Generalized Hook and Content Numbers Identities

Given a cell x in a skew diagram, the arm length a(x), the leg length ℓ(x), and therefore the ‘hook pair’ (a(x),ℓ (x)), are defined. The diagram thus determines the corresponding multiset of its hook pairs. Similarly, x ∈ Z + 2determines its ‘content pair’ (a′(x),ℓ′ (x)), which give raise to another type of multiset. Here we prove various identities between such multisets. In the special case of the hook and content numbers, these identities imply new ‘hook’ formulas for the dimensions of irreducible GL(n) representations. In the more general situation of the Jack symmetric functions Pλ(α), these identities imply new ‘α -hook’ formulas forPλ(α) (1, . . . , 1).