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The homogeneous component of degree $k$ in the free Lie algebra $\mathfrak{Lie}(x_1,\dots,x_n)$ in $n$ letters is of dimension $$g_n(k)=\frac{1}{k}\sum_{d|k}\mu(d)n^{k/d}.$$ This is also the number of Lyndon words of length $k$ in $n$ letters, and of a few other things...

Question: Is there a positive formula for this number?

As an aside,

Question: Is there a corresponding formula for the dimensions of the homogeneous components of the free Lie triple system on $n$ letters?

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Perhaps you meant the mu function instead of the phi function and n instead of m? In any case, I don't see a reason to expect a positive formula here; look at the case when n is prime. –  Qiaochu Yuan May 6 '10 at 20:44
Er, whoops; I meant the case where k is prime. You get (n^k - n)/k and there's really no sensible way around that minus sign that I can see. –  Qiaochu Yuan May 6 '10 at 20:52
Andy- positive formulae are like pornography, I know them when I see them. In general it's a sum where all the terms are positive and hopefully in bijection with known sets. –  Ben Webster May 6 '10 at 22:45
@Ben: your comment really confuses me... since for a basis of the free Lie algebra we can take Lyndon words (a very combinatorial object), the above formula can be thought of as a sum of one term which has a clear combinatorial meaning! –  Vladimir Dotsenko May 7 '10 at 5:27
I second Vladimir's comment. I'm also confused: as Mariano mentions, Lyndon words (or any Hall set in the free magma, for that matter, as in chapter II of Bourbaki's Lie groups and Lie algebras) are a basis for the free Lie algebra. Why isn't this a "positive formula"? I guess the point must really be the second question about triple systems? –  GS May 11 '10 at 10:59

1 Answer 1

This doesn't answer the question, but might still be of interest to you. Let $V$ be the $n$-dimensional vector space spanned by your $n$ letters.

The vector space $V^{\otimes k}$ has a natural $S_k$ action. There exists an $S_k$ module, which I will denote $\text{Lie}(k)$, such that the $k$th homogenous component of the free Lie algebra on $V$ is isomorphic to

$V^{\otimes k} \otimes_{S_k} \text{Lie}(k)$.

And this module has dimension $(k-1)!$. This wont help you with the dimensions you want, but I think that it's interesting.

If you want to read more then you need to learn about operads, and in particular the Lie operad.

If you just want to know the $S_k$-module structure on $\text{Lie}(k)$ then it can be given as follows: Let $C_k$ be a subgroup of $S_k$ generated by a $k$-cycle. Let $W$ be a 'primitive' representation of $C_k$. (this requires a primitive $k$th root of unity in your field). Then the module we are looking for is $W$ induced up to $S_k$.

This last bit is a bit mysterious to me.

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