Thursday, September 30, 2010

i think that i understand a bit better now the relationship between "tangent spaces of schubert varieties" and "invariant distributions on flag varieties...

one of the examples that had been bugging me for quite a while concerns the b2 grassmanian whose 2d schubert variety has a basepoint singularity. on the one hand, it seemed nearly irresistible to match up the 4 schubert varieties on the grassmanian (0d, 1d, 2d, and 3d) with the 4 submodules (0d "-", 1d "g", 2d "gh", 3d "ghi") of the "ghi" module of the "abcde" borel subalgebra (referring to a primitive picture of the b2 root system as follows:

abc
def
ghi

). but on the other hand, the basepoint singularity of the 2d schubert variety means that its zariski tangent space at the basepoint must be 3d, so the obvious straigthforward "irresistible" match-up can't be quite correct.

i think that i have this more or less straightened out now, though. you can picture the module "ghi" as the tangent space at a point of the projective light cone of pseudo-euclidean (2,3)-space. this projective light cone itself carries a (1,2)-signature conformal structure encoded as a field of light cones on its tangent spaces. so the module "ghi" is a 3d vector space equipped with a light cone, and further equipped with a favorite light ray through the origin, the remnant of a light ray in the projective light cone... the 1d submodule "g" is the favorite light ray, but what had been confusing me was the 2d submodule "gh". now it seems clear that it's "the plane tangent to the light cone and containing the favorite light ray", which is subtly different from but morally close to "the tangent space of the 2d schubert variety at the basepoint". this moral closeness seems like a big hint as to what's realy going on here...

by the way john huerta pointed out that this "plane tangent to the light cone and containing the favorite light ray can also be thought of as the pseudo-euclidean orthogonal "complement" of the favorite light ray. that seems interesting, though i'm not sure what to make of it yet...

Monday, September 27, 2010

i've been taking a pretty unsystematic approach (as usual) to the relationship between "geometric orientations" and invariant distributions, in part because in the case of flag varieties (as usual) an unsystematic approach seems to pay off, due to the humanly accessible rich-but-not-too-rich combinatorial flavor of the subject.

a more systematic approach though might include exploring the different possibilities more thoroughly. for example consider the case of the homogeneous space g/h of what i call "frames", where h is the trivial subgroup. then the atomic invariant distributions are the points of the projective space of the lie algebra, while the geometric orientations are essentially the elements of g ...

perhaps one of the distincive aspects of flag varieties is their "almost doubly transitive" nature... i just tried googling on "almost doubly transitive" and there were some hits but i couldn't tell right away how relevant they were...

Saturday, September 25, 2010

so in exploring the idea that invariant distributions on a homogenous space g/h (for example the flag variety of a nice simple algebraic group g) serve as "infinitesimalizations" of "geometric orientations between points of g/h" (aka "(h,h) double cosets"), i got somewhat confused by the case g = pgl(2) and h = gl(1), where there seems to be an excess of geometric orientations and a shortage of invariant distributions to serve as their infinitesimalizations...

the tautological action of pgl(2) on the projective line is "sharply triply transitive" in the sense that a "frame" for the geometry is equivalent to a non-degenerate ordered triple ("infinity", "0", "1") of points on the line:

first, nailing down "the point at infinity" increases the structure from that of a projective line to that of an affine line (with adjoined point at infinity); or in other words reduces (or "breaks") the symmetry group from pgl(2) down to the group "al(1)" of affine-linear transformations of the affine line.

second, nailing down "the origin" further increases the structure from affine line to vector line; or, breaks the symmetry from al(1) down to gl(1).

finally, nailing down "1" increases the structure from vector line to the completely rigid structure of a "scaled" vector line; or, breaks the symmetry from gl(1) down to nothing.

in other words, there's a natural 4-input function f on any projective line, valued in the standard projective line, with x |-> f("infinity", "0", "1", x) being the isomorphism that takes "infinity" to infinity, "1" to 1, and "0" to 0. f is a semi-famous function known as "cross-ratio" (probably because there's an algebraic formula for it involving ratios... maybe in some sort of cross-like arrangement?). f in fact encodes the structure of being a projective line (over the unspoken base field...).

since 2+2 = 4, you can use cross-ratio to help classify the geometric orientations between nondegenerate 2-tuples of points of the projective line. so there's a lot of such orientations, in particular a continuous family of them because cross-ratio is continuous-valued.

on the other hand, there's not a lot of invariant distributions on the homogeneous space of nondegenerate 2-tuples on the projective line. there's no "non-trivial" ones. so what happened in this particular case to the general idea of studying geometric orientations by studying their corresponding "infinitesimal geometric orientations" (aka invariant distributions), that seemed so useful in the case of flag varieties?

well, i thought that i understood this pretty well at one point, back here, and that the explanation was mostly common sense but involved a certain amount of staring at pictures and/or using some mild "geometric inutition", and that the upshot was cautionary but not overly cautionary; that there are situations where the microscopic geometric orientations tell you a lot about the macroscopic ones and situations where they don't, and that it's probably a good idea to try to understand why flag varieties are one of the cases where they tell you a lot.

i think that the basic explanation of "what goes wrong" in the case of nondegenerate 2-tuples on the projective line is that there are a lot of geometric orientations to a 2-tuple p that don't allow you to get anywhere near p, or in other words whose corresponding infinitesimal geometric orientation is trivial (namely the zero invariant distribution). i think that the picture that i wound up staring at was of the level curves of the function g(x,y)=xy, and how most of them are hyperbolas that don't get anywhere near the central point (0,0). but i should try to work this out again more carefully sometime to see if i really understand it.

i have a vague feeling that so-called "geometric invariant theory" might say something interesting about the different flavors of cases here. but so might the study of orbit stacks of algebraic groups acting on tangent bundles of homogeneous varieties, which (coincidentally...?) i just started thinking about. i think that i was annoyed at first that "geometric invariant theory" stole that name, but perhaps the potentially more interesting subjects from which i was imagining it got stolen would better be called "geometric covariant theory" or something...

meanwhile, even in the case of flag varieties, where the relationship between macroscopic and corresponding microscopic geometric orientations seems perhaps the nicest, there's a lot of interesting peculiarities to the correspondence that i don't understand yet. a lot of this is presumably to be explained in terms of schubert varieties being singular or tangent to each other at the basepoint, but some of the peculiarities still seem pretty peculiar to me so far. (is either or both of "singularity" and "tangency" a special case of each other??)
consider the orbit stack of the action of an algebraic group g on the tangent variety of some homogeneous variety g/h where h is a nice closed subgroup... for example the case where g/h is the flag variety of g...

so consider a simple lie algebra g, modulo a borel b, as a rep of b ...

hmm, googling on "affine schubert" seems to indicate that that's people talking about schubert calculus for grassmanians of loop groups (or something like that), rather than about relating the action of a simple lie group on the tangent bundle of the flag variety to the action on the square of the flag variety...

let's try considering b2 for example... the tangent space of the "line" grassmanian can be thought of as pseudo-euclidean (2,1)-space, i think... but what about the tangent space of the flag variety??

