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arxiv: 2406.19715 · v3 · submitted 2024-06-28 · 🧮 math.CO

A conjectural basis for the (1,2)-bosonic-fermionic coinvariant ring

John Lentfer This is my paper

Pith reviewed 2026-05-23 23:48 UTC · model grok-4.3

classification 🧮 math.CO
keywords coinvariant ringsbosonic-fermionic variablessymmetric groupmonomial basisHilbert seriesFrobenius seriesZabrocki conjecturesegmented permutations
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The pith

A conjectural monomial basis for the (1,2)-bosonic-fermionic coinvariant ring has the cardinality matching Zabrocki's dimension conjecture.

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

The paper constructs the first explicit conjectural monomial basis for the coinvariant ring R_n^(1,2) of the symmetric group acting on one set of commuting variables and two sets of anticommuting variables. This set interpolates between the modified Motzkin path basis for the (0,2) case and the super-Artin basis for the (1,1) case. A sympathetic reader cares because the construction proves that the basis elements number exactly 2^{n-1} n!, confirming the dimension predicted by Zabrocki, and it supplies an explicit combinatorial formula for the Hilbert series. The work further equates this proposal to an independent conjecture phrased in segmented permutations through a weight-preserving bijection and extends some character computations while adding a conjecture for the type B_n analog.

Core claim

We give the first conjectural construction of a monomial basis for the coinvariant ring R_n^(1,2) for the symmetric group S_n acting on one set of bosonic and two sets of fermionic variables. Our construction interpolates between the modified Motzkin path basis for R_n^(0,2) and the super-Artin basis for R_n^(1,1). We prove that our proposed basis has cardinality 2^{n-1}n!, aligning with a conjecture of Zabrocki on the dimension of R_n^(1,2), and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for R_n^(1,2) and show equivalence to conjectures in terms of segmented Smirnov words via a weight-preserving bijection. We extend some sign-2

What carries the argument

The conjectural monomial basis obtained by interpolating the modified Motzkin path basis and the super-Artin basis.

If this is right

  • The dimension of R_n^(1,2) equals 2^{n-1} n!.
  • The Hilbert series of the ring admits a direct combinatorial count from the graded pieces of the proposed basis.
  • The conjectural Frobenius series is equivalent to the one expressed via segmented Smirnov words.
  • Results previously known for the sign character extend to hook characters together with an explicit formula for the m_μ coefficients.
  • The type B_n analog admits a conjectural monomial basis of cardinality 4^n n!.

Where Pith is reading between the lines

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

  • A proof that these monomials are independent would simultaneously settle the dimension conjecture for this specific ring.
  • The explicit bijection with segmented permutations may allow algebraic identities in the ring to be translated into purely combinatorial statements about those permutations.
  • The interpolation pattern used here suggests a possible route for constructing bases in other (k,l)-bosonic-fermionic coinvariant rings.
  • The extension to hook characters indicates that the full graded character table might be accessible through similar combinatorial bookkeeping.

Load-bearing premise

The selected monomials are linearly independent in the quotient ring obtained by dividing out the ideal of positive-degree invariants.

What would settle it

For n=3 an explicit linear-algebra computation of the dimension of R_3^(1,2) that yields a number other than 24, or the discovery of a nontrivial linear dependence relation among the 24 proposed monomials inside that ring.

Figures

Figures reproduced from arXiv: 2406.19715 by John Lentfer.

