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arxiv: 1907.07616 · v1 · pith:SE7KSPFPnew · submitted 2019-07-17 · 🧮 math.RT · math.CO

Plethysms of symmetric functions and representations of SL₂(mathbb{C})

Rowena Paget , Mark Wildon This is my paper

Pith reviewed 2026-05-24 19:57 UTC · model grok-4.3

classification 🧮 math.RT math.CO
keywords plethysmSchur functorrepresentations of SL(2)plane partitionsq-binomial identitiesirreducible modulessymmetric functions
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The pith

Isomorphisms ∇^λ Sym^ℓ E ≅ ∇^μ Sym^m E of SL₂(ℂ)-representations are classified for conjugate partitions, rectangles, two-row or two-column shapes, and hooks.

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

The paper determines the conditions under which Schur functors applied to symmetric powers of the two-dimensional natural module for SL₂(ℂ) produce isomorphic representations. This classification matters because it identifies when different labelings by partitions yield the same module, which simplifies calculations in representation theory and symmetric function theory. The authors achieve complete classifications in several families of partitions by connecting the problem to the enumeration of plane partitions and proving a supporting q-binomial identity. They also obtain as a corollary a determination of all cases in which such a module is irreducible.

Core claim

We classify all isomorphisms ∇^λ Sym^ℓ E ≅ ∇^μ Sym^m E when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when ℓ = m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇^λ Sym^ℓ E is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon's enumeration of plane partitions, and prove a new q-binomial identity in this setting.

What carries the argument

The connection between the representation isomorphisms and MacMahon's enumeration of plane partitions, together with an associated new q-binomial identity that equates the relevant multiplicities or dimensions.

If this is right

  • When λ and μ are conjugate partitions the isomorphism holds under the classified conditions on the sizes ℓ and m.
  • When one partition is a rectangle the isomorphism occurs precisely for the pairs identified in the classification.
  • For partitions with at most two rows or two columns the modules are isomorphic if and only if the pairs satisfy the complete list given.
  • The module ∇^λ Sym^ℓ E is irreducible exactly in the cases determined from the skew partition result.
  • When ℓ equals m there is a partial list of isomorphisms between different shapes.

Where Pith is reading between the lines

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

  • If the plane partition connection extends beyond the studied families it could yield classifications for partitions with more rows.
  • The new q-binomial identity may have applications in other areas of enumerative combinatorics involving symmetric functions.
  • Knowing the irreducible cases allows direct construction of bases or characters for those specific modules without further decomposition.
  • The results suggest that similar plethysm isomorphisms for other classical groups might be approachable via analogous combinatorial counts.

Load-bearing premise

The connection to plane partition enumeration and the new q-binomial identity is sufficient to detect every isomorphism in the families considered without undetected exceptions.

What would settle it

An explicit pair of partitions λ, μ with at most two rows where the dimensions or characters of ∇^λ Sym^ℓ E and ∇^μ Sym^m E match but the classification does not list them as isomorphic, or a listed pair where the modules differ.

Figures

Figures reproduced from arXiv: 1907.07616 by Mark Wildon, Rowena Paget.

