Irreducible Symmetric Group Characters of Rectangular Shape

where χ(1) denotes the dimension of the character χ and (n)k = n(n − 1) · · · (n − k + 1). Thus [8, (7.6)(ii)][12, p. 349] χ(1) is the number f of standard Young tableaux of shape λ. Identify λ with its diagram {(i, j) : 1 ≤ j ≤ λi}, and regard the points (i, j) ∈ λ as squares (forming the Young diagram of λ). We write diagrams in “English notation,” with the first coordinate increasing from top to bottom and the second coordinate from left to right. Let λ = (λ1, λ2, . . .) and λ ′ = (λ1, λ ′ 2, . . .), where λ ′ is the conjugate partition to λ. The hook length of the square u = (i, j) ∈ λ is