Induced operators on symmetry classes of tensors

LetV be ann-dimensional inner product space,T_i,i=1,...,k, k linear operators onV, H a subgroup ofS_m (the symmetric group of degreem), a character of degree 1 andT a linear operator onV. Denote byK(T) the induced operator ofT onV(H), the symmetry class of tensors associated withH and . This note is concerned with the structure of the setK_{, m}^H (T₁,...,T_k) consisting of all numbers of the form traceK(T₁U₁...T_kU_k) whereU_i,i=1,...k vary over the group of all unitary operators onV. For V=ⁿ or ⁿ, it turns out thatK_{, m}^H (T₁,...,T_k) is convex whenm is not a multiple ofn. Form=n, there are examples which show that the convexity of_{, m}^H (T₁,...,T_k) depends onH and .The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for his valuable advice and encouragement.