Tournaments, flags, and spinors

It is observed that the dimension of the space of invariant spinors on the manifold $\mathrm{S}\mathrm{U}(n)\mathrm{/}{T}^{n-1}$ of full flags in ${\mathbb{C}}^n$ equals the number $\mathrm{R}\mathrm{T}(n)$ of labelled regular tournaments of order $n$, for $n$ odd. This space is decomposed under a double cover of the Weyl group, and realized as a pair of eigenspaces, first for the Dirac operator, then for a twisted version of it which acts equivariantly; this is used to show that multiplicities of irreducible representations are even for $n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.