Publication

Abstract : The Permanent Dominance Conjecture is currently the most actively pursued conjecture in the theory of permanents. If A is an n × n matrix, H is a subgroup of Sn and χ is a character of H then the generalized matrix function fχ(A) is defined as fχ(A) = ∑ σ∈H χ(σ) n∏ i=1 aiσ(i). If H = Sn and χ is irreducible then fχ is called an immanant. If H = Sn and χ is the principal or trivial character then fχ is called permanent. The permanent dominance conjecture states that perA ≥ fχ (A) χ(1n ) for all A ∈ Hn, where 1n denotes the identity permutation in Sn and Hn denotes the set of all positive semidefinite Hermitian matrices. The specialization of permanent dominance conjecture to immanants has been proved true for n ≤ 13. In this paper, we have proved that a matrix with an even number of non-positive rows and the other rows non-negative satisfies the permanent dominance conjecture. We prove that the specialization of the conjecture to immanants is satisfied by certain partitions of a natural number n. Also, we classify the partitions of a natural number n, which may not satisfy the conjecture.