  We give a Lie-theoretic explanation for the convex polytope which
parametrizes the globally smooth solutions of the topological-antitopological
fusion equations of Toda type (tt$^*$-Toda equations) which were introduced by
Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions
can be parametrized by monodromy data of a certain flat
$SL_{n+1}\mathbb{R}$-connection. Using Boalch's Lie-theoretic description of
Stokes data, and Steinberg's description of regular conjugacy classes of a
linear algebraic group, we express this monodromy data as a convex subset of a
Weyl alcove of $SU_{n+1}$.
