The Anti-Arithmetic Product of Symmetric Functions

The arithmetic product $\boxdot$ is a bilinear operation on the ring of symmetric functions over $\mathbb Q$ 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 $(p_\lambda)$. It is known that this operation preserves the subring $\mathbf{Symm}_{\mathbb Z}$ of symmetric functions with integer coefficients, and is Schur-positive on pairs of Schur functions. Indeed, it corresponds to induction of representations from $S_n \times S_m$ to $S_{nm}$. In this work, we consider the anti-arithmetic product $\diamondsuit$, which is defined by the same formula but with $\gcd$ and $\operatorname{lcm}$ interchanged. We show that it, too, preserves $\mathbf{Symm}_{\mathbb Z}$, 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 $\lambda$-ring morphism from the character ring of $S_{mn}$ to that of $S_m \times S_n$. Now, a 1975 theorem of Boorman shows that the representation ring of $S_n$ is generated by the natural permutation representation $M_n = \mathbb Q^n$ as a $\lambda$-ring. Thus, the proof of integrality boils down to only computing the image of $M_{mn}$ under this adjoint, which can be done using Möbius 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 $\lambda$-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.