The three-variable asymptotic maximal-rank conjecture for Schur polynomials

Let $clambda=(\lambda_1,\lambda_2,\lambda_3)$ be a partition with at most three parts, let $s_\lambda$ be the corresponding Schur polynomial, and let

$$A=\mathbf{k}[x_1,x_2,x_3]/(x_1^d,x_2^d,x_3^d).$$

Define

$$F=\{(\lambda_1,\lambda_2,\lambda_3)\mid \lambda_3\neq 0\text{ or }\gcd(\lambda_1+2,\lambda_2+1)>1\}\cup\{(s,s,0)\mid s\geq 5\},$$

and

$$P=\{(2,2,0),(3,3,0),(4,4,0)\}.$$

Three-variable Schur maximal-rank conjecture. For all sufficiently large $d$, the following holds: if $\lambda\in F$, then $s_\lambda$ fails to be a maximal rank element on $A$; if $\lambda\in P$, its maximal-rank behavior is periodic, with maximal rank precisely when $d\equiv 2,4,5\pmod 6$ for $\lambda=(2,2,0)$, when $d\equiv 6\pmod 8$ for $\lambda=(3,3,0)$, and when $d\equiv 8\pmod{10}$ for $\lambda=(4,4,0)$; and if $\lambda\notin F\cup P$, then $s_\lambda$ is a maximal rank element.

This conjecture proposes an asymptotic classification in three variables. The paper presents it as a computation-based prediction; its general validity for all sufficiently large $d$ remains open.