On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Iliev, Valentin. "On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem." Serdica Mathematical Journal 26.2 (2000): 155-166. <http://eudml.org/doc/11486>.

@article{Iliev2000,
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.},
author = {Iliev, Valentin},
journal = {Serdica Mathematical Journal},
keywords = {Induced Monomial Representations of the Symmetric Group; Enumeration; monomial representations of symmetric groups; Pólya enumeration theorem; symmetric function; symmetric group},
language = {eng},
number = {2},
pages = {155-166},
publisher = {Institute of Mathematics and Informatics},
title = {On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem},
url = {http://eudml.org/doc/11486},
volume = {26},
year = {2000},
}

TY - JOUR
AU - Iliev, Valentin
TI - On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem
JO - Serdica Mathematical Journal
PY - 2000
PB - Institute of Mathematics and Informatics
VL - 26
IS - 2
SP - 155
EP - 166
AB - 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.
LA - eng
KW - Induced Monomial Representations of the Symmetric Group; Enumeration; monomial representations of symmetric groups; Pólya enumeration theorem; symmetric function; symmetric group
UR - http://eudml.org/doc/11486
ER -