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JOURNAL OF
ALGEBRAIC
COMBINATORICS
  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

Galois groups of multivariate Tutte polynomials

Adam Bohn , Peter J. Cameron and Peter Müller
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

DOI: 10.1007/s10801-011-0332-2

Abstract

The multivariate Tutte polynomial $\hat{Z}_{M}$ of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable v e to each element e of the ground set E. It encodes the full structure of M. Let v={v e } e\in E , let K be an arbitrary field, and suppose M is connected. We show that $\hat{Z}_{M}$ is irreducible over K(v), and give three self-contained proofs that the Galois group of $\hat{Z}_{M}$ over K(v) is the symmetric group of degree n, where n is the rank of M. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.

Pages: 223–230

Keywords: tutte polynomial; multivariate tutte polynomial; matroids; graphs; Galois theory

Full Text: PDF

References

Cameron, P.J., Morgan, K.: Algebraic properties of chromatic roots. Submitted Kung, J.P.S.: Twelve views of matroid theory. In: Combinatorial \& Computational Mathematics: Present and Future, Pohang, the Republic of Korea, 15-17 February 2000, p. 56 (2001) CrossRef Lang, S.: Algebra. Addison-Wesley, Menlo Park (1984) Merino, C., de Mier, A., Noy, M.: Irreducibility of the Tutte polynomial of a connected matroid. J. Comb. Theory 83, 298-304 (2001) CrossRef Morgan, K.: Algebraic aspects of the chromatic polynomial. PhD thesis Oxley, J.G.: Matroid Theory, Oxford Science Publications. Clarendon/Oxford University Press, New York (1992) Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Surveys in Combinatorics, vol. 327, pp. 173-226 (2005) CrossRef



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