so suppose that we have a total flag f1 and a partial flag x1 and another partial flag x2 of the same type "constructed from f1 and x1". then x2 belongs to the schubert variety given as the closure of the bruhat class "like x1 to f1", and we can ask what's the zariski tangent space of that schubert variety at x2, and there should be some nice answer visible in the root system...

abc
def
ghj

borel abcde; down-sets of j>h>f,g = [],[f],[g],[fg],[fgh],[fghj]

f = destabilizer of "a" ??
g = destabilizer of "b" ??

b special - [f]
d - [fg]
f - [fgh] ????????
h generic - [fghj]

14 24 34
13 23
12

total flag variety 6d

partial flag varieties

point,line 5d - 34
point, plane 5d - 23
line, plane 5d - 12
point 3d - 24,23,34
line 4d - 12,34
plane 3d - 13,12,23

down-sets extending [24 23 34] = [24 23 34],[24 23 34 12],[24 23 34 12 13],[24 23 34 12 13 14]

down-sets extending [12 34] = [12 34],[12 34 23],[12 34 23 13],[12 34 23 24],[12 34 23 13 24 ],[12 34 23 13 24 14]

?? 0d,1d,2d,2d,3d,4d ??...

??corresponding to the 6 "line" faces of the weylotope??

??but so then what _about_ the singularity of that 2d shubert variety of b2??

??somehow... it doesn't "look singular" from the total flag variety viewpoint... ???or something??? .....

hmm, the "forgetting inner coordinates" idea might be screwed up because of affine vs projective here... the imperfect functoriality of the process of taking the projective space of a vector space... or something...

but anyway... i'm idly considering the possibility that when we think of the g2 incidence geometry as that of "a fermion rolling on a projective plane of 3 times the radius", the projective plane in question can be identified with a grassmanian of the long root subalgebra... hmm, but there are two dual ones of those to choose from, and it seems inelegant to have to so choose...

hmm, remember that it's the g2 "line" grassmanian that's most directly related to the long root subalgebra... ?? ...

stuff that i want to try to remember to tell john huerta about...

general stuff about invariant distributions and tangent spaces of schubert varieties and so forth, but then also stuff more particularly about g2 and the octonions and the rolling ball geometry...

14d irrep inside exterior square of 7d....

long roots subalgebra....
??sa... "forgetting inner coordinates" ....or something... and so forth...
(??whole bit about trying to give nice presentation of (multi?)homogeneous coordinate algebra of partial flag variety based very directly on weight diagram... or something like that...??...)

??sa coadjoint orbit in long root subalgebra?? os... asf...

6 special lines connecting the 6 special points...
one-parameter group associated to line...


cartan subalgebra...


??also maybe mention "pattern avoidance" idea?...

so, even though we still haven't completely clarified in our own mind what we're doing here, let's try looking at the invariant distributions on the a3 total flag variety as down-sets in a certain poset again, and let's try to identify the integrable ones, and enumerate the ones extending a given integrable one...

the down-sets:

[],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13 12 23],[13 12 23 34],[24 23 34],[12 24 23 34],[13 12 23 24 34],[14 13 24 12 23 34]

the integrable ones:

[],[12],[23],[34],...???...

so i think that part of what's going on here is that among the invariant distributions the integrable ones have a very basic special place, and it's important to work out how they fit in. and this to some extent justifies and motivates focusing on the total flag variety and treating the partial ones as just arising from the integrable invariant distributions, though i'd still like to clarify some more the way (probably perfect sensible once you understand what's going on, unlike me so far) in which _not_ quite invariant distributions seem to be entering here.

from a certain viewpoint perhaps an integrable invariant distribution corresponds to "a schubert variety whose logical intension uses only the equality predicate" (or something like that... or from a slightly different viewpoint, only equivalence predicates...). a leaf of the foliation that you get by integrating the distribution can be thought of as a "residual partial flag" (or something like that... ??or maybe i mean a "residual geometry" ??? .... try to straighten this out...)... ??and can also be thought of as a schubert variety?? ...

back to the a2 example:

ab=
cde
=fg

borel subalgebra abcd, down-sets of g>e,f = [],[e],[f],[ef],[efg]

among these [],[e],[f],[efg] are integrable... ??

so let's consider for example the g2 root system:

===a===
b=c=d=e
=f=g=h=
j=k=m=n
===p===

with borel subalgebra abcdefg; the invariant distributions on the flag variety are the submodules wrt this subalgebra of the complementary hjkmnp, which it seems reasonable to guess are essentially the down-sets in the poset p>n>m>k>h,j; namely [],[h],[j],[hj],[khj],[mkhj],[nmkhj],[pnmkhj].

now let's consider those of these submodules that contain h, or those that contain j. there are six of each... ??...

for comparison purposes let me try b2 here as well...

abc
def
ghj

borel abcde, down-sets in j>h>f,g = [],[f],[g],[fg],[hfg],[jhfg]

containing f = [f],[fg],[hfg],[jhfg]
containing g = [g],[fg],[hfg],[jhfg]

hmm... ???....

might as well do a2 as well...

ab=
cde
=fg

borel abcd, down-sets in g>e,f = [],[e],[f],[ef],[efg]

containing e = [e],[ef],[efg]
containing f = [f],[ef],[efg]


i'm somewhat confused at the moment about how the b2 calculation above relates to our previous alleged calculations about the singularity of the 2d schubert variety on one of the b2 grassmanians... ??...

so let's try understanding all the coadjoint orbits and/or conjugacy classes in sl(2,reals); seems like it should be pretty easy in terms of stuff that i've already learned about...

actually i mean to focus on projective coadjoint orbits...
also i probably meant to try something like sl(3,r) rather than sl(2,r), though sl(2,r) seems like a good idea too... also sl(2,c) and sl(3,c)... i mean the complex projective coadjoint orbits in those cases, i think...

consider for example the projective coadjoint orbit of the element

0 -1
1 0

of sl(2,r).

i guess that i should think of the groups here as groups rather than as lie algebras for purposes of trying to make precise exactly what i mean by "coadjoint orbit"...

a b
c -a

a^2+bc = -1

??

