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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0What do you mean "notation for the decomposition"? Didn't you stipulate that it is indecomposable? What do you mean 1\20 and 20/1? Where is this 20 coming from? You mention dimension... is $V$ also a vector space? – 2012-06-14
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0@rschwieb Sorry the question (which I've now editted) was very ambiguous. Basically, I have a finite group $G$ and a finite field $k$, and $V$ is a 21-dimensional module over the group algebra $kG$ (21-dimensional when considered as a vector space over $k$). $V$ has a single submodule of dimension 1, which gives rise to a single quotient of $V$ of dimension 20. In such a situation, I've seen some people write $V=20/1$ and some people write $V=1\backslash 20$. Is there a preferable way to denote the module $V$. – 2012-06-15
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0I still don't know what you mean "denote the decomposition", and to address your last comment, isn't the most preferable way to denote the module $V$ just simply "$V$"? – 2012-06-15
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0@rschwieb: Dear rschweib, "Denote the decomposition" means "denote the indicated socle filtration". Regards, – 2012-06-15
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0@MattE Thank you, that certainly makes a little more sense :) – 2012-06-15
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0I found the slashes to be confusing (you are building up a module, not quotienting it), and so would write 20.1 if the 1 was the socle. (Atlas notation for extensions). However the slash makes it a little more clear whether you have ascending or descending factors listed, so might be better. I agree with Matt E: no notation is very standard. Even submodule lattice diagrams have contradictory meanings. – 2012-06-15
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0@MattE Sorry for the ambiguous terminology – 2012-06-16
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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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0I think the authors of the papers in question are using this convention so that if they have an arbitrary module $V$, with composition series $0=V_{0}\subset V_{1}\subset\cdots\subset V_{n}=V$, with corresponding composition factors $W_{i}=V_{i}/V_{i-1}$, then they can illustrate the structure of $V$ by writing either: $V=W_{n}/W_{n-1}/\cdots/W_{1}$; or $V=W_{1}\backslash W_{2}\backslash\cdots\backslash W_{n}$. So my question could be summarised as which of these two descriptions of $V$ in terms of composition factors is more preferable? – 2012-06-15
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0@DavidWard Neither is standard as far as I know... Is there a reason they avoid "$W_n\supseteq W_{n-1}\supseteq\dots \supseteq W_1$"? – 2012-06-15
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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