⇤Math diary
Jun 10, 2023: Weyl groups
Today my partner asked what I was thinking about. I was thinking about the story I outlined in entry 15, “Morse theory on coadjoint orbits”, and how it connects to representation theory. My partner is also in math, but not my field, so I couldn’t describe things in full generality with all its many moving parts. Instead I described the simplest case, $SU(2)$. I talk about the seemingly coincidental relationships between different fields, nod knowingly, and say “this holds in general”. I’ve reproduced my explanation below. It’s fairly friendly, and though it can’t contain the full force of the math, it should get the vibes.
Prologue: groups as symmetries
We want to understand rotations in 3D space, or the lie group $SO(3)$. Let’s think of a rotation as a symmetries of a sphere centered at the origin, the possible orientations of a beachball held at a point in space. For the most part, we care about things up to conjugation. For our beachball conjugation comes from a change in perspective, looking at the rotation of your beachball from a different vantage. We notice that all rotations have two fixed points at antipodal points on the sphere, and we can always change perspective to make these the top and bottom. For our mental image, think of these fixed points as the confluence points of the wedges of the beachball, held betwixt your fingers like the north and south pole of a globe.
There are a circle of rotations who fix these two points, parametrized by a single angle. This is a subgroup $S^1 \subset SO(3)$. Notably, $S^1$ is abelian. In fact, there are no other abelian subgroups of $SO(3)$ which contain this copy of $S^1$. Try to add any other rotation, and it will necessarily introduce noncommunativity. We call $S^1$ the maximal abelian subgroup, or in the language of lie groups, a maximal torus (tori are groups $(S^1)^n$, so $S^1$ is the lamest torus). Since every rotation fixes two points, every rotation belongs to some maximal torus.
But we are not done with our conjugation shenanigans. Though every rotation fixes two points, we don’t know which is the north pole and which is the south! This extra ambiguity comes from reflection around the equatorial plane, a rotation which sends the maximal torus $S^1$ to itself. We call the residual conjugacy action the Weyl group. In this case, it’s $\mathbb{Z}_2$.
All conjugacy-invariant things on a group $G$ can be reduced to its maximal torus $T$ with the residual conjugacy action, the discrete Weyl group $W$ acting on $T$. This vastly simplifies the problem, because tori are by far the easiest lie group. Passing to infinitesimal actions, the land of Lie algebras. $T$ becomes a simple vector space $\mathfrak{t}$ with a Weyl group action. All the structure of the group reduces to that of the Weyl group, which can be encoded combinatorially. This gives rise to the structure theory of Lie groups, their representations, and many more things. It’s one of two main tricks of Lie theory (the other being free passage between global and infinitesimal transforms, the relationship between Lie groups and their Lie algebra).
I didn’t get to finish. I wanted to go on to explain the Hamiltonian generating this flow, its gradient flow, and the morse theory of that. The critical points of the height function (who generates this flow) are in bijection with the Weyl groups. Thinking of this as a morse function, we use this to build the topology of the sphere. This decomposes the space into cells, one per element of the Weyl group. These are called the schubert cells. The point is, this gives a nice algebraic approach to the topology of flag manifolds.