??a generic coadjoint orbit in sl(2,c) is equivalent as an affine variety to the "apartment" variety?? ??or something??... hmm...this sounds like something that we might have known at some point, or perhaps should have known... because it vaguely reminds me somehow of how an element in the compact form gives not just a partial flag but also a "complementary" partial flag... or something...
hmm, some confusion here...

??the projective space of the adjoint representation can be thought of as the space of "1-parameter subgroups" of the group.... ????.....

so what about the "downward homogeneization" of an ideal in a graded commutative algebra?? hmm...

??apartment variety as open subvariety of (flag variety)^2 ...??

ok, so i think that part of what i was getting confused about is that there's a lot of "projective coadjoint orbits" that are just quasi-projective varieties rather than projective... or something like that...

but let's look at the projective coadjoint orbit of

0 0 0
1 0 0
0 1 0

in sl(3,c) ... or something...

clearly we get a flag from this, as the kernels or images of the powers. but we also get isomorphisms between the three associated grades of the flag, i think. so the coadjoint orbit seems to be 5 dimensional, with stabilizer the nilpotent part of the borel. so the projective coadjoint orbit is i guess a 4-dimensional quasi-projective variety... or something...

well, all of this is tending to make me suspect that in fact the coadjoint partial flag variety is generally going to be the only partial flag variety that gets an invariant contact distribution from being a projective coadjoint orbit... which seems good because it fits with our naive guesses coming from staring at the root systems...

hmm, googling on "tangent space of a schubert variety" doesn't give that many hits but it gives a few suggestive ones (that of course i only glanced at and didn't really read yet because as usual i feel that i need to try to figure things out for myself or i won't understand them, for the most part)... in particular a buzzword "pattern avoidance" showed up... i think that i have a vague memory of hearing of that before (could be misremembering though) in connection with coxeter complexes and so forth (or something...)... in the back of my mind i think that i was even guessing that it might show up here, even though i have almost no idea what it means yet...

on the other hand, it does somewhat sound like too complicated an idea to be showing up in this context... i have the feeling that this tangent space of schubert variety stuff is basically very simple and doesn't particularly need any complicated ideas...

1234

1243


123
132
213
231
312
321

so it seems like it should be easy to find a partial flag variety with no invariant (under the "whole" group, that is) contact distribution...

so for example, let's consider the projective plane as a partial flag variety of a2...
it seems pretty clear that this has no nontrivial invariant distribution at all...

now is there any projective coadjoint orbit of the split real form (for example...) that's equivalent as a homogeneous space to the projective plane??

??hmmm, perhaps not??

on the other hand, what about the total flag variety of a3? has it got an invariant contact distribution? perhaps there _is_ a coadjoint orbit equivalent to it??

0 0 0
1 0 0
0 1 0

????.....

??so what _are_ the projective coadjoint oribts of a2 like in general??

it seems like an obvious conjecture that if you classify the schubert varieties up to nth order, then they fall into as many classes as the permutations "with n zig-zags"... or something like that; that's not quite the right way to say it, i think.

let's return to the down-sets in the "interval containment" order on 12,13,14,23,24,34:

[],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13],[13 34],[24],[12 24],[13 24],[14]

i want to see the cardinalities of the down-sets now, so instead of just listing the generators of the down-set i'll list all the elements:

[],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13 12 23],[13 12 23 34],[24 23 34],[12 24 23 34],[13 12 23 24 34],[14 13 24 12 23 34]

so it looks like we have

1 0
3 1's
3 2's
3 3's
2 4's
1 5
1 6

let's compare that to the coefficients of (1)(1+x)(1+x+x^2)(1+x+x^2+x^3) (which gives the level sizes in the kaleidoscope group 4!):

1 0
3 1's
5 2's
6 3's
5 4's
3 5's
1 6

1234

2134 - 1324 - 1243

2314 3124 - 2143 - 1342 1423

3214 4123 - 2341 2413 - 3142 1432 ??????

3241 2431 4213 - 3412 4132 ??????

3421 4231 4312

4321

??are we dealing here with a murphy congruence maybe?? perhaps not...

we're tentatively assuming that none of the schubert varieties here are singular...

we still need to clarify the relationship between "zariski tangent spaces of schubert varieties" and "invariant distributions" and related ideas... i'm getting confused about _what_ group the distribution is invariant wrt, for example... ???....

calculating the zariski tangent space of a schubert variety at the basepoint (or something...) seems in some vague sense "reciprocal" to calculating the attaching map attaching a bruhat cell to the more special ones that it attaches to...

is the "rolling and/or spinning" distribution on the g2 "point" grassmanian the zariski tangent space of any schubert variety?? of course since it's defined in terms of lie bracket from the pure rolling distribution, it has a sort of "geometric interpretation" in terms of nelson's "parallel parking" metaphor, but does it have some sort of more direct "incidence geometry" interpretation?? not sure exactly what i think the rules should be here as to what qualifies as such an interpretation...

this is part of the larger question of relationships between schubert varieties (possibly generalized in certain ways) and invariant distributions in general...

let's try listing a bunch of situations where n! counts something and catalan(n) counts some "smaller" related thing...

n! counts permutations, while catalan(n) counts permutations which "don't zig-zag too much"... ??...

n! counts a-series schubert varieties, catalan(n) counts their tangent spaces at the basepoint... or something.

n! is the dimension of the a-series "hecke algebra", catalan(n) is the dimension of its image acting on... something... related to "quantum su(2)" ....

i was just talking to chris rogers, trying to understand some ideas that he was trying to explain to me about symplectic geometry and bundles of various things and so forth, and at first i thought that i was only slightly confused but as the discussion progressed it seemed more and more like i'm profoundly confused about certain things... some of which i may have been confused about for decades. and i'm definitely still confused, though there might be some hope of getting unconfused in part through realizing how confused i am.

there are definitely some aspects of symplectic geometry that i understand quite, and it's actually pretty peculiar how such understanding and such confusion could coexist so close together.

i think that part of this goes back to when i lost a notebook several decades ago that contained some ideas that i worked out... that i'm not sure i ever managed to fully reconstruct...

i should try to describe the confusion...

let's see, suppose that i have a smooth manifold equipped with a bundle of oriented affine real lines and a smooth connection on the bundle (hopefully it's fairly obvious what i mean by this...)... then (??possibly with the help of some "decentness" assumption that i'm perfectly willing to make and as usual don't want to even think about at the moment... maybe paracompactness or something??) i might as well trivialize the bundle (because of the contractibility of the automorphism group of the fibers) and think of the connection as living on the trivial bundle... where it can be thought of as a 1-form... whose exterior differential 2-form is not merely closed but (pretty much by definition) exact... so a symplectic manifold whose structure 2-form is non-exact certainly isn't going to arise this way...

