Polynomiality of Subdimensions of Diagonal Harmonics and a Sharp Stability Bound

A sequence of representations $V_n$ of the symmetric group $S_n$ is called representation (multiplicity) stable if, after some $n$, the irreducible decomposition of $V_n$ stabilizes. In particular, Church, Ellenburg and Farb (2015) showed that for fixed $a$ and $b$, the space of diagonal harmonics $DH_n^{a,b}$ exhibits this behavior, with its dimension eventually stabilizing to a polynomial in $n$. Building on this result, we use the Schedules Formula by Haglund and Loehr (2005) to obtain an explicit combinatorial polynomial for the dimension of the bigraded spaces $DH_n^{a,b}$. This derivation not only yields the dimension formula but also produces a new sharp stability bound of $a + b$, and determines the exact degree of the dimension polynomial, which is also $a + b$.