Divisor lattices and Schur positivity in cyclic induction

Source abstract

We classify Schur positivity in two families of symmetric functions $f_n^T$ introduced by Sundaram. For $k\ge2$, $f_n^{\{1,k\}}$ fails to be Schur-positive exactly when $n=k$ and $k$ is an even integer greater than $2$. For $T_k=\{k^i:i\ge0\}$, $f_n^{T_k}$ fails to be Schur-positive exactly when $k$ is an even integer greater than $2$ and $n=k^a$ for some $a\ge1$. In every exceptional degree, the coefficient of $s_{(1^n)}$ is $-1$, and all other Schur coefficients are nonnegative. Together with Sundaram's product implication, these classifications settle her Conjectures~1--3. We also recast Sundaram's plethystic identities in divisor-indexed coordinates and classify nonnegative Foulkes coordinates in each fixed degree and globally. If a composite $k$ divides $n$, both $f_n^{\{1,k\}}$ and $f_n^{T_k}$ have a negative Foulkes coordinate even when they are Schur-positive. Extending Hou's bounded-interval argument, we prove a uniform criterion for real divisor weights for $n\ge18$. If $1\in T$ and the Möbius weight $ψ^T$ satisfies the required bounds, $f_n^T$ is Schur-positive for every odd $n\ge18$. For even $n\ge18$, it fails exactly when $n\in T$ and $n/2\notin T$; then $s_{(1^n)}$ indexes the unique negative Schur coefficient, equal to $-1$. Finally, for $k=q^s$ with $q$ prime, $s\ge2$, $k\mid n$, and $n>k$, we derive major-index residue inequalities and classify all equality cases. For $n\ge18$, we also obtain a uniform quantitative estimate for partitions whose first row and first column each have length at most $n/2$.