposition 6.3.
standard math Classification of finite irreducible complex reflection groups (Shephard-Todd).
Used to enumerate coincidental and non-coincidental groups and their exponents/coexponents.
standard math Koike's Poincaré series formula for G(de,e,n) (Theorem 3.17).
Cited from [20]; used to compute Hilbert series for monomial groups and to show failure of the formula when e > 1.
standard math Kirillov-Pak / Molchanov / Thibon / Gyoja-Nishiyama-Shimura hook-content formula for S_n (Theorem 3.3).
Cited from [14, 19, 24]; building block for type A and G(d,1,n) computations.
ad hoc to paper The Mathematica computations checking Conjecture 4.1' for H3, G25, G26, G32 are correct.
Section 7 states these checks were done but no code or output is included; the central theorem for these groups depends on them.

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Cite this review

Pith. "Pith review of Invariant theory for coincidental complex reflection groups." pith.science (2026). https://pith.science/paper/2BR2URIH

@misc{pith2026190802663,
  author       = {Pith},
  title        = {Pith review of: Invariant theory for coincidental complex reflection groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2BR2URIH}},
  note         = {Machine review of arXiv:1908.02663}
}

abstract

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family $G(d,1,n)$. We highlight consequences for the $q$-Narayana and $q$-Kirkman polynomials, giving simple product formulas for both, and give a $q$-analogue of the identity transforming the $h$-vector to the $f$-vector for the coincidental finite type cluster/Cambrian complexes of Fomin--Zelevinsky and Reading.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [30]

Saito, T

K. Saito, T. Yano, J. Sekiguchi, On a certain generator s ystem of the ring of invariants of a ﬁnite reﬂection group. Comm. Algebra 8 (1980), 373–408

work page 1980

[2] [1]

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D. Armstrong, Generalized noncrossing partitions and c ombinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202 (2009), no. 949

work page 2009

[3] [2]

Armstrong, V

D. Armstrong, V. Reiner and B. Rhoades, Parking spaces. Adv. Math. 269 (2015), 647–706

work page 2015

[4] [3]

Bessis and V

D. Bessis and V. Reiner, Cyclic sieving of noncrossing pa rtitions for complex reﬂection groups. Ann. Comb. 15 (2011), 197–222

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Matrix Factorisations Arising From Well-Generated Complex Reflection Groups

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Fomin and N

S. Fomin and N. Reading, Generalized cluster complexes a nd Coxeter combinatorics. Int. Math. Res. Not. (2005), no. 44, 2709–2757

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