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It's a common observation in Lie theory that Cartan matrices and the Killing form are named after the wrong people; they were discovered by Killing and Cartan, respectively. I remember learning about many other examples of this phenomenon, but can't think of too many at the moment. Wikipedia has some examples here and here, but I'm curious about more obscure examples.

Bonus points for an interesting story behind why the concept was incorrectly named. Concepts that were deliberately named in honor of another mathematician don't count.

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closed as no longer relevant by Kevin H. Lin, quid, Qiaochu Yuan, Zev Chonoles, Gjergji Zaimi Jul 9 '11 at 3:12

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By the way, in the mathematical community "Stiegler's law" is often referred to as "Arnol'd's law", inclusive of the corollary "Arnol'd's law applies to Arnol'd's law as well". –  pasquale zito May 10 '10 at 20:24
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To further complicate things, there is also Whitehead's law: "Everything of importance has been said before by someone who did not discover it." –  bhwang May 10 '10 at 21:38
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Oh gosh, I could not imagine that there are SO many wrong names. Perhaps some day there will be a big important Brandenburg theorem, of course just because another one has proven it. ;-) –  Martin Brandenburg May 10 '10 at 23:09
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Not that I have a problem with the question per se, but "the wrong people" is pretty ambiguous. The first person to study something might not be the most deserving -- often a crucial application or popularizations trumps the actual innovation. Nor is it necessarily the case that the intent of the naming was to honor the inventor -- frequently the naming is done for reasons of analogy ("Euler systems" come to mind). –  Cam McLeman May 11 '10 at 1:10
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Stigler's law is called Boyer's Law by H.C. Kennedy in "Who Discovered Boyer's Law?" (Amer. Math. Monthly vol. 79 1972, 66--67). It says that "Mathematical formulas and theorems are usually not named after their original discoverers." The label Boyer's law was chosen because Boyer gave many examples of this phenomenon in his book A History of Mathematics. –  KConrad Sep 8 '10 at 17:12

56 Answers 56

I was once told that Riemann's integral is due to Darboux, while Lebesgue integral is due to Borel. Riemann invented the Cauchy integral instead.

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Are you sure that Darboux did not invent Darboux integral, and Riemann actually did invent Riemann integral? The definitions are different, although equivalent. –  VA. May 12 '10 at 3:19
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Part of the confusion might be that textbooks such as Rudin develop the theory of the Riemann integral by using the Darboux integral. I also remember being told that Rudin goes on to assume that properties proven for the Darboux integral hold for the Riemann integral (and maybe even the Cauchy integral) without proving that this works. Sneaky. –  Qiaochu Yuan May 13 '10 at 2:41

Chow varieties were invented by Van der Waerden (Chow was his student). And Hilbert schemes were invented by Grothendieck (who called them Hilbert schemes himself, however).

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If you search for almost any eponymous topic in Wikipedia, you'll find that it was first studied by someone else. For example, the Gaussian distribution (according to Wikipedia) was first studied by de Moivre. It seems that in many cases, naming the body of work was given to the person who first applied its study to some other field (using the earlier example, Gauss used the distribution in astronomy).

The common story goes that L'Hôpital bought "the rights" to L'Hôpital's rule, as he was a nobleman and not a mathematician by trade, although I am not sure about the veracity of that story.

Although I am no expert on the history of Mathematics, it seems as though ideas or formulae assumed their names from certain mathematicians due either to a.) the more notable application or publication of the theory or b.) attribution by mathematicians of a later generation to pay tribute to (or garner attention from) the work of their predecessors.

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L'Hopital hired Johann Bernoulli to teach him the calculus, and as allowed by their contract, wrote a book under his own name containing what he had learned, some of which were Bernoulli's original results. Wikipedia has references. –  Nate Eldredge May 10 '10 at 22:03
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Damn,that was the one I was going to post! –  Andrew L Aug 7 '10 at 19:00

Pythagoras' Theorem apparently predates Pythagoras.

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And similarly much of Euclid predates Euclid. –  Tom Smith May 10 '10 at 20:51
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But so far as we know, Pythagoras (or one of his followers) was the first to prove it, or even to attempt to do so. –  Charles Staats May 11 '10 at 2:39

I don't know if this is a real example, but it led to a nice gem in a recent abstract on the arXiv: "Glaisher's correspondence goes back to Euler."

