Timeline for Understanding a quip from Gian-Carlo Rota
Current License: CC BY-SA 4.0
36 events
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| Jul 24, 2023 at 0:08 | comment | added | Tom Copeland | 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 web.archive.org/web/20070705103720/http://www.rota.org/hotair/…) | |
| Mar 23, 2023 at 13:14 | comment | added | Tom Copeland | 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. | |
| Oct 27, 2021 at 17:37 | comment | added | Konrad Swanepoel | The quote is from 1937. The forced migrations were over a long period 1930-1952: 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… | |
| Oct 27, 2021 at 17:17 | comment | added | Tom Copeland | @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.) | |
| Oct 27, 2021 at 16:00 | comment | added | Konrad Swanepoel | 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. en.wikiquote.org/wiki/…. | |
| Aug 23, 2021 at 20:00 | history | edited | Tom Copeland | CC BY-SA 4.0 |
New quote from Rota
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| Aug 22, 2021 at 17:55 | comment | added | Tom Copeland | (cont) Hazewinkel writes: It seems unlikely that there is any object in mathematics richer and/or more beautiful than this one ... | |
| Aug 22, 2021 at 17:54 | comment | added | Tom Copeland | 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. | |
| Jun 16, 2021 at 8:56 | comment | added | Tom Copeland | 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. arxiv.org/pdf/math/0012213.pdf | |
| May 27, 2021 at 18:46 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| May 27, 2021 at 17:36 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Introduced Rota's statement as noted in my comments
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| May 27, 2021 at 17:26 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added Knutson's comments on symmetric functions and lambda-rings
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| Apr 30, 2021 at 20:50 | comment | added | Tom Copeland | See also the intro to Unitary Symmetry and Combinatorics by Louck, in particulat the discussion of boson polynomials. | |
| Apr 28, 2021 at 17:52 | comment | added | Tom Copeland | @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 mathoverflow.net/questions/73711/the-concept-of-duality/… | |
| Apr 28, 2021 at 16:23 | comment | added | Ellie K |
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
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| Apr 26, 2021 at 5:58 | comment | added | Tom Copeland | More in "$\lambda$-Rings and the Representation Theory of the Symmetric Group" by Donald Knutson. | |
| Aug 10, 2020 at 11:36 | comment | added | Tom Copeland | 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. | |
| Jul 23, 2020 at 15:30 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Removed link to (now) pay site
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| May 23, 2020 at 21:27 | comment | added | Tom Copeland | 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. | |
| Feb 27, 2020 at 13:44 | comment | added | Tom Copeland | Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.) | |
| Feb 3, 2020 at 20:18 | comment | added | Tom Copeland | Related mathoverflow.net/questions/111770/… | |
| Feb 3, 2020 at 18:02 | comment | added | Tom Copeland | 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. | |
| Jan 26, 2020 at 1:59 | comment | added | Tom Copeland | 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)" arxiv.org/pdf/math/0411536 | |
| Nov 11, 2019 at 21:07 | comment | added | Tom Copeland | see also mathoverflow.net/questions/127730/… . | |
| Nov 9, 2019 at 13:00 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Wrong name corrected
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| Nov 9, 2019 at 0:20 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Further notes on interpreting Arnold
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| Jul 25, 2017 at 18:01 | comment | added | Tom Copeland | 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 5, 2017 at 5:03 | comment | added | Tom Copeland | The Lost Cafe by Rota displays somewhat the breadth of his interests and his concern about plunderings of original research. | |
| May 26, 2017 at 1:15 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Added relevant note by Lascoux
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| May 24, 2017 at 16:39 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Added a link on Rota and lambda rings
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| May 24, 2017 at 15:44 | history | edited | Tom Copeland | CC BY-SA 3.0 |
Edit per comment
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| May 24, 2017 at 7:16 | comment | added | Ellie K | 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 6:11 | comment | added | Tom Copeland | "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 3:02 | comment | added | Tom Copeland | Metropolis and Rota wrote about Witt vectors and necklace algebras, which are discussed in the refs above. | |
| May 24, 2017 at 2:09 | comment | added | Tom Copeland | 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 oeis.org/A263916. | |
| May 23, 2017 at 20:15 | history | answered | Tom Copeland | CC BY-SA 3.0 |