Criteria of positivity for linear maps constructed from permutation pairs

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In this paper, we show that a D-type map ΦD:Mₙ→ Mₙ with D=(n-2)Iₙ+Pπ₁+Pπ₂ induced by a pair {π₁,π₂} of permutations of (1,2,..., n) is positive if {π₁,π₂} has property (C). The property (C) is characterized for {π₁,π₂}, and an easy criterion is given for the case that π₁=πᵖ and π₂=πq, where π is the permutation defined by π(i)=i+1 mod n and 1≤ p<q≤ n.