Ultradiscrete Pl\"ucker Relation Specialized for Soliton Solutions

We propose an ultradiscrete analogue of Pl"ucker relation specialized for soliton solutions. It is expressed by an ultradiscrete permanent which is obtained by ultradiscretizing the permanent, that is, the signature-free determinant. Using this relation, we also show soliton solutions to the ultradiscrete KP equation and the ultradiscrete two-dimensional Toda lattice equation respectively.