@TheMaskedAvenger This is definitely not what is going on. The modifications you are proposing would yield Lyndon words for the associated ordering. Nyldon words behave very differently. For example, the Nyldon words $101$ and $1011$ defy this kind of description.

@darijgrinberg: sorry, no. I was not saying that. I was just pointing out that things that seem to happen in a 2 letter alphabet are unlikely to be true for bigger alphabets. For example, certain palindromes are Nyldon, but cycling the first letter to the end will not make them Lyndon Honestly, these Nyldon words are baffling.

@PerAlexandersson, I don't think this coincidence in a 2 letter alphabet generalizes. The thing about Lyndon words is that they are filled with patterns. For example, every Lyndon work looks like $w=w_1^kw_1'i$, where $w_1$ is Lyndon, $w_1'$ is a (possibly empty) left factor of $w_1$ and $i$ is a letter such that $w_1'i>w_1$ (this was proved by Leclerc). In contrast, Nyldon words seem to be pattern avoiding.

I don't have Bourbaki with me. Is $\varpi_1$ short or long?

Just to focus in, is your question about when US universities adopted linear algebra in their core curriculum? And, if this can be established, who were the advocates of this that made it happen?

The permutation representation of $S_n$ is an $n$-dimensional vector space $V$ with basis $x_1,\ldots,x_n$. The symmetric algebra is the ring of symmetric functions in $n$-variables. The standard $n-1$ dimensional representation (call it $W$) is the subspace of $V$ with basis $x_i-x_{i+1}$. Now, $S(W)\subset S(V)$ consists of shift invariant symmetric functions.

Wouldn't `Schur functions' be the canonical choice?

are you specifically interested in equation (4.2) in the link? This seems much more tractable than the general problem.

@Alex R. I seem to have added the assumption that $d_1=n-1$ in my answer. As you point out, it is much more complicated if this is not the case. I think that if $d_1=n+k-1$, then something like I wrote holds if $d_1-d_2=k_1+k$ (with $G=0$ for $>k_1+k)$.