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Title: On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem
Authors: Iliev, Valentin
Keywords: Induced Monomial Representations of the Symmetric Group
Issue Date: 2000
Publisher: Institute of Mathematics and Informatics
Citation: Serdica Mathematical Journal, Vol. 26, No 2, (2000), 155p-166p
Abstract: Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.
ISSN: 1310-6600
Appears in Collections:Volume 26 Number 2

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