Understanding a quip from Gian-Carlo Rota

I think Abdelmalek Abdesselam and William Stagner are completely correct in their interpretation of the words "Behind" and "one immutable source" as describing one theory, the theory of symmetric functions, being the central core of another.

The issue that led to this question instead comes from misunderstanding this sentence:

Today it is K-theory yesterday it was categories and functors, and the day before, group representations.

The listed objects are not a list of theories. If they were, he would say "category theory" and "representation theory". Instead, it is a list of different languages, or as Rota calls them, jargons. The function of this sentence is to explain what jargons he is referring to in the previous sentence.

If we delete it, the paragraph still makes perfect sense, but lacks detail:

In all mathematics, it would be hard to find a more blatant instance of this regrettable state of affairs than the theory of symmetric functions. Each generation rediscovers them and presents them in the latest jargon. [...] Behind these and several other attractive theories stands one immutable source: the ordinary, crude definition of the symmetric functions and the identities they satisfy.

The "theories" in question are not K-theory, category theory, and representation theory but rather the theory of symmetric functions expressed in the languages of K-theory, category theory, and representation theory. For instance presumably one of them is the character theory of $GL_n$, expressed in the language of group representations.

The reason I am confident in this interpretation is nothing to do with grammar but rather the meaning and flow of the text. The claim that symmetric function theory is the source of three major branches of mathematics seems wrong, but if correct, it would be very bizarre to introduce it in this way, slipped in the end of a paragraph making a seemingly less shocking point, and then immediately dropped (unless the quote was truncated?). One would either lead with it, or build up to it, and in either case then provide at least some amount of explanation.

Thus instead I (and Joel, and Vladimir) interpret it as making a less dramatic claim.

Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.)

Perhaps the Wiki on the Adams operation and "Formal groups, Witt vectors, and free probability" by Friedrich and McKay provide a quick intro to the connections the OP is questioning.

"Today the jargon is that of K-theory, yesterday it was that of categories and functors, and, the day before, group representations." All three jargons are used in the refs above (and those in my comments), serving to present different perspectives on, or even generalizations of, the basic, originally discovered relations among the symmetric functions. Territorial instincts may compel some camps to claim the superiority (and even priority) of their insights, or approach, which is probably what Rota decries even though he was certainly guilty of this same behavior.

(Read the introductory paragraph of "Alphabet Splitting" by Lascoux: ... meals were followed by long discussions about the comparative merits of algebraic structures, Gian Carlo for his part tirelessly asking me to repeat the definition of λ-rings that he copied each time in his black notebook with a new illustrative example.)

Added Nov. 8, 2019

I'm fairly convinced that Rota expressed exactly what he meant to express--that the identities/properties of the symmetric polynomials lie at the foundations of "these and several other attractive theories." See the refs and comments in the MO-Q "Canonical reference for Chern characteristic classes," in particular, "Characteristic classes and K-theory" by Randal-Williams, the linked Wikipedia article on Chern classes, the relevant sections in "Manifolds and Modular Functions" by Hirzebruch et al., and the Wikipedia article on the Splitting Principle.

Added 5/27/21:

Donald Knutson, in "$lambda$-Rings and the Representation Theory of the Symmetric Group," states, "the notion of $lambda$-ring is built upon the classical Fundamental Theorem of Symmetric Functions," that much of classical algebra is based on this theorem, and "the general definition (of a $lambda$-ring) is somewhat complicated ... and will be best understood by first analyzing one manifestation of the ring Z, its appearance in the simplest example of K-theory." In addition, "the main technical tool (In proving the Fundamental Theorem) is the notion of $lambda$-ring, first introduced by Grothendieck in 1956 ... in an algebraic-geometric context, and later used in group theory by Atiyah and Tall ... ."

Edit 8/23/2021:

From "Ten lessons I wish I had learned before I started teaching differential equations" by Rota:

I have always felt excited when telling the students that even though there is no formula for the general solution of a second order linear differential equation, there is nevertheless an explicit formula for the Wronskian of two solutions. The Wronskian allows one to find a second solution if one solution is known (by the way, this is a point on which you will find several beautiful examples in Boole’s text). ... every differential polynomial in the two solutions of a second order linear differential equation which is independent of the choice of a basis of solutions equals a polynomial in the Wronskian and in the coefficients of the differential equation (this is the differential equations analogue of the fundamental theorem on symmetric functions, but keep it quiet).

I am not a native speaker of English and moreover belong to the ethnic group that is known to mess up the articles, but I certainly don't feel that the sentence "Behind these and several other attractive theories stands one immutable source" necessarily implies that the theory of symmetric functions is "THE" core of those theories. Many mathematical theories are connected to symmetric functions and their various structures, most notably the plethysm, and indeed experts do develop their own language and notation for those structures, sometimes reinventing the bicycle, and that, in my opinion was what Rota wanted to express, in his usual provocative way.

Rota is not around anymore, so we can't go and ask him what he meant. My guess is that he is referring to the $lambda$-ring structure of symmetric functions which is related to plethysm and the composition of Schur functors (that's the representation theory connection as well as the category theoretic one regarding polynomial functors). This $lambda$-ring structure plays a role in $K$-theory as explained, e.g. see "Riemann-Roch Algebra" by Fulton and Lang. For other references see, e.g,

1. This set of notes by Darij Grinberg.
2. Donald Yau's "Lambda-Rings" book
3. This survey article about big Witt vectors by Hazewinkel.

I think you are misinterpreting the quote. In the last sentence, the word "source" does not mean "source of these theories (K-theory, categories, group representations", but "source of the theory of symmetric functions". Rota is not claiming that K-theory, etc. have "at their core the ordinary, crude definitions of symmetric functions", but that "the theory of symmetric functions", tautologically, does. What he is saying, it seems, is that the beautiful and rich theory of symmetric functions can be and has been developed without the need of modern fashionable abstract theories.