Friday, September 24, 2010

just a silly thought:

is there some interesting way to interpret a simple lie group as the configuration space of a "cartan-rolling" system in which one copy of the flag manifold "rolls" in some away on another copy of it? or something?

or is it easy to see that the group has the wrong homotopy type for this??
so let's try taking some nice simple examples of graded commutative algebras with maximal ideals corresponding to mildly (or something...) stacky points, and then try to understand the symmetric monoidal algebroid of fp graded modules of the quotient algebra by the square of the ideal. or something like that.

for example let's try "the orbit stack of gl(1) acting on k^1". so we have one generating grade g, and one generating element x in grade g. and let's consider the ideal generated by x, and the square of that ideal. and so it seems like this is an example that we've thought about before, where the graded modules end up being z-graded chain complexes, with the usual convolutional tensor product but without the usual sign-flip in the symmetry maps. and this is weirdly suggestive in a number of ways but we're not really sure what to make of it yet...

let's try another example. one generating grade g, generators x in grade 2 and y in grade 3. maximal ideal generated by x. a graded module of the quotient algebra is ...

hmm, so wait, that's not quite i wanted this last example to be... i don't want to just set x to zero; i also want to invertibilize y... or something like that... in which case a graded module now seems to be... a triple of vector spaces... with the tensor product being essentially the usual convolutional tensor product of "co-modules" of z/3, with the usual symmetry... and now if we instead mod out by the square of the ideal, while still invertibilizing y, then the graded modules now seem to be "chain complexes of period 3" ... again with the expected tensor product but the slightly unexpected symmetry.

hmm...

Thursday, September 23, 2010

do the torsors of a group g form a closed category (in the sense of eilenberg and kelly, or something...)?? they don't form a monoidal category unless g is abelian ... ????...

if so then what about categories enriched over it?? for example itself... ???...

what about the free cartesian closed category on an object x equipped with an isomorphism (??perhaps with some nice "coherence" property?) to x^x? what about some sort of combinatorial-game interpretation of this??

hmm, there's definitely some confusion here... between internal and external hom, for one thing... the external hom from one torsor to another is naturally a torsor, but ... ???? ??or something??

i guess that there's lots of examples of a category c equipped with a factorization of its hom bifunctor through itself that don't give a closed category; for example lots of examples where c is discrete... is that correct?

maybe this is "worse" than that, though... i'm still somewhat confused here...
hmm, maybe i'm finally beginning to understand the relationship between "cartan geometries" and a generalized sort of "rolling" that derek tried to explain to me...

when derek explained to me the definition of "cartan geometry modeled on the homogeneous space g/h", i found it a useful conceptual simplification to reformulate it as what i call an "h-flat g-torsoroid". a g-torsoroid is similar to a g-torsor in that it's a manifold of the same dimension as the lie group g and equipped with a "maurer-cartan t_1(g)-valued 1-form", except that the 1-form isn't required to be flat. so infinitesimally the torsoroid has a g-valued "distance" or "displacement" function, but unlike a torsor the displacements don't satisfy the triangle equality (aka the homomorphism law for the principal action of g). being "h-flat" though makes it possible to "mod out by h", giving a quotient space (that i'll call "x") that resembles g/h but is wobblier.

like i say, i found that to be a useful simplification that helped me understand what a cartan geometry really is. on the other hand though it may have delayed my understanding of what derek meant by "generalized rolling". in the standard approach to cartan geometries the quotient space x is the central focus, and the torsoroid arises as its "h-frame" bundle. my approach was to jettison x because i could reconstruct it from its h-frame bundle, but because of that i also jettisoned a convenient piece of structure on x, namely the t_1(g)-valued 1-form that it manages to inherit from its h-frame bundle.

so i think that derek's point is that you can interpret this 1-form as a connection on the trivial g-torsor bundle over x, and take the "horizontal distribution" of that connection and interpret it as a "generalized rolling distribution" on the total space of that trivial bundle. and this really does generalize the classical sort of "rolling distribution" that i've been studying in connection with g2 and the octonions and the rolling ball.

i think that part of what confused me here all along is that i didn't manage to distinguish clearly enough between two ways of getting an ehresmann connection from a cartan connection: you can get one on the h-frame bundle of x (and this might be the way that allegedly led to ehresmann's invention of ehresmann connections?), but pehaps more importantly, you can get one on the trivial g-torsor bundle over x, and this is the way that gives a "generalized rolling distribution" as its horizontal distribution.

derek sometimes mentions the (actually pretty visually intuitive) idea that "rolling" of one surface on another doesn't require that _either_ surface be homogeneous. i'm not sure to what extent derek or others may have worked out this idea, but anyway it suggests viewing the configuration space of the "generalized rolling" system in a more symmetrical way wrt the roles played by the homogeneous surface and the non-homogeneous one. there's the "point of contact" on each of the two surfaces, and there's the h-torsor of structure-preserving isomorphisms between the tangent spaces of the two contact points... ??...

it's funny that i didn't understand this stuff until i decided that i wanted to try to generalize the "rolling" interpretation of the g2 incidence geometry to other simple lie groups...
derek told me about how the quasi-familiar concept of "ehresmann connection" was apparently created by taking a more involved concept of "cartan connection" and isolating a less involved part of it (or something...) ... but on the other hand we seem to be running into certain situations (for example in connection with an ordinary "rolling ball") where an ehresmann connection (or something...) can be used to create a distribution... so i wonder how that might relate to the idea of the "cartan-rolling distribution" of a cartan connection... ??or something...

hmm, these ideas aren't fitting together for me at the moment... i'm getting confused about the idea of a torsoroid vs the idea of the configuration space for a "rolling" system... ???...
so in trying to understand the idea of the "cartan-rolling distribution" coming from a "cartan geometry", we should in particular consider the ("constant curvature" ?) case of a homogenous cartan geometry... perhaps exploiting special algebraic techniques that become available in this case... ??and perhaps becoming particularly relevant in connection with invariant distributions on partial flag varieties??

to what extent do _all_ distributions arise as "cartan-rolling distributions"? and to what extent can the original cartan geometry be recovered from the distribution? i guess that that those are roughly surjectiveness and injectiveness questions, respectively... ??is there maybe a third question in that family somewhere?

to what extent do continuous parameters occur in the classification of graded nilpotent lie algebras?? (or something...) there's stuff here that confuses me... concerning the radius-ratio parameter in the rolling ball geometry, and so forth....

perhaps derek would prefer to call a "cartan-rolling distribution" a "generalized hamster-ball rolling distribution" or something...
so in connection with the relationship between graded nilpotent lie algebras (and so forth...) and distributions, what sort of special role do the graded lie algebras generated by grade 1 play?
john huerta told me (or reminded me, or something...) that f4 is supposed to be in some sense "the automorphism group of the octonionic projective plane"...

