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This is a simple terminology question: I want to know if the involution $z \mapsto z^{-1}$ on Laurent polynomials (over some ring, I happen to be working over $\mathbb{Z}$ but that's not important) has a special name.

My motivation is perhaps a little unusual for this site. I'm doing some computations that involve manipulating Laurent polynomials and, being a lazy sort of fellow, I'm letting the computer do it. Being extra lazy, I don't particularly want to learn a new programming language to do this so I'm using Perl as it's the only one that I know. However, there my laziness stops as whilst there's a Perl module for ordinary polynomials there isn't one for Laurent polynomials. Still, it wasn't hard to adapt it to Laurent polynomials so I did and the program is chugging away churning out these computations to its heart's content. In writing the methods (meaning, things you can do to a Laurent polynomial), most already have obvious names (add, subtract - actually called sub_, mul(tiplication), and so forth) but I don't know one for the obvious involution $z \mapsto z^{-1}$. inv sounds a little to easy to mistake for inverse.

So, is there a name for this? If not, would anyone like to suggest one (preferably with an unambiguous shortening - I've already gotten fed up of typing monomial every time)?

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By the way, I wiki-hammered. –  Ben Webster Dec 10 '10 at 20:36
Ryan: but it's not complex conjugation: $z^{-1}$ and $\overline{z}$ are two different things, aren't they? Ben: I was anticipating a few votes-to-close, but the wiki-hammer? I realise that my "would anyone like to suggest one" sounds a bit like it should be CW, but I was hoping that there would be a single, definite answer and would expect to wait to see if there were before wikifying. –  Andrew Stacey Dec 10 '10 at 20:58
They're the same when $z$ is a unit complex number. So I see no harm in calling it conjugation. I think I've seen this convention in some knot theory texts. It's also advantageous since it allows for notational sloppiness $z^{-1} = \overline{z}^1$. When you're computing Alexander polynomials and if you have sloppy handwriting, this is a major advantage. –  Ryan Budney Dec 10 '10 at 21:27
Ryan: my main issue with conjugation is that if the coefficient ring happens to be $\mathbb{C}$ then I would expect conjugation to act on the coefficients as well. –  Andrew Stacey Dec 10 '10 at 21:32
I don't much care either way, but I don't think that this is the type of question that should be wiki-hammered. –  Theo Johnson-Freyd Dec 10 '10 at 23:21
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5 Answers

up vote 5 down vote accepted

I would call it the antipode. If your base ring is commutative, then the Laurent polynomials are the coordinate ring of the multiplicative group, and the antipode gives you the inversion on the group scheme.

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I'm going for "antipode". "Bar" is too close to conjugation for my liking. –  Andrew Stacey Dec 20 '10 at 10:36
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I think the answer to the original question is that there is no special name for the involution (otherwise it would have turned up by now). My first encounter with it was in the 1979 Kazhdan-Lusztig paper on Hecke algebras, where they use a bar notation and combine this involution on Laurent polynomials with the inversion in a given Coxeter group to get an action on the Hecke algebra of that Coxeter group. The bar notation makes it unnecessary to invent a name for the involution on Laurent polynomials, but "bar involution" will certainly do.

By the way, a ring of Laurent polynomials (say over $\mathbb{Z}$) provides a nice nontrivial example for a graduate algebra course, even though it rarely if ever occurs in textbooks. It's natural to ask what are the prime ideals and factor rings, etc. Most often the examples of commutative rings which students see are too boring and predictable to motivate the ideal machinery.

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Nice example of an interesting ring! –  drbobmeister Dec 10 '10 at 21:58
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I'm with David here. It's "the bar involution," a very hard involution to write the symbol for alone.

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But bar is a perfectly good command in a Perl script! –  Theo Johnson-Freyd Dec 10 '10 at 23:20
Theo: fits very well with foo! –  Andrew Stacey Dec 11 '10 at 19:20
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"Argument involution"? I have never heard a name for the map, and I'm sure I would use "the $z\leftrightarrow z^{-1}$ map" as the name, in a paper.

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Well, to offer a somewhat garbled paraphrase T.S. "Old Possum" Elliot (see for example, http://www.americanpoems.com/poets/tseliot/5536):

You may think at first I'm as mad as a hatter

But The Naming of Things is a difficult matter,

It isn't just one of your holiday games!

Having so established my credentials in this matter, I present the following:

Think of the map $z \to z^{-1}$ as swapping charts on the Riemann sphere. (I'm OK with here, right? I mean, I haven't looked carefully at this stuff in quite awhile.) Think of the $z$-coordinate as corresponding to the North Pole. Then the $z^{-1}$- coordinate goes with the South Pole.

Now, like Ben Webster, I gots out my lil' ol' wiki hammer, yes I did, but just gave the great wiki mountain a tiny tap: what broke off was: http://en.wikipedia.org/wiki/South_Star.

The North Star is Polaris; the South Star is one Sigma Octantis.

Therefore, reminiscent, of the notion that the map $z \to z^{-1}$ is somewhat of a swap twixt north and south, how about calling your involution something based on Sigma Octantis?


SigmaOctantisInvolution----way to long to type

SigOct or sigoct----shorter, not too long, you'll not likely forget it

gee, what about

$\Sigma$ or perhaps $\sigma$ ----hard to do on a keyboard; BUT there is

"sigma or perhaps just "sig"---beginning to look like "real math"!

Ah ha! Abara K'Dabara--I create as I speak! From now on, I call the involution $z \to z^{-1}$ on the algebra of Laurent polynomials $\sigma$. You could call it sig for short; if that's a reserved word in Perl use a variant like sigoct etc. Or perhaps even better, for a function in a computer language, siginv--for the sigma involution, though I'd probably just go with sig if I could.

BTW, this question inspired one of my own: http://mathoverflow.net/questions/48994/humorous-curious-unusual-names-for-mathematical-entities


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@Ben Webster--will, if not $\sigma$, maybe $\beta$--for "bar" involution. –  drbobmeister Dec 10 '10 at 22:54
BTW, my "inspired question" was promptly closed. –  drbobmeister Dec 11 '10 at 5:21
@Andrew Stacey: so what name did you finally decide on? –  drbobmeister Dec 11 '10 at 5:22
Second thoughts--the involution $z \to z^{-1}$ is somewhat peculiar to Laurent polynomials and series; the ordinary "bar" symbol can be confused with (complex) conjugation; so maybe in the light of Scott Carnahan's answer this involution should be called $\alpha$ (for antipode) or perhaps $\lambda$-for Laurent. –  drbobmeister Dec 13 '10 at 4:18
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