"Twisted" Dimension
One may interpret the character of a representation as the "twisted" dimension of a vector space. Treating the character as a function of the elements of the group χ(g), its value at the identity is the dimension of the space, since Accordingly, one can view the other values of the character as "twisted" dimensions.
One can find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant is the graded dimension of an infinite-dimensional graded representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.
Read more about this topic: Character Theory
Famous quotes containing the words twisted and/or dimension:
“Then one will say, ‘He is not dead, maybe,
Who was mortality’s unshaken lover
Who loved the spring upon the Tennessee,
The hushed fall and, again, the coming clover.’
None will recall, not knowing, the twisted roads
Where the mind wanders till the heart corrodes.”
—Allen Tate (1899–1979)
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)