Finite geometry and permutation groups: some polynomial links

• Published in: Rendiconti di matematica e delle sue applicazioni (1981) Vol. 26; no. 3-4; pp. 339 - 350
• Main Author: Peter J. Cameron
• Format: Journal Article
• Language: English
• Published: Sapienza Università Editrice 01.01.2006
• Subjects: linear code; matroid; permutation group; projective space; tutte polynomial
• ISSN: 1120-7183; 2532-3350

Any set of points in a finite projective space PG(n, q) defines a matroid which is representable over GF(q). The Tutte polynomial of the matroid is a two-variable polynomial which includes a lot of numerical information about the configuration of points. For example, it determines the weight enumerator of the code associated with the point set, and hence the cardinalities of hyperplane sections of the set. Another polynomial used in enumeration is the cycle index of a permutation group, which includes information about the number of orbits of the group on various configurations. This is the subject of a well-developed theory. The aim (not yet realised) of the research reported here is to combine the Tutte polynomial of a matroid with the cycle index of any group acting on the matroid to obtain a more general polynomial which tells us about the number of orbits of the group on configurations counted by the Tutte polynomial. The paper includes an introductory exposition of all these topics.