s . Further, let h $^{∗}$( s ) = l ( s ) + α (1 + a ( s )) and h$_{∗}$ ( s ) = l ( s ) + 1 + αa ( s ) . Finally, let c ( κ, α ) = ∏ s ∈ κ h$_{∗}$ ( s ), c $^{′}$( κ, α ) = ∏ s ∈ κ h $^{∗}$( s ), and w ( κ, α ) = c ( κ, α ) c $^{′}$( κ, α ). We will need the following lemmata. Lemma A1 . For the torus scalar product of C ( α ) κ with itself, we have Proof: Macdonald (1995, Chapter VI, § 10) establishes the orthogonality of “ P ” normalizations of Jack polynomials, P ( α ) κ , with respect to the torus scalar product. His formula (10.37) gives an explicit expression (up to a constant that can be evaluated using (10.38)) for 〈 P ( α ) κ , Q ( α ) κ 〉 α , where Q ( α ) κ = c ( κ,α ) c ′ ( κ,α $_{)}$P ( α ) κ (see (10.16)). On the other hand, (see, for example, Table 6 of Dumitriu et al, 2007). Substituting this expression in Macdonald’s formulae, we get (34). □ P ( α ) κ = c $^{′}$( κ, α ) α | κ | | κ | ! C ( α ) κ (35) 23 〈 C ( α ) κ , C ( α ) κ 〉 α = ( α | κ | | κ | ! ) 2 w ( κ, α r ∏ j =1 ( Γ ( r − j +1 α ) Γ ( 1 α ) Γ ( 1+ r − j α ) ) ∏ s ∈ κ r + a $^{′}$( s ) α − l $^{′}$( s ) r +( a $^{′}$( s )+1) α l $^{′}$( s ) − 1 . (34) Figure 3: The Ferrers diagram of partition [4 , 3 , 3 , 1 , 1]. S =[4,3,3,1,1], l( )=5, | |=12, a(s)=2, l(s)=3, a’(s)=0, l’(s)=1.