Coalgebra dual to the symmetric algebra

Question

I'm interested in the coalgebra dual to the symmetric algebra $S(V)$ on some finite dimensional vector space $V$. I'd like to know explicitly what the comultiplication map is.

Answer 1

I recently ran into this question as well and tried to google search for a reference. Let $S^*(V)$ denote the symmetric algebra on $V$. It is given in degree $n$ by $V^{\otimes n}/\Sigma_n$.

Let $S_n(V)$ denote the symmetric n-tensors in $V$, e.g. $S_n(V)=(V^{\otimes n})^{\Sigma_n}$ (here we do not take quotients, but fixed points).

We then have a canonical evaluation map $S^n(V^*)\otimes S_n(V)\to \mathbb{F}$ given by $[f_1\otimes\ldots f_n](v_1\otimes \ldots \otimes v_n)=f_1(v_1)\ldots f_n(v_n)$. Note if we write a symmetric tensor as a linear combination of elementary tensors, then the summands need not be symmetric. So the upper notation is a bit sloppy.

It is easy to check that this evaluation map is well defined. Let $e_1,\ldots,e_n$ be a basis of $V$ and $f_1,\ldots,f_n$ be the dual basis. Then $f_1^{i_1}\ldots f_n^{i_n}$ is a basis for $S^n(V^*)$, where $i_*$ runs though all multi-indices adding up to $n$.

For such a multiindex $i$, we can also write down an element of $S_n(V)$ as follows. Let $S$ be a system of representatives of $\Sigma_n/(\Sigma_{i_1}\times \ldots \times \Sigma_{i_n})$ and then look at the vector $$E_{i}:=\sum_{\sigma\in S}\sigma (e_1^{\otimes i_1}\otimes \ldots e_n^{\otimes i_n})$$. Then these bases are dual to each other.

For example, if $\dim(V)=2$ and $n=2$, we get the elements $e_1\otimes e_1,e_1\otimes e_2+e_2\otimes e_1,e_2\otimes e_2$ as a basis for $S_2(V)$.

Next the coproduct should just be the restriction of the coproduct on the tensor algebra, i.e. $$S_n(V)\to \bigoplus_{r+s=n}S_r(V)\otimes S_s(V)$$ sending $E_i$ to $\sum_{k+l=i}E_k\otimes E_l$. Here the sum runs over all possibilites to write $i$ as a sum of two other multiindices.