Lie Groups in Nature
Okay, everyone else has mentioned this, so I suppose I should too: A team considering the one-dimensional Ising model of ferromagnetism has found some evidence they claim is linked to certain symmetry; an action of the Lie group on the space of states which preserves some interesting property or another. I can’t read the full paper myself, but John Baez discusses it a bit at the beginning of the most recent This Week’s Finds.
Now those of you who have been reading for a while might be thinking, “Eee-eight.. doesn’t that sound familiar from somewhere?” Well, actually those of you who have been reading for a while are probably experienced mathematicians and know all about to begin with but work with me here, people.
So, yes, we have talked about here! Back in the spring of 2007 as this whole project was just getting off the ground, the Atlas of Lie Groups project announced that they’d completed calculating the Kazhdan-Lusztig-Vogan polynomial for the split real form of . Immediately, I gave a quick overview of the idea of a Lie group, a Lie algebra, and a representation; a rough overview for complete neophytes of what, exactly, had been calculated; and an attempt to explain why we should care. All with the promise of more information eventually.
Of course it was the fall of 2008 before I even defined a group representation in the main exposition, so some of this information has been a long time coming. But still, what progress we’ve made! Of course, we’re not up to the point of really understanding Zuckerman’s three lectures, but all in good time.
I have, though, been inspired to think about a nice little toy geometry problem to talk about as my coverage of Riemann integration winds down.