9
$\begingroup$

Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He introduces the permutohedra like so:

Let us recall the definition of the permutohedron (alias zylchgon).

I had not heard this alternate name for permutohedra before, and tried to figure out the origin, but couldn't find anything. Sadly, Loday is no longer alive, so I can't ask him. Does anyone know the origin of "zylchgon"?

$\endgroup$
  • $\begingroup$ This is almost assuredly not the reason, but: the Chow polytope of a 0-dimensional variety (i.e., a set of points) is a Minkowski sum of standard simplices, i.e., a generalized permutohedron. So possibly 0-dimensional = "zilch"? $\endgroup$ – Sam Hopkins Oct 19 '18 at 18:11
  • 2
    $\begingroup$ Aha: it seems to have originated in section 6 of Carlsson and Milgram's chapter, "Stable homotopy and iterated loop spaces", maths.ed.ac.uk/~v1ranick/papers/carlmilg.pdf . "We now introduce a family of combinatorial cells which do just this, the Zilchgons, (also called permutahedra by combinatorialists), $C(n)$." I'm still not sure where the "zilchgon" terminology comes from. $\endgroup$ – Nathaniel Bottman Oct 19 '18 at 18:25
  • 2
    $\begingroup$ in physics the "zilch current" is a concept from electromagnetism, so called because it has "zilch" (zero) physical significance; the name originated in a 1964 paper (as discussed here on page 241) --- who knows, it may have inspired Milgram... $\endgroup$ – Carlo Beenakker Oct 19 '18 at 20:07
  • 5
    $\begingroup$ incidentally, the first instance of "zilchgon" I have found is a 1974 paper by Milgram, Unstable Homotopy from the Stable Point of View . Milgram cites an earlier 1966 paper as the origin of the concept, but I did not find the name there. $\endgroup$ – Carlo Beenakker Oct 19 '18 at 20:18
  • 2
    $\begingroup$ Hi Nate! Have you considered emailing Gunnar? Sometimes he writes back, and he's as likely to know the answer as anyone else: math.stanford.edu/~gunnar $\endgroup$ – Vidit Nanda Oct 19 '18 at 22:36

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.