Criteria Of Positivity For Linear Maps Constructed From Permutation Pairs

In this paper,we introduce a notion of property(C)for permutation pairs and show that a D-type map ΦD:Mnâ†'Mn with D=(n-2)In+PÏ€1+PÏ€2induced by a pair{Ï€1,Ï€2) of permutations of(1,2,...,n)is positive if{Ï€1,Ï€2)has property(C).The property(C)is characterized for any{Ï€1,Ï€2),and an easy criterion is given for the case that Ï€1=Ï€P and Ï€2=Ï€q,where Ï€ is the permutation defined by Ï€(i)=i+1mod n and1≤p<q≤n. We studied the case n=4intensively,and give a simple necessary and sufficient condition for{71,Ï€2}to have the property(C).Furthermore,a necessary and sufficient condition for a, D-type map Φπ11,Ï€2on4×4matrices constructed from a pair of arbitrary permutations {Ï€1,Ï€2] to be positive is obtained.