?? structure preserved by unsplit maximal torus .... ???? .....
Sunday, April 29, 2012
?? well, one source of confusion about possibility of fitting gl(2,f_3) into mckay correspondence is ... ?? "binary tetrahedral group" (=?= "e6" ???) only covers _rotations_ of tetrahedron .... ??? .....
?? on the other hand, what about e7 ?? ... ??? still doesn't seem to have right number of irreps .... ?? what about number of even irreps ??? .... ?? still seems wrong ?? .... ?? ....
?? hmm, maybe extra dot is key here ??? ... ?? neighbors of extra dot as "spinor" rep ??? ....
?? e7 with extra dot .... ??? ....
?? ok, this seems like it might make sense .... ?? also mesh with child's drawing stuff ??? ..... ???? .....
?? on the other hand, what about e7 ?? ... ??? still doesn't seem to have right number of irreps .... ?? what about number of even irreps ??? .... ?? still seems wrong ?? .... ?? ....
?? hmm, maybe extra dot is key here ??? ... ?? neighbors of extra dot as "spinor" rep ??? ....
?? e7 with extra dot .... ??? ....
?? ok, this seems like it might make sense .... ?? also mesh with child's drawing stuff ??? ..... ???? .....
?? gl(2,f_2) ... ??? ....
?? split maximal torus ??? .....
?? hmm, unsplit maximal torus here as maybe anomalous ??? .... ?? because quadratic extension of f_2 as anomalous ..... ???? .....
?? but is there something funny about split maximal torus here as well ?? ... ??? ....
?? cuspidal vs non-cuspidal here .... ????? .....
?? trivial rep and 2d irrep as non-cuspidal here ... other 1d rep as cuspidal ...
?? making it seem like split maximal torus here should be z/2 or something .... ??? ....
?? bit todd pointed out about .... ?? bn-pair "n" here (?? ... ??? 2 vs 3 ??? ....) as not precisely normalizer of ..... ???? ..... ????? .....
?? actual maximal abelian subgps of 3! ... ?? z/2 and z/3 ..... ????? ..... ?? z/3 as unsplit maximal torus, maybe ??? .... ?? relationship to other 1d rep ..... ?????? ........
?? z/2 as "point" stabilizer here ???? ... ?? abelian parabolic .... ???? .....
?? maybe lots of anaomalies here ... unclear how related to each other ... ?? try looking at gl(2,f_3) instead ... ??? ....
?? some anomalies which might become less anomalous on "passing to algebraic closure" / "treating as algebraic group" .... ??? .....
?? split maximal torus ??? .....
?? hmm, unsplit maximal torus here as maybe anomalous ??? .... ?? because quadratic extension of f_2 as anomalous ..... ???? .....
?? but is there something funny about split maximal torus here as well ?? ... ??? ....
?? cuspidal vs non-cuspidal here .... ????? .....
?? trivial rep and 2d irrep as non-cuspidal here ... other 1d rep as cuspidal ...
?? making it seem like split maximal torus here should be z/2 or something .... ??? ....
?? bit todd pointed out about .... ?? bn-pair "n" here (?? ... ??? 2 vs 3 ??? ....) as not precisely normalizer of ..... ???? ..... ????? .....
?? actual maximal abelian subgps of 3! ... ?? z/2 and z/3 ..... ????? ..... ?? z/3 as unsplit maximal torus, maybe ??? .... ?? relationship to other 1d rep ..... ?????? ........
?? z/2 as "point" stabilizer here ???? ... ?? abelian parabolic .... ???? .....
?? maybe lots of anaomalies here ... unclear how related to each other ... ?? try looking at gl(2,f_3) instead ... ??? ....
?? some anomalies which might become less anomalous on "passing to algebraic closure" / "treating as algebraic group" .... ??? .....
?? "every rep is induced, just not all the way" ... ???? ....
?? "central character" vs "parameterization by characters" .... ???? ....
?? irrep parameterized by character of split maximal torus as induced ... ?? but is the inducement from that torus, or from something related but somewhat larger ?? .... ????? .......
?? unsplit maximal torus of real lie group ....... ????? .... ?? "cayley" (?? ...) representation of (c,*) on underlying real vsp of c ..... ????? .....
?? cayley representation of (f_q^n,*) on underlying f_q-vsp of f_[q^n] .... ????? .....
