

A169657


The classical Lie superfactorial of type Dr ~ SO(2r): When a Lie group G is simply laced, the classical Lie superfactorial sf_G is the product of s! where s belongs to the multiset E of exponents of G. Here G=Dr.


3




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,... g1} s! , with g = 2r2.
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



FORMULA

a(n) = sqrt(BarnesG(2*n)*Gamma(n)) / 2^((n1)/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 GlaisherKinkelin 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^((n1)/2), {n, 3, 10}] (* Vaclav Kotesovec, Apr 19 2024 *)
Table[Det[Table[i^(2*j), {i, 1, n1}, {j, 1, n1}]], {n, 3, 10}] (* Vaclav Kotesovec, Apr 19 2024 *)


CROSSREFS

A000178 gives sf_G for G=Ar=SU(r+1). A169667 gives sf_G for G=E6, E7, E8. A169668 describes sf_G for nonsimply laced series.


KEYWORD

easy,nonn


AUTHOR



STATUS

approved



