Numerology of square equations

Abstract:

In the present work, which is a sequel of the paper “Can one see the signs of structure constants?”, we describe how one can see the form and the signs of the senior Weyl orbit of equations on the highest weight orbit directly in the weight diagram of microweight representations and adjoint representations for the simply-laced case. As special cases, the square equations we consider include the vanishing of second order minors, Plücker equations in polyvector and adjoint representations of classical groups, Cartan equations in spin and half-spin representations, Borel–Freudenthal equations defining the projective octave plane $\operatorname {E}_6/P_1$, and most of the equations defining Freudenthal’s variety $\operatorname {E}_7/P_7$. In view of forthcoming applications to the construction of decomposition of unipotents in the adjoint case, special emphasis is placed on the senior Weyl orbit of equations for the adjoint representations of groups of types $\operatorname {E}_6$, $\operatorname {E}_7$, and $\operatorname {E}_8$. This orbit consists of 270, 756, or 2160 equations, respectively, and we minutely discuss their form and signs. This generalizes Theorem 3 of the preceding paper “A third look at weight diagrams”, where we considered microweight representations of $\operatorname {E}_6$ and $\operatorname {E}_7$.