Timeline for How to correctly name "irreducible subrepresentation of an indecomposable representation"
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2013 at 19:54 | comment | added | Nithilher | Thank you, @Julian and @Noah for bringing the notion `socle' to my attention. And many thanks to @Jim for explaining to me that my language is mathematically imprecise. In fact, the situation I mentioned in my question is only a special case of indecomposable representations. The misleading wording happened because the typical situation we encounter in my physics context is, that all (relevant) representations can be written as direct sums of irreducible representations. By the way, I decided to use the long phrase given in my question. | |
| Nov 14, 2012 at 16:05 | comment | added | Jim Humphreys | P.S. In most settings the language of "representations" is equivalent to the language of "modules" (over some associative algebra), in which case it's shorter to write "simple submodule" than "irreducible subrepresentation". On the other hand, physicists and sometimes others often shorten things by writing "irrep" and the like. But the word 'indecomposable" has no obvious substitute except the shorthand "indec" rarely used except in lectures. | |
| Nov 14, 2012 at 15:53 | comment | added | Jim Humphreys | Your definition of indecomposable isn't the one normally used in mathematics: instead, this word refers to a module which can't be written as the direct sum of two proper submodules (nothing being said about the irreducibility of submodules). Aside from this, the long quoted description in your header is probably the best you can do in general. Naturally you can always make up new words, but if you use established terms you need to accept their standard meanings. | |
| Nov 14, 2012 at 12:28 | comment | added | Noah Snyder | Are you saying that this irreducible subrep is unique? In that case it is just the socle. | |
| Nov 14, 2012 at 10:08 | comment | added | Julian Kuelshammer | "direct sum of all irreducible subrepresentations" is called socle, but I guess "indecomposable direct summand of the socle" is not shorter. | |
| Nov 14, 2012 at 9:53 | history | asked | Nithilher | CC BY-SA 3.0 |