(As far as I know Glaisher generalized Euler's bijection, which is why he gets the eponym -- in addition some people say "Euler-Glaisher" instead.)

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Huh? I don't follow. Euler did not have "bijective proof" as a concept. It was Glaisher who invented the bijection, but if I recall correctly never published it - Sylvester did it and attributed it to him some time later. You are probably referring to Euler's "odd vs. distinct" partitions theorem. Using modern "involution principle" technology one can convert analytic proofs into bijective, and in this case the one-line Euler's proof becomes Glaisher's bijection (this is O'Hara's theorem - see my survey on partitions and a recent paper on O'Hara algorithm, joint with Konvalinka). –  Igor Pak May 10 '10 at 21:50

According to Wikipedia, Markov's inequality is due to Chebyshev, and Chebyshev's inequality is due to Bienaymé.

On top of that, Hölder's inequality was first proved by Rogers, and Jensen's inequality by Hölder. What a mess!

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Related mess: Lipschitz and Hölder functions. –  Victor Protsak May 13 '10 at 0:11
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Would it be terminally geeky to point out that these attributions to Markov et al form a chain? –  smci Jan 18 '11 at 19:56

Burnside's Lemma, which asserts that the number of orbits of a group action is the average number of fixed points, was known to Cauchy. Burnside himself even attributed it to Frobenius in his book.

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I like the fact that this lemma is sometimes called "Burnside's Lemma" and sometimes "The lemma that is not Burnside's". –  Konrad Waldorf May 10 '10 at 21:31

Euler's nine point circle was never discussed by Euler. This is an error of the "argument by authority" type: Catalan propagated that incorrect attribution made by another scholar, the "learned Terquem", without checking it himself.

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And the Bianchi identities are due to Ricci (according to Levi-Civita).

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  1. Cauchy–Riemann equations were known to d'Alembert and Euler.

  2. Two-dimensional Voronoi diagrams were used by Descartes, three-dimensional - by Dirichlet. Also should be noted, that this construction has several other names in physics: Wigner–Seitz cells, Thiessen polygons.

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Banach algebras should probably be called Gelfand Algebras, or something similar. I'm not sure of the history here, but presumably the "Banach" is attached because this is the study of "complete" normed algebras. I don't believe that Banach actually did much work on algebras (as opposed to Banach spaces).

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The most amazing example I know is the Cayley formula which was discovered by Carl Borchardt nearly 30 years earlier. Not only Cayley knew about this, in his paper he specifically wrote that this formula is due to Borchardt, and all he wants to do is give a new simple proof (without determinants as in the matrix tree theorem).

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The "Lichnerowicz formula" relating the square of the Dirac operator to the Laplacian has been proved more than 30 years earlier by Schrödinger.

See: E. Schrödinger, Dirac'sches Elektron im Schwerefeld, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1932, 105-128 (1932).

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The Pell equation was named so because Euler thought that John Pell was responsible for some key results involving this equation. While Pell was a notable mathematician, he had essentially no connection to the equation. The common belief is that Euler mistook Pell for Lord Brouncker who indeed had a number of results related to the "Pell" equation.

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I thought some Indian mathematicians had something to do with it. See André Weil's Number Theory, from Hammurapi to Legendre. –  Chandan Singh Dalawat May 13 '10 at 2:49

The Frobenius automorphism associated to a prime ideal in a Galois extension of number fields was actually developed by Dedekind, who wrote about it (and the associated ramification groups, later found by Hilbert) in a letter to Frobenius on June 8, 1882. Frobenius published this construction in a paper in 1896. Some citations:

  1. Frobenius, Collected Works, Vol. 2, pp. 719--733.

  2. van der Waerden, Modern Algebra, Vol. 1 (Spring 1966), p. 203.

  3. Zassenhaus, Canadian Math. Bulletin 18 (1975), p. 448.

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Bézout's theorem

This result was discovered first by Newton in 1665. Even though MacLaurin (1720) and Leonhard Euler gave proofs, the theorem is usually attributted to Etienne Bézout who much later (1776) gave an incorrect proof of the result.

