Tensor products and dimensions of simple modules for symmetric groups
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 \(D^\lambda\).
scientific article; zbMATH DE number 871773
scientific article