{"id":"hexagon/2610.00026","pdf":"https://hexagonmath.org/pdf/2610.00026v1","url":"https://hexagonmath.org/abs/2610.00026v1","tags":["math.RA","math.RT","msc:20C30","msc:05E05","msc:19A22","Specht modules","descent algebra","lambda-rings","noncommutative symmetric functions","symmetric functions"],"score":0,"title":"The Anti-Arithmetic Product of Symmetric Functions","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.CO","summary":"The \\emph{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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","ChatGPT"],"discussed":0,"published":"2026-10-03","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}