?? maximal torus as not central .... center of simple group as trivial ... ?? must be forgetting some key idea here ..... ????? .... ?? ok, one key idea is gp center vs ring center .... ???? ... but bump _is_ talking only about gp centers here, right ?? .... ???? .....
?? hmm, they make it sound like they're talking about gp centers, but it seems like they're secretly really talking about ring centers .... ???? .....
?? center of eneveloping algebra / gp algebra .... ??? vs .... ??? of quotient thereof ... ?? associated with rep ??? .... ???? .... ?? or of just plain assoc alg .... ??? .... ???? ..... ??? ....
?? "central character" vs "parameterization by characters" .... ???? ....
?? irrep parameterized by character of split maximal torus as induced ... ?? but is the inducement from that torus, or from something related but somewhat larger ?? .... ????? .......
?? unsplit maximal torus of real lie group ....... ????? .... ?? "cayley" (?? ...) representation of (c,*) on underlying real vsp of c ..... ????? .....
?? cayley representation of (f_q^n,*) on underlying f_q-vsp of f_[q^n] .... ????? .....
?? maximal torus as not central .... center of simple group as trivial ... ?? must be forgetting some key idea here ..... ????? .... ?? ok, one key idea is gp center vs ring center .... ???? ... but bump _is_ talking only about gp centers here, right ?? .... ???? .....
?? hmm, they make it sound like they're talking about gp centers, but it seems like they're secretly really talking about ring centers .... ???? .....
?? center of eneveloping algebra / gp algebra .... ??? vs .... ??? of quotient thereof ... ?? associated with rep ??? .... ???? .... ?? or of just plain assoc alg .... ??? .... ???? ..... ??? ....
Saturday, April 28, 2012
[for david yetter]
hi...
thanks for trying to help me get accepted at ksu, but it looks like it isn't going to work out ...
i hope that it's clear why it makes no sense for me to accept the compromise proposal that i've been offered ... for me to teach under those conditions is a guarantee of failure, while at the same i know that if given a fair chance under more reasonable conditions i can demonstrate that i can be a highly competent and successful teacher ....
hi...
thanks for trying to help me get accepted at ksu, but it looks like it isn't going to work out ...
i hope that it's clear why it makes no sense for me to accept the compromise proposal that i've been offered ... for me to teach under those conditions is a guarantee of failure, while at the same i know that if given a fair chance under more reasonable conditions i can demonstrate that i can be a highly competent and successful teacher ....
?? "multiplicative group of k[x]/x^2=5" (for example ...) as algebraic group .... ?? to what extent and in what context we've thought about stuff like that recently ... ?? .... ?? part of "alg comm ring" bit ?? .... ?? ...... ??? ..... ?? in part hoped to relate zeta function to l-function ... ??? ..... ??? .....
?? asking this now because of reading bump .... ??? .... "unsplit torus" ... algebraic group over f_q that "splits" to give torus over f_[q^n] ... ?? more abstractly, alg gp over z ..... ????? ...... ???? .....
?? confusion here about ... ??? naive size count ..... ??? .... (q-1)^n vs q^n-1 .... ???? ......
?? gaussian integers .... ???? ..... reciprocity law .... splitting pattern .... ???? ..... ???? .... ???? .... counting solutions ....
?? counting homomorphisms from gaussian integers, vs counting dimension of space of homomorphisms from .... ???? "formal gaussian integers" .... ???? ..... ????? ...... ?????? ....... ????? ...... zeta vs l .... ????? .... ???? ....... ???? .....
?? studying reps of gl(2,f_q) .... "unsplit torus" intrudes .... ??? .... gl(1,f_[q^2]) -> gl(2,f_q) .... ???? ....
??? studying reps of gl(2,z) ... ??? gl(1,z[i]) -> gl(2,z) ...... ???? ..... ???? .....
?? gl(1,z[i]) -> gl(2,z) as algebraic stabilizer subgroup of what structure on what stuff ??? ..... ???? ... hmmm .... ??? stuff here as "2d object" ???? ..... ???? .... stabilized structure as .... ????? ......
vs gl(1,z^2) -> gl(2,z) .... ????? ...... ???? .... ?? "splitting" ... ??? bit about "complex splitting" / "complex idempotent" .... ??? .... ?? .... ??? level slip ??? .... ??? ..... ???? ...... "splitting" here as "generic flag-pair" ??? .... ???? ..... ????? ..... ?? "twisted splitting" ??? ..... ???? .....
