There is an extensive philosophical literature on representations. Much if it is aimed at understanding mental representations, which isn't our concern here, but some of the arguments presented are intended to clarify the nature of representations generally.
In what way do representations represent what they represent? I've argued that (i) a representation represents something only in the context of a representational system, in which its correspondence to a target is picked out. I've also argued that (ii) representational systems arise, or are devised, because they allow operations to be performed with less work, in an inclusive sense, than they can be performed without them. These arguments are similar to those presented by Ruth Millikan in her Language, Thought, and Other Biological Categories (MIT Press, 1984) [Millikan's terminology is different; she reserves the term "representation" for situations in which something has as "its focused proper function to precipitate acts of identification of the referents of its elements (p 198)." For example, a bee dance, clearly a representation in our sense, is not a representation for Millikan, because its purpose is not to identify any referent.]Allowing for difference in terminology, Millikan argues that (i) what representations refer to is determined by the function it serves in the (biological) system in which it is embedded, and that (ii) this function is seen in the adaptive value of the representation in an evolutionary sense. My arguments (i) and (ii) are extensions of these beyond the domain of biological systems.
A conflicting interpretation of representation is sought by Cummins in Representations, Targets, and Attitudes (MIT Press, 1996), where he argues for a "conception of representational content" that is "independent of use or functional role" (pp. 85-86). Cummins's central worry about meaning-as-use theories of representation is making sense of error in representations. We want to talk about a representation being "wrong", but if its meaning is determined by its use, how can a representation ever be "wrong", rather than meaningless?
Cummins's account of "wrong" representations relies on assigning "content" to representations, "independent of use or functional role", and comparing the content with the "target" of the representation, the thing that it is meant to refer to in a given situation. A representation is "wrong" if there is a mismatch, as when a map of Paris is used when New York is the target. But, as we've seen earlier, computational representations, like those using blocks of bits, simply can't be assigned content out of context. So a different account of content, and "wrongness", has to be offered, at least for these cases.
In the account I am suggesting, we identify a "wrong" representation when we analyze a system of things and relationships as if it were a representational system, but then find that the crucial requirement, that mapping operations into the representation domain and then back has to save work, doesn't hold. For example, a map of Paris is "correct" as a representation when seen in a representational system for navigating in Paris, but "wrong" in a system for navigating in New York: we don't save any work trying to get around Paris using a map of New York.
This kind of usage for "wrong" is common in talking about complex systems that partially satisfy some description. Rather than refusing to apply the term "representation" at all to a system that only partially meets the definition, we apply the term with a qualifier, "wrong" in this case, allowing us to describe both the points of agreement with the definition and the discrepancy. (Viewing the application of a term like "representation" as itself an instance of representation, one can say that using the term, with qualifications, provides value as long as the value gained by applying the term outweighs the complication involved in managing the qualifications. If a system shares very little with a full-fledged, working representational system, describing it as a representational system won't be useful. But there is no bright line separating appropriate from inappropriate usage, any more than there is bright line separating "insufficiently precise" measurements from "sufficiently precise" ones.)
This treatment is consistent with Millikan. Allowing as before for difference in terminology, for Millikan a representation is "wrong" when it fails to perform its proper (biological) function.