Here are four (seemingly) different uses of the word conjugate:

  1. Complex conjugates are a concrete instance of the idea of conjugacy in field extensions.

  2. In group theory, there's the idea of conjugacy classes

  3. In probability theory, there are conjugate distributions

  4. Then there's also the convex conjugate of a function

What do they have in common? What is the most general idea of a conjugate?

  • Why would they have anything in common? The same term could have arisen several times independently; or, more probably, the term was carried over from one area to the next through analogy. But there's no reason to expect any overarching formal concept of conjugacy. – Yuval Filmus Mar 6 '15 at 5:59
  • @YuvalFilmus Maybe this is a better way to phrase the question: When the mathematicians who invented each of those definitions named their idea, why did they choose the word "conjugate" as opposed to some other word? Maybe there's a historical or linguistic context? – Mike Izbicki Mar 6 '15 at 6:07
  • Or it could be chance. Indeed, perhaps some of these concepts have (or have had) several competing names. Perhaps in other languages there is no cognate of "conjugate" common to all these terms. – Yuval Filmus Mar 6 '15 at 6:10
  • When something is called a "conjugate", it sometimes suggests that "the conjugate of the conjugate is the thing you started with." This is true for the complex conjugate and for the convex conjugate (under certain assumptions). – littleO Mar 6 '15 at 6:33
  • @littleO That's also typically true of things called "dual". – Mike Izbicki Mar 6 '15 at 7:14

Galois Theory says that conjugate subfields correspond to conjugate subgroups, where conjugate subfields are as in field extensions and conjugate subgroups are as in group theory.

  • This is a nice connection between types 1 and 2, but it doesn't really answer the question since it doesn't address the other types of conjugacy. – Mike Izbicki Mar 7 '15 at 22:40
  • Do you know anything about the historical development of these names? My guess would be that complex conjugates came first, then the idea of conjugates in a field and group conjugates were created about the same time with the development of galois theory. – Mike Izbicki Mar 7 '15 at 22:42
  • 1
    Sorry, can't help. "Conjugate" comes from the Latin (perhaps via the French), and essentially means "conjoined" --- thus, it gets used when you have two or more things of the same type that are often seen in each other's company. It may not go any deeper than that. – Gerry Myerson Mar 8 '15 at 5:56

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