OFFSET
3,1
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 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 given sequence is the Lie superfactorial of type Dr: r --> sf_{Dr} = (g/2)! Product_{s in 1,3,5,... g-1} s! , with g = 2r-2.
If G is exceptional of type E, the Lie superfactorial does not define an infinite sequence (see A169667).
If G is not simply laced, i.e., (Br, Cr, G2, F4) cases, the Lie superfactorial is also simply related to the product of factorials s! where s belongs to the multiset E of exponents of G. See sequences A169668.
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 r the rank of G, gamma its Coxeter number, 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 ).
LINKS
Robert Coquereaux, Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups, arxiv:1003.2589 [math.QA], 2010.
Robert Coquereaux, Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups, arXiv preprint arXiv:1209.6621 [math.QA], 2012-2013. - From N. J. A. Sloane, Dec 29 2012
FORMULA
From Vaclav Kotesovec, Apr 19 2024: (Start)
a(n) = sqrt(BarnesG(2*n)*Gamma(n)) / 2^((n-1)/2).
a(n) ~ 2^(n^2 - n + 17/24) * Pi^(n/2) * n^(n^2 - n/2 - 1/24) / (sqrt(A) * exp(3*n^2/2 - n/2 - 1/24)), where A is the Glaisher-Kinkelin constant A074962.
(End)
MATHEMATICA
sfD[r_] := Factorial[(2 r - 2)/2] Product[Factorial[s], {s, 1, (2 r - 2) - 1, 2}]
Table[Sqrt[BarnesG[2*n]*Gamma[n]] / 2^((n-1)/2), {n, 3, 10}] (* Vaclav Kotesovec, Apr 19 2024 *)
Table[Det[Table[i^(2*j), {i, 1, n-1}, {j, 1, n-1}]], {n, 3, 10}] (* Vaclav Kotesovec, Apr 19 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Robert Coquereaux, Apr 05 2010
STATUS
approved