(hmm, so what's the automorphism group of the quaternionic projective plane???...)

(whoops, i just realized that i'm unsure about whether it's supposed to be projective _plane_ here or projective _line_... should try to straighten this out...)

so this raises a bunch of interesting questions and ideas about ways that f4 might interact (or something...) with g2, and about geometric/"mechanical" interpretations of the f4 incidence geometry (including ideas about "cartan-rolling distribution"...), and so forth...
i'm confused about "closedness" of distributions... coming from trying to think about naive ideas about "rolling without skidding" vs "rolling with neither spinning nor skidding" (or something...) ... vague feeling that contrasting topologies on a single space are relevant here, which sounds annoyingly pathological at first, though maybe it won't be so bad since you can think of foliations as a case of that (i think...) ...
so given an h-flat g-torsoroid (or something...), can we get some sort of distribution on it (or something... ??hmm, perhaps on the associated "cartanian g/h geometry"??...) by taking its "excess flatness subalgebra" (or something...)? if so then does this deserve the name "the cartan-rolling distribution" (or something)??

(i forget whether "distributions" are usually allowed to vary in dimension from point to point...

also, what about ideas about "the symbol lie algebra of a lie algebra filtration" here, and so forth? something about nilpotence here and so forth...)

which reminds me that i should try to organize a list of questions to ask derek wise about...

1 the above question...

2 something about graded nilpotent lie algebras and their relationship to distributions...

3 do you know anything about "cartan's method of curvature invariants"? (or whatever the name was that bor and montgomery (i think it was them...) called that method...)

4 something about relationships between "de sitter relativity" and segal's cosmology... or something...

5 have you thought about the (degenerate?) case of cartan geometries modeled on an arbitrary manifold considered as a homogeneous space of its diffeomorphism group?

6 something about invariant contact distribution on coadjoint partial flag variety...

Wednesday, September 22, 2010

so what about the relationship between "blowing up a sub-variety" and "getting a legendrian submanifold of t*(x) from an arbitrary submanifold of x"?? and so forth... ??...

Tuesday, September 21, 2010

so i guess that we can ask how does "long root sub-system of a root system" get along with "folding of a dynkin diagram" ... or something... ??been bumping into some example involving folding d4 to get g2, but also including d4 as long root subsystem of f4 ...
does the triality symmetry (or something...) of d4 extend in some interesting way to f4, along the inclusion of d4 as the long root sub-system of f4??

i also vaguely wonder whether this might relate to some semi-mysterious comment of mckay's that i never quite understood...
so what about the vague general idea of ... ??trying to apply rational homotopy theory to the study of algebraico-geometric theories (or something...) in a "straightforward" way... besides possible more peculiar ways i might vaguely have in mind ... ???...

what about "extra structure (or something...) that the (unstabilized and/or stabilized...) fundamental infinity-groupoid of a space gets when the space becomes a complex algebraic variety"?? and so forth... (??something about "gaga" here...??...)
maybe try to get huerta interested in stack ideas (and so forth... or at least my interpretation of them..) by connecting with "super-geometry" ideas... though also try to understand that other approach that he was describing...??...

??also maybe try to get him interested in this "renormalization" stuff again... this idea of "[differential calulus as special case of renormalization] as [ideal power filtration as special case of filtration]" or something...

also mention the idea of introducing g2 transformations into the mathematica animation of octonionic rolling, in the hope that watching the change in rotation speed gives some insight into the symmetry... or something... ??also try similar idea in for example b2 "gimbal ring" case ... (some aspects of this case get me confused about the whole idea... something about "rigidity" of the gimbal ring in some sense... ??...) ??? and so forth...
seems like a somewhat obvious thought but i'm not sure i remember saying it outloud before... ideal power filtration as a special case of filtration seems sort of like "differential calculus as a special case of renormalization technique" ... or something... ??...
so what about specializing the idea of "generalized legendrian submanifold" to the case where there are no non-trivial invariant distributions (or something... ??relationship to "hermitian symmetric space" and so forth... ??something about grassmanian vs partial flag variety... ??...) ?? so maybe arbitrary submanifolds qualify as generalized legendrian in such a context?? seems like a case where "geometric quantization" develops a somewhat "classical" flavor... ??or something...
we know some stuff about the space of figures of a given type t incident to a given figure f of some other type t'... mainly from the viewpoint of "what a t'-figure looks like in the t-picture"... but also as a special case of a schubert variety (??which perhaps indicates a way or ways that what i'm about to suggest should be generalized... let f be a (perhaps partial) flag instead of a figure... but then also going beyond incidence...)...

??but can we sharpen this to ... "thinking of the grassmanian of t-figures as a projective variety in its canonical (??...) way, knowing what sort of projective variety a t'-figure 'appears as', as a subvariety of it"??

for example, consider the g2 case, with t being "point" and t' being "line"... we know that a t'-figure appears as a 2d subspace of the canonical vector space in which the t-figures appear as 1d subspaces. or projectively, a t-figure is a point and a t'-figure is a curve of degree 0 (??or something like that??)... is it always about as simple as that, and are there nice systematic ways (from "staring at the dynkin diagram" or "staring at the root system" or something...) of figuring out how it is??

(i have a vague memory of some discussion of stuff vaguely like this in that book "spin geometry"...)

i'm also vaguely thinking about how this idea gets along with the relationship between the incidence geometry coming from a root system and that coming from its long root sub-system... and so forth...

