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The Anti-Arithmetic Product of Symmetric Functions

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.RA — Rings and Algebrasmath.RT — Representation Theory

Contributed by Darij Grinberg ↗

Version 1 / Oct 03, 2026 / CC0 1.0

Abstract

The \emph{arithmetic product} ⊡\boxdot is a bilinear operation on the ring of symmetric functions over Q\mathbb Q defined by \[ p_\lambda \ \boxdot\ p_\mu = \prod_{i,j}p_{\operatorname{lcm}(\lambda_i,\mu_j)}^{\gcd(\lambda_i,\mu_j)} \] on the power-sum basis (pλ)(p_\lambda). It is known that this operation preserves the subring SymmZ\mathbf{Symm}_{\mathbb Z} of symmetric functions with integer coefficients, and is Schur-positive on pairs of Schur functions. Indeed, it corresponds to induction of representations from Sn×SmS_n \times S_m to SnmS_{nm}. In this work, we consider the \emph{anti-arithmetic product} ♢\diamondsuit, which is defined by the same formula but with gcd⁡\gcd and lcm⁡\operatorname{lcm} interchanged. We show that it, too, preserves SymmZ\mathbf{Symm}_{\mathbb Z}, even though it lacks the Schur positivity property. This was conjectured on MathOverflow in 2014. Our method is a more intricate variant of the representation-theoretical interpretation of ⊡\boxdot. The anti-arithmetic product does not correspond to an operation on actual representations; but its adjoint can still be described as a map on characters, which (as we show) is a λ\lambda-ring morphism from the character ring of SmnS_{mn} to that of Sm×SnS_m \times S_n. Now, a 1975 theorem of Boorman shows that the representation ring of SnS_n is generated by the natural permutation representation Mn=QnM_n = \mathbb Q^n as a λ\lambda-ring. Thus, the proof of integrality boils down to only computing the image of MmnM_{mn} under this adjoint, which can be done using M\"obius inversion in the representation ring. This work is aimed at readers familiar with representation rings and the representation theory of symmetric groups. No prior knowledge of λ\lambda-rings is presumed. Three self-contained proofs of Boorman's theorem are given in the appendices, two of them lifting the theorem to the noncommutative symmetric functions (or Solomon's descent algebra), recovering a result of Schocker.

Provenance statement

This work was written by GPT-6 based on a proof found by GPT-5.5. It has then been extensively edited and fully proofread by myself. Also available on my website: https://www.cip.ifi.lmu.de/~grinberg/algebra/ariprod-gpt.pdf / https://www.cip.ifi.lmu.de/~grinberg/algebra/ariprod-gpt.tex . Will probably upload to the arXiv too.

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Version history

  1. v1Submitted by Darij GrinbergInitial depositCurrentOct 03, 2026