??...i think that the confusions bugging me here are very basic... and not specifically about just symplectic geometry but something more general... again it seems surprising that you can be this confused about some things without being confused about a whole lot more.... ????....

it seems like the process of taking the zariski tangent space of a schubert variety at the basepoint in the a-series case gives us a map from n! (the schubert varieties on the flag variety) to catalan(n) (the invariant distributions on the flag variety), so i started trying to connect this with other situations where there's a map from n! to catalan(n) ...

(it's not obvious to me yet whether the map that we're getting is surjective in general, though i'm guessing that it probably is.)

a permutation of the spaces between a string of factors gives a "bracketing" of the string, by using the permutation to determine the "precedence" of the joining operators corresponding to the spaces. (moreover, this process might play a significant role in certain aspects of higher-dimensional category theory...) so i decided to check whether permutations sharing a bracketing are essentially the same as schubert varieties sharing a zariski tangent space. and the answer that i seem to be getting so far is no.

that's somewhat of a disappointment, but it reminded me of a similar disappointment that i think i vaguely remember... which raises the possibility of trying to get those two disappointments to "cancel out"...


so let's consider the action of the group n! on the subsets of n.... ??....

i got an e-mail out of the blue the other day from laurens gunnarsen, who i'd never heard of before. the subject heading was "a wild idea" and i think that i was pretty sure that it was spam and was about to delete it when i decided to read it for a laugh and was surprised to find out that it was a serious message.

apparently gunnarsen is an associate of some kind of someone named jonah sinick, who had some sort of interaction with john baez which i'm imagining had something to do with how gunnarsen came to contact me.

gunnarsen and/or sinick have some sort of ideas/plan/project involving the teaching of mathematics. so do i, of course, though unfortunately mine are pretty much all a lot closer to the idea stage than to the project stage...

the very rough idea that i've gotten so far about what they're trying to do is that they're more or less giving up on "the school system" and instead trying to find individual young students who show some sort of mathematical promise (which in practice for them might mean mainly children of rich silicon-valley-type parents; if i'm not misremembering too badly then i think that they actually even mentioned something like that somewhere) and offering them some sort of high-quality exposure to mathematical ideas and to the process of becoming a mathematician (or possibly some other type of creative user of mathematics). i think they intend that the experience should be interesting and exciting for the student, and in a way that genuinely arises from the mathematics. they mentioned drawing some kind of inspiration from the "suzuki method" of music teaching but i don't know very much about that method.

well, i probably didn't convey the idea that well. my description sounded pretty generic whereas i probably already got some more specific idea of what they intend. but not that much more specific, especially since i think that you can usually tell a lot more from watching someone actually teach than from hearing them describe their teaching philosophy.

i'm pretty eager to talk to gunnarsen, but i need to prepare first, to organize my thoughts about my own teaching ideas as well as about theirs.

the way that larry harper describes his generalized directed graph morphisms involves the idea that "vertexes can get mapped either to vertexes or to edges". i'll have to think about that. one thought is that maybe they're all getting mapped to edges, with the ones that look like they're going to vertexes actually going to degenerate edges. that might not be on the right track though; it might actually be a case of v -> v+e, just as harper says.

is there a nice way to get a dimensional theory from a kac-moody lie algebra, generalizing from the case of a finite-dimensional simple lie algebra? and so forth...

so let's proceed to the alleged calculation of the modular lattice of invariant distributions on the b2 flag variety.

abc
def
ghj

take abcde as flag stabilizer borel. now we want (we think) the down-sets in the poset
where j dominates h which in turn dominates f and g, namely [],[f],[g],[fg],[fgh],[fghj].

hmm. i don't feel at the moment too much like trying to guess which schubert varieties these might be the zariski tangent spaces of. not yet.

let's try the case a3. we want the down-sets in a certain poset whose elements are 12,13,14,23,24,34. maybe it's just the "interval containment" order? it should be easy to actually see what the correct order is, but instead let's just use this guess for now. the down-sets are [],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13],[13 34],[24],[12 24],[13 24],[14]. (i'm just listing the "generators" of the down-set.) hmm, suspiciously i seem to have just listed 14 such down-sets, which of course was my main guess as to what would happen here, what with the catalan numbers showing up in so many places. but is there any obvious nice way to biject these down-sets with something else that we know of offhand that's counted by the catalan numbers?

perhaps given a "partially bracketed string", you can get a down-set in the interval poset of the string by taking the intervals which "don't violate the bracketing" in some hopefully obvious sense... or something like that... ?? hmm, i don't think that that actually makes sense yet.

so let's try more or less explicitly (but not too carefully) calculating the poset (perhaps a modular lattice or something like that?) of invariant distributions on the flag variety, in for example the a-series case to start with.

let's consider a2, with the root system labeled as in the following low-tech picture:

-a-b-
c-d-e
-f-g-

let's take abcd as the borel subalgebra stabilizing a flag, thus efg is the tangent space at that basepoint of the flag variety, and let's guess that the stabilizer-invariant subspaces of the tangent space are the down-sets in the poset "e<-g->f". there seem to be 5 such down-sets [],[e],[f],[ef],[efg] and it seems reasonable to guess that these correspond to the tangent spaces of the 6 schubert varieties based at the basepoint, with the two 2d ones doubling up here as we've already decided that they seem to be tangent to each other. on the other hand if this guess is correct then it seems that none of the schubert varieties are singular at the basepoint, as the dimensions of the tangent spaces (the cardinalities of the basis sets such as card([ef])=2) seem to match the naive dimensions of the schubert varieties.

