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In a similar fashion, any row or column i of F with a zero value may be eliminated if the corresponding value of x i is not desired. A reduced K may be reduced again. As a note, since each reduction requires an inversion, and each inversion is an operation with computational cost O(n 3), most large matrices are pre-processed to reduce ...
The space of complex-valued class functions of a finite group G has a natural inner product: , := | | () ¯ where () ¯ denotes the complex conjugate of the value of on g.With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table:
In particular, when F = C, every such character value is an algebraic integer. If F = C and χ is irreducible, then [: ()] () is an algebraic integer for all x in G. If F is algebraically closed and char(F) does not divide the order of G, then the number of irreducible characters of G is equal to the number of conjugacy classes of G.
If is a character of a finite group (or more generally a torsion group) , then each function value () is a root of unity, since for each there exists such that =, and hence () = = =. Each character f is a constant on conjugacy classes of G , that is, f ( hgh −1 ) = f ( g ).
Furthermore, a class function on is a character of if and only if it can be written as a linear combination of the distinct irreducible characters with non-negative integer coefficients: if is a class function on such that = + + where non-negative integers, then is the character of the direct sum of the representations corresponding to .
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
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The number of these irreducibles is equal to the number of conjugacy classes of G. The above fact can be explained by character theory. Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and |G ...