%I #16 Aug 27 2022 03:14:06

%S 6,720,3628800,1316818944000,52563198423859200000,

%T 327312129899898454671360000000,

%U 428017682605583614976547335700480000000000

%N The product of factorials s! where s belongs to the multiset of exponents of the Lie groups G=Br or G=Cr. Also 2^r times the classical Lie superfactorial of type Br ~ SO(2r+1). Also 2^{r(r-1)} times the Lie superfactorial of type Cr ~ Sp(2r).

%C To every simple Lie group G one can associate both a quantum and a classical superfactorial of type G.

%C 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.

%C 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.

%C 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.

%C The superfactorial of type Dr [nonascii characters here] SO(2r) defines the infinite sequence A169657.

%C If G is exceptional of type E, the Lie superfactorial defines a sequence with only three terms, see sequence A169667.

%C 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.

%C 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!

%C 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!

%C If G = F4, sf_{F4} = 1/2^{12} 1! 5! 7! 11! = 5893965000 (a sequence with only one term).

%C If G = G2, sf_{G2} = 1/3^{3} 1! 5! = 40/9 (a sequence with only one term).

%C 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.

%C 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 ).

%H R. Coquereaux, <a href="http://arxiv.org/abs/1003.2589">Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups</a>, arxiv:1003.2589

%H R. Coquereaux, <a href="http://arxiv.org/abs/1209.6621">Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups</a>, arXiv preprint arXiv:1209.6621, 2012. - From _N. J. A. Sloane_, Dec 29 2012

%F Product_{s \in 1,3,5,.., 2r-1} s!

%F a(n) ~ 2^(n^2 + n + 5/24) * n^(n^2 + n/2 - 1/24) * Pi^(n/2) / (sqrt(A) * exp(n*(3*n+1)/2 - 1/24)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Mar 05 2021

%t Product[Factorial[s], {s, 1, (2 r - 1), 2}]

%Y A000178 gives sf_G for G=Ar=SU(r+1). A169657 gives sf_G for G=Dr~SO(2r). A169667 gives sf_G for G=E6, E7, E8.

%K easy,nonn

%O 2,1

%A _Robert Coquereaux_, Apr 05 2010