# What does $\langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu}$ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part.

The $Y^l_m$ function is as usually defined for real spherical harmonics, as here.

Early in the paper, however, the author states:

$$\langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu}$$

Not for slack, but purely as background information, I'm a computer scientist. I like to understand math as much as I absolutely minimally need can. I only have a vague understanding of what this means - it is the presentation of a group.

But practically what does this mean for the $\delta$ function when it is used later in the paper? For example, later in the paper (in the actual recurrence relations 6.3-6.6) we see use of $\delta_{m1}$. Does it means $Y_{lm}$ equals $\delta_{lm}$?

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Kroenecker delta? – Pedro Tamaroff Apr 19 '12 at 23:58
Physicists often use $\langle \cdot \mid \cdot \rangle$ for the inner product. This comes from Dirac's bra-ket notation. It does NOT mean a group presentation here. – Henry T. Horton Apr 20 '12 at 0:02

Inner products are denoted by $\langle\cdot,\cdot\rangle$ and Kronecker deltas by $\delta_{ab}$.
Wow! Ok, that makes sense, that should be equal to 1 when $l=\lambda$ and $m=\mu$, ie the spherical harmonics are an orthonormal basis, as the text above the relation says ("The real spherical harmonics form an orthonormal complete basis"). – bobobobo Apr 20 '12 at 0:11
I'll assume then, that any usage of $\delta$ is the Dirac delta function then, and $\delta_{m1}=1$ when $m=1$, $0$ otherwise. – bobobobo Apr 20 '12 at 0:12