(at some point we should also think about logical combinations of schubert varieties...)

then we should try to test these guesses some more, as well as generalize them.

is there some fact about _all_ projective varieties transitively acted on by quasi-simple algebraic group being partial flag varieties that we haven't been exploiting as much as we should? or something... ??something about coadjoint orbits vs orbits of arbitrary irreps.... ??something about whether there are _other_ partial flag varieties besides the coadjoint one that occur as coadjoint orbits, and whether such have natural contact distributions, and how such distributions might show up in the root system?? ...and so forth... ???....

not sure how much overkill it might be, but it might be interesting to apply "deformation cohomology" ideas to calculation of the tangent space of a schubert variety at the basepoint.

huerta... ideas about making g2/octonion/rolling ball stuff explicit...

baez... idea about adjunction between hilbert spaces and kaehler varieties...
getting it to work maybe, or something, but also something about perspecitve it might give on "geometric quantization" (or something... ??what about different flavors here?? and so forth... ??something about "orbit method" or something?? also something about "rho shift" and so forth...?? ??something in particular about flag varieties and borel-weil and so forth??) ... maybe "turning the crank" perspective, or something...

gunnarsen... mathematica.... whiteboard... various technical (or something...) problems...


ideas about tangent spaces of schubert varieties, and so forth... something about guessing from among candidates of appropriate invariance, vs more systematic approach...??or something...

the ideas about the natural contact distribution on the coadjoint partial flag variety of a simple lie algebra that we've been thinking about recently have a slightly peculiar flavor to them that's bugging me a bit. this peculiar flavor probably isn't unique to this context; we've probably run into it elsewhere as well. the vague idea that i'm trying to get at here is that... there's some sort of peculiar interaction here between different ways of thinking about "base spaces vs total spaces of line bundles" (or something like that), and various kinds of structures on line bundles... connections in the case of symplectic geometry... holomorphic structure in the case of projective varieties... ??...

so i think now that the natural contact distribution on the coadjoint partial flag variety of an a-series simple lie algebra is given by the zariski tangent space of _either_ of the two schubert varieties "their point lies on our hyperplane" or "our point lies on their hyperplane", the two varieties being tangent to each other at the basepoint of the coadjoint partial flag variety. in the a2 and a3 cases it seems possible to visualize the tangency between the two schubert varieties in a number of ways. in the a2 case it seems almost obvious from staring at the weylotope and thinking about the bruhat genericity order on the schubert varieties corresponding to the vertexes, in that these schubert varieties are 2-d and each dominate both of the 1-d schubert varieties "their point is our point" and "their line is our line".

i'm not sure yet what to make of the general phenomenon of tangency between schubert varieties...

anyway, at least i seem to have resolved some of what was confusing me here; the "naturality" of the contact distribution seems to require the "self-duality" of the contact distribution, but i was skeptical about whether it really is self-dual. probably this self-duality is supposed to be conceptually intuitively obvious from a contact geometry viewpoint. it may be that the self-duality more or less amounts to "the envelope principle" stating that a hypersurface is generally the envelope of its tangent hyperplanes, where "envelope" is defined in terms of intersections of infinitesimally nearby tangent hyperplanes (or something like that). and of course the envelope principle is related to "legendre transform" which is all about the relevant kind of "duality". or something like that.

is there an interesting relationship between the contact distributions on the coadjoint partial flag varieties of a simple lie algebra and of its "long root subalgebra"? or something...

??how does the usual obvious contact distribution on the projective cotangent bundle of projective n-space get along with "point-hyperplane duality" (or something...)?? ... i'm confused about this...

so... projective varieties (for example...) are "pseudo-stacky" in a certain sense...in the sense that, being non-affine, they're inadequately represented by the commutative rings of functions over them; instead you can represent them by the "categorified cpmmutative rings" of coherent sheaves over them... does this idea maybe repeat at a higher level in a simple-minded way? there are some specific examples of stacks that i've wondered whether they're such that they're inadequately represented by the coherent sheaves over them, but adequately represented by "the next thing up"; without however being actually "2-stacky" (in some sense...), and thus just "pseudo-2-stacky" or something like that... i think that i have some fairly specific idea of what "the next thing up" here (and "2-stacky" and so forth) is, though i'm not sure how systematically it forms part of a pattern...

so let's consider for example the coadjoint partial flag variety of sl(2,reals), as the projective coadjoint orbit of a highest weight vector in the coadjoint representation... perhaps we can get away with using the complexified coadjoint representation here... yes, seems like it... just so i don't have to worry about to what extent stuff like "highest weight vector" makes sense using the uncomplexified coadjoint representation... the group itself remains uncomplexified though... so then we want to try to figure out where, if anywhere, this orbit fits into the picture given by the orbit space of the root system under the kaleidoscope group... the "if anywhere" issue seems significant here... ??....

first let me try to see whether there's some standard "normal form" theorem for linear operators on finite-dimensional vector spaces over a field that i more or less know that's relevant here...

hmm, first let's consider the characteristic polynomial of a 2-by-2 matrix with real entries as an invariant of the matrix...

so given real numbers a,b, consider the affine real algebraic variety of matrixes m=

c d
e f

satisfying m^2 = m*a + b ... or something like that...

actually not too much like that; i was getting mixed up here between merely satisfying a polynomial p and having it as characteristic polynomial... or something...

g h
h j

g+j=1
gj-hh=0

g(1-g)=hh

g-gg=hh

sorry, just trying a few things...

anyway, constraining trace and determinant tends to cut down from 4 dimensions c,d,e,f to 2 ...

c d
e -c

determinant = -(cc+de)

hmmm... setting determinant to zero here as (quadratic) homogeneous constraint...

??so the coadjoint partial flag variety is here the _only_ projective coadjoint orbit that's contact rather than symplectic??? or something??? ???what's going on here???

??hmm, is there something here about "homogeneous poisson ideal" of the enveloping poisson algebra ??? or something???

??so in the split real a-series case, an adjoint orbit is closed under scalar multiplication precisely in case all the eigenvalues of the operator are zero?? ??is that correct??

