Characterization of uniform representation multiplicity stability from dimensions and character values

Determine whether a sequence of \(S_n\)-modules \((V_n)_n\) is uniformly representation multiplicity stable whenever \(\dim(V_n)\) agrees eventually with a polynomial of degree \(K\) and the character values \(\chi^{V_n}(\sigma\circ(n-k\;n-k+1\;\cdots\;n))\) stabilize for every \(\sigma\in S_k\) and every \(k\leq K\).

Background

The paper observes that uniform representation multiplicity stability implies eventual polynomiality of dimensions and stabilization of specified character values. It asks whether the converse can be obtained from only the eventual dimension polynomial and character stabilization for permutations augmented by a long cycle, rather than requiring full polynomial character formulas in cycle-count variables. A positive answer would provide a converse to the cited representation-stability theorem under fewer character-value conditions.

References

Suppose that for large n, \dim(V_n)=p(n) for a degree K polynomial p and the character values $$\chi{V_n}(\sigma\circ (n-k~~~n-k+1~\cdots ~n))$$ stabilize for all \sigma \in S_k and all k\leq K. Is (V_n)_n URMS?

— Monomial stability of Frobenius images  (2503.04950 - Borisov, 6 Mar 2025) in Question environment in Section 7, “Further work”