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The Vandermonde Determinant/Matrix. Apparently Vandermonde never explicitly discussed his eponymous determinant. According to Lebesgue in his survey of Vandermonde work, it was probably due to somebody misinterpreting Vandermonde's notation.

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Q: Who proved the Cayley-Hamilton Theorem?

A: Frobenius!

We now have the interesting question: Is this a maximal example of Stigler's law? That is, can we find distinct persons A, B, and C who are given credit for a result proven by D? Or A and B who are given credit for a result proven by C and D?

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And how long a directed cycle can we find in the graph where there is an edge from P to Q if P discovered/proved something attributed to Q? –  Saul Glasman May 11 '10 at 9:00
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What do you mean? That Frobenius was the first to prove it in full generality? Please supply a reference, as Cayley and Hamilton at least had proofs in dimension 2 and 3, if I recall correctly. –  Guntram May 12 '10 at 22:41

In my first algebra book the Eisenstein criterion for irreducibilty of a polynomial is named Schönemann criterion and is left as an exercise. This is confusing when all others are talking about the Eisenstein criterion ;-). In fact, here is a quote from Wikipedia:

The criterion is named after Ferdinand Eisenstein. It was published by T. Schönemann in Crelle's Journal 32 (1846), p. 100, and was popularized by Eisenstein in Crelle's Journal 39 (1850), pp. 166-169. Eisenstein's application of this theorem was to polynomials with coefficients in Z[i], not Z.

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Our linear algebra professor had a joke he told every year at the same spot in the lectures, for some 30 or 40 years. He'd say in an absolutely dry voice and facing the blackboard: "And this is the Cauchy–Bunyakovsky–Schwarz inequality, named like this because it was first proved by Lebesgue". Apparently, Cauchy just did it just as an inequality for sums (ie findim spaces), and Bunyakovsky and Schwarz independently as an inequality for integrals (ie for L2).

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I wouldn't really call this "naming it after the wrong person". Lots of analytic facts come from natural generalizations of observations from finite dimensional vector space to $\ell_p$ spaces then to $L^p$ spaces. –  Willie Wong May 11 '10 at 8:16

To expand on Pasquale's comment, here's a quote from Arnold's article:

Similarly to the fact that America does not carry Columbus's name, mathematical results are almost never called by the names of their discoverers.

In order to avoid being misquoted, I have to note that my own achievements were for some unknown reason never expropriated in this way, although it always happened to both my teachers (Kolmogorov, Petrovskii, Pontryagin, Rokhlin) and my pupils. Prof. M. Berry once formulated the following two principles:

The Arnold Principle. If a notion bears a personal name, then this name is not the name of the discoverer.

The Berry Principle. The Arnold Principle is applicable to itself.

Perhaps somebody knows which results by Kolmogorov et. al. he is thinking of.

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Stokes's theorem was stated by William Thomson (Lord Kelvin) in a letter to Stokes. The letter is reproduced on the cover of Spivak's Calculus on Manifolds. I believe the theorem was named after Stokes because he frequently put it on the Tripos exam in Cambridge.

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According to wikipedia: Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. –  Regenbogen May 15 '10 at 17:08

The Cantor-Schroeder-Bernstein theorem was proved by Dedekind; this link is to Dedekind's collected works and there is an informative note at the end.

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Nakayama's lemma was first proved by Krull in a special case, and by Goro Azumaya in the general case.

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I think that's why many people prefer to call it NAK lemma. –  VA. May 12 '10 at 2:51
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It's not so clear. In the historical notes to his book, Nagata says that in the case N=0 and M an ideal, the lemma was given by Krull, and that the general case is in a paper of Azumaya. But Nagata learned it from Nakayama and Azumaya when he was an undergraduate. Since some mathematicians were calling it Nakayama's lemma, he asked Nakayama who had this formulation first, to which Nakayama responded that he did not remember whether Nakayama or Azumaya was the first. Probably it should be considered folklore, and the name "Nakayama's lemma" seems appropriate. –  JS Milne May 12 '10 at 11:34
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"History is a myth men agree to believe."-Napoleon Bonaparte –  Andrew L May 13 '10 at 14:32

Farey fractions were introduced by C. Haros. John Farey asked a question about them that reached Cauchy, and Caucy then attributed the question and result to Farey, and the rest is history.

