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I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.

Where can I find this reference?

UPDATE: Some have questioned my use of the word "noteworthy." I really just want this information for as many groups as possible. Also, the Atlas of Finite Group Representations might very well be an example of the desired reference, but which of the cataloged representations are actually faithful? And are there other references available that give the information I want for other groups?

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Have you tried GAP? gap-system.org –  S. Carnahan May 15 '13 at 1:51
    
@S. Carnahan: No, I haven't, but having read a bit about GAP, it seems as though it would reference the Atlas of Finite Group Representations to address my queries. Unfortunately, it isn't clear to me that this reference offers the information I need... –  Dustin G. Mixon May 15 '13 at 2:04
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Perhaps you need to specify more precisely exactly what information you are looking for, and for which which groups. –  Derek Holt May 15 '13 at 8:11
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It's also worth observing that the smallest dimensional faithful representation is not always irreducible (for a noncyclic abelian group for example) so, even if you know the character table, then the computations are not always completely routine. –  Derek Holt May 15 '13 at 11:01
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To answer your final query, all of the representations in the Atlas, apart from the trivial representation, are faithful for the group that they are listed under. You could also try this webpage (which I found with a google search): maths.manchester.ac.uk/~jm/wiki/… –  Derek Holt May 15 '13 at 14:45

4 Answers 4

up vote 10 down vote accepted

Dustin, if you know the character $\chi$ of a representation $\rho$ of a finite group, it is easy to see whether it is faithful or not. Your representation is faithful if and only if for every $g \in G$, $\chi(g)=\chi(e)$ implies $g=e$. For if $g \in G$, and $\chi(e)=n$ is the dimension of your representation $\rho$, $\chi(g)$ is the sum of the $n$ eigenvalues of $\rho(g)$, which are roots of unity, hence $|\chi(g)| \leq n$ with equality if and only if all eigenvalues are the same, that is if and only if $\rho(g)$ is scalar, and then $\chi(g)=n$ if and only if $\rho(g)=1$.

So each time you have a table of characters for a group, you know which of its irreducible representations is faithful, and you can easily find the one of smallest dimension. You can find tables of characters in many places, like in the atlas of finite groups already mentioned for certain groups (apparently close to simple groups, so not the one for which your question is the most interesting), or for some noteworthy groups on wikipedia, or on sage by typing G.table_of_characters() if G is your group, etc...

Now as Derek said in comments, that doesn't really answer your initial question of finding the faithful representation of smallest dimension, which may not be irreducible. For this you just have to test with the above criterion the various sums (with repetitions) of irreducible characters, by increasing order of dimension, until you find one that is faithful, which in general should to take too long. This would be easy to program in sage, say...

This is a naive approach of a non-specialist. It is very possible that there are more clever ways to find the smallest dimension of a faithful representation. This suggests many questions -- What can be said about groups such that there smallest faithful representation is irreducible, for instance ? Or, what is the dimension of the smallest faithful representation of, say, $GL_2(\mathbb Z/\ell^n \mathbb Z)$ when $\ell$ is a fixed prime and $n$ varies? In this instance the table of characters has been recently determined, but it is quite complicated and the naive method suggested above seems impractical.

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There is no need of the «with repetitions» part: the smallest faithful reps are always multiplicity free. –  Mariano Suárez-Alvarez May 15 '13 at 23:44

How about the Atlas of Finite Groups ?

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Allow me to put my ignorance on display: Where in the Atlas can I find the information I want? –  Dustin G. Mixon May 15 '13 at 1:37
    
10 seconds of playing with it shows me "Choose the family of group", then "Representations", and you should look at the first line of "matrix representations". –  Allen Knutson May 15 '13 at 2:24
    
Which of these representations are faithful?? –  Dustin G. Mixon May 15 '13 at 2:26
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But the Atlas of Finite Groups only has information about groups that are "close" to being nonabelian simple. If all of the "noteworthy finite groups" have this property, then fine, but it would be no use at all for the many other types of interesting finite groups. –  Derek Holt May 15 '13 at 8:09

It may be useful to note that if $M$ is a subgroup of the finite group $G$ and $M$ contains no non-identity normal subgroup of $G$ then the permutation character of $G$ afforded by the action on the cosets of $M$ is faithful, so subtracting the trivial character gives a faithful complex character of degree $[G:M]-1.$ This usually won't be the minimal degree of a faithful character, but it is a handy upper bound for the minimal degree.

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In addition to the atlas of finite groups I would look for more specific results on minimal faithful representations of finite groups.

1.) Here is a nice tlak with slides from Saunders: caul.cii.fc.ul.pt/GSConf2011/Slides/saunders.pdf‎.

2.) here is a nice talk with slides from Lötscher: http://jones.math.unibas.ch/~loetsche/FaithfulRepresentations-Slides.pdf

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