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New references at symmetric function and new stub noncommutative symmetric function. An (unfinished?) discussion query from symmetric function moved here:
David Corfield: Why does Hazewinkel in his description of the construction of $\Lambda$ on p. 129 of this use a graded projective limit construction in terms of projections of polynomial rings?
John Baez: Hmm, it sounds like you’re telling me that there are ’projections’
$\Lambda_{n+1} \to \Lambda_n$given by setting the $(n+1)$st variable to zero, and that Hazewinkel defines $\Lambda$ to be the limit (= projective limit)
$\cdots \to \Lambda_2 \to \Lambda_1 \to \Lambda_0$rather than the colimit
$\Lambda_0 \to \Lambda_1 \to \Lambda_2 \to \cdots$Right now I don’t understand the difference between these two constructions well enough to tell which one is ’right’. Can someone explain the difference? Presumably there’s more stuff in the limit than the colimit.
Mike Shulman: I think the difference is that the limit contains “polynomials” with infinitely many terms, and the colimit doesn’t. That’s often the way of these things.
Actually, on second glance, I don’t understand the description of the maps in the colimit system; are you sure they actually exist? What exactly does it mean to “add in new terms with the new variable to make the result symmetric”?
David Corfield: The two constructions are explained very well in section 2.1 of the Wikipedia article.
Mike Shulman: Thanks! Here’s what I get from the Wikipedia article: the projections are easy to define. They are surjective and turn out to have sections (as ring homomorphisms). The ring of symmetric functions can be defined either as the colimit of the sections, or as the the limit of the projections in the category of graded rings. The limit in the category of all rings would contain too much stuff.
I gave elementary symmetric function its own brief entry and cross-linked a bit more.
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