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It seems to me that Lagrange's theorem may well be one of the most prominent examples of the phenomenon under discussion.

According to J. J. Rotman,

the theorem was inspired by work of Lagrange (1770), but it was probably first proved by Galois.

Curiously enough, the Wikipedia article adscribes the first complete proof of the theorem to Pietro Abbati Marescotti.

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Cartan discovered the Killing form, and Killing discovered the Cartan matrix.

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That's the example I used in the original post :) –  Qiaochu Yuan May 13 '10 at 1:44
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Aw, nuts, that's what I get for not reading things carefully. :) –  David Hansen May 13 '10 at 2:08

Pell's equation

By a mistake of Euler, the Diophantine equation y^2 - Ax^2 = 1 has been erroneously known as "Pell's equation"; but, in fact, the English mathematician John Pell (1611-1685) did no more than copy it down in his papers, from Fermat's letters of 1657 and 1658.

For an extensive historical account on "Pell's equation", see Sir Thomas L. Heath, Diophantus of Alexandria : A Study in the History of Greek Algebra (Dover Pub., New York, 1931-1963, 552 pages), Supplement, Section II: "Equation y^2 - Ax^2 = 1, pp. 277-292. Particularly in page 285, after a presentation of the history of the equation up to Fermat's time (including citations to Pythagoreans, Archimedes, Diaphanous, and the Indian solution), one can read that:

" ... Fermat rediscovered the problem and was the first to assert that the equation x^2 - Ay^2 = 1, where A is any integer not a square, always has an unlimited number of solutions in integers. His statement was made in a letter to Frénicle of February, 1657 (cf. Oeuvres de Fermat, II, pp.333-4). Fermat asks Frénicle for a general rule for finding, when any number not a square is given, squares which, when they are respectively multiplied by the given number and unity is added to the product, give squares. If, says Fermat, Frénicle cannot give a general rule, will he give the smallest value of y which will satisfy the equations 61y^2 + 1 = x^2 and 109y^2 + 1 = x^2 ? (Footnote 3: Fermat evidently chose these cases for their difficulty; the smallest values satisfying the first equation are y=226153980, x=1766319049, and the smallest values satisfying the second are y=15140424455100, x=158070671986249)." And, after a extensive quotation of Fermat's letter, in page 286, one can read that: "The challenge was taken up in England by William, Viscount Brouncker, first President of the Royal Society, and Wallis (Footnote 1: An excellent summary of the whole story is given in Wertheim's paper "Pierre Fermat's Streit mit John Wallis" in Abhandlungen zur Gesch. der Math., IX. Heft (Cantor-Festschrit), 1899, pp.557-576). See also H. Konen, Geschichte der Gleichung t^2-Du^2=1, Leipzig (S. Hirzel), 1901). At first, owing apparently to some misunderstanding, they thought that only rational, and not necessarily integral solutions were wanted, and found of course no difficulty in solving this easy problem. Fermat was, naturally, not satisfied with this solution, and Brouncker, attacking the problem again, finally succeeded in solving it. The method is set out in letters of Wallis (Footnote 2: Oeuvres de Fermat, III, pp.457-480, 490-503) of 17th December, 1657, and 30th January, 1658, and in chapter XCVIII of Wallis' Algebra; Euler also explains it fully in his Algebra (Footnote 3: Part II, chap. VII), wrongly attributing it to Pell (Footnote 4: This was the origin of the erroneous description of our equation as the "Pellian" equation. Hankel (in Zur Geschichte der Math. im Alterthum und Mittlelalter, p.203) supposed that the equation was so called because the solution was reproduced by Pell in an English translation (1668) by Thomas Brancker of Rahn's Algebra; but this is a misapprehension, as the so-called "Pellian" equation is not so much as mentioned in Pell's additions (Wertheim in Bibliotheca Mathematica, III, 1902, pp.124-6); Konen, pp.33-4 note). The attribution of the solution to Pell as a pure mistake of Euler's, probably due to a cursory reading by him of the second volume of Wallis' Opera where the solution of the equation ax^2 + 1 = y^2 is given as well as information as to Pell's work in indeterminate analysis. But Pell is not mentioned in connexion with the equation at all (Eneström in Bibliotheca Mathematica, III, 1902, p.206)."