Figure 1
Figure 1. Figure 1: The basis B (0,2) 3 . Each modified Motzkin path is labeled with its corre￾sponding monomial. Next, we recall the super-Artin basis, defined by Sagan and Swanson. Let χ(P) be 1 if the proposition P is true, and 0 if the proposition P is false. Let θT denote the ordered product θt1 · · · θtk for any subset T = {t1 < · · · < tk} ⊆ {1, . . . , n}. For any T ⊆ {2, . . . , n}, define the α-sequence α(T) = (α1(T… view at source ↗
Figure 2
Figure 2. Figure 2: The basis B (1,1) 3 . The α-sequence is shown as the outline of all boxes, and those xi used for a particular basis element are shaded in gray. In this paper, we assume familiarity with the basics of symmetric function theory (see for example [25, Chapter I] or [29, Chapter 7]). Let mλ, eλ, hλ, pλ, and sλ denote respectively the monomial, elementary, complete homogeneous, power-sum, and Schur symmetric fun… view at source ↗
Figure 3
Figure 3. Figure 3: The basis B (1,2) 3 . Below each modified Motzkin path is the outline of the generalized α-sequence, and those xi used for a particular basis element are shaded in gray. Note that the outline of the generalized α-sequence can be determined by the following rule: in each position i, the column of boxes extends up until its right side is one unit below the path above it. Conjecture 3.2. The set B (1,2) n is … view at source ↗
Figure 4
Figure 4. Figure 4: Examples of applying the slinky rule. Proposition 8.2. An element b ∈ B (1,2) n satisfies Asc(b) = {d + 1, . . . , n − 1} if and only if when b is written as b = ± Yn m=1 x αm m θ βm m ξ γm m , for some αm ∈ Z≥0 and βm, γm ∈ {0, 1}, we have that for some a ∈ {d + 1, . . . , n}, (a) αm = βm = γm = 0 for all m ∈ {1, . . . , d + 1}; (b) βm = 0 and αm−1 < αm + γm for all m ∈ {d + 2, . . . , a}; (c) βa = 0 and … view at source ↗
Figure 5
Figure 5. Figure 5: A diagram showing the three regions for some b with Asc(b) = {d + 1, . . . , n − 1}. The boxes shaded in crosshatch indicate where there is a minimal structure of boxes which must be filled in, and then above the minimal structure, boxes may be filled in, subject to certain increasing/decreasing conditions. some facts on such a b. Region 0 will consist of only up-steps with no contribution to the x-degree.… view at source ↗
Figure 6
Figure 6. Figure 6: All possible paths for Region 1, when d+1 = 5 and n−d−1−k = 4, along with each path’s minimal structure shaded in crosshatch. If f = 4, the contribution of that minimal structure is q ( 1 2 ) [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A demonstration of deleting the minimal structure from the maximal staircase, and letting the remaining boxes fall by gravity. This will always result in a rectangular shape. upon deletion of the minimal structure MS(1), and letting the remaining boxes fall by gravity becomes: • the part of the staircase that is filled must weakly increase. See [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: An element x1x 3 2x 2 5x6θ3θ4ξ3ξ6 in B (1,2) B6 , represented by a type B modified Motzkin path and partially filled-in staircase. Conjecture 9.6. The set B (1,2) Bn is a basis for R (1,2) Bn . Define stairB q (π) := Y k∈β(T(π),S(π)) [k + 1]q, where T(π) and S(π) are determined by which elements in {1, . . . , n} appear as indices for θi and ξi respectively in the weight of the modified Motzkin path π. As … view at source ↗
read the original abstract

We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $S_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim-Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan-Swanson (2024) and proven by Angarone et al. (2025). We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for $R_n^{(1,2)}$. We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2024) on $R_n^{(1,2)}$ in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the $m_\mu$ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type $B_n$, and show that it has cardinality $4^nn!$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a conjectural monomial basis for the (1,2)-bosonic-fermionic coinvariant ring R_n^{(1,2)}. It proves that the proposed set has cardinality exactly 2^{n-1}n!, matching Zabrocki's dimension conjecture, constructs a weight-preserving bijection to segmented Smirnov words equating the Hilbert series conjecture with that of Iraci-Nadeau-Vanden Wyngaerd, extends results on the sign character to hook characters, gives a formula for the m_μ coefficients of the conjectural Frobenius series, and conjectures an analogous basis for the type B_n ring with cardinality 4^n n!.

Significance. If the linear independence of the proposed monomials holds, the work supplies the first explicit conjectural monomial basis for R_n^{(1,2)}, interpolating between the modified Motzkin path basis of Kim-Rhoades for (0,2) and the super-Artin basis for (1,1). The unconditional proofs of cardinality, the bijection, the hook-character extensions, and the m_μ formula stand on their own combinatorial arguments and provide combinatorial expressions for the Hilbert and Frobenius series while connecting the conjecture to segmented Smirnov words.

minor comments (2)
  1. [§2] The definition of the proposed basis monomials in §2 could be accompanied by a small-n table (e.g., n=3) listing the monomials explicitly to aid verification of the cardinality count.
  2. [final section] The statement of the type-B conjecture in the final section would benefit from a brief comparison table contrasting the (1,2) and type-B conditions on the monomials.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper explicitly frames its monomial construction as conjectural and does not claim to derive linear independence from the ring relations. It unconditionally proves that the proposed set has cardinality exactly 2^{n-1}n! (matching Zabrocki's independent dimension conjecture) via direct combinatorial counting, and exhibits an explicit weight-preserving bijection to segmented Smirnov words that equates two separate conjectures on the Hilbert series. All supporting results (cardinality, bijection, hook-character extensions, m_μ coefficients) are self-contained combinatorial arguments that do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. The only open claim is the conjectural basis property itself, which is not asserted as proven.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, ad-hoc axioms, or invented entities; the work rests on standard definitions of bosonic/fermionic variables, symmetric group action, and existing combinatorial objects (Motzkin paths, Smirnov words).

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The sign character of the triagonal fermionic coinvariant ring

    math.CO 2025-01 unverdicted novelty 7.0

    Determines trigraded sign multiplicity in R_n^(0,3) proving it sums to n^2-n+1; gives explicit double-hook formula for R_n^(0,2) and discusses R_n^(0,4) and a graded refinement of another conjecture.

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