A positive formula for the dimensions of homogeneous components of free Lie algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:54:12Z http://mathoverflow.net/feeds/question/23782 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23782/a-positive-formula-for-the-dimensions-of-homogeneous-components-of-free-lie-algeb A positive formula for the dimensions of homogeneous components of free Lie algebras Mariano Suárez-Alvarez 2010-05-06T20:39:34Z 2011-09-01T13:22:12Z <p>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 <a href="http://en.wikipedia.org/wiki/Lyndon_word" rel="nofollow">Lyndon words</a> of length $k$ in $n$ letters, and of a few other things...</p> <p><strong>Question:</strong> Is there a <em>positive</em> formula for this number?</p> <p>As an aside,</p> <p><strong>Question:</strong> Is there a corresponding formula for the dimensions of the homogeneous components of the <a href="http://en.wikipedia.org/wiki/Triple_system" rel="nofollow">free Lie triple system</a> on $n$ letters?</p> http://mathoverflow.net/questions/23782/a-positive-formula-for-the-dimensions-of-homogeneous-components-of-free-lie-algeb/25982#25982 Answer by James Griffin for A positive formula for the dimensions of homogeneous components of free Lie algebras James Griffin 2010-05-26T09:45:40Z 2010-05-26T09:45:40Z <p>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.</p> <p>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</p> <p>$V^{\otimes k} \otimes_{S_k} \text{Lie}(k)$.</p> <p>And this module has dimension $(k-1)!$. This wont help you with the dimensions you want, but I think that it's interesting.</p> <p>If you want to read more then you need to learn about operads, and in particular the Lie operad.</p> <p>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 $S_k$.</p> <p>This last bit is a bit mysterious to me.</p>