For more information about "Pell's equation", see Harold M. Edwards, The Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Springer-Verlag, New York, 1977, 410 pages), pp. 25-33. Particularly in page 33 one can read that

"This problem of Fermat is now known as "Pell's equation" as a result of a mistake on the part of Euler. In some way, perhaps from a confused recollection of Wallis's Algebra, Euler gained the mistaken impression that Wallis attributed the method of solving the problem not to Brouncker but to Pell, a contemporary of Wallis who is frequently mentioned in Wallis's works but who appears to have had nothing to do with the solution of Fermat's problem. Euler mentions this mistaken impression as early as 1730, when he was only 23 years old, and it is included in his definitive Introduction to Algebra written around 1770. Euler was the most widely read mathematical writer of his time, and the method from that time on has been associated with the name of Pell and the problem that it solved --- that of finding all integer solutions of y^2 - Ax^2 = 1 when A is a given number not a square --- has been known ever since as "Pell's equation", despite the fact that it was Fermat who first indicated the importance of the problem and despite the fact that Pell had nothing whatever to do with it."

See also André Weil, Number Theory : An approach through history - From Hammurapi to Legendre (Birkhäuser, Boston, 1984, xv+375 pages), in many different pages. In particular, at page 174, one can read that:

"Pell's name occurs frequently in Wallis's Algebra, but never in connection with the equation x^2 - Ny^2 = 1 to which his name, because of Euler's mistaken attribution, has remained attached; since its traditional designation as "Pell's equation" is unambiguous and convenient, we will go on using it, even though it is historically wrong."

Raul Nunes ( raul_nunes@uol.com.br ) NEST Nunes' Exposition of Scientific Truths ( http://www.geocities.com/raulnunes )

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There was a paper published in 2006 entitled "Simpson's Paradox in the Farey Sequence". The paradox is not Simpson's nor is the sequence Farey's. Bonus points.

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I think the Kazhdan-Lusztig Conjectures are due to Vogan.

EDIT.

True or false, the claim is mainly based on the very first two paragraphs of


[II] Irreducible characters of semisimple Lie groups II. The Kazhdan-Lusztig conjectures. David A. Vogan, Jr. Duke Math. J. Volume 46, Number 4 (1979), 805-859. --- The link

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077313724

gives a universal access to the first page, which contains the two paragraphs in question. In case you don't have access to the full paper, here is a scan of the references (to completely understand the two paragraphs):

http://www.iecn.u-nancy.fr/~gaillard/vogan_ref.pdf


Here are two more references:


[I] Irreducible characters of semisimple Lie groups I, David A. Vogan, Jr., Duke Math. J. Volume 46, Number 1 (1979), 61-108.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077313255


[KL] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae, Volume 53, Number 2, 165-184.

http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN356556735_0053&DMDID=dmdlog14


I would summarize things as follows.

Step 1. In [I] Vogan made a certain conjecture.

Step 2. [II] and [KL] were written simultaneously. Each paper cites the other. In [KL] Kazhdan and Lusztig also made a certain conjecture. When he learned this, Vogan immediately (or at least very fast) proved that the "Step 1 conjecture" implies that of Kazhdan and Lusztig. (He even showed that the "Step 1 conjecture" generalizes that of Kazhdan and Lusztig.)

But, again, the best is to read carefully the first two paragraphs of [II]. Vogan explains this much more clearly than I, and it's always better to hear things from the horse's mouth.

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The answer is incorrect, since the KL 1979 paper (Invent. Math.) was the first to state the KL Conjecture based on Hecke algebras and KL polynomials. Kazhdan and Lusztig were motivated by questions about singularities of Schubert varieties, Springer's representations of Weyl groups, Jantzen's work on Verma modules, etc. Vogan was approaching similar machinery from the direction of real Lie group representations, which led him after the KL paper to complete parts of his own program using "KLV polynomials". See Steven Kleiman's history of intersection homology (arXiv) for the role of GM. –  Jim Humphreys Aug 7 '10 at 16:00

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