A counterexample to Stanley's conjecture on Smith forms of Jucys–Murphy elements

We disprove Stanley's conjectured Smith-form formula for Jucys–Murphy elements on integral Specht modules: for shape $(3,1,1)$ and $k=3$, the Smith form is $(1,1,1,1,2,2)$ instead of $(1,1,1,1,1,4)$. A second example, of shape $(5,2)$, disproves his conjectural remark that every invariant factor is a product of some of the eigenvalues. For $n\ge3$, we prove in writing that the formula holds at $k=3$ precisely for partitions with at most two rows or at most two columns. The classification holds on every full-rank lattice in the rational Specht module stable under the permutations of $\{1,2,3\}$. Stanley's formula was based on Darij Grinberg's computations. Grinberg also clarified the integral lattice, and Alexander Chervov asked about the conjecture's status. The elements first appeared in Alfred Young's work and were rediscovered by Jucys and Murphy; Vershik and Okounkov give their common eigenbasis and tableau-content eigenvalues. The proof combines those eigenvalues with an integral group-ring identity and explicit integer matrices. Lean verifies the finite matrix calculations and supporting algebraic identities; independent programs identify the matrices with the Specht-module actions outside Lean.

Written proof, with Lean and computer checks; not yet reviewed by experts. Produced by AI models under the author's direction.