$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Tor{Tor}$Let's suppose $V$ is a $k$-vector space equipped with its standard (left) $\GL (V)$-action. The shuffle algebra is the graded dual of the tensor algebra with the standard Hopf algebra structure, and for ease I'm just going to denote this shuffle algebra by $T^\bullet (V)$.

Notice that the shuffle algebra is equivariant with respect to the induced $\GL (V)$-action, by which I mean $g (v_1 \cdot v_2) = (g v_1) \cdot (g v_2)$ for any $g \in \GL(V)$, $v_i \in T^\bullet (V)$. The *decomposables* of $T^\bullet (V)$ are going to be all elements in the image of the multiplication map $T^+ (V) \otimes_k T^+ (V) \to T^+ (V)$ ($-^+$ means positive degree); this is evidently a $\GL (V)$-representation by the equivariance mentioned above. The *indecomposables* are the elements that are not decomposable, unsurprisingly.

My question is this: is it known what the indecomposable elements of the tensor algebra are as a $\GL (V)$-representation? Or, maybe a better way to say this: is there a "nice" (interpretation up to you) description of the representation theoretic structure of the indecomposables?

I can give one possible description that I've come up with but isn't totally satisfying (and likely hard to read): the shuffle algebra may be viewed as the Tor-algebra $\Tor_\bullet^R (k,k)$ of the residue field $k$ over the ring $R = S_\bullet (V) / (V)^2$ (that is, take a polynomial ring and mod out by the irrelevant ideal squared). This is a so-called Golod ring and hence its homotopy Lie algebra is free on (a shift of) $\textrm{Tor}_\bullet^{S_\bullet (V)} (R , k)^*$, which is well-known to be equivariantly isomorphic to a direct sum of Schur modules corresponding to hook shapes $(2,1^i)$. Since the Ext algebra is the universal enveloping algebra of the homotopy Lie algebra in this case, we deduce that the indecomposable elements correspond to a plethysm problem for Schur modules, which I would like to avoid.

Anyway, any help is appreciated.

Markdown`*Markdown*`

, not $\textit{MathJax trickery}$`$\textit{MathJax trickery}$`

, to italicise. I have edited accordingly. $\endgroup$