On the Hu - Hurley -

Suppose m and n are integers such that 1 m n, and H is a subgroup of the symmetric group S m of degree m. Deene the generalized matrix function associated with the principal character of the group H on an m m matrix B = (b ij) by d H (B) = X 2H m Y j=1 b j(j) ; and deene the generalized numerical range of an n n matrix A associated with d H by It is known that W H (A) is convex if m = 1 or if m = n = 2. Hu, Hurley and Tam made the following conjecture: Suppose H = S m , 2 m n with (m; n) 6 = (2; 2). Let A 2 M n be a normal matrix. Then W H (A) is convex if and only if A is a multiple of a Hermitian matrix. In this note, counterexamples are given to show that the conjecture is not true when m < n. Some techniques are developed to show that the conjecture is valid under more restrictive assumptions.