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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Tutte polynomials and bicycle dimension of ternary matroids

Author: François Jaeger
Journal: Proc. Amer. Math. Soc. 107 (1989), 17-25
MSC: Primary 05B35
MathSciNet review: 979049
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Abstract: Let $ M$ be a ternary matroid, $ t\left( {M,x,y} \right)$ be its Tutte polynomial and $ d\left( M \right)$ be the dimension of the bicycle space of any representation of $ M$ over $ {\text{GF}}\left( 3 \right)$.

We show that, for $ j = {e^{2i\pi /3}}$, the modulus of the complex number $ t\left( {M,j,{j^2}} \right)$ is equal to $ {\left( {\sqrt 3 } \right)^{d\left( M \right)}}$. The proof relies on the study of the weight enumerator $ {W_\mathcal{C}}\left( y \right)$ of the cycle space $ \mathcal{C}$ of a representation of $ M$ over $ {\text{GF}}\left( 3 \right)$ evaluated at $ y = j$. The main tool is the concept of principal quadripartition of $ \mathcal{C}$ which allows a precise analysis of the evolution of the relevant invariants under deletion and contraction of elements.

Soit $ M$ un matroïde ternaire, $ t\left( {M,x,y} \right)$ son polynôme de Tutte et $ d\left( M \right)$ la dimension de l'espace des bicycles d'une représentation quelconque de $ M$ sur $ {\text{GF}}\left( 3 \right)$.

Nous montrons que, pour $ j = {e^{2i\pi /3}}$, le module du nombre complexe $ t\left( {M,j,{j^2}} \right)$ est égal à $ {\left( {\sqrt 3 } \right)^{d\left( M \right)}}$. La preuve s'appuie sur l'étude de l'énumérateur de poids $ {W_\mathcal{C}}\left( y \right)$ de l'espace des cycles $ \mathcal{C}$ d'une représentation de $ M$ sur $ {\text{GF}}\left( 3 \right)$ pour la valeur $ y = j$. L'outil essentiel est le concept de quadripartition principale de $ \mathcal{C}$ qui permet une analyse précise de l'évolution des invariants concernés relativement à la suppression ou contraction d'éléments.

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PII: S 0002-9939(1989)0979049-1
Article copyright: © Copyright 1989 American Mathematical Society

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