Formal Versions of Conway's and Norton's Conjectures
The first conjecture made by Conway and Norton was the so-called "moonshine conjecture"; it states that there is an infinite-dimensional graded M-module
with for all m, where
From this it follows that every element g of M acts on each Vm and has character value
which can be used to construct the McKay–Thompson series of g:
The second conjecture of Conway and Norton then states that with V as above, for every element g of M, there is a genus zero subgroup K of PSL2(R), commensurable with the modular group Γ = PSL2(Z), such that is the normalized main modular function for K.
Read more about this topic: Monstrous Moonshine
Famous quotes containing the words formal, versions, conway, norton and/or conjectures:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (1564–1616)
“The assumption must be that those who can see value only in tradition, or versions of it, deny man’s ability to adapt to changing circumstances.”
—Stephen Bayley (b. 1951)
“Gentlemen, I give you a toast. Here’s my hope that Robert Conway will find his Shangri-La. Here’s my hope that we all find our Shangri-La.”
—Robert Riskin (1897–1955)
“Every cloud must have a silver lining.”
—George A. Norton (1880–1923)
“Our conjectures pass upon us for truths; we will know what we do not know, and often, what we cannot know: so mortifying to our pride is the base suspicion of ignorance.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)