During the development of Enlighten, Chris Doran and I did some work on the Spherical Harmonic representation of irradiance. Since the Geomerics website is no more, I’ve rehashed the material that used to be there into a series of blog posts here:

The first post explains the (simplified!) definitions of SH that we use for lighting.

The second post explains the conversion between SH radiance and SH irradiance, and the improved non-linear model we developed for L1 irradiance.

The third post mentions a curiosity about the antipodally-exact nature of L1.

Let me know if you find these useful, or if you have any questions!