# Definition Given a [[Group Representation|matrix representation]] of a [[group]], the character is the [[trace]] of the matrix representation. $ \chi^{(\Gamma_j)}(g) = \text{trace}\left[D^{(\Gamma_j)}(g)\right] $ where $g$ is a [[group element]]. From the definition, it follows that a representation will have $h$ characters, one for each element of the group, where $h$ is the [[group order|order]] of the group. Note that the [[trace]] is invariant under [[Similarity Transformation|similarity transformations]], thus the character remains invariant. # Properties The [[Orthogonality Relations for Character#First Orthogonality Relation|Orthogonality Theorem for Character]] implies that the [[Primitive Character|primitive characters]] of an [[Irreducible Representation]] form a set of [[Orthogonality|orthogonal]] vectors in [[group element|group-element]] space, the space spanned by $h$ vectors, one for each element of the [[group]], which is also called [[Hilbert Space]]. ==What does this even mean?== Character tells us if a representation is [[Irreducible Representation|irreducible]] or not. If a representation is reducible, then the characters are not primitive, and they do not obey the [[Orthogonality Relations for Character|orthogonality relations]]. Furthermore, they tell us whether or not we have found all the [[Irreducible Representation|irreducible representations]]. From [[Group Representation#The Vector Space of Representations|the note on the vector space of representations]].