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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.

William Stagner
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    A hint to decoding Rota: Rota said a lot of intentionally provocative things. – Sam Hopkins May 23 '17 at 14:59
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    There's also a dearth of real scholarly evidence in many of his diatribes/rants. – Todd Trimble May 23 '17 at 15:28
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    @ToddTrimble: Isn't that what diatribes are for? :-) – R.P. May 23 '17 at 15:45
  • This MO post might be relevant https://mathoverflow.net/questions/66075/the-half-life-of-a-theorem-or-arnolds-principle-at-work – Liviu Nicolaescu May 23 '17 at 15:51
  • @René Touché! Of course you're right. :-) – Todd Trimble May 23 '17 at 18:17
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    $K_0$ 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. – darij grinberg May 23 '17 at 18:29
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    @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). – Denis Nardin May 23 '17 at 23:51

5 Answers5

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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.

Abdelmalek Abdesselam
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Will Sawin
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    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. – Abdelmalek Abdesselam May 23 '17 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. – Will Sawin May 23 '17 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. – Will Sawin May 23 '17 at 21:14
  • I don't know how many articles by Rota you read, but from my reading of his work I would tend to opt, when in doubt, for the most provocative interpretation...:) – Abdelmalek Abdesselam May 23 '17 at 21:20
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    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). – Christian Remling May 23 '17 at 21:29
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    @Christian: "100%", "THE correct interpretation", "perfectly obvious", "very misleading", "correct answer: nothing"... That's a bit much don't you think? – Abdelmalek Abdesselam May 24 '17 at 13:26
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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.

R.P.
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Joël
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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.
Abdelmalek Abdesselam
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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).

