The Q-invariant and shape-invariant generation conjecture for ${\rm Inv}_{R\overline{e}}$

Let $m,n$ be positive integers, let $\operatorname{LSym}$ be the ring of loop symmetric functions, and let ${\rm Inv}_{R\overline{e}}={\rm Inv}_{\overline{e}}\cap\operatorname{Frac}(\operatorname{LSym})$. Let $Q$-invariants and shape invariants $S_i$ be as defined from the loop symmetric-function and geometric RSK constructions.

Q-invariant and shape-invariant generation conjecture. The field ${\rm Inv}_{R\overline{e}}$ is generated by the $Q$-invariants and shape invariants. Moreover, every element of $\mathbb{C}[x_i^j]\cap{\rm Inv}_{R\overline{e}}$ is a polynomial in reduced $Q$-invariants $\widetilde{Q}_{i}^{(j)}$ and ratios $S_i/S_{i+1}$, with $S_{\min(m,n)+1}=1$.

The conjecture refines the known field-generation result for the common invariant field by predicting generators for its polynomial part as well. The paper notes algebraic independence and matching transcendence degree as supporting evidence, while the corresponding ring-generation assertion is not established.