??so given a linear operator, consider the flag obtained by the kernels of its powers... ?? ...or something...

so let's consider the coadjoint partial flag variety of so(n) and try to see whether we understand the alleged natural contact distribution on it.

so the coadjoint partial flag variety here is the space of rank 2 partial isometries e->f where e,f are euclidean spaces of dimensions m,m+1 respectively if n=2m+1, and of m,m respectively if n=2m, i think.

i'm not seeing it yet...

??so the coadjoint parabolic subalgebra is somewhat more "photogenic" in the root system picture than the other parabolics are, because its "boundary hyperplane" is orthogonal to the highest weight of the coadjoint irrep itself rather than to that of some other irrep; and this extra photogenicness is essentially the manifestation of the invariant contact distribution on the coadjoint partial flag variety, coming from the symplectic nature of coadjoint orbits... ??or something like that...

but to what extent, or when, do parabolic subalgebras generally have such "boundary hyperplanes" ?? perhaps maximal parabolics always do, with the boundary orthogonal to a highest weight lying on an extreme ray of the weyl chamber. but the coadjoint parabolic isn't generally maximal. hmm, and i guess that in fact there's really only an unambiguous "boundary" hyperplane when the parabolic is maximal. nevertheless the coadjoint parabolic has a special pseudo-boundary hyperplane, namely the mirror orthogonal to a long root.

(does the mirror orthogonal to a short root also bound a parabolic, and is there anything specially interesting about that parabolic??)

??so in "complexified contact geometry" (or something like that), is a "complexified lagrangian subvariety" (or something like that) some sort of "generalized complex codimension 1 (but thus real codimension 2) subspace of configuration space" ??? ????.....

so it seems a reasonable wild guess that the contact distribution on the coadjoint partial flag variety x is given by the zariski tangent space of the "schubert variety" (not sure whether it actually officially qualifies as one, but it more or less is one) given by "the second most special x/x orientation" ... or something like that... on the grounds that that orientation corresponds precisely in some sense to "the corner heisenberg, minus the exact corner" (or something like that)... though certain annoying "convention mismatches" that i've hinted at make me somewhat less optimistic about this guess than i might otherwise be... anyway, it seems like a good idea to check this guess on the a-series case... and be quite prepared to modify it if it doesn't seem to be working out...

so... in the a-series case the contact distribution has something to do with "moving the coadjoint partial flag in such a way that the point slides along the original hyperplane" ... ??so what sort of bruhat class (or whatever) does this amount to?? ...

hmmm, there seems to be some tricky stuff going on here, and in particular my guess above doesn't seem quite correct ... ???....

what about whether the contact distribution is the zariski tangent space of the supremum of all of the schubert varieties except the most generic?? (or something like that.) it's not clear to me that this is guaranteed to work, because (??among other things??) for all i know the tangent cone might be un-snug in the zariski tangent space, which might be co-dimension 0 ... ??....

??so it might be interesting to study the homogeneous coordinate algebra of the coadjoint partial flag variety as a "graded poisson algebra" ... ???... or something like that...

so let r1 be a commutative ring, r2 a commutative r1-algebra,and x an r2-algebroid, equipped with for each "tame" r1-algebroid morphism m : [a,_r1-mod_] -> [b,_r1-mod_] a corresponding r2-algebroid morphism m' : [a,x] -> [b,x], and for each natural transformation between m1 and m2 a corresponding one between m1' and m2' ... and so forth.

for example if r1 = r2 = z and "tame" is defined the right way, then i think that x is an abelian category...

now consider [c,x] where c is some algebroid. in general this needn't have the same structure... ???....

??a projective coadjoint orbit might correspond to a 1-parameter family of coadjoint orbits of the same dimension as itself, or to a single coadjoint orbit of 1 dimension higher?? ??or something like that??

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. ...

ok, so after just talking to chris rogers about it a bit, the story of the invariant contact distribution on the "coadjoint partial flag variety" of a simple lie algebra seems to be getting clearer.

when i first started noticing these invariant contact distributions and the heisenberg subalgebras associated with them, before i realized that they were associated with the symplectic structure on coadjoint orbits, i referred to these heisenberg subalgebras as "hidden heisenbergs". only now am i finally realizing just how un-hidden they are; they always lie right at any "corner" of a root system, being the killing-orthogonal complement of the stabilizer of the extremal weight space (thought of as a point of the coadjoint partial flag variety) at the opposite corner. (it looks like you can think of a simple lie algebra as sort of "pasted together" from the heisenberg subalgebras at its corners plus the cartan subalgebra in the middle, but i don't really know what to make of that yet.) the fact that they're so un-hidden suggests of course that they must already be well-known and well-understood, so i'm a bit surprised that i don't think that i've heard about them.

hmm, is a simple lie algebra always generated by the heisenberg algebra associated with a long root and the copy of sl(2) assocated with it, and is there some nice way of presenting the simple lie algebra arising from this?

one question that bugs me a bit at the moment is this: the coadjoint partial flag variety of a simple lie algebra x is a quotient space of a certain coadjoint orbit; what (if any) irrep of x arises from "geometric quantization" of that coadjoint orbit??

also... ??we can interpret this coadjoint orbit as an adjoint orbit (because of the nondegeneracy of the killing form, right?) and then obtain a conjugacy class by exponentiation... hmm, i was going to try to relate this conjugacy class to yet another partial flag variety, but... again i'm being sloppy about different forms of the lie algebra... compact vs complex vs split real.... ??.... it seems pretty clear that this kind of sloppiness is causing confusion with the previous question too...

i've been thinking a bit more about "the doctrine of abelian categories" and about incorporating tensor product structure into the doctrine as well... i should try to describe some of my thoughts about this stuff... some things seem trickier than i realized... tricky in the sense of i'm not sure whether i see how to get useful results.... ??...