???? "lattice" ???? ...... ???? .... "extra dot" .... ???? brown vs pressley and segal .... ???? ......
?? asking this now because of reading bump .... ??? .... "unsplit torus" ... algebraic group over f_q that "splits" to give torus over f_[q^n] ... ?? more abstractly, alg gp over z ..... ????? ...... ???? .....
?? confusion here about ... ??? naive size count ..... ??? .... (q-1)^n vs q^n-1 .... ???? ......
?? gaussian integers .... ???? ..... reciprocity law .... splitting pattern .... ???? ..... ???? .... ???? .... counting solutions ....
?? counting homomorphisms from gaussian integers, vs counting dimension of space of homomorphisms from .... ???? "formal gaussian integers" .... ???? ..... ????? ...... ?????? ....... ????? ...... zeta vs l .... ????? .... ???? ....... ???? .....
?? studying reps of gl(2,f_q) .... "unsplit torus" intrudes .... ??? .... gl(1,f_[q^2]) -> gl(2,f_q) .... ???? ....
??? studying reps of gl(2,z) ... ??? gl(1,z[i]) -> gl(2,z) ...... ???? ..... ???? .....
?? gl(1,z[i]) -> gl(2,z) as algebraic stabilizer subgroup of what structure on what stuff ??? ..... ???? ... hmmm .... ??? stuff here as "2d object" ???? ..... ???? .... stabilized structure as .... ????? ......
vs gl(1,z^2) -> gl(2,z) .... ????? ...... ???? .... ?? "splitting" ... ??? bit about "complex splitting" / "complex idempotent" .... ??? .... ?? .... ??? level slip ??? .... ??? ..... ???? ...... "splitting" here as "generic flag-pair" ??? .... ???? ..... ????? ..... ?? "twisted splitting" ??? ..... ???? .....
???? "lattice" ???? ...... ???? .... "extra dot" .... ???? brown vs pressley and segal .... ???? ......
?? "maximal torus" .... ?? "toric variety" .... ??? ..... galois aspect of toric variety .... ???? ..... gl(1) as torus .... gl(1) and "abelian langlands" .... ???? .... ?? torus as variety vs as alg gp ... ??? ...
?? kaleidoscope as fan .... ???? ..... maximal torus ..... ????? .......
??bike path to(wards) san bernardino .... ????? ......
?? kaleidoscope as fan .... ???? ..... maximal torus ..... ????? .......
??bike path to(wards) san bernardino .... ????? ......
a?? so having gotten some of the numerology a bit straighter let's try thinking a bit more about gl(2) .... ??? .....
q=2
6 = (1^2)*1 + (1^2 + 2^2)
q=3
48 = (2^2)*3 + (1^2 + 3^2)*2 + (4^2)*1
q=4
180 = (3^2)*6 + (1^2 + 4^2)*3 + (5^2)*3
?? hmmm ... ?? seemingly obvious naive numerological guess about which part of middle term lives over pgl(2) .... ?? seems to not work ??? .....
?? pgl(2,f_3) .... ???? .... ??? acting faithfully and transitively on 4 points of projective line .... ???? just 4! ?? ....
4 ?? 1
31 ?? 3
22 ?? 2 ?????
211 ?? 3
1111 ?? 1
gl(2,f_2) 3 irreps ...
gl(2,f_3) 8 irreps ...
gl(2,f_4) 15 irreps ....
... ???? .....
?? in principle it's just the cuspidals here (?? gl(2,f_q) for generic q ...) that we don't "understand" so far ?? ....
??? tensoring with 1d reps here (...) ... ??? ... ?? whether this puts some order into the variety of cuspidals here ... ?? ....
?? hmmm, trying to understand these 1d reps better ... ?? hmm, by 2 approaches ....
1 abelianization ...
2 ?? "vector bundle over projective line assigning to 1d subspace x f(x)#g(v/x) where f and g are functors from _1d f_q-vsp_ to _1d c-vsp_" .... ??? .... ?? then understanding "q-braiding" here and modding out by it .... ??? .... "hecke operator / algebra ..." .... ????? .....
?? abelianization and determinants ???? ....
?? abelianization of gl(n,field) vs of pgl(n,field) ?? .... ??? .....
?? abelianization of gl(n,field) as gl(1,field) ??? ..... ????? ....
?? "algebraic k-theory" ... ???? ...... ?????? ..... ???? ....
