kaleidoscope toric variety ... ?? basepoints of the torus orbits ?? ...

?? also the semigroup (?? ...) structure .... ??????.....

?? points of "finite order" wrt the semigroup structure ?? ... in real or complex spectrum ... ?? "unitary" .... ???? ..... "compact order" ????? ...... ?? real forms .... ???? ....

?? vague memory ... simple lie alg ... certain monoidal cats with same objects ?? .... ?? comparison functor between them ??? ....

?? quiver reps .... ?? ....

?? trying to make multi-graded commutative algebra whose grades include not just fd irreps of simple lie alg but also inf-d ones .... ??....

?? derived ... ???? ....

?? projective vs complete for toric varieties ... ??? .... structure vs property ?? .... ?? specific line bundle ... ??? ...."canonical" ... ??? .... ??? ..... ... no good ... ??? .... ???? .... ??? "off-center center" ... serre duality ... ???? generalizing some understanding from flag variety case .... ???? .... ?? cohomology .... ???? .....

?? so given dimensional theory, consider .... ???? ..... commutative monoid obtained as follows :

?? take ab gp of iso classes of dimensions ...

?? take subset of "occupied" dimensions ...

?? take submonoid generated by it ...

??? .....

??? conceptual / geometric interpretation supposed to be ... ?? affine toric variety (?? ...) given by .... ???? ..... ????? .....

??? well, this started out as just a straightforward attempt to extract an affine toric variety in the form of a commutative monoid from a [torus acting on an affine variety, equipped with a point of the variety ...] in the form of a [dimensional theory equipped with a model ... ??? ...] ... ?? but then the model started seeming a bit superfluous to me ... though lack of it may in fact be screwing up / complicating conceptual /geometric interpretation ... ??? ....

??? "picard stack" and toric variety .... ????? ......

??? "jacobian ..." .... ?? ... "albanese ..." .... ???? ....

??? elliptic curve .... ???? .....

?? underlying real variety of complex variety .... ????? .....


?? kaleidoscope case?? .... ??? hmmm, as what more or less prompted the idea in the first place ??? .... ?? and maybe it's in fact tying in in a nice way here .... ???? .... ???? ......

?? issue of how monoid algebras can sometimes be twisted .... ???? .....

?? hodge theory for flag varieties ??? .... ?? as maybe somewhat trivial in some way ???

?? nice description of how to extract commutative monoid corresponding to affine toric variety coming from orbit of torus action on affine variety ??? ...

?? full orbit decomposition of g/p wrt t .... ??? ...

?? singularity of schubert variety and schubert variety as element of cohomology ring ... ???? ....

?? "class class" ... ??? ....

?? "n-order" and sesqui-clever truncation ... for example in cobordism context ... ?? for certain purposes here important _not_ to coarsen for proset to poset ??? ... "cobordant" as pre-order .... ????? ....

"special-relativistic space-time geometry is "round" in the sense that a combination of two "boosts" (the kind of shift in perspective that you experience when you step onto a fast-moving train) is itself not necessarily just a boost anymore."

?? so ... train riding train ... non-commutative ... ??? .... ???

?? relationship between ... ??? philosophy of "complex numbers as conceptual mistake" and ... ??? comparative rehabilitation of quaternions, octonions, ... ??? ....

?? vs other progressions / developments .... ??? ...

?? bruhat cell as "affine" vs toric variety as built out of _torus_es ... ???? ....

?? trying to visualize quotient of (??classic donut ...) torus by funny winding circle ... ?? ...

?? alternative real forms of toric varieties .... ???? .... ?? combinatorial classification ??? ... ??? ....

?? ...

?? besides blowdowns of kaleidoscope toric varieties there's also all sorts of sub-varieties (and sub-quotients ...) that should be good to understand ... ??? ...

?? in fact, idea of "bottom-up" approach here, vs top-down ?? ... ??? very "disjunctive" ... ?? "analytic" / "frankenstein" ... ???? maybe more conjunctive (????? "synthetic" ????) approach afterwards ???? ... ????? ......

?? given affine toric variety, consider .... ?? its torus lattice, and .... all Z-gradings of that ... or all quotients of it .... and toric varieties coming form these, all organized together .... ???? ....

?? kaleidoscope case ... ?? ...

?? weight lattice grading vs highest weight lattice grading ... ??? .... ?? is there some straightforward involution switching these ??? .... ?????

a2 kaleidoscope toric variety as not completely trivial ?? ... blowdowns as trivial, but whole thing not completely ... ??? so actually maybe very good case to try to understand pretty fully ... ???

?? b2 ... long root subalgebra and kaleidoscope toric variety blow-down (?? non-singular one ??).... ??? ...

?? g acts homogeneously on g/b, of course .... and t maps to g ..... ?????? orbit decomposition of g/b wrt t ???? ...... (g,t)-double cosets .... ?? flag-apartment orientations ... ??? but which one to pick ?????.......

?? maybe does seem somewhat promising though ?? ... ?? "all the flags whose orientation towards the standard apartment is ..." .... ??? ....

??? orientation towards standard apartment .... giving orientations towards parts of standard apartment ... ??? .... ??? .... logic ... disjunction, conjunction .... ???? ....

?? also ... _of_ parts, towards parts ... ???? .....

?? irreducibility of toric variety ?? .... disjunction vs conjunction ... ??? ....

?? blow-downs ... ?? same torus, so still orientations towards the standard apartment ... ?? but now you only look at part of the flag ...

?? some orbit closure of t -> g -> perm(g/h) with h general parabolic now instead of borel ???

?? flag-apartment orientations ... ?? classification of flag triples and/or flag [n+1]-tuples ... ??? .... ?? showing up in .... ?? hecke algebras and/or cohomology algebras of flag varieties ... ??? .... multiplication operations therein ... ???....

?? presentation as commutative algebra of field of rational functions ... ?? sort of obvious, but ... perhaps particularly nice version ??? .... ?? relationship to "partial fraction basis" ..... ????? .....

1/(x-a) * 1/(x+1/a) = 1/() .... ??/ ......