A169667 The classical Lie superfactorial of types E6, E7, E8.

23361421521715200000, 19403468278119790545603479218421760000000000, 12389761771281087987161913865011039548629176646031786340025309566313679656889905840128000000000000000000000

Offset: 1

Author

Robert Coquereaux, Apr 05 2010

Keywords

• easy
• fini
• full
• nonn
• bref

Comments

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=E6, E7, E8. When G is exceptional of type E (this case), the Lie superfactorial does not define an infinite sequence: it has only three terms.

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

Since there are only three simply laced exceptional Lie groups, the r --> sf_{Er} sequence has only three terms.

If G is not simply laced, i.e. Br, Cr, G2 or 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 sequence 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 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 ).

Links

• R. Coquereaux, Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups, arxiv:1003.2589 [math.QA], 2010.
• R. 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

Crossrefs

A000178 gives sf_G for G=Ar=SU(r+1). A169657 gives sf_G for G=Dr~SO(2r). A169668 describes sf_G for non-simply laced cases.

Formula

sf_{E6} = 1! 4! 5! 7! 8! 11!.

sf_{E7} = 1! 5! 7! 9! 11! 13! 17!.

sf_{E8} = 1! 7! 11! 13! 17! 19! 23! 29!.