フック長の公式
Note that l i {\displaystyle l_{i}} is the hook length of the first box in each row of the Young Diagram. Transforming this expression into the form n ! ∏ h λ ( i , j ) {\displaystyle {\frac {n!}{\prod h_{\lambda }(i,j)}}} claimed by the hook-length formula is a fairly simple exercise in combinatorics: For any given i = 1 , 2 , ... , k {\displaystyle i=1,2,\ldots ,k} , one has to argue that l i ! = ( ∏ j > i ( l i − l j ) ) ⋅ ∏ c h λ ( c ) {\displaystyle l_{i}!=\left(\prod _{j>i}(l_{i}-l_{j})\right)\cdot \prod _{c}h_{\lambda }(c)} , where the latter product ranges over all cells c {\displaystyle c} in the i {\displaystyle i} -row of the Young diagram of λ {\displaystyle \lambda } .
the inner product with p 1 ( n ) = h 1 ( n ) {\displaystyle p_{1^{(n)}}=h_{1^{(n)}}} is K λ 1 ( n ) {\displaystyle K_{\lambda 1^{(n)}}} , because ⟨ m μ , h ν ⟩ = δ μ ν {\displaystyle \langle m_{\mu },h_{\nu }\rangle =\delta _{\mu \nu }} . Note that K λ 1 ( n ) {\displaystyle K_{\lambda 1^{(n)}}} is equal to d λ {\displaystyle d_{\lambda }} . Hence
dim  V λ = d λ . {\displaystyle \dim V_{\lambda }=d_{\lambda }.}
An immediate consequence of this is
∑ λ ⊢ n ( f λ ) 2 = n ! {\displaystyle \sum _{\lambda \vdash n}\left(f^{\lambda }\right)^{2}=n!}
The above equality is also a simple consequence of the Robinson–Schensted–Knuth correspondence.
The computation also shows that:
If λ is a partition of some integer p, a reverse plane partition of n with shape λ is obtained by filling in the boxes in the Young diagram with non-negative integers such that the entries add to n and are non-decreasing along each row and down each column. The hook lengths h 1 , ... , h p {\displaystyle h_{1},\dots ,h_{p}} can be defined as with Young tableau. If πn denotes the number of reverse plane partitions of n with shape λ, then the generating function can be written as
∑ n = 0 ∞ π n x n = ∏ k = 1 p ( 1 − x h k ) − 1 {\displaystyle \sum _{n=0}^{\infty }\pi _{n}x^{n}=\prod _{k=1}^{p}(1-x^{h_{k}})^{-1}}
Stanley discovered another formula for the same generating function.
( x 1 + x 2 + ⋯ + x k ) n = ∑ λ ⊢ n s λ f λ . {\displaystyle (x_{1}+x_{2}+\cdots +x_{k})^{n}=\sum _{\lambda \vdash n}s_{\lambda }f^{\lambda }.}
Which is the expansion of p 1 ( n ) {\displaystyle p_{1^{(n)}}} in terms of Schur functions using the coefficients given by the inner product, because ⟨ s μ , s ν ⟩ = δ μ ν {\displaystyle \langle s_{\mu },s_{\nu }\rangle =\delta _{\mu \nu }} . The above equality can be proven also checking the coefficients of each monomial at both sides and using the Robinson–Schensted–Knuth correspondence or, more conceptually, looking at the decomposition of V ⊗ n {\displaystyle V^{\otimes n}} by irreducible G L ( V ) {\displaystyle GL(V)} modules, and taking characters. See Schur–Weyl duality.