The notion of Cartesian symmetry classes is introduced in [T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl. 292 (1999) 281. In this paper, we discuss these classes and compute the dimensions of these classes in terms of the fixed point character of S m. Also, we give a formula for the dimension of Cartesian symmetry class V χ(G) in terms of the rank of an idempotent matrix M(χ). Some properties of generalized trace functions of a matrix are concluded.
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