(hmm, what about the idea of "a schubert variety in the total flag variety which doesn't constrain the t-figure in the flag and can thus be (sort of...) thought of as a schubert variety (or as a bundle over such...) on the partial flag variety where t is ignored"?? perhaps my point here is that this is the same as a schubert variety in the total flag variety that's larger than the schubert variety "aside from their t-figure all their figures are the same as ours", whose "infinitesimal analog" is "an invariant distribution on the total flag variety that's larger than ..." (or something...), which we discussed recently but perhaps without sufficient conceptual clarification...)

??hmm, so what about the pretty simple idea that ... when you give an embedding of a variety into a grassmanian (or something...), you're hinting at some axiomatization of the corresponding algebraico-geometric theory where the basic stuff is "a geometry of the residual type wrt that grassmanian" (or something...)... ?? ... simply generalizing the case where the grassmanian is a projective space and you're interpreting the projective variety as (approximately) the moduli stack of models of a dimensional theory... probably in the case at hand this will work out to some fairly tautological idea, but that's probably not such a bad thing...
??so is there some nice generalization of "zariski tangent space of an algebraic variety" where the variety is generalized to a stack and the tangent space is a chain complex or something??? ???

but then what about zariski tangent space vs tangent cone?? ... and so forth... ??...

??hmm, so what _about_ the idea of "a filtered algebraic-geometric theory" and so forth?? and how it relates to stuff like "algebraic stack of poisson algebras as normal stack to substack of commutative algebras inside ambient stack of associative algebras" ??? ...

hmm, something about the way that... a filtration on an operad (for example...) can cut down to a quotient operad (if it's the ideal power filtration of the corresponding operad ideal... i guess that we still have to get around to thinking about the more general case of a filtration that's not just an ideal power filtration...), but not more finely, to something really point-like... for that it seems like you have to go to a richer doctrine... ??hmm, does that really make sense??? ....

(what about "intended environment" for operad doctrine?? or something...)


??what about a normal cone (or something...) as sort of like "a tangent cone to a somewhat generic point" ?? ??or something???

anyway, so one thing that we should probably do is to try to work out how the whole big "blowing-up of a sub-stack" (or something...) story should reduce in some special case (or something) to some concept of "zariski tangent space of a point of an algebraic stack" (or something), which might be some sort of chain complex or something... ??...

we should really try, for example, taking the dimensional theory of "cubo-quadratic algebras" and considering on it the ideal power filtration of the ideal corresponding to gauss's lemniscate; and looking particularly closely at the first (and also zeroth??) associated grade... or something... ??...
hmm, so can we give a nice geometric interpretation of the long root subsystem of a non-simply-laced root system x in terms of the "folding symmetry" producing x from a simply-laced system??
hmm, it might be interesting to watch the mathematica animation of octonionic rolling when viewed under various sorts of g2 tranformations... because of how the rolling speed gets affected... ??...

hmm... ??something about "residual geometry" and "residual stuff" and ideals of borel subalgebra and "nilpotent radical" and invariant distributions on flag varieties and so forth ... ??...

Monday, September 20, 2010

so is there some general way of obtaining a distribution (on what??) from a cartan geometry? by "cartanian rolling" or something like that???
so is there some idea here about "generalized legendrian submanifolds as cartanian geometries of _residual_ type" ?? or something like that??

"generalized legendrian submanifold" as something like "submanifold satisfying all of the invairant differential relations satisfied by figure subspaces of given type" or something??
so i should start making a list of mathematical ideas to tell/ask gunnarsen about... mainly from the point of view of ideas that admit some sort of interesting way of explaining or dramatizing them... but maybe other viewpoints too...

1 g2/octonions/rolling ball mathematica animation

2 "relativistic robotwars" game

3 kummer's chemistry analogy

4 "masquerade" game...

5 "new kid on the block" interpretation of pascal's triangle...

6 commutative diagrams and related formalisms...

7 finite difference calculus...

8 ordinal "epsilon zero" (or whatever it's called) and game trees and so forth... hydra and so forth...

9 "cross-dimensional cavalieri's principle"

10 fundamental theorem of calculus... ??and stokes's thm?? (?sp?) ...and so forth...
so what about "geodesically non-holonomic rolling" or something like that??

i've been describing the g2 incidence geometry as involving "a fermion rolling on a projective plane of three times the radius", but actually i'm not sure exactly which systems if any would deserve such a description. for example the way that i have the mathematica animation set up at the moment, you see the fermion roll off the edge of the hemisphere, and simultaneously reappear at the antipodal edgepoint, _without_ flipping its fermionic bit; but why shouldn't flipping it be another reasonable option? i guess that part of the point is that when you're already admitting "fermionic global non-holonomy" (or something) it tends to interfere with the decision as to what wrapping or unwrapping of the base space you're using... ??or something?? ??is there really some principled way to decided that the morally correct description is "the large ball is just projective while the small one is genuinely spherical, and the connection between their frames is fermionic"?? ??or something??

??what about the possibility that although "octonionic rolling" is the only ("ungeneralized"...??) rolling ball system with extra _continuous_ symmetry, there might be other such systems with interesting discrete symmetry, and maybe some interesting way to classify these, perhaps even in a way that places the anomalous g2 continuous symmetry of octonionic rolling in an interesting context??

??so what about the idea of a numerological investigation of invariant distributions on partial flag varieties of exceptional lie groups to see whether the numerology of the distributions might match that of any relatively obvious "naively geometrically natural non-holonomic distribution" such as, say, "a 4d ball rolling on a 3d plane" (in some sense... i'm vaguely imagining some sort of "cartanian rolling" along the lines of the "cartanian geometry" ideas that derek wise has tried explaining to me, though that might be somewhat overkill as well as not the only sort of possibility...)?? to what extrent might this (or something...) be ruled out by what some people know about "prolongations of graded nilpotent lie algebras" (or something like that...), or by the failure of g2 to fit into larger root systems, for example??

??so what's the long root subsystem of f4?? this being the only other root system with such a sybsystem, to speak of (??) ... what about whether this might give hints about some interesting "generalized rolling" interpretation of the f4 incidence geometry??

is it something like a2 X a2 ?? no wait, maybe it's so(7)... no, i guiess that it's obviously so(8)...

so what _is_ the numerology of the "4d ball rolling on a 3d plane" distribution??

??9d configuration space?? ... ??...

??so _does_ a ball have to overcome some sort of friction in order to spin rather than roll?? (or in addition to rolling??) doesn't really seem like it, intuitively... ??...

??so what about the idea of a 3d ball rolling on a 1d wire?? or something??

???so what about some sort of generalization of "legendrian submanifold" (or something) that would include smooth non-geodesic rolling trajectories of the octonionic rolling geometry? ... ??again i'm imagining some vaguely cartanian ideas here... ??... ??something about "torsoroids" and so forth??

