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On a new approach to Williamson’s generalization of Pólya’s enumeration theorem. (English) Zbl 0956.05009

Let \(W\) be a subgroup of the symmetric group \(S_d\). The Pólya enumeration theorem [G. Pólya, Acta Math. 68, 145-254 (1937; Zbl 0017.23202)] counts the \(W\)-orbits in \({\mathbb{N}}_0^d\) provided with weights and shows that the corresponding symmetric function in a countable set of variables is expressed in terms of the cyclic index of the group \(W\) (giving the decomposition of the elements of \(W\) as product of independent cycles). This theorem was generalized in [S. G. Williamson, J. Comb. Theory, Ser. A 11, 122-138 (1971; Zbl 0222.05008)] for arbitrary one-dimensional characters of \(W\) in such a way that the theorem of Pólya is obtained for the trivial character. The main purpose of the paper under review is to present a new point of view on these classical combinatorial results. The author gives a new proof in terms of the Schur-Macdonald theory of “invariant matrices” which can be traced back to the thesis of Schur [I. Schur, Gesammelte Abhandlungen. Band I. 1-70 (Springer-Verlag, Berlin-Heidelberg-New York) (1973; Zbl 0274.01054)]. The corner stone of the latter theory is the equivalence between the categories of the finite dimensional representations of the symmetric group and the homogeneous polynomial functors of degree \(d\) on the category of finite dimensional vector spaces and, comparing with the approach of Williamson, the proposed approach seems to have some conceptual advantages.

MSC:

05A15 Exact enumeration problems, generating functions
20C30 Representations of finite symmetric groups
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