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

Question

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)?

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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?

Answer 1

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.

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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 $a_{i,j}$.