Figure 1
Figure 1. Figure 1: The statistics R(λ), E(λ), S(λ) and partitions EP(λ), SP(λ). Two boxes have their content indicated. We begin with three equivalent conditions for the existence of infin￾itely many plethystic equivalences between distinct partitions λ and µ. We then prove a fourth equivalent condition, namely that µ = λ 0 and EP(λ) = SP(λ) 0 , obtaining Theorem 5.5 and, a fortiori, Theorem 1.3. Proposition 5.2. Let λ and µ… view at source ↗
Figure 2
Figure 2. Figure 2: The content of the partition obtained from λ by delet￾ing a part of size R(λ) and inserting it as a new column is C(λ)+1 is obtained by adding 1 to the content of each box of [λ]; this can be seen here by comparing the shaded and unshaded boxes. We remark that the bound in (ii) is tight: for example, by the Hermite reciprocity seen in §1.2, if λ = (n) and µ = (n + 1) where n ∈ N, then λ n+1∼n µ and n + 1 ≥… view at source ↗
Figure 3
Figure 3. Figure 3: shows the partitions λ and µ. Clearly if E = S and κ = ν 0 then λ 0 = µ and so H(λ) = H(µ). Conversely, suppose that H(λ) = H(µ). The hook lengths outside the two thick rectangles in [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graphs of ch(d) (a b) in each of the cases in Lemma 8.8 [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Young diagram of a skew partition π/π? with π ? shaded in grey is shown. Deleting the hatched boxes leaves the Young diagram of the proper skew partition λ/λ? , where [λ ? ] consists of the shaded unhatched boxes. a way that does not depend on Littlewood–Richardson coefficients, or the complete reducibility of representations of SL2(C). Generalizing Lemma 2.7, we have Φ∇λ/λ? Sym`E (1, q) = sλ/λ? (1, q,… view at source ↗
Figure 6
Figure 6. Figure 6: Partition of the pyramids for λ`m and η`m. The im￾portant row and column numbers are indicated; an entry involves m if and only if it is in the shaded region. where in each case the notation c . . . c m indicates m consecutive entries of c. Similarly, the first 8 rows of the pyramid Q for ηm` are 1 1 1 2 2 1 + m 1 2 1 `−9 . . . 1 3 2 2 3 4 3 + m 2 + m 3 3 2 `−10 . . . 2 4 3 4 5 5 + m 4 + m 4 + m 4 4 3 `−11… view at source ↗
read the original abstract

Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda \!\mathrm{Sym}^\ell \!E \cong \nabla^\mu \!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\lambda$ and $\mu$ are conjugate partitions and when one of $\lambda$ or $\mu$ is a rectangle. We give a complete classification when $\lambda$ and $\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^\lambda \!\mathrm{Sym}^\ell \!E$ is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon's enumeration of plane partitions, and prove a new $q$-binomial identity in this setting.

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 / 3 minor

Summary. The paper classifies isomorphisms ∇^λ Sym^ℓ E ≅ ∇^μ Sym^m E of SL_2(ℂ)-representations (E the natural module) in the cases where λ, μ are conjugate partitions, where one is a rectangle, where both have at most two rows or columns, and where both are hooks; it gives a partial classification when ℓ = m. As a corollary of a general result on Schur functors on skew partitions it determines all cases in which ∇^λ Sym^ℓ E is irreducible. The proofs combine representation-theoretic techniques with combinatorial arguments that make explicit use of MacMahon's plane-partition enumeration and a new q-binomial identity.

Significance. If the claimed classifications are exhaustive, the work supplies a systematic extension of the results of King and Manivel, together with a concrete link between plethysm isomorphisms and plane-partition counting that may be useful for further combinatorial representation theory. The new q-binomial identity and the irreducibility corollary constitute concrete, verifiable contributions.

minor comments (3)
  1. The statement of the new q-binomial identity (presumably in §3 or §4) would benefit from an explicit display of the identity together with a short verification that it is indeed new and not a direct consequence of existing q-series identities.
  2. Tables or explicit lists enumerating the isomorphisms for partitions of small weight (e.g., weight ≤ 6) would make the classification statements easier to check and would strengthen the claim of exhaustiveness.
  3. [Introduction] The notation ∇^λ for the Schur functor is introduced without recalling its precise definition in terms of Young symmetrizers or Weyl modules; a one-sentence reminder in the introduction would aid readers from adjacent fields.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper classifies isomorphisms of Schur functors applied to symmetric powers using representation theory and combinatorics, explicitly connecting to MacMahon's external enumeration of plane partitions and proving a new q-binomial identity. It generalizes prior results of King and Manivel (distinct authors) and scopes claims to specific cases (conjugates, rectangles, hooks, two-row/column partitions) without any described reduction of a central result to a self-defined parameter, fitted input renamed as prediction, or load-bearing self-citation chain. All load-bearing steps rest on independent external combinatorial facts and standard tools, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are indicated in the abstract; the work rests on standard background facts about Schur functors, symmetric powers, and SL2 representations.

axioms (1)
  • standard math Standard properties of Schur functors, symmetric powers, and finite-dimensional representations of SL2(C)
    Invoked throughout the classification statements.

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Reference graph

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