Horn inequalities for nonzero Kronecker coefficients

Nicolas Ressayre (ICJ)

arxiv: 1907.07931 · v1 · pith:4D7RDP7Anew · submitted 2019-07-18 · 🧮 math.AG · math.RT

Horn inequalities for nonzero Kronecker coefficients

Nicolas Ressayre (ICJ) This is my paper

Pith reviewed 2026-05-24 19:54 UTC · model grok-4.3

classification 🧮 math.AG math.RT

keywords Kronecker coefficientsHorn inequalitiesLittlewood-Richardson coefficientspartitionssymmetric groupsrepresentation multiplicitiestensor product decompositions

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The pith

Nonzero Kronecker coefficients satisfy the essential Horn inequalities on their partitions.

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

The paper extends the essential Horn inequalities, already known for Littlewood-Richardson coefficients, to the setting of Kronecker coefficients. It proves that whenever the Kronecker coefficient attached to three partitions is positive, those partitions must obey the essential Horn inequalities. The argument relies on the classical Littlewood-Murnaghan embedding that realizes Littlewood-Richardson coefficients as special cases of Kronecker coefficients. A reader cares because the result supplies concrete linear inequalities that any triple with positive Kronecker coefficient must satisfy, narrowing the possible support of these multiplicities. The work therefore gives a necessary condition for positivity that applies directly to decompositions of tensor products of symmetric-group representations.

Core claim

By a classical result the Kronecker coefficients extend the Littlewood-Richardson coefficients. The nonvanishing of a Littlewood-Richardson coefficient implies linear inequalities on the triple of partitions, called Horn inequalities. The paper shows that the essential Horn inequalities also hold for every triple of partitions whose Kronecker coefficient is nonzero.

What carries the argument

The Littlewood-Murnaghan embedding of Littlewood-Richardson coefficients into Kronecker coefficients, which transfers the facet structure of the Horn cone.

If this is right

Any triple with nonzero Kronecker coefficient lies inside the Horn cone cut out by the essential inequalities.
The support of the Kronecker coefficient function is contained in the set defined by those inequalities.
The same linear conditions that are necessary for positive Littlewood-Richardson coefficients remain necessary for positive Kronecker coefficients.

Where Pith is reading between the lines

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

Computation of Kronecker coefficients for small partitions can begin by checking the essential Horn inequalities as a fast filter.
If analogous embeddings exist for other families of coefficients, the same transfer argument may apply.
Tables of known Kronecker coefficients can be scanned to locate any triples that come close to violating the inequalities.

Load-bearing premise

The classical Littlewood-Murnaghan embedding preserves the facet structure of the Horn cone so that the essential inequalities transfer directly.

What would settle it

Three partitions whose Kronecker coefficient is positive but that violate at least one essential Horn inequality.

read the original abstract

The Kronecker coefficients and the Littlewood-Richardson coefficients are nonnegative integers depending on three partitions. By definition, these coefficients are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of a Littlewood-Richardson coefficient implies linear inequalities on the triple of partitions, called Horn inequalities. In thispaper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.

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.

Desk Editor's Note private letter to a colleague

The paper extends essential Horn inequalities to nonzero Kronecker coefficients via the Littlewood-Murnaghan embedding, but the transfer to the full Kronecker cone rests on an unverified assumption about facet preservation.