?? is "projectivization" always finer than "abelianization" ??? .... ?? meaning ... ?? ... when you abelianize gl(n,field), does it always happen that the constants get killed off ?? ..... ???? .... ?? meaning, maybe (?? ...) that the determinant of a constant is ...... ???? ......
gl(2,f_3) .... ????
?? here it really is true that the constants are determinant 1 ??? ..... ???? ....
?? but that seems like an extremely special case .... ?? ....
?? did i ask the question anywhere close to correctly ?? ... ???? ...
?? maybe we had an original intuition here (?? ...) that was right-track ... ?? that _of course_ determinant wouldn't survive too well to projective level .... ??? .... ??? .....
?? for some reason i feel tempted to say "determinants of constants form an obstruction to determinant surviving to projective level" .... ???? does that make _any_ sense ??? .... ???? ....
?? take a look at 1d irreps of gl(3,f_q) as somewhat classified in recent post here ... ??? ... ??? see if fits with "determinant" idea ??? .... ???? ....
?? yes, does seem to fit fairly straightforwardly ?? ... only 1d irreps that show up seem to be "applying trivial 3-box young diagram to gl(1)-cuspidals" .... ???? .... ??? hmm .... ??? idea that .... ?? abelianization of gl(n) is gl(1) via determinant (?? which idea maybe i'm beginning to remember some subtleties to ... ?? maybe only in beyond-field case ?? ... anyway never mind that for a moment ... ?? ...) as embodied here as "process of applying trivial n-box young diagram to gl(1) irrep" ... ??? .... ???? .... ????? ... ?? suggestive somehow ?? ... ???? .....
?? maybe enough progress here to satisfy original goal of being ready to take a look at what bump says about this stuff ... ?? though not yet running into any obvious way in which kinds of things bump seemed to be about to say ("induced rep from maximal torus rep" ?? ... ??? ...) show up ..... ???? .....
?? size numerology of maximal toruses here ??? ...
?? field = f_q .... ??? but ... ??? trying to see where field = c (or r ....) fit in here .... ??? .....
?? "maximal torus" in gl vs pgl vs sl case .... ???? .....
?? diagonal torus of gl(n) ..... ???? .....
q=2
diagonal torus 1 ... gl(2) 6
?? so 6d induced rep ?? ... ?? but maybe breaks into smaller irreps ??? ...
?? ok, so when bump talks about "induced reps", maybe they're really talking about "corresponding" irreps .... ?? how much does that help to straighten stuff out here ??? ..... ????? .....
?? so one stupid guess that doesn't quite make sense yet is something like ... ?? ... ?? bump is setting aside the cuspidal reps (my first term above ... ?? ....) for the moment (?? ...), and associating my second term with "unsplit torus", and my third term with "split torus" .... ...... ?? really not seeing it yet .... ????? .....
q=3
diagonal torus 4 .... gl(2) 48
??? 12 ..... ???? .... 4*3 ???
q=4
180 / 9 = 20 = 5*4
15*12 = 5*3*4*3 ..... ????? .....
?? "generic flag pair" .... ????? ......
?? vector bundle over [flag variety]^2 ??? .....
??? alleged other sort of "maximal torus" ..... ????? .....
?? splitting of unsplit torus ..... ????? ......
?? trying to associate "vector bundle over [flag variety]^2" with my third term as ... ?? seeming rather iffy and relying on special coincidences ... gl(2) vs gl(n) for more general n .... ???? .... ???? ....
?? mckay correspondence for binary tetrahedral group ..... ???? .... ?? is that e6 ?? .... bipartite graph .... ???? ...
?? having some trouble trying to get mckay correspondence numerology to work here .... ????? ... (actually having trouble at the moment getting it to work anywhere, so don't hold it against here too much .... ?? ...)
?? elt of gl(2,f_q) that .... ?? arises from .... ??? putting structure on [f_q]^2 making it into an f_[q^2] ... ??? and then .... ??? .... ?? picking generator for cyclic group gl(1,f_[q^2]) ... ?? and considering group it generates ... ??? hmmm, i guess that that's pretty much like saying ... ?? consider gl(1,f_[q^2]) -> gl(2,f_q) arising in hopefully obvious way .... ???? ....
?? so one stupid guess is that that's what an "unsplit (?? maximal ... ?? ....) torus" is ... ??
?? "ramified torus" ???? .... ???? gl(1,f_q[t]/t^2) -> gl(2,f_q) .... ?????
