a binary relation r between sets x and y induces a "galois connection" between the power-sets p(x) and p(y), which in turn induces a "galois correspondence" between f(x) and f(y), where f(x) is the subcollection of p(x) containing just those subsets fixed under the process of traveling back and forth via the connection, and similarly for f(y).

an interesting example is when x is the root system of a simple lie algebra and y is the vertexes of the corresponding coxeter complex, with x1 r y1 iff "the one-parameter group corresponding to x1 preserves the geometric figure corresponding to y1". the resulting galois correspondence is interestingly analogous to the original galois correspondence (between subfields of a field and and subgroups of its galois group).

for example for the simple lie algebra sl(n), a root is a non-constant map 2->n while a figure is a non-constant map n->2. the binary relation is that the composite 2->n->2 is ....

recently i've begun thinking of the figures concretely as extremal weight spaces in irreps, besides more abstractly as just "geometric figures". because of this, certain things that previously seemed just a matter of convention now seem more fixed, and one of these is bugging me because it seems to be working out with a convention opposite to the one that i think i find most aesthetically pleasing. ...