dg.differential geometry - Representations of SU(2) and tensors on SU(2)

A cop-out answer is to read Chapters 22-3 (maybe 21 also) of Lam's Topics In Contemporary Mathematical Physics. These chapters are viewable in Google Books at

http://books.google.com/books?id=bXq_M3_t1voC&dq=Topics+in+Contemporary+Mathematical+Physics

The treatment is accessible to undergraduates. I have included notes and errata below.

---

Chapter 21
p. 223: NB. Regarding equation (21.19), note that the Young diagrams ${6,5,4,1}$, ${6,4,4,2}$, and ${6,4,4,1,1}$ aren't included on the RHS because they have more than $n = 3$ rows.

p. 228. For Corollary 21.1, see also Theorem 19.19 in Fulton and Harris. At the bottom of the page, other legitimate Young diagrams with two rows should be included.

p. 229. At the top of the page, legitimate Young diagrams of the form (e.g.) ${r_1, r_2,1}$ should be included. Regarding the first full paragraph, see section 19.5 of Fulton and Harris for Weyl's ``associated" diagrams.

p. 230: The reference to equation (20.71) should be to (20.72); just below, note that $dim([mu_1]) = tbinom{mu_1 + n - 1}{n - 1}$ equals the expressions given for $n=3$. The ``Completely" after equation (21.50) should be decapitalized. It may be worth highlighting (or elaborating on) the reference to spin.

Chapter 22
p. 234: In the first line of equation (22.3), the term $S_{2j}(e_1^{(2j)})$ should be $S_{2j}(e_1^{2j})$. In the text immediately following, some kind of reference should be made along the lines of the sentence beginning ``Note that in the above equation..." in the paragraph after equation (22.9).

p. 235: NB. $u' equiv left(
begin{smallmatrix}
u^1 & u^2
end{smallmatrix}
right) left(
begin{smallmatrix}
a & b \
-overline{b} & a
end{smallmatrix}
right)$, cf. (22.22).

p. 237: In equations (22.17) and (22.19), factorial symbols need to accompany all the terms under the square root. The reference to equation (22.25) near the end of the page should be to (22.15).

p. 238: In equations (22.24-5), factorial symbols need to be added for the terms in the square roots.

p. 239: quqntum" should readquantum".

p. 240: NB. $a = e^{-iphi/2}$ and $b=0$ implies equation (22.30).

p. 241: There should be a mention to the effect that there exists $pi$ such that the eighth equality of equation (22.42) holds.

p. 242: Factorial symbols need to go in the square root in equation (22.47).

p. 245: In the penultimate sentence of the middle paragraph, a reference could profitably be made to (22.35).

p. 246: I skipped these exercises, but the theorem numbers appear to be low by 1.

Chapter 23
p. 249: Regarding the first sentence of the first full paragraph, note that $P^1$ is invariant under $SO(3)$, and the invariance of $P^k$ follows. Equation (23.13) should be $H^k equiv ker(nabla^2)|_{P^k}$.

p. 250: NB. At the bottom of the page, draw a correspondence $2theta rightsquigarrow phi$. Note that
begin{equation}
D(R_3(2theta)) cdot begin{cases}
x pm iy \
z
end{cases} = begin{cases}
(x cos 2theta - y sin 2theta) pm i(x sin 2theta + y cos 2theta) \
z
end{cases} = begin{cases}
(x pm iy)e^{pm 2itheta} \
z
end{cases}.
end{equation}

p. 251: The unnumbered equation and its preceding text is not quite right: see (the results of) exercise 23.2.

p. 252: Regarding the comment for p. 251 and exercise 23.2, note that $frac{x pm iy}{r} = e^{pm iphi}sin theta$ and $frac{z}{r} = cos theta$. It follows that $Y_2^0 = sqrt{frac{5}{16pi}}left( 2 left(frac{z}{r}right)^2 - left(frac{x+iy}{r}right)left(frac{x-iy}{r}right) right), quad Y_2^1 = -sqrt{frac{15}{8pi}} left(frac{z}{r}right)left(frac{x+iy}{r}right), quad Y_2^2 = -sqrt{frac{15}{32pi}} left(frac{x+iy}{r}right)^2;$
$Y_3^0 = sqrt{frac{7}{16pi}}left( 2 left(frac{z}{r}right)^3 - 3left(frac{z}{r}right)left(frac{x+iy}{r}right)left(frac{x-iy}{r}right) right), quad Y_3^1 = -sqrt{frac{21}{64pi}} left(frac{x+iy}{r}right) left( 4 left(frac{z}{r}right)^2 - left(frac{x+iy}{r}right)left(frac{x-iy}{r}right) right),$
$Y_3^2 = sqrt{frac{105}{32pi}} left(frac{z}{r}right)left(frac{x+iy}{r}right)^2, quad Y_3^3 = -sqrt{frac{35}{64pi}} left(frac{x+iy}{r}right)^3.$

p. 253: Before equation (23.44) reference equations (18.5-6) and discussion preceding them; also see (23.52-3). For equation (23.46), include $J^3|El0rangle = 0$. The reference to (22.42) should be to (22.41).