?? gl(1,z/p^2) -> ????? ..... ?? analog for q in place of p ??? .... ????? .... ??? ...
?? split torus and (q-1)^2, vs unsplit torus and q^2-1 ??? ..... ?? anything like that show up in our classification yet ??? .... ???? ......
?? hmm, almost getting a numerological glimmer of ... ?? cuspidal : unsplit :: non-cuspidal : split .... ???? ... try testing it out a bit further ... ?? or maybe generically .... ??? .....
(q^2-1)*(q^2-q) = (q-1)^2*(q*(q-1)/2) + (1^2 + q^2)*(q-1) + (q+1)^2*((q-1)*(q-2)/2)
?? the glimmer argument is something like ... ?? ... dividing left-hand side by unsplit torus size gives q^2-q, and factors of that get squared in first term ... ?? more specifically, the factor q-1 does .... ??? whereas dividing left-hand side by split torus size gives q^2+q, and factors of that get squared in last two terms .... ??? .... ?? so _is_ this a good hint of what bump's talking about ??? .... ???? .....
?? well, bump does say this :
The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method.
... so yeah, it seems like we're on the right track here .... though in retrospect maybe it should have been somewhat obvious ... ?? ...
?? counting characters here .... ??? also trying to develop correspondences with real and complex cases .... ???? ....
?? not quite getting the non-squared numbers to correspond nicely to numbers of characters yet ..... ??? ..... hmmm, but ... ?? might be possibilities .... ??? "kaleidoscope folding" ?? .... ??? ....... ??? ..... ?? or sub rather than quotient ?? .... ?? "... chamber ..." ... ???? .... ??? hmmm, try adding together the raw counts for the second and third terms .... ??? .... ?? does seem to have some relationship to "(q-1)^2", while raw count for first term seems to relate to "q^2-1" .... ???? ..... ???? .....
?? wait a minute ... ?? might be right track, but some confusion here ... ?? between "parameterization by" as given by inducement (? which sort of seemed to work numerologically ... ?? ...), vs bit about those parameterized by nonsplit toruses as "_not_ induced" .... ???? .....
q=2
6 = (1^2)*1 + (1^2 + 2^2)
q=3
48 = (2^2)*3 + (1^2 + 3^2)*2 + (4^2)*1
q=4
180 = (3^2)*6 + (1^2 + 4^2)*3 + (5^2)*3
?? hmmm ... ?? seemingly obvious naive numerological guess about which part of middle term lives over pgl(2) .... ?? seems to not work ??? .....
?? pgl(2,f_3) .... ???? .... ??? acting faithfully and transitively on 4 points of projective line .... ???? just 4! ?? ....
4 ?? 1
31 ?? 3
22 ?? 2 ?????
211 ?? 3
1111 ?? 1
gl(2,f_2) 3 irreps ...
gl(2,f_3) 8 irreps ...
gl(2,f_4) 15 irreps ....
... ???? .....
?? in principle it's just the cuspidals here (?? gl(2,f_q) for generic q ...) that we don't "understand" so far ?? ....
??? tensoring with 1d reps here (...) ... ??? ... ?? whether this puts some order into the variety of cuspidals here ... ?? ....
?? hmmm, trying to understand these 1d reps better ... ?? hmm, by 2 approaches ....
1 abelianization ...
2 ?? "vector bundle over projective line assigning to 1d subspace x f(x)#g(v/x) where f and g are functors from _1d f_q-vsp_ to _1d c-vsp_" .... ??? .... ?? then understanding "q-braiding" here and modding out by it .... ??? .... "hecke operator / algebra ..." .... ????? .....
?? abelianization and determinants ???? ....
?? abelianization of gl(n,field) vs of pgl(n,field) ?? .... ??? .....
?? abelianization of gl(n,field) as gl(1,field) ??? ..... ????? ....
?? "algebraic k-theory" ... ???? ...... ?????? ..... ???? ....
?? is "projectivization" always finer than "abelianization" ??? .... ?? meaning ... ?? ... when you abelianize gl(n,field), does it always happen that the constants get killed off ?? ..... ???? .... ?? meaning, maybe (?? ...) that the determinant of a constant is ...... ???? ......
gl(2,f_3) .... ????
?? here it really is true that the constants are determinant 1 ??? ..... ???? ....
?? but that seems like an extremely special case .... ?? ....
?? did i ask the question anywhere close to correctly ?? ... ???? ...