??assuming for the moment that some idea of "cartan geometry modeled on a non-homogeneous space" makes sense, then consider the case where the automorphism group of the non-homogeneous space is "frobenius integrable" ... ??is the geometry then automatically flat?? or something??

??any relationship between "shield" (as in "wall of shields") and "hoops"/"gimbal rings"?? ??perhaps not, since a gimbal ring seems like an "extended material object with persistent individuality of each of its points" ...?? or something??

Sunday, September 19, 2010

??so what about meta-modules of the meta-rigoid of fp bi-modules of fp rigoids?? or something...

also the analogous question for semi-lattices in place of commutative monoids... ??
so consider the symmetric monoidal finitely cocomplete algebroid t given as the opposite of the algebroid of fg free modules of the base field (or something...) ... ???what is this the algebraico-geometric theory of??? ... or something...

consider the algebraico-geometric morphism from the initial algebraico-geometric theory to t...

the morphisms invertibilized by this functor are... ??what??... ??is there something here about matrixes of determinant 1 ??? ???or something like that?? ??but what about non-square matrixes??? hmmm...

hmm, now i'm getting a bit confused about the relationship of this to stuff that we thought we almost understood about the algebraico-geometric theory corresponding to a "galois stack", or something like that... ??...

??something about normal vs non-normal extensions and property vs structure... or something ... ???... "axioms" ... ??...

??hmm, something about invertibilizing morphisms between free modules vs between not neessarily free ... ???or something??...
??hmm, what about something about "matrixes between matrixes" in the sense of, given a domain r1 X g1 matrix m1 and a co-domain r2 X g2 matrix m2, a g1 X g2 matrix "preserving the relators" ... i was going to suggest trying to generalize determinants to this context, but that doesn't seem quite right... ??something about "witness for preservation of each basic relator" as giving commutative square of matrixes??


let me remind myself that i think my original motivation for starting to think about this was to experiment with the idea of "extending distributivity by distributivity" or something like that... ??...
??can we think of an abelian category x as a meta-module of a certain meta-ringoid... ??or something... ??meta-ringoid with one object for each fp ringoid r... realized wrt the meta-module as ringoid of ringoid morphisms from r to x...

?? i guess that this is supposed to be analogous to thinking of an abelian group as a module of the ringoid of finite matrixes of integers... or something... (??hmm, something about bi-module as meta-matrix?? [note added later: ??bad level slip right here??] abelian category as meta-module of the meta-ringoid of fp bi-modules of fp ringoids... ??and then later something about meta-algebras of the meta-operad of multi-modules, analogous to algebras of the operad of tensors ...??or perhaps we should be using symmetric tensors and meta-symmetric multi-modules .... ????or something...) ... part of the point being that you could also get away with for example the morita-equivalent ringoid (not closed under direct sums) corresponding to the ring of integers... and analogously you could get away with various morita-equivalent smaller meta-ringoids, or something...

(hmm, so what about "embedding theorems" here, and their decategorification??... and/or maybe categorification...)

??according to pattern i'm vaguely imagining, something like a "symmetric monoidal abelian category" would then be a meta-algebra of a meta-operad... or something like that... ??...

??hmmm... consider the subringoid of the ringoid of fp abelian groups generated by finite limits from z ... ???.... ??hmm, i have a vague memory of recently thinking about situations where the projective modules are closed under limits... ???or maybe it was even under colimits??? no, it was more like... the subringoid of projective modules has its _own_ colimits... ??or something like that?? is this related to that?? ... ??....

some aspects of attempted analogy here that i'm very vague and maybe confused about ... ??ringoid as analog of set?? ringoid with direct sums as analog of commutative monoid or something??

can/should the meta-ringoid here be "morita-equivalent" to a one-object meta-ringoid??

??in the ringoid of abelian groups, each fg free one is an absolute colimit from the full sub-ringoid containing just z...

??in the meta-ringoid of abelian categories, is the module category of an fp ringoid some sort of "absolute 2-colimit" (or something...) of the full sub-meta-ringoid containing just the ringoid of abelian groups???

hmm, i'm vaguely imagining trying to get this to connect with something about... fg free ab gp as being in some sense "cohomologically trivial", and also module category of an fp ringoid as being in some sense "cohomologically trivial" ... ??or something?? ??does this make any sense??... module category of an fp ringoid ct module category of a commutative ring... ??...

aside from bad level slips that may have crept in here... ??some possible patterns other than "meta-module of meta-ringoid" that might or might not be more salient here... meta-[algebraic theory]?? ...cateogorified monad ... ??and so forth... ??...
notes for discussion with alex today...

??trying to concretely identify limit/colimit reversal on free ab cat on for example walking object??

??something about indecomposable module as cyclic?? no wait, need irreducible for that??? hmm, how much might that complicate certain answers??...

hmm, well then what _about_ trying to approach the classification problem via the irreducibles instead of the indecomposables?? some general principle (??_how_ general??) that makes this work better than you might have at first hoped?? something about jordan-hoelder braiding??

?? ext(z/p,x) vs x tensor z/p ... ??

??issue of weighted colimits in abelian categories... and so forth...

...lots of other stuff...
what about modules of the algebroid of coherent sheaves over a nice variety or stack (or something), vs some sort of sheaves of modules of module algebroids... ?? or something...

??what about simultaneously extending tensor product by usual distributivity over colimits, and limits by "abelian cat distributivity"?? or something?? ... ??something about "extending distributivity by distributivity" or something???....

??what about algebraic-geometric morphisms with algebraic part preserving limits?????? and so forth, or something... ??also something about "flatness" ....

what about "toric case" of some stuff here??...

what about stuff here (or something...) and "symmetric tensors ct polynomials" (or something...) and "strongly n-dimensional object" ... ?? ... or something... ???? ... ????something about "the doctrine of all multi-additive functorial operations on the algebroid of vector spaces of a finite field" ... ??and so forth...



??what about freely adjoining limits and/or colimits to a symmetric monoidal category, and also algebroidizing it??? ... or various sub-sets of that or something... ??? ...

??what about abelian group objects in a symmetric monoidal complete category?? whether there's a good notion of tensor product of them??? ...

??what about stuff here (or something...) and "schur functors" (or something...)? ...

