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Tensor products and dimensions of simple modules for symmetric groups. (English) Zbl 0849.20008
Let \(K\) be an algebraically closed field of characteristic \(p>0\) and let \(D^\lambda\) be the simple module of the symmetric group \(S_n\) over \(K\), where \(\lambda\) is a \(p\)-regular partition of \(r\). Then, the dimension of \(D^\lambda\) for \(\lambda\) with at most \(n\) parts are the same as the multiplicities of direct summands of \(E^{\otimes r}\) where \(E\) is the natural \(n\)-dimensional module for \(\text{GL}_n(K)\). The problem of finding the dimension of \(D^\lambda\) is a major open problem; thus that the author has been able to calculate recursive formulae for these multiplicities in the case \(n=2\) leading to a determination of their generating functions is of some interest. She shows that these are rational functions and remarkably, explicit expressions in terms of Chebyshev polynomials are obtained. These in turn lead to explicit formulae for the dimensions of the \(D^\lambda\).

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
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