|
|
COMMENTS
|
To every simple Lie group G one can associate both a quantum and a classical superfactorial of type G.
The classical Lie superfactorial of type G , denoted sf_G, is defined as the classical limit (q-->1) of the quantum Weyl denominator of G.
If G is simply laced (ADE Dynkin diagrams), i.e., Ar,Dr,E6,E7,E8 cases, the integer sf_G is the product of factorial s!, where s runs over the multiset of exponents of G.
The usual superfactorial r --> sf[r] is recovered as the Lie superfactorial r --> sf_{Ar} of type Ar [nonascii characters here] SU(r+1), sequence A000178.
The superfactorial of type Dr [nonascii characters here] SO(2r) defines the infinite sequence A169657.
If G is exceptional of type E, the Lie superfactorial defines a sequence with only three terms, see sequence A169667.
If G is not simply laced, i.e., Br (this case), Cr (this case), G2 or F4, the Lie superfactorial differs by simple pre-factors from the product of factorials of exponents.
If G=Br ~ SO(2r+1), the pre-factor is 1/2^r and r --> sf_{Br} = (1/2^r) Product_{s \in 1,3,5,.., 2r-1} s!
If G = Cr ~ Sp(2r), the pre-factor is 1/2^{r(r-1)} and r --> sf_{Cr} = (1/2^{r(r-1)}) Product_{s \in 1,3,5,.., 2r-1} s!
If G = F4, sf_{F4} = 1/2^{12} 1! 5! 7! 11! = 5893965000 (a sequence with only one term).
If G = G2, sf_{G2} = 1/3^{3} 1! 5! = 40/9 (a sequence with only one term).
The classical Lie superfactorial of type G enters the asymptotic expression giving the global dimension of a monoidal category of type G at level k, when k is large.
Call gamma the Coxeter number of G, r its rank, Delta the determinant of the fundamental quadratic form, and dim(G) its dimension, the asymptotic expression reads : k^dim(G) / ((2 pi)^(r gamma) Delta (sf_G)^2 ).
|
