Consider a matrix chosen uniformly at random from the group . (Precisely, is a random variable distributed according to Haar measure.) The expected value of is zero, because is a measure preserving symmetry of . The expected value of is . (A nice exercise.) In general, the expected value of is zero, and the expected value of is the Catalan number . This is the sequence that starts , , , , , , …, where that first is the expected value of .
I learned that fact from Hyperelliptic curves, -polynomials, and random matrices, a very interesting
paper whose main substance I will completely ignore. That paper also discusses what happens when you replace with . The case is so easy we can work it out in detail: must be of the form , and is chosen uniformly at random from the interval . So and the expected value of is
Now, combinatorialists know that the Catalan numbers are closely related to properties of the symmetric groups . And the sequence is known to play the same role for the symmetry groups of the hypercubes. In fact, we call the symmetric groups the “Coxeter groups of type ” and the symmetry groups of the hypercubes the “Coxeter groups of type “. There is a growing philosophy that the sequence should be thought of as the Catalan numbers of type .
Now, there is one more infinite sequence of Coxeter groups — the groups of type . The Catalan numbers of type are ; the first few are , , , , …. Is there some random variable such that and ? And, if so, can we view
as the pushforward of some kind of Haar measure?
UPDATE: I think I lose. Let’s write . Then
I think that should implies that , with probability .
But then, if we want the expected value of to be , then we need to have with probability . And that implies .
UPDATE: Well, this idea seems dead for now (although I’d be curious to here if anyone can revive it.) I encourage readers to check out the comments, though: Greg Kuperberg and Michael Lugo both bring interesting perspectives.