??so what _about_ whether the invariant contact distribution on the coadjoint partial flag variety has a nice uniform "incidence geometry" interpretation in general?? in terms of the zariski tangent space of some sort of schubert variety or something??

larry harper asked me about a certain concept of morphism between directed graphs that he's trying to invent, motivated in a certain way by the idea of "circuit elements connected in series or in parallel analyzable as a single compound circuit element" (or something like that); he wants to find out to what extent it's already been invented. i told him that i'd think about it a bit, and that there ought to be some nice ways of thinking about it...

somehow what harper's trying to do vaguely reminds me of joyal's category of n-stage trees... or something like that... ???...

??perhaps we should try developing harper's category as fibered over a category of vertex sets and partial maps... ??or something like that??

might there be some interesting concept of "quaternionic manifold" amounting to a hyperkaehler mamifold equipped with some certain sort of involution??? or something like that???? ... somewhat silly idea, but...

??so... ??the coadjoint partial flag variety is a codimension 1 quotient space of the coadjoint orbit containing a(/the?? os???) highest weight vector... ??so it makes some sense for the former to be contact while the latter is symplectic ... ???os??? .... asf os...

??this is vaguely reminding me of stuff about treating the poisson operad as a graded operad and certain poisson algebras as graded models of it...???os???.... in connection with coadjoint rep ... os... asf os...

???w_a_ sa "a vs the" highest weight vector here???? os??? what sort of ambiguity is or isn't this relating to concerning "coadjoint partial flag variety" ????? ???os??? .... asf os....

??sa co-/adjoint partial flag kaehler variety ... ??sa langlands duality here???
??sa co-/root system as co-/adjoint weylotope degeneration .... ???os??? .... asf os...

(hmm, now i think that i may have made a silly verbal slip here; that "adjoint vs co-adjoint" doesn't particularly connect with "root vs co-root"...)


??sa possibility of other (??"even more" ?? os???) partial flags appearing as legendre submanifolds in co-adjoint partial flag kaehler variety ??? os??? .... asf os... ??does this actually happen anywhere besides the a-series?? ??more generally maybe we're dealing with sub-legendrian manifolds or something?? ??is that an interesting concept??

??is there anything interesting going on with schubert varieties and legendrian singularities here, or something like that??? not sure whether this idea actually makes any sense but for a moment i thought that it might...

??sa 3d rep of politically varyingly defined territory??...

??bit about "[2:51:13 AM] walter smith: a highest-weight co-adjoint orbit seems to have 2 reasons to "be a flag variety" in some sense..." os... asf os... ???sa idea that "_all_ partial flag varieties are coadjoint partial flag varieties" .... in a certain sense ... ???os???.... asf os...

??so is the partial flag variety in the projective space of an irrep a branched cover (or something like that??...) of the projective space of the "outer ring" of the irrep?? ??or something like that??

so are we now saying that the "inner coordinates" of a figure are _not_ quite completely redundant?? os??? ... asf os...

??is there some sort of interesting deligne-mumford stack here somewhere???? os??? ... asf os .... ????.....

hmmm.... some more belated realizations of apparently obvious stuff.... first of all, there's some sort of "coincidence" that the invariant contact distributions on partial flag varieties generally seem to be on those ones associated with the _adjoint_ representation... and working out why leads to some obvious/fun/tricky stuff... or something like that...

consider "the coadjoint orbit of a highest weight vector" ... or something like that.... ????.....

??so what about relationships between "incidence geometry for partial flags" and "geometric quantization" here ??? .... and so forth ... ??.....

so given a partial flag x, let's consider how the stabilizer subgroup of x in the automorphism group g of the flag building (or perhaps in some related group g'??...) intersects the conjugacy classes of g (??or of g'?? ...or something like that...we should also consider coadjoint orbits and so forth here)...

i think that there's some tendency towards... the stabilizer subgroup intersecting just those conjugacy classes that are shaped like the partial flag variety in which x lives, and intersecting each such class in precisely one element, so that the stabilizer subgroup is actually a classification of those such classes... except that this is almost certainly wildly over-(not to mention mis-)stating the case... (for example consider the conjugacy class of the identity element...)... the vague intuition that's leading me in roughly this direction has to do with the idea of "a structure canonically inducing an automorphism of itself" that i learned about from david joyce's papers on quandles... and with the special case of classical grassmanians as hermitian symmetric spaces (which idea we've been running into again lately as a very degenerate example of the theory of invariant distributions on partial flag varieties... ??maybe that's again related to what we're trying to get at here??...)

(??some of what we're thinking about here could be explored in a very austere "pure group-theoretic" context... starting with an arbitrary coset and seeing how it intersects the conjugacy classes of the group... not sure how this might go... again might relate to joyce's (and/or freyd's and/or yetter's...) ideas...??)

(there's also another (??) whole complex of vaguely remembered ideas that this is reminding me of...???... from several years ago.... the idea of "classes of conjugacy classes" and so forth... and how this relates to stuff like "geometric quantization vs [borel-weil theorem and/or verma modules and/or certain other ideas that i can't quite articulate right at the moment]" (??resp "coadjoint orbits vs partial flag varieties" and so forth...) and "k-a-n decomposition" (and/or stuff like that) and hecke operators and young diagrams and so forth... ??vaguely remembered conversation with allen knutson... hmmm.... i think that part of it had to do with partial flag varieties that correspond to different subsets of the dot-set of the dynkin diagram, but which are "equivalent via hecke operator" (or something like that... (??i guess that this also relates to stuff like understanding the kernel of the homomorphism from the burnside ri(n)g to the "green ri(n)g", though i don't think that it was stored in my memory that way just now...) ...of course all of this is tied in (in my memory at least, but probably in more than just that) with weird stuff about hecke operators and general sorts of "braids" (for example in an artin-brieskorn-coxeter sense...hmmm, this is leading me towards yet other whole huge complexes of ideas... "cardinal braid" ... (??for example "kleinian"??) singularity... milnor... "multi-homogeneous..." ... hyperplane quandle... schubert calculus... ?????...... ??????....)

