A new proof of a theorem of Littlewood

Abstract

In this paper we give a new combinatorial proof of a result of Littlewood [D.E. Littlewood, The Theory of Group Characters, 2nd ed., Oxford University Press, 1950], p. 124: S μ ( 1 , q , q 2 , … ) = q n ( μ ) ∏ s ∈ μ ( 1 − q h μ ( s ) ) , where S μ denotes the Schur function of the partition μ , n ( μ ) is the sum of the legs of the cells of μ and h μ ( s ) is the hook number of the cell s ∈ μ .