The Cone of Class Function Inequalities for the 4‐By‐4 Positive Semidefinite Matrices

A function f from the symmetric group S n into R is called a class function if f (σ) = f (τ) whenever τ is conjugate to σ. Let d f be thegeneralized matrix function associated with f , mapping the n ‐by‐ n positive semidefinite Hermitian matricesto R. For example, if f (σ) = sgn (σ), then d f ( A ) = det A . We consider the cone K n of those f for which d f ( A ) ⩾ 0 for all n ‐by‐ n positive semidefinite Hermitian matrices. For n = 4 we show that K n is polyhedral, and explicitly find the extreme rays. Equivalently, f belongs to K n if d f ( A ) ⩾ 0 for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order n = 4. We obtain similar results for 4‐by‐4 real positive semidefinite matrices. 1991 Mathematics Subject Classification : 15A15, 15A45, 15A48.