anyway, to try to get back to the vague humble idea that i started with here... motivated in part by trying to visualize elements of g2 as explicit automorphisms of the geometry of a fermion rolling on a projective plane of 3 times the radius... let's consider some configuration x of this rolling fermion... perhaps we should identify this with the octonion "(j,j)"... and let's consider the stabilizer of x... so this is a 9-dimensional parabolic subgroup... or something like that... ??so am i really suggesting that there's a 9-dimensional space of conjugacy classes that are shaped like the configuration space in which x lives?? ...i think that that is what i was suggesting, and that it's not probably not completely wrong-headed, but screwed up in certain ways... if we considered a total flag instead of a partial flag here, then it's stabilizer would be an 8-dimensional borel subgroup, and... sorry, i think that i'm just trying to remind myself here of some basic numerology concerning dimension of the space of coadjoint orbits...

while i'm trying to straighten that out, let me also mention another source of confusion here... i'm being guided by certain intuitions concerning how partial flag varieties of the _complex_ form of a simple lie group relate to conjugacy classes of the _compact_ form... which for some (mysterious to me at the moment...) reason gives a very simple version of one of the games that i'm trying to play here... not sure how much more complicated the version that i'm trying to play is... ??...

ok, sorry, there's a lot of mistakes here, and/or obvious omissions that i forgot about... i'll try to fix some of this...

we should clarify to ourself how the g2 incidence geometry axioms work... not quite correct to say that given a generic pair of configurations there's a unique way of getting from one to the other in 3 rolls...

??so maybe we should identify the "line" grassmanian inside the projective space of the 14-dimensional irrep of g2 with certain 1-parameter automorphism groups of the octonions canonically associated with the "lines"... perhaps this is obvious from a certain point of view, but it seems good to try to state it explicitly here... because of the way that it meshes with our recent interest in thinking about partial flags more concretely, in terms of somewhat visible 1d subspaces of an irrep... especially the "extremal weight spaces" visibly corresponding to the partial flags in the favorite apartment...

??does it make sense to think of the 14d irrep as consisting of "anti-symmetric matrixes" acting on the split octonions as derivations preserving the relevant inner product?? or something like that??

??hmm, are we suggesting here that the "line" grassmanian has some sort of quandle structure?? or something like that?? this doesn't seem to be the "hermitian symmetric space" case, but... ??maybe it's related to that case in some vague way... ??....

well, of _course_ there's a quandle structure here in some sense, what with the relationship between partial flag varieties and conjugacy classes... but do we really understand what's going on here??...

??so... ??we should try to develop some nice ways of understanding the relationship between the symmetric monoidal category of representations of the simple lie group (or something like that) and the one of coherent sheaves over the flag variety (as well as some other related ones...) by means of... certain diagrammatic techniques involving the root system...which i'm vaguely imagining at the moment but too vaguely to articulate very clearly right now... of course we've thought about things sort of like this before, but... i think that i have certain ideas now about things to try here... for example about thinking of general partial flags as something sort of like "superpositions" (or something like that...) of the ones in the favorite apartment...
??...

one more ingredient to try to incorporate into the picture is the way in which the poisson algebra of the coadjoint orbit occurs as the scaling limit of... something or other... ??...

projective variety : conical singularity :: multi-projective variety : ???

projective variety : conical singularity :: ??? : more general singularity

projective variety : "multi-dimensional theory" :: conical singularity : ???

??so should we be thinking of an irrep of a simple lie algebra as the zariski tangent space of some sort of singularity?? or something like that?? with the codimension of the tangent cone in the zariski tangent space being something like the "inner" part of the irrep?? and what about something about bott's part of borel-bott-weil here?? is there some idea here about higher cohomology groups and "obstructions" carving out the tangent cone from the zariski tangent space?? of course we've made limited progress in understanding such ideas before, but i don't remember trying to connect it with borel-bott-weil in this way...

hmm, are the "inner coordinates" of a partial flag "redundant"?? ??because existentially quantified over, though perhaps actually unique as well??

so there's certain bunches of 6 1d subspaces of the 7d irrep of g2 that seem significant... what's the relationship between these?? ??what about "highest weight vector" and "change of which chamber counts as highest..." ?? or something...

also have we got a nice way of thinking of a g2 flag as some sort of "folded ordinary flag on an 8d vsp" ?? ...and so forth... ??....

the walking short exact sequence 2

so suppose that we have a morphism m : x -> y of short exact sequences, and suppose that we consider the functor from short exact sequences to base objects given by "homs from x, modulo homs that extend to y" ... or something like that... ??then to what extent can we recover m from this functor??

we should also try to clear up the situation with stuff like "the doctrine with syntactic 2-category given by the concrete operations on the algebroid of finite-dimensional vector spaces over field k" or something like that... probably just comes down to mostly obvious concept of "abelian category with all objects semi-simple" (or something like that... then also "tensored" version and so forth), but good to clarify
(categorified) "equational" nature of such characterization...

so is there a barr-beck distributivity between the "finite limits" and "finite colimits" monads on the 2-category of algebroids (or something like that) that gives rise to the "abelian category" monad?

taking "op-modules" is freely adjoining colimits... taking the opposite of the modules algebroid is freely adjoining limits...

so let x be an algebroid, and consider the algebroid of fp modules of the fp module algebroid of x, compared to the algebroid of fp modules of the fp op-module algebroid of x. are these algebroids opposite to each other??

fp modules of the fp right-module algebroid, compared to fp modules of the left-module algebroid...

so let f be an fp module of the fp module algebroid of x. then let's try to define an fp module g of the fp op-module algebroid of x as follows:

g evaluated at an fp module m of x^op should be ... m tensored over something with something ... or something ...

hmm, perhaps the point is that there should be some very straightforward morita equivalence here; let's try to make that explicit...

or wait a minute... is the alleged morita equivalence here contravariant, and does that actually make any sense??....

hmm, perhaps this (the existence of the invertible distributivity natural functor here) is obvious if we believe the alleged description of free abelian categories that we think we heard...

if we take the standard pseudo-euclidean quadratic form on the imaginary split octonions and then restrict it to the linear subspace carved out by "x*k=0" for k a given non-zero point on the light cone, then is the restriction of the quadratic form to the subspace the square of a linear form? and can this explain how the projective light cone for the restriction could be a projective plane? ??or something like that??

