# Understanding a quip from Gian-Carlo Rota

In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of the mathematical community. He argues that a particularly egregious symptom of this tendency is the cyclical rediscovery of forgotten mathematics by young mathematicians who are unlikely to realize that their work is fundamentally unoriginal. My question is about his example of this phenomenon.

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. Today it is K-theory yesterday it was categories and functors, and the day before, group representations. Behind these and several other attractive theories stands one immutable source: the ordinary, crude definition of the symmetric functions and the identities they satisfy.

I don't see how K-theory, category theory, and representation theory all fundamentally have at their core "the ordinary, crude definition of the symmetric functions and the identities they satisfy." I would appreciate if anyone could give me some insight into these alleged connections and, if possible, how they exemplify Rota's broader point.

• A hint to decoding Rota: Rota said a lot of intentionally provocative things. May 23, 2017 at 14:59
• There's also a dearth of real scholarly evidence in many of his diatribes/rants. May 23, 2017 at 15:28
• @ToddTrimble: Isn't that what diatribes are for? :-) May 23, 2017 at 15:45
• $K_0$ of a ring is a $\lambda$-ring, and of course the free $\lambda$-ring on one generator is the ring of symmetric functions... so K-theory in some sense has its origins in the symmetric functions. Though I don't know how much of this sticks around in the higher $K_i$'s. May 23, 2017 at 18:29
• @darijgrinberg The $\lambda$-structure of $K_*(X)$ is kind of important (e.g. rationally the splitting of $K_*(X)$ in eigenspaces for the Adams operations is precisely the splitting in rational motivic cohomology groups) but not really "central", in the sense that you can develop large swaths of the theory without mentioning it (although it will probably lurk in the backgroud). May 23, 2017 at 23:51

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.

• You may be right but we can't know for sure since, as I said in my answer, Rota is not around anymore. Anyway, I don't see the point of trying to spin what he said into a watered-down version regarding the (I think intentional) provocative aspects. May 23, 2017 at 21:09
• @AbdelmalekAbdesselam I don't think we disagree as much as we seem to. I agree with both what you wrote in your answer and in your comment. May 23, 2017 at 21:11
• @AbdelmalekAbdesselam Regardless, I have tried to say something in response to the mathematical question by mentioning a connection that is slightly different from those previously described. I will try to expand on it later. May 23, 2017 at 21:14
• This is, in my opinion at least, 100% the correct interpretation. I think this becomes perfectly obvious if one reads the complete quote, which is actually not at all aimed at $K$ theory, categories and functors etc., but rather at the "hero worship" mentioned parenthetically in the OP. The paragraph quoted in the OP is, as it were, tacked on at the end of the piece when all the main points have already been made, and it's very misleading to now wonder what R may have wanted to say about $K$ theory (correct answer: nothing). May 23, 2017 at 21:29
• @Christian: "100%", "THE correct interpretation", "perfectly obvious", "very misleading", "correct answer: nothing"... That's a bit much don't you think? May 24, 2017 at 13:26

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.

• I wouldn't be so sure. The sentence "Behind these and several other attractive theories stands one immutable source" is quite clear in this regard. May 23, 2017 at 16:08
• One says "the source of a river", not "the source behind a river".
– Joël
May 23, 2017 at 18:51

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.
• Sadly, neither Fulton-Lang nor my notes actually handle symmetric functions, much to their joint detriment. (I wrote most of my notes when I was just starting to get familiar with the subject; sadly, it shows. What my notes actually do is build up the basics of $\lambda$-rings from the power-series point of view in detail.) Donald Yau's "$\lambda$-rings" book and Hazewinkel's "Witt vectors, part I" are far better sources for this. May 23, 2017 at 18:35
• See the Lascoux ref in my answer. May 24, 2017 at 17:23
• @Tom: Thanks. One can also find Lascoux's article at phalanstere.univ-mlv.fr/~al/ARTICLES/RotaMarEnglish.ps.gz May 24, 2017 at 17:41

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.)

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.

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).

• "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 above refs, serving to present different perspectives on, or even generalizations of, the basic, originally discovered relations among the symmetric functions. Territorial instincts may compell some camps to claim the superiority of their insights, or approach, which is probably what Rota decries even though he was certainly guilty of this same behavior. May 24, 2017 at 6:11
• Tom, you might want to consider including your third comment in your answer. Rota's sentence structure is poor, and leads to ambiguity. In your comment, you clarify what he said without changing the meaning of his original phrase. The rest of your comment responds directly to the question in its entirety (there are two parts: what Rota meant in the quoted paragraph AND about the connections between symmetric functions to the three terms). Combined, this is a good answer! May 24, 2017 at 7:16
• For a nice presentation of the Adams operation in K-theory and relations to the Faber or Newton polynomials, see p. 221 of "A geometric introduction to K-theory" by Dugger (math.uoregon.edu/~ddugger/kgeom.pdf). Jul 25, 2017 at 18:01
• Feb 3, 2020 at 20:18
• Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.) Feb 27, 2020 at 13:44

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.