フック長の公式
For example, if w = ( 154 ) ( 238 ) ( 6 ) ( 79 ) {\displaystyle w=(154)(238)(6)(79)} then τ ( w ) = ( 3 , 3 , 2 , 1 ) {\displaystyle \tau (w)=(3,3,2,1)} .
Since the identity permutation e {\displaystyle e} has the form e = ( 1 ) ( 2 ) ⋯ ( n ) {\displaystyle e=(1)(2)\cdots (n)} in cycle notation, τ ( e ) = 1 + 1 + ⋯ + 1 = 1 ( n ) {\displaystyle \tau (e)=1+1+\cdots +1=1^{(n)}} . Then the formula says
dim  V λ = χ λ ( e ) = ⟨ s λ , p 1 ( n ) ⟩ {\displaystyle \dim V_{\lambda }=\chi ^{\lambda }(e)=\langle s_{\lambda },p_{1^{(n)}}\rangle }
Considering the expansion of Schur functions in terms of monomial symmetric functions using the Kostka numbers
s λ = ∑ μ K λ μ m μ , {\displaystyle s_{\lambda }=\sum _{\mu }K_{\lambda \mu }m_{\mu },}
e., no repetition), such that each row and each column form increasing sequences. For each cell of the Young diagram in coordinates ( i , j ) {\displaystyle (i,j)} (that is, the cell in the i {\displaystyle i} th row and j {\displaystyle j} th column), the hook H λ ( i , j ) {\displaystyle H_{\lambda }(i,j)} is the set of cells ( a , b ) {\displaystyle (a,b)} such that a = i {\displaystyle a=i} and b ≥ j {\displaystyle b\geq j} or a ≥ i {\displaystyle a\geq i} and b = j {\displaystyle b=j} .
For the following we need at least as many variables as rows in the partition, so from now on we work with n {\displaystyle n} variables x 1 , ⋯ , x n {\displaystyle x_{1},\cdots ,x_{n}} .
Each term Δ ( x ) s λ {\displaystyle \Delta (x)s_{\lambda }} is equal to
a ( λ 1 + k − 1 , λ 2 + k − 2 , ... , λ k ) ( x 1 , x 2 , ... , x k ) = det [ x 1 l 1 x 2 l 1 ... x k l 1 x 1 l 2 x 2 l 2 ... x k l 2 ⋮ ⋮ ⋱ ⋮ x 1 l k x 2 l k ... x k l k ] {\displaystyle a_{(\lambda _{1}+k-1,\lambda _{2}+k-2,\dots ,\lambda _{k})}(x_{1},x_{2},\dots ,x_{k})=\det \left[{\begin{matrix}x_{1}^{l_{1}}&x_{2}^{l_{1}}&\dots &x_{k}^{l_{1}}\\x_{1}^{l_{2}}&x_{2}^{l_{2}}&\dots &x_{k}^{l_{2}}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{l_{k}}&x_{2}^{l_{k}}&\dots &x_{k}^{l_{k}}\end{matrix}}\right]}