?? maybe we had an original intuition here (?? ...) that was right-track ... ?? that _of course_ determinant wouldn't survive too well to projective level .... ??? .... ??? .....
?? for some reason i feel tempted to say "determinants of constants form an obstruction to determinant surviving to projective level" .... ???? does that make _any_ sense ??? .... ???? ....
?? take a look at 1d irreps of gl(3,f_q) as somewhat classified in recent post here ... ??? ... ??? see if fits with "determinant" idea ??? .... ???? ....
?? yes, does seem to fit fairly straightforwardly ?? ... only 1d irreps that show up seem to be "applying trivial 3-box young diagram to gl(1)-cuspidals" .... ???? .... ??? hmm .... ??? idea that .... ?? abelianization of gl(n) is gl(1) via determinant (?? which idea maybe i'm beginning to remember some subtleties to ... ?? maybe only in beyond-field case ?? ... anyway never mind that for a moment ... ?? ...) as embodied here as "process of applying trivial n-box young diagram to gl(1) irrep" ... ??? .... ???? .... ????? ... ?? suggestive somehow ?? ... ???? .....
?? maybe enough progress here to satisfy original goal of being ready to take a look at what bump says about this stuff ... ?? though not yet running into any obvious way in which kinds of things bump seemed to be about to say ("induced rep from maximal torus rep" ?? ... ??? ...) show up ..... ???? .....
?? size numerology of maximal toruses here ??? ...
?? field = f_q .... ??? but ... ??? trying to see where field = c (or r ....) fit in here .... ??? .....
?? "maximal torus" in gl vs pgl vs sl case .... ???? .....
?? diagonal torus of gl(n) ..... ???? .....
q=2
diagonal torus 1 ... gl(2) 6
?? so 6d induced rep ?? ... ?? but maybe breaks into smaller irreps ??? ...
?? ok, so when bump talks about "induced reps", maybe they're really talking about "corresponding" irreps .... ?? how much does that help to straighten stuff out here ??? ..... ????? .....
?? so one stupid guess that doesn't quite make sense yet is something like ... ?? ... ?? bump is setting aside the cuspidal reps (my first term above ... ?? ....) for the moment (?? ...), and associating my second term with "unsplit torus", and my third term with "split torus" .... ...... ?? really not seeing it yet .... ????? .....
q=3
diagonal torus 4 .... gl(2) 48
??? 12 ..... ???? .... 4*3 ???
q=4
180 / 9 = 20 = 5*4
15*12 = 5*3*4*3 ..... ????? .....
?? "generic flag pair" .... ????? ......
?? vector bundle over [flag variety]^2 ??? .....
??? alleged other sort of "maximal torus" ..... ????? .....
?? splitting of unsplit torus ..... ????? ......
?? trying to associate "vector bundle over [flag variety]^2" with my third term as ... ?? seeming rather iffy and relying on special coincidences ... gl(2) vs gl(n) for more general n .... ???? .... ???? ....
?? mckay correspondence for binary tetrahedral group ..... ???? .... ?? is that e6 ?? .... bipartite graph .... ???? ...
?? having some trouble trying to get mckay correspondence numerology to work here .... ????? ... (actually having trouble at the moment getting it to work anywhere, so don't hold it against here too much .... ?? ...)
?? elt of gl(2,f_q) that .... ?? arises from .... ??? putting structure on [f_q]^2 making it into an f_[q^2] ... ??? and then .... ??? .... ?? picking generator for cyclic group gl(1,f_[q^2]) ... ?? and considering group it generates ... ??? hmmm, i guess that that's pretty much like saying ... ?? consider gl(1,f_[q^2]) -> gl(2,f_q) arising in hopefully obvious way .... ???? ....
?? so one stupid guess is that that's what an "unsplit (?? maximal ... ?? ....) torus" is ... ??
?? "ramified torus" ???? .... ???? gl(1,f_q[t]/t^2) -> gl(2,f_q) .... ?????
?? gl(1,z/p^2) -> ????? ..... ?? analog for q in place of p ??? .... ????? .... ??? ...
?? split torus and (q-1)^2, vs unsplit torus and q^2-1 ??? ..... ?? anything like that show up in our classification yet ??? .... ???? ......
?? hmm, almost getting a numerological glimmer of ... ?? cuspidal : unsplit :: non-cuspidal : split .... ???? ... try testing it out a bit further ... ?? or maybe generically .... ??? .....
