Chang Number

In mathematics, the Chang number of an irreducible representation of a simple complex Lie algebra is its dimension modulo 1 + ''h'', where ''h'' is the Coxeter number. Chang numbers are named after , who rediscovered an element of order ''h'' + 1 found by .

showed that there is a unique class of regular elements σ of order ''h'' + 1, in the complex points of the corresponding Chevalley group. He showed that the trace of σ on an irreducible representation is −1, 0, or +1, and if ''h'' + 1 is prime then the trace is congruent to the dimension mod ''h''+1. This implies that the dimension of an irreducible representation is always −1, 0, or +1 mod ''h'' + 1 whenever ''h'' + 1 is prime.

Examples

In particular, for the exceptional compact Lie groups ''G''₂, F4, E6, E7, and E8 the number ''h'' + 1 = 7, 13, 13, 19, 31 is always prime, so the Chang number of an irreducible representation is always +1, 0, or −1.

For example, the first few irreducible representations of G2 (with Coxeter number ''h'' = 6) have dimensions 1, 7, 14, 27, 64, 77, 182, 189, 273, 286,...
These are congruent to 1, 0, 0, −1, 1, 0, 0, 0, 0, −1,... mod 7 = ''h'' + 1.

References

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*{{Citation , authorlink=Victor Kac , last1=Kac , first1=Victor G , title=Algebra, Carbondale 1980 (Proc. Conf., Southern Illinois Univ., Carbondale, Ill., 1980) , publisher=Springer-Verlag , location=Berlin, New York , series=Lecture Notes in Math. , doi=10.1007/BFb0090559 , mr=613179 , year=1981 , volume=848 , chapter=Simple Lie groups and the Legendre symbol , pages=110–123, isbn=978-3-540-10573-2

Representation theory