TY - JOUR
T1 - Circular planar electrical networks
T2 - Posets and positivity
AU - Alman, Joshua
AU - Lian, Carl
AU - Tran, Brandon
N1 - Publisher Copyright:
© 2014 Elsevier Inc.
PY - 2015/5/1
Y1 - 2015/5/1
N2 - Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We first construct a poset EPn of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. We then answer various enumerative questions related to EPn, adapting methods of Callan and Stein-Everett. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of Postnikov, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy.
AB - Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We first construct a poset EPn of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. We then answer various enumerative questions related to EPn, adapting methods of Callan and Stein-Everett. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of Postnikov, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy.
KW - Electrical positroid
KW - Laurent phenomenon
KW - Network response
KW - Planar graph
KW - Poset
KW - Resistor network
UR - https://www.scopus.com/pages/publications/84920288767
U2 - 10.1016/j.jcta.2014.11.004
DO - 10.1016/j.jcta.2014.11.004
M3 - Article
AN - SCOPUS:84920288767
SN - 0097-3165
VL - 132
SP - 58
EP - 101
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
ER -