A New Plethysm Formula for Symmetric Functions

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This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for μ ⊢ b,

where the sum is over semistandard tableaux T of weight ab, ω is a root of unity, and majμ(T) is a major index like statistic on semistandard tableaux.

An Sb-representation, denoted Sλ,b, is defined. In the special case when λ ⊢ b, Sλ,b is the Specht module corresponding to λ. It is shown that the character of Sλ,b on elements of cycle type μ is

where the sum is over semistandard tableaux T of shape λ and weight ab. Moreover, the eigenvalues of the action of an element of cycle type μ acting on Sλ,b are {ωmajμ(T):T}. This generalizes J. Stembridge's result [11] on the eigenvalues of elements of the symmetric group acting on the Specht modules.

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