= Square-sum identity for shifted partition coordinates
{title2=$\sum_{\ell_i\ge0,\ \sum_i\ell_i=m(m+1)/2}\Delta(\ell)^2/\prod_i(\ell_i!)^2=1$}
Repeated coordinates contribute zero. Sorting a distinct tuple and subtracting the staircase gives a partition of $m$, and its $m!$ orderings have identical squared summands. The <beta-set hook-product identity> turns each summand into $(\dim S^\lambda/m!)^2$. Summing and using the <Artin–Wedderburn theorem> for $\mathbb CS_m$ gives $m!\sum_{\lambda\vdash m}(\dim S^\lambda)^2/(m!)^2=1$. This is a normalized <group algebra> dimension identity.
Back to article page
