On the Hu-Hurley-Tam conjecture concerning the generalized numerical range Original Research Article

Suppose m and n are integers such that 1less-than-or-equals, slantmless-than-or-equals, slantn, and H is a subgroup of the symmetric group Sm of degree m. Define the generalized matrix function associated with the principal character of the group H on an m×m matrix B=(bij) by image dH(B)=∑σset membership, variantH∏j=1mbjσ(j),and define the generalized numerical range of an n×n matrix A associated with dH by image WH(A)={dH(V*AV):V is n×m suchthat V*V=Im}.It is known that WH(A) is convex if m=1 or if m=n=2. Hu, Hurley and Tam made the following conjecture: Suppose H=Sm, 2less-than-or-equals, slantmless-than-or-equals, slantn with (m,n)≠(2,2). Let Aset membership, variantMn be a normal matrix. Then WH(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