(q^2-1)*(q^2-q) = (q-1)^2*(q*(q-1)/2) + (1^2 + q^2)*(q-1) + (q+1)^2*((q-1)*(q-2)/2)
?? the glimmer argument is something like ... ?? ... dividing left-hand side by unsplit torus size gives q^2-q, and factors of that get squared in first term ... ?? more specifically, the factor q-1 does .... ??? whereas dividing left-hand side by split torus size gives q^2+q, and factors of that get squared in last two terms .... ??? .... ?? so _is_ this a good hint of what bump's talking about ??? .... ???? .....
?? well, bump does say this :
The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method.
... so yeah, it seems like we're on the right track here .... though in retrospect maybe it should have been somewhat obvious ... ?? ...
?? counting characters here .... ??? also trying to develop correspondences with real and complex cases .... ???? ....
?? not quite getting the non-squared numbers to correspond nicely to numbers of characters yet ..... ??? ..... hmmm, but ... ?? might be possibilities .... ??? "kaleidoscope folding" ?? .... ??? ....... ??? ..... ?? or sub rather than quotient ?? .... ?? "... chamber ..." ... ???? .... ??? hmmm, try adding together the raw counts for the second and third terms .... ??? .... ?? does seem to have some relationship to "(q-1)^2", while raw count for first term seems to relate to "q^2-1" .... ???? ..... ???? .....
?? wait a minute ... ?? might be right track, but some confusion here ... ?? between "parameterization by" as given by inducement (? which sort of seemed to work numerologically ... ?? ...), vs bit about those parameterized by nonsplit toruses as "_not_ induced" .... ???? .....
?? trying to figure out new plan now that ksu plan seems to have fallen through ...
?? stony brook ?? ...
?? baez / ucr ?? ... seems like real bad idea ....
?? huerta ??? ....
?? robin cockett ?? ....
?? categories mailing list .... ??? other vaguely similar ideas about possible places to "advertise" ... ??? .....
?? stony brook ?? ...
?? baez / ucr ?? ... seems like real bad idea ....
?? huerta ??? ....
?? robin cockett ?? ....
?? categories mailing list .... ??? other vaguely similar ideas about possible places to "advertise" ... ??? .....
Friday, April 27, 2012
6 = 1 + 4 + 1
48 = 1 + 9 + 1 + 9 + ...
28 ... ???? ....
15*12 = (1 + 16)*3 + 129
?? 129 = 3 * 43 .... ????? ....
?? 2d f_3 vsp .... ???
?? 4 1d subspaces ...
3d irrep ...
?? 2 cuspidals of gl(1,f_2) ?? ....
?? green convolution of 2 1d cuspidals ..... ????? .....
?? "take 2d f_3 vsp v, take 1d subspace x, take a-structure on x and b-structure on v/x" .... ????? ....
0
[]
1
[1]
2
[2] [11]
3
[3] [21] [111]
4
[4] [31] [22] [211] [1111]
5
[5] [41] [32] [311] [221] [2111] [11111]
??? gl(2) ....
?? cuspidal ??
?? green convolution of 2 different gl(1) cuspidals ...
?? apply 2-box young diagram (?? either [2] or [11] ?? ...) to gl(1) cuspidal ... ???? ....
c2 ....
c1 X c1 - c1 ... ????
c1 X 2 ... ???? ....
??? hmmmm .... q=2 .... ???? cj = 1 so c1 X c1 - c1 = 0 ??? .... ??? ...
6 = 1^2 + 2^2 + 1^2 .....
48 = 1^2 + 3^2 + 1^2 + 3^2 + 4^2 + 12 .... ???? 12 = (2^2)*3 ??? ....
15*12 = (1^2 + 4^2)*3 + (5^2)*3 + 54 ..... ???? 54 = (3^2)*6 ??? ....
24*20 = (1^2 + 5^2)*4 + (6^2)*6 + 160 .... ???? 160 = (4^2)*10 ??? ....
48*42 = (1^2 + 7^2)*6 + (8^2)*15 + (6^2)*21 ?????? .....
2016 = 300 + 960 + 756
?? so naive guesses seem to be checking out here .... ??? ....
7*6*4 = 168
26*24*18 = 11232
63*60*48 = 181440
?? "monocuspidal" :
gl(3) cuspidal .... ??? ....
gl(1) cuspidal with 3-box young diagram applied ... ??? ...
