What is so "plactic" about the plactic monoid? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:46:33Z http://mathoverflow.net/feeds/question/76254 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76254/what-is-so-plactic-about-the-plactic-monoid What is so "plactic" about the plactic monoid? Oliver 2011-09-24T01:49:01Z 2012-01-22T04:09:51Z <p>The <a href="http://en.wikipedia.org/wiki/Plactic_monoid" rel="nofollow"><em>plactic monoid</em></a> 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. </p> <p>I believe the plactic monoid was introduced by Knuth, but without that name. Lascoux and Schützenberger named it "<em>le monoïde plaxique</em>" in a French paper (1981) of the same name. (DISCLAIMER: I have never seen that paper; perhaps my second question is answered in it.)</p> <p>Several questions: </p> <p>1) How did <em>plaxique</em> $\rightarrow$ <em>plactic</em>? (This isn't the most obvious Anglicization; note that the <a href="http://www.ams.org/mathscinet-getitem?mr=646486" rel="nofollow">MathSciNet entry</a> for the original Lascoux/Schützenberger paper translates the title as <em>"Plaxic'' monoids</em>.) Who introduced the latter form of the word and why?</p> <p>2) What is <em>plactic/plaxique</em> supposed to mean? As far as I know neither was a word in their respective languages before being applied to the word <em>monoid/monoïde</em>. I am entertaining an etymology from Greek $\pi \lambda \alpha \xi$ "flat surface," but I don't find it very compelling.</p> http://mathoverflow.net/questions/76254/what-is-so-plactic-about-the-plactic-monoid/76396#76396 Answer by mhum for What is so "plactic" about the plactic monoid? mhum 2011-09-26T07:50:41Z 2011-09-26T08:07:16Z <p>You can find the original Lascoux &amp; Schützenberger paper <a href="http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1981-1PlaxiqueNaples.pdf" rel="nofollow">here</a>. My French (especially mathematical French) is not great, so I haven't been able to determine how the term "<em>plaxique</em>" comes in. However, I can observe that L&amp;S first introduce <em>la congruence plaxique</em> and define <em>le monoïde plaxique</em> 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?</p> <p>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 "<em>plaxus</em>" 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.):</p> <blockquote> <p>an unattested post-classical Latin form *plaxus , alteration of classical Latin <em>plexus</em> , past participle of <em>plectere</em> to plait, interweave, twine (see plexus n.)</p> </blockquote> <p>So, perhaps "<em>plaxique</em>" is meant to invoke a sense of intertwining or weaving? I could see how that could apply to the congruence relation. </p> <p>Bonus fun fact: Plaxico Burress makes an appearance in the OED in a citation for the entry "return date":</p> <blockquote> <p>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.</p> </blockquote> http://mathoverflow.net/questions/76254/what-is-so-plactic-about-the-plactic-monoid/76425#76425 Answer by quim for What is so "plactic" about the plactic monoid? quim 2011-09-26T13:54:21Z 2011-09-26T13:54:21Z <p>This is wild speculation, stemming only from the italian abstract to <a href="http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1981-1PlaxiqueNaples.pdf" rel="nofollow">Lascoux &amp; Schützenberger</a>. 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 <a href="http://mathoverflow.net/questions/58339/origin-of-terms-flag-flag-manifold-flag-variety" rel="nofollow">question</a> about the origin of <em>that</em> name...</p> http://mathoverflow.net/questions/76254/what-is-so-plactic-about-the-plactic-monoid/86352#86352 Answer by Oliver for What is so "plactic" about the plactic monoid? Oliver 2012-01-22T04:09:51Z 2012-01-22T04:09:51Z <p>The English translation of <em>Symmetric Functions, Schubert Polynomials and Degeneracy Loci</em> by Laurent Manivel contains the following footnote on the phrase "<em>plactic ring</em>":</p> <blockquote> <p>From the Greek $\pi \lambda \alpha \xi$, flat place, stone plate, tablet. This terminology is due to Lascoux and Sch\"utzenberger.</p> </blockquote> <p>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.</p>