Tom Copeland
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  • An important role is played by the Faber polynomials, which provide transformations among the symmetric functions---elementary, complete, and power--the Newton / Waring / Girard identities. See also the Dress and Siebeneicher paper referenced in https://oeis.org/A263916. – Tom Copeland May 24 '17 at 02:09
  • Metropolis and Rota wrote about Witt vectors and necklace algebras, which are discussed in the refs above. – Tom Copeland May 24 '17 at 03:02
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    "Today is K-theory, yesterday it was 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. – Tom Copeland May 24 '17 at 06:11
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    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! – Ellie Kesselman May 24 '17 at 07:16
  • The Lost Cafe by Rota displays somewhat the breadth of his interests and his concern about plunderings of original research. – Tom Copeland Jul 05 '17 at 05:03
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    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 (http://math.uoregon.edu/~ddugger/kgeom.pdf). – Tom Copeland Jul 25 '17 at 18:01
  • see also https://mathoverflow.net/questions/127730/what-is-the-correspondence-between-combinatorial-problems-and-the-location-of-th . – Tom Copeland Nov 11 '19 at 21:07
  • The big Witt vectors are represented by Symm, the Hopf algebra of symmetric functions, arguably the most beautiful and rich object in present day mathematics. Immediately related are lambda and beta rings; for instance because the universal lambda ring on one generator is again Symm. --Hazewinkel in "Hopf algebras: their status and pervasiveness (as of Oct. 2004)" https://arxiv.org/pdf/math/0411536 – Tom Copeland Jan 26 '20 at 01:59
  • E.g., see p. 13 of "The moduli space of curves and Gromov-Witten theory" by Vakil: "It is a miraculous “fact” that everything else you can think of seems to lie in this subring. For example, the following generating function identity determines the λ-classes from the κ-classes in an attractive way, and incidentally serves as an advertisement for the fact that generating functions (with coefficients in the Chow ring) are a good way to package information ..." The author then gives a generating function for the λ-classes as Chern classes. – Tom Copeland Feb 03 '20 at 18:02
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    Related https://mathoverflow.net/questions/111770/cycling-through-the-zeta-garden-zeta-functions-for-graphs-cycle-index-polynomi – Tom Copeland Feb 03 '20 at 20:18
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    Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.) – Tom Copeland Feb 27 '20 at 13:44
  • See the section The Splitting Principle in Ch. 2 of "The K-book: An introduction to Algebraic K-theory" by Charles A. Weibel to again see relations to sym fct. – Tom Copeland May 23 '20 at 21:27
  • P. 63 of "Vector Bundles and K-Theory" by Hatcher (Version 2.2) discusses the relations among Adams operations, symmetric polynomials, Newton identities (introducing the Faber polynomials), and the splitting principle. – Tom Copeland Aug 10 '20 at 11:36
  • More in "$\lambda$-Rings and the Representation Theory of the Symmetric Group" by Donald Knutson. – Tom Copeland Apr 26 '21 at 05:58
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    This is lovely, Tom! Le teorie vanno e vengono ma le formule restano. It evokes the same feeling as realizing that Ohm's Law, V=IR remains valid in the classical AND quantum worlds – Ellie Kesselman Apr 28 '21 at 16:23
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    @EllieKesselman, nice, thanks. (Main article: "Ohm's Law Survives to the Atomic Scale" by Weber et al.) Wonder whether there is an enlightening connection to the duality expressed in https://mathoverflow.net/questions/73711/the-concept-of-duality/262691#262691 – Tom Copeland Apr 28 '21 at 17:52
  • See also the intro to Unitary Symmetry and Combinatorics by Louck, in particulat the discussion of boson polynomials. – Tom Copeland Apr 30 '21 at 20:50
  • From "K-Theory Past and Present" by Michael Atiyah: "K-theory may roughly be described as the study of additive (or abelian) invariants of large matrices. ... Examples of abelian invariants are traces and determinants." And, of course these are the purview of symmetric function theory. https://arxiv.org/pdf/math/0012213.pdf – Tom Copeland Jun 16 '21 at 08:56
  • From "Schur Functors and Categorified Plethysm" by Baez, Moeller, and Trimble: the ring of symmetric functions, denoted Λ ... shows up in many guises throughout mathematics. For example: • It is the Grothendieck group of the category Schur.• It is the subring of Z[[x1, x2, . . . ]] consisting of power series of bounded degree that are invariant under all permutations of the variables. • It is the cohomology ring H∗(BU), where BU is the classifying space of the infinite-dimensional unitary group. – Tom Copeland Aug 22 '21 at 17:54
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    (cont) Hazewinkel writes: It seems unlikely that there is any object in mathematics richer and/or more beautiful than this one ... – Tom Copeland Aug 22 '21 at 17:55
  • It might be that Rota based his quote on a well-known quote of Stalin: "The leaders come and go, but the people remain." At least it has the same form. https://en.wikiquote.org/wiki/Joseph_Stalin#:~:text=The%20leaders%20come%20and%20go%2C%20but%20the%20people%20remain. – Konrad Swanepoel Oct 27 '21 at 16:00
  • @KonradSwanepoel, was that before or after he forced migrations of whole ethnic groups? (I bet the same had been said since the first city-states evolved. Too bad Stalin and the likes haven't cycled through faster.) – Tom Copeland Oct 27 '21 at 17:17
  • The quote is from 1937. The forced migrations were over a long period 1930-1952: https://en.m.wikipedia.org/wiki/Population_transfer_in_the_Soviet_Union Looking at the context of the quote, he is addressing Soviet mining bosses, explaining how great communist economic leaders are as opposed to capitalist ones… – Konrad Swanepoel Oct 27 '21 at 17:37
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    Pg. 25 of "Symmetric Functions and Hall Polynomials" by Macdonald has a mapping between the symmetric functions and those of the free $\lambda$-ring in one variable. – Tom Copeland Mar 23 '23 at 13:14
  • Another example of Rota's motif: "Partial rediscoveries of this fact are still being published every few years by mathematicians who haven't done their reading." (From Light Shadows: Remembrances of Yale in the Early Fifties by Rota at https://web.archive.org/web/20070705103720/http://www.rota.org/hotair/light.html.) – Tom Copeland Jul 24 '23 at 00:08
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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.

Vladimir Dotsenko
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