If $V$ is a 21-dimensional indecomposable module for a group algebra $kG$ (21-dimensional when considered as a vector space over $k$), which has a single submodule of dimension 1, what is the most acceptable notation for the decomposition of $V$, as I have seen both $1\backslash 20$ and $20/1$ used (or are both equally acceptable)?
Notation for an indecomposable module.
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0@JackSchmidt Thanks for your (as ever) valuable advice. – 2012-06-16
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My feeling is that this notation is not sufficiently standard for you to use either choice without explanation, hence whichever choice you make, you should signal it carefully in your paper. Given that, either choice looks fine to me.
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0Thanks for the answer. All of the answers and comments clarify that there is no standard notation, and hence as you say it is desirable to define things at the start of any paper. – 2012-06-15
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$V=1\backslash 20$? The 20/1 and 1\20 notation "for a module" looks nonsensical and bad for a few reasons, but then again math is a huge subject, so I could just be suffering from limited experience.
Is the idea of $1\backslash 20$ to say "$V$ has one submodule of codimension 20"? If so I imagine they have a better way of writing this than "module=number\number".
If $V$ had a single submodule $S$, I would talk about the quotient $V/S$, and its dimension $\dim (V/S)=20$.
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0Thanks for all of your comments. I think the reason that they don't use the notation ''$W_{n}\supseteq W_{n-1}\supseteq \cdots \supseteq W_{1}$'' is that they then go on to use the modules in subscript form to index objects which contain sub-objects and they have reserved the subset notation for the latter. – 2012-06-15