I don't know what this symbol means

Question

I somehow made it to grad school without coming across this symbol:

$\left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array}\right) $

Here, $l_i$ and $m_i$ are all integers, and I think maybe the value of this expression is either 0 or 1 depending on the values.

What's the name of this symbol, and what is its value?

Answer 1

This notation is often used to write a permutation $\sigma$ so that $l_i \mapsto m_i$.

Answer 2

This usually represents the function $f$ with domain $\{l_1,l_2,l_3\}$, such that $f(l_1)=m_1$, $f(l_2)=m_2$, and $f(l_3)=m_3$.

($l$ is a horrible name for a variable. It looks like a $1$ when you draw it. $\ell$ is better.)

Answer 3

Since spherical harmonics was mentioned then this is most likely the Wigner $3j$ symbol.

This symbol is useful if you want to descibe the product of two spherical harmonics expanded in a series of spherical harmonics

$$Y^{m_1}_{l_1}Y^{m_2}_{l_2} = \sum_{l,m}\sqrt{\frac{(2l_1+1)(2l_2+1)(2l+1)}{4\pi}}\begin{pmatrix} l_1 & l_2 & l \\[8pt] 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l\\ m_1 & m_2 & m \end{pmatrix}\overline{Y}_{m}^l$$

and it also appears in integral formulas of spherical harmonics like for example:

$$\int Y_{l_1m_1}(\theta,\varphi)Y_{l_2m_2}(\theta,\varphi)Y_{l_3m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\[8pt] 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix}$$