Saturday, September 18, 2010

so suppose that x is a ringoid, with for every fp ringoid r and fp r-module m a ringoid morphism f_m : [r,x] -> x, tw ... ???

should f be thought of as a weighted colimit operation..... ??


hmmm,so what _about_ weighted colimits????? in the "abelian" world ... ???or something?? ??have we been overlooking something important here??? _does_ am abelian category automatically have weighted limits and colimits, or something????

Friday, September 17, 2010

notes for discussion with alex this morning...

??problem too easy??

"the abelian category axiom" as a kind of distributivity...

example of ext(z/2,x) ... generalizing...

"multi-functorial operations" ...or something like that...

??what sort of compatibility relations do we get this way between tensor product and limits?? or something like that ... ??perhaps something "cohomological" or something like that??

Thursday, September 16, 2010

so given a ringoid r (perhaps restricted to be finitely presented or something like that?) and a fp _fp r-module_-module, can we give a corresponding syntactic construction in the doctrine of abelian categories? and so forth...

then consider, instead, a ringoid equipped with, for each n-tuple x of ringoids and each fp _fp x-n-module_-module, a corresponding multi-functorial operation...

back to the case n=1...

r ringoid (??perhaps fp or something??)

m fp _fp r-module_-module

x ab cat

f : r -> x

then we want to obtain an object of x ...

??for each generator g of m, evaluate f at the object where g lives ... ??

??for each relator r of m, evaluate f at the morphism where r lives ... ??

??take the colimit in x of the diagram ... ??...

???is this some sort of weighted colimit of the diagram f with weighting given by ... ?? ??or something like that??

wait a minute, i think that there are some level slips or something here...

let's go back and try specializing to some example that we think we can understand...

r is the walking object; that is, the ring z as a one-object ringoid.

m has one generator g at the z-module z, and one relator r = "g*2".

x is some abelian category, and f is essentially some object of x.

the resulting object of x is, we think, the cokernel of the scalar 2 acting on the object f ... or something like that ...

perhaps we want to kan-extend f... and then kan-extend the kan-extension the other way... ??...
some of the ideas that gunnarsen told me about related to basic algebraic number theory topics such as ideal class groups of quadratic number fields, and certain "graphical" methods of demonstrating them. i haven't really grasped yet what these graphical methods are doing, but they vaguely reminded me at a naive visual (and/or tactile, etc) level of methods that i developed myself in trying to understand and explain approximately the same topics. so i started to try to remind myself of the methods that i developed, which is what the unfinished proof below is about. (later i started to think that the visual resemblances that i noticed were probably just superficial; in any case i still want to understand gunnarsen's methods and how they relate to various ideas that i've explored.)

here's a "proof" that a prime p of the form 4n+1 is a sum of two squares:

(i'll illustrate the proof with pictures corresponding to the case p=5.)

take a square grid:

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(picture may have distorted aspect ratio)

and inside of it mark off a similar but sparser grid with nodes spaced p units apart:

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now in "clock arithmetic" modulo p, there must exist a square root of -1, because the multiplicative group is cyclic of order 4n ...
consider the "truth table" monad on the category of sets; that is the monad whose algebras are boolean algebras (with their usual underlying set). in how many ways can this be expressed as a barr-beck composite monad?

boolean algebras can be thought of as boolean rings... does this give one such way?
"and distributing over xor" or something like that...

but are there any other ways??
so let's consider for example the functor f : _ab gp_ -> _ab gp_ given by x |-> ext(z/2,x). we want to express this as a syntactic construction that makes sense in any abelian category; i guess that this is "cokernel of the endomorphism 2 on x".

for example how is f realized on the opposite of the category of abelian groups?? hmm, it seems to be "kernel of 2", which is just "homming from z/2" ....???....


is there an analog of the "extension" interpretation of f in the general case?

for another example let's consider g given by x |-> tor(x,z/2) (or whatever that notation should be...).

back to f... let's consider f as a module of the ringoid of fp abelian groups... or something like that... try to give a nice presentation of it... ??so take one generator g at the object z, and one relator "2g" ... ??is that correct??

Tuesday, September 14, 2010

i sent gunnarsen a response, mentioning that i've been writing about his messages here, and suggesting a chat on thursday. he said that thursday will probably work and he responded to a few of my comments.

he picked up on my vague phrase "the mathematically disenfranchised", emphasizing that elitism based on student parental wealth isn't the kind of elitism he prefers. that's part of what i was getting at, but not all.

sometimes i think that there's no such thing as anti-elitism; rather there are just competing visions of who counts as elite. it's pretty clear that i have such a vision even if the details of it aren't so clear. i don't know yet how my elitism might compare to gunnarsen's, or how much that might matter.

gunnarsen again emphasized the suzuki method of music teaching as some sort of model for what he has in mind. i don't know much about this method yet. gunnarsen's e-mail had an attachment that i didn't read yet where maybe he says more about it.

sometimes i hear teachers in subject area x complain that x-teaching has generally failed to keep up with advances in teaching methods pioneered in subject area y, while many conscientious y-teachers are in turn thoroughly dissatisfied with the general standards of y-teaching. in particular i sometimes hear this with y = music.
often music teaching succeeds in getting students really involved in the subject in a way that teachers in other subject areas may envy, but conscientious music teachers often despair at the dominance of rote imitation over creativity in many music classes.

for some reason i'm also reminded now of something i heard recently from some russian mathematician who said that dealing with "western"-trained mathematicians was like listening to musicians who learned how to play on instruments that were slightly but audibly out of tune.

i'm just rambling here; i have to meet gunnarsen in a few minutes and i don't have time to express my thoughts carefully right now.

i'm keeping an open mind so far about what gunnarsen and/or sinick may have in mind.

i have strong opinions about mathematics and about how it should be taught. for me to agree with practically anyone about such things enough to enter into some sort of joint venture with them might take a minor miracle.

i'll see how it goes.
let k be a field. consider the 3-object k-algebroid obtained as the full sub-algebroid of the algebroid of morphisms of k-modules, with the 3 objects "0->1", "1->1", and "1->0".

consider the modules of this algebroid...

consider the algebroid algebra of this algebroid...

Monday, September 13, 2010

so let's fix a particular 2d null subalgebra of the split octonions, and consider the 2d g2 schubert variety based at it, and try to make explicit a nice map down to the space of axises through the rolling ball... or something like that...
i'm getting somewhat confused about the relationship between "(generalized...) de sitter space" and "the conformal compactification of pseudo-euclidean (t,s)-space" ...
i have a pretty strong feeling that there _is_ a relationshship... hmmm... ??something about non-degenerate hyperboloid vs projective degenerate hyperboloid, or something?? ??something about the general idea of obtaining some sort of space-time (or something... metrically or conformally structured, perhaps...) as some sort of "hyperboloid" in a larger space-time ... ???or something like that??

