Root systems and diagram calculus. II. Quadratic forms for the Carter diagrams

Abstract

For any Carter diagram $\Gamma$ containing 4-cycle, we introduce the partial Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from $\Gamma$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent. The linkage diagrams connected under the action of dual partial Weyl group (associated with $B_L$) constitute the linkage system, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. For Carter diagrams $E_6(a_i)$ and $E_6$ (resp. $E_7(a_i)$ and $E_7$; resp. $D_n(a_i)$ and $D_n$), the linkage system has, respectively, 2, 1, 1 components, each of which contains, respectively, 27, 56, $2n$ elements. Numbers 27, 56 and $2n$ are well-known dimensions of the smallest fundamental representations of semisimple Lie algebras, respectively, for $E_6$, $E_7$ and $D_n$. The 8-cell "spindle-like" linkage subsystems called loctets play the essential role in describing the linkage systems. It turns that weight systems also can be described by means of loctets.