And then for the interesting part: when n = 1, 2 or 4, and seemingly in no other cases, all the even moments are *integers*.

The reason is that these are the dimensions in which the spheres are *groups*. We can see that the even moments are integers because they are differences of dimensions of certain representations of these groups. So, Rogier Brussee and Allen Knutson pointed out that if we want to broaden our line of investigation, we can look at other compact Lie groups. That’s what I’ll do today.

has even moments that are *central binomial coefficients*, while the distance function

has even moments that are *Catalan numbers*.

Thinking about these two cases suggests some clues that may help us figure out the general pattern.

The irreducible representations of are indexed by integers. Imagine doing a discrete-time random walk on the integers, starting at 0, where at each time you take one step either left or right. How many ways are there to get back to 0 after steps?

So we’re getting central binomial coefficients.

The irreducible representations of are indexed by *nonnegative* integers (twice the physicists ‘spins’). Imagine doing a discrete-time random walk on the nonnegative integers, starting at 0, where at each time you take one step either left or right. How many ways are there to get back to 0 after steps?

So we’re getting Catalan numbers!

In short, the even moments of the distance function for seem to be counting walks from the trivial representation to itself in the set of irreducible representations of

There’s something about this that’s obvious from my post… but there’s also something nonobvious! The nonobvious part is that the square of the distance function is the character of a virtual representation; tensoring with this virtual representation over and over again generates a kind of random walk on the set of irreducible representation, but it’s *not* the ‘obvious’ random walk I just described!

You’ll see this from how hard Greg worked to show that the even moments for

are Catalan numbers.

So, there’s something sneaky going on, which I don’t understand. But we can still go ahead and try to figure out the even moments of the distance function for , which will be related to a random walk in the set of irreducible representations for the Weyl chamber, which looks like 1/6th of a triangular lattice.

(Or, for a less thrilling but probably instructive example, .)

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as special cases *compact Lie groups equipped with a faithful unitary representation*.

If we have a compact Lie group with a faithful unitary representation on , we can say

that is, it sits inside the algebra of matrices, which has a nice Euclidean distance function on it:

Moreover has a unique translation-invariant probability measure on that’s a Borel measure, called ‘normalized Haar measure’.

So, the distance from the identity element is a function on and we can try to compute its mean, variance, and higher moments.

And I believe if we normalize things correctly (which I may have just done), we will often get *sequences of integers* this way.

It would be nice to understand these. I think the first informative new case would be , with its usual representation on (I was using complex representations, but in this case working gives the same answer as if we’d used )

However, before even trying this, I got sidetracked by some pleasant thought about the two cases we already did!

]]>Ultimately all this is part of the age-old puzzle: “what are negative numbers, really?” Or, in more concrete terms, “I know what it’s like to have 2 apples; what’s it like to have -2 apples?” And of course many answers are known, but the most popular ones don’t say what it means to *peel* -2 apples.

I have worked on this stuff:

• John Baez, The mysteries of counting, 14 July 2005.

I don’t yet see a use for the more high-brow approaches to virtual representations in our puzzles here. *However*, I would like to generalize our puzzles from “spheres in Euclidean space that happen to be groups” to “groups that happen to live in Euclidean space”… and if we do that, we may ultimately see some advantages of a high-brow approach. I don’t know.

• Jim Hefferon, Linear Algebra, available free online at http://joshua.smcvt.edu/linalg.html/.

If you already know this material and want to go further, there are many directions to go—so many, in fact, that it’s hard for me to make a recommendation unless you narrow things down a bit!

]]>At some point in one’s experience one drops the exclamation point from “it turns out to be a power of 2!”, later from “it turns out to be a binomial coefficients!”, and I may be ready to lose it from “it turns out to be a Catalan number!”.

On the other hand, factorials never lose their power to thrill.

I’m really intrigued by that 5/6 factor, though, which would be a reasonable thing to measure for any compact group G + unitary representation V.

That sounds fun. We thought we were studying spheres—and in Part 2 we actually will—but there’s probably a lot more fun to be had studying compact groups with unitary representations! We actually looked a couple of questions that could generalize to those.

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