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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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    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.2012-04-20

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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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    Yes, otherwise it would be infinity!2012-04-20