What are...intertwiners?

Goal.
Explaining basic concepts of representation theory in an intuitive way.
This time.
What are...intertwiners? Or: Maps between representations.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
Representation theory is a vast field, and a representation could be of a group, an associative algebra, a Lie algebra etc., and these even in flavors. These subfields are similar, but then also different in nature. So to be precise, this video series is mostly about finite dimensional representations of finite groups and monoids.
Slides.
www.dtubbenhaue...
Website with exercises.
www.dtubbenhaue...
Representation theory.
en.wikipedia.o...
en.wikipedia.o...
Representation.
en.wikipedia.o...
mathworld.wolf...
www.mathphysic...
Intertwiner.
en.wikipedia.o...
en.wikipedia.o...
Pictures used.
en.wikipedia.o...
Some books I am using (I sometimes steal some pictures from there).
link.springer....
users.metu.edu...
link.springer....
math.mit.edu/~...
link.springer....
link.springer....
www.cambridge....
link.springer....
Mathematica.
arxiv.org/abs/... (Mostly for representations of Lie groups and algebras)
Magma.
magma.maths.us...
magma.maths.usy...
SageMath.
doc.sagemath.o...
sagecell.sagem...
#representationtheory
#algebra
#mathematics

[@like_that4966]

Thx so much! I just found your channel and you got a series on just everything of what I wanna learn. You're the best!

[@VisualMath]

Thanks for the feedback, I am glad that you like the videos, friend ☺

[@culpritgene]

I do not get the arithmetic around 3:12-4:18. In the Vector Space V' (bottom one) linear representation M'_g does not seem to hold: M'_g \cdot (1,1,1) = (2,-1,0) eq (0,1,0) M'_g \cdot (0,0,1) = (1,-1,-1) eq (1,1,1) And the Linearized rep M'_g for me is expressed by other matrices, e.g. one with -2 at the left bottom corner. Similarly for M'_h: M'_h \cdot (1,1,1) = (2,0,-1) eq (0,0,1) At the 3:12 you also say "g... just fixes that vector (you highlight (1,1,1) ... this vector is just a common eigenvector for g and h" I get how it is in the Vector Space V, but in V' it does not seem to be the case - i.e. (1,1,1) is no longer a common Eigenvector and instead it should be (1,2,2). Is it some fundamental misunderstanding (on my side) on the way Linearized Actions should be applied to elements on the targeted set expressed as directions in the Vector Space? Or can it be just a typo in the matrices?

[@VisualMath]

Thanks for reaching out ☺ I think the confusion is that I wanted the matrices at the bottom on the basis {(1, 1, 1), (0, 1, 0), (0, 0, 1)}, which means that (1,1,1) ↔ (1,0,0). And (1,0,0) is indeed a common eigenvector of the bottom g and h. Does that make some sense?

[@Moeman-mf5yq]

Hilariously, it seems group theory is a natural extension of linear algebra. This is the equivalent to coordinate transformations!

[@VisualMath]

Or the other way around, as suggested by representation theory 😉

[@dmit10]

So representations are not matrices, as you stated in the first video but linear spaces, the matrices act upon, right?

[@VisualMath]

Yes and no: (A) Representations are a coherent collection of matrices, one for each group element, such that the matrices satisfy the group multiplication. In this sense they are matrices. In this case they are usually called representations (you represent the group via matrices). (B) On the other hand, you can thin of a representation as a vector space with a group action, as you suggest. In this case they are usually called modules. (A) and (B) are equivalent: if you know one of them you can construct the other “canonically”. Hope that clarifies the situation!

[@dmit10]

Can an intertwiner e.g. act from 3d to 2d?

[@VisualMath]

I do not understand the question, sorry. Maybe you can try to rephrase it?