Abstract
LetK be an algebraically closed field of characteristic,p>0 and letD λ be the simple modules of the symmetric groupS r overK where λ is a p-regular partition ofr. The dimensions ofD λ for λ with at mostn parts are the same as the multiplicities of direct summands ofD ⊗r whereE is the natural module for the groupGL n (K). Whenn=2 we determine generating functions for these multiplicities and hence for the dimensions ofD λ for all partitions λ with two parts. These can be expressed as rational functions of Chebyshev polynomials; and we obtain explicit formulae for the coefficients.
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Erdmann, K. Tensor products and dimensions of simple modules for symmetric groups. Manuscripta Math 88, 357–386 (1995). https://doi.org/10.1007/BF02567828
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DOI: https://doi.org/10.1007/BF02567828