read the letter

The central new statement is that any triple of partitions with positive Kronecker coefficient g(λ,μ,ν) must satisfy the essential Horn inequalities that define the facets of the Littlewood-Richardson cone. The argument rests on the classical embedding that realizes every LR coefficient as a Kronecker coefficient for suitably padded partitions, then claims the inequalities carry over directly. This is a clean, if modest, extension rather than a re-derivation from scratch. The abstract is explicit about the claim and the prerequisite result, which is a point in its favor for a short note. The embedding itself is standard and correctly cited, so the logical step from LR nonvanishing to the inequalities is solid for those special triples. The soft spot is exactly where the stress-test flags it: the embedding lands inside the Kronecker cone, and nothing in the given abstract shows that the inequalities remain necessary (let alone facet-defining) for Kronecker triples that lie outside the image. If a triple violates an essential Horn inequality yet still has g > 0, the claim fails. The paper would need an independent argument or explicit verification that no such counterexample exists; the abstract gives no sign of one. Without the full manuscript I cannot check whether that gap is closed, but on the stated reasoning alone the support for the general case is incomplete. This is the kind of targeted extension that belongs in a specialized journal or as a short note. A reader already working on Kronecker positivity or Horn-type cones would find it useful to have on record, but it does not reorganize the area. I would send it to peer review so that a referee can verify the transfer step on the actual proofs; the question is well-posed even if the current evidence is thin.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that the essential Horn inequalities (those defining the facets of the Littlewood-Richardson cone) are satisfied by any triple of partitions with nonzero Kronecker coefficient g(λ,μ,ν). The argument invokes the classical Littlewood-Murnaghan embedding, which realizes every Littlewood-Richardson coefficient as a Kronecker coefficient for suitably padded partitions, to transfer the inequalities from the LR case to the Kronecker case.

Significance. If the central claim holds, the result supplies a concrete set of linear inequalities that are necessary for nonzero Kronecker coefficients. This would give a partial description of the Kronecker cone and could be useful for computational checks or further structural results in the representation theory of symmetric groups. The reliance on a classical external embedding is explicitly credited as the mechanism.

major comments (1)

[Abstract] Abstract (paragraph 3) and the main argument: the claim that the essential Horn inequalities transfer directly rests on the assumption that the Littlewood-Murnaghan embedding preserves necessity outside its image. No independent argument is supplied showing that violation of an essential Horn inequality forces g(λ,μ,ν)=0 for Kronecker triples not arising from LR coefficients. If the embedded triples lie in the relative interior of the Kronecker cone, facet-defining inequalities for LR need not remain necessary for the larger set.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this subtlety in the transfer argument. We address the concern directly below.

read point-by-point responses

Referee: [Abstract] Abstract (paragraph 3) and the main argument: the claim that the essential Horn inequalities transfer directly rests on the assumption that the Littlewood-Murnaghan embedding preserves necessity outside its image. No independent argument is supplied showing that violation of an essential Horn inequality forces g(λ,μ,ν)=0 for Kronecker triples not arising from LR coefficients. If the embedded triples lie in the relative interior of the Kronecker cone, facet-defining inequalities for LR need not remain necessary for the larger set.

Authors: We agree that the manuscript invokes the embedding to connect the settings but does not supply a separate argument establishing necessity for Kronecker triples outside the image. The embedding alone shows containment of the LR cone in the Kronecker cone (after padding) but does not automatically guarantee that LR facet inequalities remain necessary for the larger cone. We will revise the paper by adding an explicit subsection that proves directly that violation of any essential Horn inequality forces the Kronecker coefficient to vanish, using the semigroup structure of the Kronecker coefficients and the fact that the inequalities are preserved under the relevant limits and stabilizations. The abstract will also be updated for precision. This constitutes a major revision. revision: yes

Circularity Check

0 steps flagged

No circularity detected; extension rests on classical external embedding

full rationale

The paper invokes the classical Littlewood-Murnaghan result (external, non-self-cited) to realize LR coefficients as special cases of Kronecker coefficients, then claims the essential Horn inequalities transfer. No step reduces a claimed prediction or facet to a fitted parameter, self-defined quantity, or load-bearing self-citation chain inside the paper. The derivation chain is therefore independent of its own outputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on one classical domain assumption and no free parameters or invented entities.

axioms (1)

domain assumption Littlewood-Murnaghan result that Kronecker coefficients extend Littlewood-Richardson coefficients
Invoked in abstract to justify the extension

pith-pipeline@v0.9.0 · 5616 in / 1046 out tokens · 16875 ms · 2026-05-24T19:54:31.427914+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By a classical Littlewood-Murnaghan’s result the Kronecker coefficients extend the Littlewood-Richardson ones... we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Kron(e+1,f+1,e+f+1) is a finitely generated semigroup... the cone Q≥0 Kron(...) is a closed convex polyhedral cone. The inequalities (7) and (8) are essential.

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.