A Decomposition into irreps of compact Lie group

  • A
  • Thread starter Thread starter Orodruin
  • Start date Start date
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2024 Award
Messages
22,800
Reaction score
14,854
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 rewritten as
$$
\langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{C \in [G]} n(C) \chi(C) \chi_0(C)^*,
$$
where ##[G]## is the set of conjugacy classes of ##G## and ##n(C)## is the number of elements in ##C##.

This should generalise to compact Lie groups with the character inner product given by
$$
\langle \chi, \chi_0\rangle = \frac{1}{V_G} \int_{g\in G} \chi(g) \chi_0(g)^* d\mu_G
$$
where ##V_G = \int_{g\in G}d\mu_G## and ##d\mu_G## is the Haar measure on ##G##. The corresponding expression on the conjugacy class level should be
$$
\langle \chi, \chi_0\rangle = \frac{1}{V_G} \int_{C\in [G]} \chi(C) \chi_0(C)^* d\mu_{[G]}
$$
with ##d\mu_{[G]}## being a measure on the conjugacy classes determined by integrating the Haar measure over each conjugacy class. Right?

My particular problem is doing this for ##SO(3)##. More explicitly, the conjugacy classes of ##SO(3)## should be parametrised by the rotation angle ##\theta## (with the conjugate elements being rotations around different axis by the same ##\theta##). I am especially looking for the function ##f(\theta)## such that
$$
\langle \chi, \chi_0\rangle = \frac{1}{V_G} \int_{0}^\pi \chi(\theta) \chi_0(\theta)^* f(\theta) d\theta.$$

Any insights?
 
Physics news on Phys.org
fresh_42 said:
I have found the formula
$$
\int_{\operatorname{SU}(2)}f(\theta)\,d\theta = \dfrac{2}{\pi}\int_0^\pi (f\circ e)(t)\sin^2(t)\,dt
$$
including a proof, but in German. Maybe it helps anyway.

Source: https://wwwold.mathematik.tu-dortmund.de/~lschwach/SS10/Bachelor-Seminar/Averim.pdf
I think I found my issue. I was trying ##\sin^2(\theta)##, but when mapping from SU(2) to SO(3), the ##t## here is actually ##\theta/2##?

Using ##\sin^2(\theta)## I obtained that the fundamental representation should contain the trivial one once, which would be absurd. Using ##\sin^2(\theta/2)## gives zero as expected.

It also gives the correct result (2) for the representation ##\operatorname{Sym}^2(\operatorname{Sym}^2 V)## (with ##V## being the fundamental representation), which is what I was really looking for.
 
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