Diagrams illustrating the Projection Theorem in Weight Space and the Dynkin indices of Semisimple Subgroups of Semisimple Groups.

by Anthony P. Stone

These diagrams illustrate the paper, A. P. Stone, "Semisimple subgroups of semisimple groups", Journal of Mathematical Physics, 11, 29-38 (1970).

It is hoped to add more explanation and diagrams, but two basic results derived in the paper are: if G' is a semisimple subgroup of a semisimple group G, then the weights of irreducible representations of G project orthogonally on to the weights of irreducible representations of G', and the Dynkin index (written as a superscript) of G' as a subgroup of G may be calculated from the geometry of the weight space by the extremely simple formula:


                   squared length of longest root of G

    Dynkin index = ------------------------------------ ,

                   squared length of longest root of G'

This extends the concept of Dynkin index from simple groups to semisimple groups. The projection result gives the solution to part of problem 5-3 (p.149) in Brian R. Judd, Operator Techniques in Atomic Spectroscopy, Princeton University Press, Princeton, NJ, 1998 (first published by McGraw-Hill in 1963).

. . A2 to A1(1) . . A2 to A1(4)