We study graph parameters whose associated edge-connection matrices have exponentially bounded rank growth. Our main result is an explicit construction of a large class of graph parameters with this property that we call mixed partition functions. Mixed partition functions can be seen as a generalization of partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe and V. F. R. Jones, Graph invariants related to statistical mechanical models: examples and problems, J. Combin. Theory Ser. B 57 (1993), no. 2, 207–227.] and they are related to invariant theory of orthosymplectic supergroup. We moreover show that evaluations of the characteristic polynomial of a simple graph are examples of mixed partition functions, answering a question of de la Harpe and Jones. (NOTE. Some of the results of this paper were announced in an extended abstract: G. Regts and B. Sevenster, Partition functions from orthogonal and symplectic group invariants, Electron. Notes Discrete Math. 61 (2017), 1011–1017. Unfortunately that reference contains a mistake; we will comment on that below).
Publié le :
DOI : 10.4171/aihpd/100
Mots-clés : Partition function, graph parameter, orthogonal group, symplectic group, orthosymplectic Lie super algebra, circuit partition polynomial, connection matrix
@article{AIHPD_2021__8_2_179_0, author = {Regts, Guus and Sevenster, Bart}, title = {Mixed partition functions and exponentially bounded edge-connection rank}, journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D}, pages = {179--200}, volume = {8}, number = {2}, year = {2021}, doi = {10.4171/aihpd/100}, mrnumber = {4261669}, zbl = {1465.05068}, language = {en}, url = {http://archive.numdam.org/articles/10.4171/aihpd/100/} }
TY - JOUR AU - Regts, Guus AU - Sevenster, Bart TI - Mixed partition functions and exponentially bounded edge-connection rank JO - Annales de l’Institut Henri Poincaré D PY - 2021 SP - 179 EP - 200 VL - 8 IS - 2 UR - http://archive.numdam.org/articles/10.4171/aihpd/100/ DO - 10.4171/aihpd/100 LA - en ID - AIHPD_2021__8_2_179_0 ER -
%0 Journal Article %A Regts, Guus %A Sevenster, Bart %T Mixed partition functions and exponentially bounded edge-connection rank %J Annales de l’Institut Henri Poincaré D %D 2021 %P 179-200 %V 8 %N 2 %U http://archive.numdam.org/articles/10.4171/aihpd/100/ %R 10.4171/aihpd/100 %G en %F AIHPD_2021__8_2_179_0
Regts, Guus; Sevenster, Bart. Mixed partition functions and exponentially bounded edge-connection rank. Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 2, pp. 179-200. doi : 10.4171/aihpd/100. http://archive.numdam.org/articles/10.4171/aihpd/100/
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