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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
up vote 4 down vote accepted

Inner products are denoted by $\langle\cdot,\cdot\rangle$ and Kronecker deltas by $\delta_{ab}$.

The specific relation you cite can be found on Wikipedia's spherical harmonics article, and the inner product at hand is specifically seen to be the surface integral (over the sphere) of the first argument times the complex conjugate of the second argument (the arguments are of course functions).

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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
@bobobobo: Careful, technically this is the Kronecker delta, a discrete analog of the Dirac delta. (The latter isn't a function per se, but rather a "generalized" function, which is studied in distribution theory.) – anon Apr 20 '12 at 0:14
Yes, otherwise it would be infinity! – bobobobo Apr 20 '12 at 0:16

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