フック長の公式
The complex irreducible representations V λ {\displaystyle V_{\lambda }} of the symmetric group are indexed by partitions λ {\displaystyle \lambda } of n {\displaystyle n} (for an explicit construction see Specht module) . Their characters are related to the theory of symmetric functions via the Hall inner product in the following formula
χ λ ( w ) = ⟨ s λ , p τ ( w ) ⟩ {\displaystyle \chi ^{\lambda }(w)=\langle s_{\lambda },p_{\tau (w)}\rangle }
where s λ {\displaystyle s_{\lambda }} is the Schur function associated to λ {\displaystyle \lambda } and p τ ( w ) {\displaystyle p_{\tau (w)}} is the power-sum symmetric function of the partition τ ( w ) {\displaystyle \tau (w)} associated to the cycle decomposition of w {\displaystyle w} .
Combining the two formulas for the generating functions we have
P ( x ) ( 1 − x ) ( 1 − x 2 ) ⋯ ( 1 − x n ) = ∏ ( i , j ) ∈ λ ( 1 − x h ( i , j ) ) − 1 {\displaystyle {\frac {P(x)}{(1-x)(1-x^{2})\cdots (1-x^{n})}}=\prod _{(i,j)\in \lambda }(1-x^{h_{(i,j)}})^{-1}}
Both sides converge inside the disk of radius one and the following expression makes sense for | x | < 1 {\displaystyle |x|<1}
P ( x ) = ∏ k = 1 n ( 1 − x k ) ∏ ( i , j ) ∈ λ ( 1 − x h ( i , j ) ) . {\displaystyle P(x)={\frac {\prod _{k=1}^{n}(1-x^{k})}{\prod _{(i,j)\in \lambda }(1-x^{h_{(i,j)}})}}.}