ab_[j+1] = e_j
bc_[j+1]= def_j
a_j = d_j
c_[j+1] = f_j
b_[j+1] = de_j
abc[j+1] = ef_j

still not sure that that's actually correct, but let's go ahead and try decategorifying it (i guess pretending that it really was categorified originally) into a 3-by-3 matrix of laurent polynomials... or something...

hmm, does this really make any sense?? hmm... 3 vs 6 ... simple roots vs positive... ??...

. a b c
d 1 1/q
e 1/q
f 1/q

??maybe specialize q=-1 for some reason??

. a b c
d 1 -1
e -1
f -1

i have no idea what i'm doing here but it sort of seems to be holding together for some reason...

??this matrix is involutory?? ... ???...


??so what about the possibility that it's in the weyl group somehow, and that with the q's it's in the artin-brieskorn-coxeter braid group? ...or something...

again, i wish that i could remember some of the stuff that simon willerton tried to explain to me about "verdier duality" and so forth... i have the feeling that some aspects of that might be relevant here... ??.... ??something about... ??for certain purposes, cohomology as effectively z/2-graded... ???or something like that??


. -2 1 0
. 1 -2 1
. 0 1 -2

a |-> a + -2a = -a
b |-> b + a = a + b
c |-> c

. -1 1 0
. 0 1 0
. 0 0 1

hmm... that does look similar to the "reflection functor" matrix above... more precisely it seems to be its negative... or something... ??so what's going on here??

i also wish i could remember certain ideas about how multiple meanings of "cartan matrix" are secretly related... i'm trying to remember the name of that author... ??benson??... "representations and cohomology" ?? ...