??for some reason this is reminding me of that business about... ??how did it go??... co-/adjoint orbits stable under scalar multiplication, treated projectively, getting contact structure, vs unstable ones, treated projectively, keeping something like symplectic structure... ??????? or something???.....
kirwan, p 4-5 ... mention of "variation of hodge structure" that seems to threaten to almost make sense... ??something about cohomology groups that don't interestingly vary over a moduli space but carrying a "hodge filtration" that does interestingly vary ... ???or something like that??...

also... the discussion about ideas about intersection cohomology being motivated by ideas about l2 cohomology seems interesting...
so what about the intersection cohomology (or whatever that stuff is...) of the light cone in the conformal compactification of pseudo-euclidean (1,2)-space?

Sunday, September 12, 2010

i just want to compare the b2 and g2 cases a bit, to see whether they seem to follow the same naive pattern...

for g2, the 2d schubert variety on the "point" grassmanian seems to be non-singular, and the one on the "line" grassmanian seems to be singular, where the "point" grassmanian is the one that embeds into the projective space of the smallest irrep. does this hold for b2 as well? i think so, but i should check more carefully. actually i should check both cases more carefully.
so we should try to actually explicitly develop the idea of expressing schubert varieties as "de-stacky-izations" of certain stacks...

for example the singular 2d b2 schubert variety...

so let's consider the point <(1,0,1,0,0)> in the projective light cone of pseudo-euclidean (2,3)-space...

(a,b,c,d,e)

a^2+b^2 = c^2+d^2+e^2

a=c

b^2 = d^2+e^2

????

??so is it something like... ??the schubert variety here is "the light cone in the conformal compactification of pseudo-euclidean (1,2)-space"? perhaps this is somewhat retrospectively obvious (if correct), but did we quite notice it before?? i guess that we did notice that the ambient variety is that conformal compactification, so we should have noticed that the schubert variety is the light cone... except i guess that we did notice that it's _a_ "light cone", but that we weren't completely sure whether there's a _the_ light cone here... ??or something??... i guess that there is though... ??...

we did think about the big bruhat cell... and about segal's cosmology a bit... hmm, i also have a vague memeory, i think, of thinking about how two "complementary" bruhat classifications overlap here (i found some notes about this stuff in my windows journal notebook "huerta-dolan-discussion-1", around page 36-40 or so...) ... how the big bruhat cell is the complement of one of the light cones, while the other one lies partly in it... i guess that we should try harder to visualize this though... in the (2,2) and (2,3) cases... and also the (2,4) case to the extent that it makes sense, since i think that that's where segal's cosmology lives...

hmm, in the (2,2) case, the picture of the overlapping bruhat classifications that i'm getting suspiciously reminds me of the pictures of peirce's quincunxial conformal map projection and "world map wallpaper" and gauss's lemniscate... which is a bit peculiar because that stuff is about signature (0,2) rather than (1,1)... though now i'm also thinking about penrose diagrams... ???.... (i mean the "conformal signatures" (0,2) and (1,1) here, associated with the pseudo-euclidean signatures (1,3) and (2,2) respectively...)

??so suppose that you're living in a 2d spherical spatial universe... consider the expanding light ripple of an event... coming into focus at the antipodal point of the spatial universe... ???....

Saturday, September 11, 2010

consider the 2d schubert variety on the g2 "line" grassmanian, and consider the hopefully obvious map to the space of geodesics (??) on the rolling ball... ??or something...
perhaps a better (if somewhat more obscure) name for the "hoop" in "hoop geometry" is "gimbal ring". that's what they seemed to call it when i tried looking up the name of that part of a spinning-earth-globe or gyroscope.

(the oed vacillates about whether the g should be hard or soft.)

Friday, September 10, 2010

i'm way behind schedule in replying to laurens gunnarsen - i hope that he hasn't already lost interest - so i should try here to organize my thoughts about how to respond.

i guess that the ultra-short summary is that although i probably have all sorts of reservations about whatever it is that gunnarsen and/or sinick are proposing, i'm desperate enough in many ways that i'm eager to explore the possibilities further.

my life goal for almost as long as i've had one has been to be a mathematics teacher of a sort, and in this goal i've been a comical and/or tragical failure so far. my ideas about what mathematics is and how to teach it seem different enough from more commonly accepted such ideas that i've almost always found it intolerable to make the practical concessions necessary to obtain a teaching job; to pretend to teach under such conditions would so lack value for me that i find it preferable to continue living in unemployment and poverty.

it hasn't been so one-sidedly discouraging as to cause me to give up hope in the past, though, and even now i still haven't completely given up hope. if i'd had just a few more lucky breaks or a few less personal failings, if i'd worked harder at it and concentrated more exclusively on it; most especially, if i'd ever found someone remotely sympathetic to my ideas and in a position to do something about it, then i might have found a job that would actually be worthwhile to me. conceivably i could still do so.

what's usually considered mathematics teaching strikes me as more like the opposite of teaching, and the opposite of mathematics.

the audience that i seek for my mathematical ideas is a mathematically disenfranchised class of students... one of my major points of uncertainty as to whether what i want to do can mesh with what gunnarsen and sinick want to do is to what extent the class of students that they aim to teach overlaps the class that i want to teach...


gunnarsen's "war stories".... chinese room... baez... who do i want to reach... ??.....
relationship between mathematics and "real life" and "real world problems" ...
...need for practice...

classroom teaching ... schools... feedback/interaction ... moore method.... socratic ...

Wednesday, September 8, 2010

for huerta...

ridenour... ideals in parabolic subalgebras and so forth

pgl(3) and g2... animation...

Monday, September 6, 2010

so as a potential guide to the g2 case (as well as probably lots of other reasons), let's try thinking a bit about the long root subalgebra so(2,2) of b2 = so(2,3).

so let's take a highest weight vector of the adjoint representation of so(2,3) and consider it's orbit under so(2,3), but then let's consider how that breaks up into orbits under the long root subalgebra so(2,2).

