# How to distinguish property of particular representation from property of algebraic structure?

It is common that you have some interesting object (set, group, algebra or something, whatever) which has certain properties, structure etc. You may try describe it in pure algebraic way. Sometimes You cannot find representation different from this you are working on.

How to distinguish its algebraic properties form properties arising from particular representation? Is there any systematic procedure?

Are there only trivial answers such as: "find another model/representation", or "every object may be regarded as example of different algebraic structures"?

Or maybe it is enough to describe something without notions from representation ( finite dimensionality, matrix indices etc)?

Examples: suppose you have an algebraic structure given by a set $$M$$ with some operations on it. It has a representation in a matrix algebra. Suppose that this representation has following properties:

1. for every matrix $$M$$ from the representation $$\det(M) = 1$$
2. for every matrix from the representation $$\mathrm{Tr}(M) = 0$$
3. the whole representation is a vector space of dimension $$n$$ with basis given by set of $$n$$ independent matrices
4. there is a matrix $$S$$ in the representation for which $$S^2=id$$
5. for certain matrices $$A$$, $$B$$ and $$C$$, there is the relation $$AB^2 - C = C^2$$

etc.

If we regard its as a group only property 4 will be representation independent. But if we talking about vector space obviously property 3 is crucial.

We have some freedom to choose what is important: so if we are talking about abstract group property 3 may be called particular property of representation, whilst when we are in vector spaces, property 4 may be particular. Property 2 may be called fundamental if we are talking about "matrix groups" etc. What about property 5? Is it important?

Suppose we are able to wrote relations in a way which is purely algebraic ( for example for matrices we may write relations which not must be narrowed to matrix operations, we may interpret it as a general algebraic property of more general object, as relation $$S^2 =\mathrm{id}$$). And it lead to some interesting conclusions. How to be sure, the property we are research is not only property of chosen representation? What with property 5?

Of course we may regard the same object as a group and as a vector space simultaneously. In this way however certain properties may be considered as detached one from each other. Is this the only way?

• Wouldn't the algebraic properties be just another form of representation? Or are you thinking of trying to determine things like, e.g. nilpotence from a given generator-relation presentation of some finitely generated group? If you elaborate with a few examples (even if they are made-up), I might have something to contribute toward the question. Gerhard "Ask Me About System Design" Paseman, 2010.03.27 – Gerhard Paseman Mar 27 '10 at 23:14
• What do you mean by properties arising from a particular representation? Could you give an example? Also, do you mean presentation instead of representation? You may want to look at mathoverflow.net/questions/17157/… and the preceding questions. – Douglas Zare Mar 27 '10 at 23:14
• I don't vote to close, but this question absolutely needs to be rewritten. Part (and only part!) of the problem is that, I think, English is not OP's first language. Part of the problem is that "representation" has a technical meaning, and it's not clear if OP means it or not. But the main problem is that the question doesn't make any sense whatsoever. – Theo Johnson-Freyd Mar 28 '10 at 6:24

## 1 Answer

I'm not sure I completely understand what you're asking, but here is some information that appears to be relevant.

In the context you're describing, you have two languages: the pure language L0 of groups and the augmented language L1 of groups together with a linear representation over some field (see note). You seem to be asking the following:

• When does a sentence in L1 (i.e. a property of groups with linear representations) equivalent to a sentence in L0 (i.e. a pure property of groups)?

The Robinson Consistency Theorem answers this, at least in part.

Let $$G$$ be a group and let L2 be the language of groups augmented with a constant for every element of $$G$$. Let T2 be the complete theory of $$G$$ in L2 and let T0 = T2 ∩ L0 be the complete theory of $$G$$ in L0. The difference between T0 and T2 is that statements in T2 are allowed to mention particular elements of $$G$$. So if $$x$$ is an element of order $$2$$ in $$G$$, then that fact is recorded in T2 but not in T0. However, the fact that $$G$$ has an element of order 2 is recorded in T0 since that fact does not explicitly mention any element of $$G$$.

Now let φ be any statement in the language L1 of groups with linear representations. The Robinson Consistency Theorem says that if T0 ∪ T1 ∪ {φ} is consistent then so is T2 ∪ T1 ∪ {φ}, where T1 ⊆ L1 is the theory of linear representations (see note). Stated differently, T2 ∪ T1 ⊦ φ if and only if T0 ∪ T1 ⊦ φ. (Recall that T ⊦ φ iff T ∪ {¬φ} is inconsistent.) The models of T2 are precisely the elementary extensions of $$G$$. Thus the following are equivalent:

• φ is a consequence of some purely group theoretic property of $$G$$
• φ is true for every linear representation of an elementary extension of $$G$$

Sometimes we can say more. If φ is an existential statement in L1 (i.e. φ is equivalent to a sentence of the form ∃x,y,z,...φ0(x,y,z,...) where φ0 is quantifier free) then we don't have to check all elementary extensions of $$G$$. Thus, for such φ, the following are equivalent:

• φ is true in all linear representations of $$G$$
• φ is a consequence of some purely group theoretic property of $$G$$

Of course, the Robinson Consistency Theorem is not particular to groups and linear representations of groups; the same reasoning applies in all sorts of contexts.

Note: I'm assuming that all languages are first-order (possibly with multiple sorts). There are various ways to formulate linear representations in first-order logic, but none are completely satisfactory. For fixed dimension $$n$$, one can add a sort $$F$$ for the field elements together with functions $$a_{i,j}:G\to F$$ for the entries of the matrices of the representations. Then T1 consists of all field axioms together with all required identities between the $$a_{i,j}$$.

• +1 . – Joel David Hamkins Mar 28 '10 at 4:14
• @FGD: you have such a knack for giving interesting answers to questions that seem completely opaque to me that I'm thinking of investing in a room full of monkeys, each equipped with a computer which is permanently logged on to MO. Can you handle the volume? – Pete L. Clark Mar 28 '10 at 7:15
• (Note: of course I am not comparing kakaz to a monkey. His question is obviously one written by an intelligent human being. What I'm saying is that -- unlike Francois -- my own understanding of what he's getting at in the question, and hence my ability to usefully answer it, is very close to zero.) – Pete L. Clark Mar 28 '10 at 7:20
• @Pete: LOL. Thanks, but I'll pass on the monkeys. (Unless they can also teach and grade?) – François G. Dorais Mar 28 '10 at 7:28
• Thank You very much! Indeed it is not exactly what I am asking about, but is no doubt about. You mention theorem which states about consistency when situation I describe arises ( I am also not asking about groups in particular, but about general structures). Probably it is the only possible outcome in this matter. I rework my question in a way which relate to problem of finding criteria if certain property is consequence of (for example) group axioms, and it is known that sometimes it may exists. However as group theory is undecidable, in this particular situation we have no such method. – kakaz Mar 28 '10 at 17:57