B Can the Diagram in the Article Be Interpreted as Commutative?

Stephen Tashi
Science Advisor
Homework Helper
Education Advisor
Messages
7,864
Reaction score
1,602
TL;DR Summary
Does the given example of commutative diagram use conventional notation?
I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way?

In the article:

##k[x]## is the ring of polynomials over a field k.

##V## is a vector space.

##T## is a linear transformation on ##V##

CommDiagScreenshot.jpg


I understand ##T## and ##X## as maps, but do the vertical arrows go from a map to the argument of a map?
 
Physics news on Phys.org
The arrows (all of them, vertical and horizontal) go from element to element. Their LaTeX code is \mapsto.
 
I'll understand the vertical arrow on the left this way: There is an (unnamed) isomorphism mapping ##V## to a direct sum of quotient modules. So ##g(x) + <f_i(X)>## is one element of that direct sum and it is in the coset of the ##<f_i(X)>##. So the unnamed isomorphism maps an element of the direct sum to a vector ##v## in ##V##.
 
the visual clue to what fresh and Stephen are saying is that the arrows have little tails at the beginning which are perpendicular to the arrow.
 
  • Like
Likes Stephen Tashi
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top