# What is so “plactic” about the plactic monoid?

The plactic monoid is the monoid consisting of all words from the alphabet $\mathbb{Z}^+$ modulo certain relations. It is important mainly because its elements enumerate semistandard Young tableaux.

I believe the plactic monoid was introduced by Knuth, but without that name. Lascoux and Schützenberger named it "le monoïde plaxique" in a French paper (1981) of the same name. (DISCLAIMER: I have never seen that paper; perhaps my second question is answered in it.)

Several questions:

1) How did plaxique $\rightarrow$ plactic? (This isn't the most obvious Anglicization; note that the MathSciNet entry for the original Lascoux/Schützenberger paper translates the title as "Plaxic'' monoids.) Who introduced the latter form of the word and why?

2) What is plactic/plaxique supposed to mean? As far as I know neither was a word in their respective languages before being applied to the word monoid/monoïde. I am entertaining an etymology from Greek $\pi \lambda \alpha \xi$ "flat surface," but I don't find it very compelling.

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x → ct is not that uncommon, e.g. galaxy → galactic, anorexia → anorectic; cf. also the variant spelling "connexion" of "connection" (and maybe in reverse the abbreviation "pix" for "pictures", or "pixel" for "picture element"). As for what "plactic" is supposed to mean, no idea — maybe ask Plactico Burress. – Noam D. Elkies Sep 24 '11 at 6:11
I wouldn't say there is a derivation x→ct, rather both groups may appear a result of an s or a t coming after a root ending in k. For instance, from an indoeuropean root * glak- we have the Greek γάλα, γάλακ-τος and the Latin lac, lac-tis, as well as γαλακ-σιας = γαλαξίας, the milky way. – Pietro Majer Sep 24 '11 at 8:04
@P.Majer: Yes, that looks like a better way to think about it, since it makes x/ct a special case of a larger and more familiar relation —e.g. sepsis/septic (as in tank, not polynomial!), opsin (or cyclops)/optic, neurosis/neurotic, etc. Further examples with x include apoplexy/apoplectic and praxis/practice. – Noam D. Elkies Sep 24 '11 at 13:39
In the third volume of the Leçons de mathématiques d'aujourd'hui, Xavier Viennot says: « On l'appelle le produit plaxique (ou plactique) des deux tableaux Young initiaux, parce que sa construction fait penser un peu à la tectonique des plaques. » – Maxime Bourrigan Sep 26 '11 at 14:02

You can find the original Lascoux & Schützenberger paper here. My French (especially mathematical French) is not great, so I haven't been able to determine how the term "plaxique" comes in. However, I can observe that L&S first introduce la congruence plaxique and define le monoïde plaxique as the quotient of the free monoid over the congruence. So, it seems to me that they were really thinking of the congruence as plactic/plaxic more than the monoid itself (perhaps a fine distinction?). They highlight the relevant properties of the congruence in Proposition 2.5, so maybe that provides a clue?

EDITED TO ADD: A quick scan of the OED yields no results for either "plactic" or "plaxic", but there is one result for the Latin "plaxus" under the etymology for the obsolete word "plash" (To bend down and interweave (stems partly cut through, branches, and twigs) so as to form a hedge or fence.):

an unattested post-classical Latin form *plaxus , alteration of classical Latin plexus , past participle of plectere to plait, interweave, twine (see plexus n.)

So, perhaps "plaxique" is meant to invoke a sense of intertwining or weaving? I could see how that could apply to the congruence relation.

Bonus fun fact: Plaxico Burress makes an appearance in the OED in a citation for the entry "return date":

New York Giants star receiver and gun nut Plaxico Burress breezed in and out of Manhattan Criminal Court in 15 minutes yesterday, with little happening besides the setting of a June 15 return date.

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In the italian abstract they use "monoide `a placche'", so they think of the word as deriving from "plaque". FWIW. – quim Sep 26 '11 at 11:31
My French (mathematical and otherwise) is fine. I concur - there is no introduction to why the name plaxique is chosen. It appears as if the authors assume this is already known? – Jacques Carette Sep 26 '11 at 12:19
Thanks for your good ideas and for the link to the paper especially! – Oliver Sep 27 '11 at 15:19

The English translation of Symmetric Functions, Schubert Polynomials and Degeneracy Loci by Laurent Manivel contains the following footnote on the phrase "plactic ring":

From the Greek $\pi \lambda \alpha \xi$, flat place, stone plate, tablet. This terminology is due to Lascoux and Sch\"utzenberger.

There is no evidence provided for this assertion, but the fact that the author is French makes me suspicious that he has insider information. There is no discussion of why this name was chosen or what it is meant to suggest.

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This is wild speculation, stemming only from the italian abstract to Lascoux & Schützenberger. There, "monoide a placche" is used alongside the parallel construction "varietà a bandiere" (flag varieties). At the end of the "préface," "la cohomologie des variétés drapeaux sur les corps finis" appears (again, drapeau=flag) as one of the connections worth mentioning. It may be plausible to think of a "plaque" as a "rigid flag", or a "discrete flag". Which then prompts the question about the origin of that name...

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