the walking short exact sequence

alex and i tried googling on "free abelian category" yesterday, and this led to some ideas that seem pretty much simpler than what i was expecting, and which have apparently been understood pretty well for a long time. (for example one of the references was to a paper by freyd from the long-ago la jolla conference. that's not too surprising, of course; it's just the fact that the ideas seem so easy to understand that surprised me a bit.)

i didn't really get into the details, but if i'm not too badly confused here then apparently an abelian category is essentially just "an algebroid x equipped with operations [a,x]->[b,x] just like all the finitary such operations in the case where x is the algebroid of finitely presented abelian groups". (or something like that.) perhaps it's already near-obvious that something like that should be true, but actually understanding it could be very helpful.

there's a bunch of interesting variations on this idea depending on (roughly) just how we construe the concept of "operation" here. perhaps this amounts to investigating various meta-doctrines (or something like that).

anyway, it occurred to me that this might indicate that it would be interesting to try to calculate for example the "walking short exact sequence" abelian category. we know a nice easy way to describe the "walking epi" finitely cocomplete algebroid, and that it's not abelian, so that's part of it why it should be interesting to try to calculate the "walking short exact sequence".

i was just recalling a vague memory of freyd mentioning some ideas about "the doctrine of abelian categories" (in some vague sense of "doctrine", which word he probably didn't actually use), and offhand i don't think that i remember him emphasizing this line of thought (how to formalize the idea that an abelian category is one with "all" (in some interesting sense) of the structure that the category of abelian groups has). i could be wrong about that though.

anyway, i still have to think more about how these ideas might apply to algebraic geometry. on some vague level it reminds me of johnstone's work on comparing the idea of seeing a topos as a "geometric theory" to the idea of seeing it as a theory of a richer doctrine where for example exponentiation is also preserved (or something like that).

so if my current guess is right, then the g2 "point" grassmanian is the configuration space of "a fermionic ball rolling on a projective plane of 3 times the radius". the 6 configurations in the main apartment have the ball stationed at one of the 3 "cardinal poles" (the "north-south" pole, the "east-west" pole, and the "front-back" pole", with either of 2 orientations that are 360 degrees apart.

thinking of the real g2 point grassmanian as the projective light cone in the imaginary split octonions, the 2d schubert variety in it at a basepoint k is carved out by the equation "x*k=0". can we as simply carve out the 4d schubert variety?

i guess that this raises some interesting questions about the "algebraic geometry as a form of logic" program and how the algebra of resultants (used in calculating some concept of "image of an algebraic-geometric map") relates to lawvere's analysis of existential quantification as an adjoint to substitution. thus we want the equational hull of "there exists y such that x*y=0 and y*k=0", or something like that.

i finally have the mathematica program working (at least it seems to be... might need a bit more testing still) that demonstrates how the geometry of a ball rolling on another of 3 times the radius is equivalent to the null subalgebra geometry of the split octonions.

actually, after that bit more of testing, i think that it's still not working perfectly yet. but the basic plan of starting from the most special bruhat classes and working towards the most generic does seem to be working; the correspondence seems to be straightened out for the special classes but not yet for the most generic one.

b2 schubert singularity

i've been thinking a bit more about the basepoint singularity of the 2-dimensional schubert variety on the projective light cone of a pseudo-euclidean space of signature (2,3), and there seem to be some pretty nice and retrospectively obvious ways of visualizing it that i hadn't noticed yet the last time that i tried to describe it.

one idea is this: the projective light cone of a pseudo-euclidean vector space of signature (t,s) itself carries a conformal structure of signature (t-1,s-1). (perhaps the most familiar example is t=1, s=3, with the projective light cone being our sky, whose conformal structure is revealed whenever we travel near the speed of light.) as i discussed somewhere here recently, a conformal structure is a field or "distribution" of homogeneous quadratic subspaces (called "light cones") of tangent spaces. and the light cone in the tangent space of a point p of the projective light cone of the (2,3) pseudo-euclidean vector space is the tangent cone of the 2-dimensional schubert variety built at the basepoint p. so it's a nice conical singularity, about as i was imagining. the schubert variety is "the 2-dimensional light beams emanating from a 1-dimensional light beam in the (2,3) signature geometry" (so to speak...), which infinitesimally projectively comes down to "the 1-dimensional light beams emanating from a 0-dimensional light beam in the (1,2) signature geometry", which is essentially just the light cone at the point of the projective light cone. (or something like that.)

zariski tangent spaces of schubert varieties

i've been writing a bit recently about invariant distributions on partial flag varieties and how this relates to singularities of schubert varieties and so forth. i want to try developing some of this a bit more systematically, in particular by trying to develop a systematic procedure for reading off from a root system the zariski tangent space of an arbitrary schubert variety at the "basepoint". or something like that...

??so what about "integral surfaces of higher-degree distributions" and so forth, and relationship to "pfaffian geometry" (and so forth...)?? ... hmm, but the pfaffian stuff as i understand it is just for integrable distributions, right?? ...or something like that... hmmm... simplicial interpretation.... ???there _is_ a "higher-degree" aspect, though, isn't there???.... not sure... i'll have to think about this....

so let's explore the idea of "self-duality" of the absolute galois group of a global field ... or something like that ... ??....

so do people talk about "adeles" in the case of a global field other than a number field?

so let's consider functors from the category of finite fields to the category of vector spaces... ??....

consider for example the groupoid-enriched category where the objects are the symmetric monoidal finitely cocomplete algebroids with the property of being abelian. does this have all small homotopy-limits? but moreover, if it does, then is there some conceptually intelligible account of the "logic" here??

some weird (i guess meaning unexpected to me...) aspects of the doctrine of symmetric monoidal finitely cocomplete algebroids...

??strongly non-abelian examples??

??anomalousness of alleged concept of "n-dimensional object" ??...

the concept of "dirichlet character" as we've been reading about it recently seems to defined in a rather "formal" way... well, but i guess that of course there _is_ supposed to be a much less formal treatment available, relating them to continuous characters of the absolute abelian galois group of the rationals... or something like that... but maybe the point is that if if i try to work out that less formal treatment for myself, there seem to be certain puzzles here... for example relating to this alleged concept of "primitive dirichlet character" .... ???....