Beta-set hook-product identity ($H_{\lambda }={\prod }_i{\ell }_i!/{\prod }_{i<j}({\ell }_i-{\ell }_j)$)

For ${\ell }_i={\lambda }_i+m-i$, the hook lengths in row $i$ are ${\ell }_i-q$, where $q$ runs over the non-beta numbers in $[0,{\ell }_i-1]$. Their product is ${\ell }_i!/{\prod }_{k>i}({\ell }_i-{\ell }_k)$. Multiplying over rows proves the identity and converts the hook-length formula into a Vandermonde determinant divided by factorials. Padding the partition with zero rows changes the beta set but not the hook product.

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 $\mathbb{C}{S}_m$ gives $m!{\sum }_{\lambda ⊢m}(\dim S^{\lambda }{)}^2/(m!{)}^2=1$. This is a normalized group algebra dimension identity.