I’d previously worked through a dozen or so chapters of the same Woit textbook you’ve linked as context for Representation Theory.
Given some group G, a (limear) “representation” is a homomorphism from G into the GL(V) the general linear group of some vector space.
That is, a map π:G→GL(V) is a representation iff for all elements g,h∈G,π(gh)=π(g)π(h).
Does “preferences between deals dependant on unknown world states” have a group structure? If not it cannot be a representation in the sense meant by Woit.
I can see how this could be confusing, but in mathematics, the phrase “representation theorem” is not specifically about “representation theory”. Wikipedia’s definition is quite broad:
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.
The list of examples it gives is probably more useful.
(Adding to the confusion: a famous example of a representation theorem is a corollary of Cayley’s Theorem: for every group G, there is a vector space V such that there is an injective homomorphism G↪GL(V). Ie, the theorem is that every group has a “faithful representation” in the sense of representation theory. So representation theory is built off of a famous example of a representation theorem. But as you can see from the above link, most things labelled “representation theorems” have nothing to do with “representation theory”.)
Yeah, basically this. I realize Woit’s book is not quite the right resource, but it’s just the first thing my brained returned when asked for a resource and it felt spiritually similair enough that I trusted people would get what I was pointing at.
I’d previously worked through a dozen or so chapters of the same Woit textbook you’ve linked as context for Representation Theory.
Given some group G, a (limear) “representation” is a homomorphism from G into the GL(V) the general linear group of some vector space.
That is, a map π:G→GL(V) is a representation iff for all elements g,h∈G,π(gh)=π(g)π(h).
Does “preferences between deals dependant on unknown world states” have a group structure? If not it cannot be a representation in the sense meant by Woit.
I can see how this could be confusing, but in mathematics, the phrase “representation theorem” is not specifically about “representation theory”. Wikipedia’s definition is quite broad:
The list of examples it gives is probably more useful.
(Adding to the confusion: a famous example of a representation theorem is a corollary of Cayley’s Theorem: for every group G, there is a vector space V such that there is an injective homomorphism G↪GL(V). Ie, the theorem is that every group has a “faithful representation” in the sense of representation theory. So representation theory is built off of a famous example of a representation theorem. But as you can see from the above link, most things labelled “representation theorems” have nothing to do with “representation theory”.)
Yeah, basically this. I realize Woit’s book is not quite the right resource, but it’s just the first thing my brained returned when asked for a resource and it felt spiritually similair enough that I trusted people would get what I was pointing at.