On the dimension of some modular irreducible representations of the symmetric group
For fields of characteristic zero, there is a well known formula for the degree of an irreducible representation of the symmetric group \(S_n\) corresponding to a partition \(\lambda\); namely it is the number of standard tableaux of shape \(\lambda\). The determination of the corresponding formula in characteristic \(p\) has up to now been elusive. This paper provides such a formula for certain partitions \(\lambda\). First, a different, but not new, interpretation of the standard tableaux formula as paths on the set of Young diagrams is given. This interpretation is then modified and applied to the set of \(Y_\ell(p)\) of all Young diagrams corresponding to the partition \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)\) of height \(\ell\) such that \(\lambda_1-\lambda_\ell\leq p-\ell\); that is, the degree of the \(p\)-modular representations corresponding to the partition \(\lambda\) is the number of oriented paths from \(\emptyset\) to \(\lambda\) entirely contained in \(Y_\ell(p)\). It is also possible to give a degree formula for an arbitrary partition in terms of paths in a graph. However, in this case, the graph contains multiple edges and in general, so far, it is not possible to compute these multiplicities. The results concerning \(Y_\ell(p)\) have also been proved by \textit{A. S. Kleshchev} [J. Algebra 181, No. 2, 584-592 (1996)].
scientific article; zbMATH DE number 922383