(by the way does the inclusion of g2 into so(3,4) "induce" an inclusion of the long root subalgebra sl(3) of g2 into the one so(3,3) of so(3,4)? or something? doesn't seem particularly likely, even though there _is_ a nice inclusion of sl(3) into sl(4) = so(3,3)... ??...)

so suppose that we have an orientation-preserving linear isometry x from e^2 to e^2, and a 1d subspace s of the domain e^2. then this gives us also a unique orientation-reversing linear isometry y that agrees with x on s.

hmm, so consider the hoop configurations that confine the hoop to the favorite equator, and the "lines" whose "axis" lies on that equator. this seems to correspond nicely to the so(2,2) geometry inside the so(2,3) one.

are there any elements of so(2,3) that don't preserve the equivalence relation on flags given by "share the same special antipodal pair of points on the material hoop", and if so then can we nicely visualize any of them?? are they "boost-like" or something?? hmm, is this equivalence relation maybe "generalized simultaneity" or something?? i think that it probably is; the material hoop plays the role of "time". so does it make sense to "boost by an isometry from time to space" or something??

hmm, so let's try a naively analogous approach with g2. let's try taking the "favorite equator" of the unit-length quaternions to be the imaginary ones. so is it true that the "second quaternion components" of the 6 projective octonions in the favorite g2 apartment are imaginary?? or something?? i think that it is...

??this seems to be fitting with the idea that the a2 total flag variety "has no interior coordinates". or something...

??are we getting the wrong number of constraints on the configurations here??

Sunday, September 5, 2010

notes for discussion with huerta this evening...

??does the diffeomorphism preserve the fibers? or something...

??is the diffeomorphism as sugeested so far exactly correct in between the special points??

issue of whether "initial point" should explicitly appear in diffeomorphism formula...


??borel ideals and invariant distributions on flag variety...

??canonically identify pair (1d null subalg x of split octonions, point y of projective tangent space of x) with pair (x,f(x,y)) of 1d null subalgebras?? or something?? ....
schubert calculus.... residual ct partial ... ideal ct subring.... os... asf os...

??so what _about_ cohomology rings of schubert varieties, and so forth?? i really don't get this idea about "getting poincare duality to work in singular situations" ... ??...

Saturday, September 4, 2010

so let's consider for example the root polytope of a3, and try to interpret its facial structure as embodying the incidence geometry in the coadjoint partial flag picture...

so the root polytope is a "cuboctahedron"...

hmm, but wait a minute... do we really know how to interpret the facial structure in this case??

coadjoint partial flags appear as vertexes...

??edges of two colors??

i'm confused...

??a point corresponds to a 2d space of coadjoint partial flags... namely those where the point is the point, and the hyperplane is any hyperplane through it... of which there's a 2d space... dual to 2d space of points on a hyperplane...

a line corresponds to a 3d space of coadjoint partial flags??? i'm not getting that... seems more like 2d ... ????....

Thursday, September 2, 2010

so is the apartment variety affine or just quasi-projective?? either way, what are the coherent sheaves??
so what about the relationship between the "schubert-galois correspondence" and the partial order on the negative roots coming from the positive roots? ... ?? ...

??well, so what about invariant distributions on the apartment (or the framed apartment) variety, and the integrable such, and so forth? hmmm...

there's a continuous family of apartment-apartment orientations (for example consider gl(2) and "cross-ratio"...), but only a finite set of invariant distributions here... so it may be interesting to see what happens to the zariski tangent space process here...

hmm, so what seems to be going on here is this: ...??...

hmm, i seem to have left that hanging for a while... i give somewhat of an update here.
we should also try looking at the invariant distributions on the c-series flag varieties, especially to compare to the b-series and try to understand possible dependence on dynkin diagram as opposed to mere coxeter diagram...
so i think that the partial order on the b_n roots that gives the invariant distributions on the flag variety as its down-sets is as follows:


(1,1,0,0)
=== (1,0,1,0)
(0,1,1,0) == (1,0,0,1)
=== (0,1,0,1) == (1,0,0,0)
(0,0,1,1) == (0,1,0,0) == (1,0,0,-1)
=== (0,0,1,0) == (0,1,0,-1) == (1,0,-1,0)
(0,0,0,1) == (0,0,1,-1) == (0,1,-1,0) == (1,-1,0,0)

this is for the case b_4, but the pattern continues.

(the shape of the poset is suggestive, but i'm not sure what it means yet. i'm trying to connect it with vague memories of some stuff that arnold mentioned about bratteli diagrams and young diagrams and flag varieties and bruhat classes or stuff like that... but it definitely reminds me of certain bratteli diagrams and the representation theory of su(2) and so forth...)

let's try counting the down-sets as a function of n...

2,6,20,...

is this "2n choose n"?

is it obvious why it should be that?

let's try identifying the integrable invariant distributions here...
especially the maximal integrable ones...

2 ???

4,4 ???

8,11,6 ???

16,26,22,8 ???

well, i fed "4,11,26" into sloane's encyclopedia and it led me to the "triangle of eulerian numbers":

1 1 1 1 1 1 1 1 1
1 4 11 26 57 120 247 502
1 11 66 302 1191 4293 14608
1 26 302 2416 15619 88234
1 57 1191 15619 156190
1 120 4293 88234
1 247 14608
1 502
1

with the following comment:

T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004

and perhaps some other potentially relevant and interesting comments.

on the other hand i don't quite completely get the relationship between my triangle above and theirs yet...

maybe try the next row of mine?

32,57,61,37,10 ???

1 + 1 + 2 + 4 + 8 + 16 = 32

1 + 1 + 2 + 4 + 8 + 15 + 26 = 57

1 + 1 + 2 + 4 + 7 + 11 + 15 + 20 = 61

1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 37

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10


well, so anyway, what's going on here with the zariski tangent spaces of the b2 schubert varieties??

4=[0,1,2,3] 4=[0,1,2,3]
8=[0,1,2,3,3,4,5,6] 11=[0,1,2,2,3,3,4,4,5,6,7] 6=[0,1,2,3,4,5]
16 26 22 8
32 57 61 37 10

8 vertexes of cube.... "distances" from a flag...

...i'm getting a match-up in this case... the 8 distances here are [0,1,2,3,3,4,5,6]. matching the 8 dimensions of the invariant distributions extending the maximal integrable one...

??but that's not convincing me yet that the correspondence actually goes like that... because of the business about the singular 2d b2 schubert variety...

but now let's consider the 12 edges of the cube... the 12 distances that i'm getting at the moment are [0,1,2,2,3,3,4,4,5,5,6,7]... so how does that compare to the 11 dimensions?? hmm, just as i guessed: naively it looks like the 2 5's merged into just one... ??...

Wednesday, September 1, 2010

is there a nice way to express a schubert variety as an orbit variety, with the basepoint as what would be a very stacky point ... ??or something?? ... hmmm... ??something abut span vs relation here??... and so forth... ???