?? "bicuspidal" :
gl(2) cuspidal green-convolve gl(1) cuspidal
(gl(1) cuspidal with 2-box young diagram applied) green-convolve gl(1) cuspidal
?? "tricuspidal" :
gl(1) cuspidal green-convolve gl(1) cuspidal green-convolve gl(1) cuspidal
q=2 ... ??? ...
168 = 1^2 + 1^2 + ??? ....
?? nothing, point, flag .... ???? .....
?? induced vs irrep ... categorified gram-schmidt .... ???? ....
?? nothing, point - nothing, flag - point*2 + nothing
1, q^2+q+1, (q^2+q+1)*(q+1)
1, q^2+q, (q^2+q+1)*(q-1)+1 = q^3 ??? ....
q=2 ...
1, 6, 8 ???? ....
168 = (1^2) + (1^2 + 6^2 + 8^2) + (7^2)
1 + (1 + 36 + 64) + 49
?? could it be that there are 2 3d gl(3) cuspidals here ???? ..... ????
168 = (3^2 + 3^2) + (1^2 + 6^2 + 8^2) + (7^2)
11232 = (?^2)*?? + (1^2 + 12^2 + 27^2)*2 + ((13*2)^2)*3*2 + (13^2 + 39^2)*1
11232 - (1748 + 4056 + 3380) = 3738 = 2*3*7*89 ?? .... must be arithmetic mistake somewhere .... ???? ....
?? hmmm, so maybe the mistake was ... to neglect that for the case where you use a distinct pair of gl(1) cuspidals, the pair are distinguishable ...... ??? (maybe figure out what i mean by difference between "distinct" and "distinguishable" here sometime ??? ...) ... so that 1690 should be doubled to 3380 ... ?? and now the arithmetic seems a bit more encouraging ??? .... 2048 left over ... = 2^11 .... ??? .....
??? so that could fit with 2 32d gl(3,f_3) cuspidals ?? ... ?? or 8 16d ones ??? ... ?? ... or 32 8d ..... ???? or 128 4d, or 512 2d, or 2048 1d .... ???? .... ?? which seems more plausible offhand ??? ....
2 1d gl(1,f_3) cuspidals ....
3 2d gl(2,f_3) cuspidals ...
8 16d gl(3,f_3) cuspidals ... ???? .....
...
1 1d gl(1,f_2) cuspidal ....
1 1d gl(2,f_2) cuspidal ....
2 3d gl(3,f_2) cuspidals .... ???? ....
...
q-1 1d gl(1,f_q) cuspidals ....
q(q-1)/2 [q-1]d gl(2,f_q) cuspidals .... ????? ...
.... ???? .....
181440 = (?^2)*?? + (1^2 + 20^2 + 64^2)*3 + ((21*3)^2)*6*3 + (21^2 + (21*4)^2)*6 + ((21*5)^2)*1
181440 - 13491 - 71442 - 44982 - 11025 = 40500 = 2^2 * 3^4 * 5^3
?? so .... ???? gl(3,f_4) cuspidals ... ??? maybe 5 90d ??? or 20 45d ?? .... ?? or 45 30d ?? .... or 125 18d ?? .... or 180 15d ?? .... or 405 10d ... or 500 9d ... or 1125 6d ... or 1620 5d .... or 4500 3d ... or 10125 2d ... or 40500 1d ... ???? ......
124*120*100 = (?^2)*?? + (1^2 + 30^2 + 125^2)*4 + ((31*4)^2)*10*4 + (31^2 + (31*5)^2)*12 + ((31*6)^2)*4
1488000 - 66104 - 615040 - 299832 - 138384 = 368640 = 2^13 * 3^2 * 5
342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =
33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7
?? some sort of categorified gram-schmidt here ?? .... ???? .... ?? try to get q-polynomials for various sectors here ... ??? ... ?? ...
48 = 1 + 9 + 1 + 9 + ...
28 ... ???? ....
15*12 = (1 + 16)*3 + 129
?? 129 = 3 * 43 .... ????? ....
?? 2d f_3 vsp .... ???
?? 4 1d subspaces ...
3d irrep ...
?? 2 cuspidals of gl(1,f_2) ?? ....
?? green convolution of 2 1d cuspidals ..... ????? .....
?? "take 2d f_3 vsp v, take 1d subspace x, take a-structure on x and b-structure on v/x" .... ????? ....
0
[]
1
[1]
2
[2] [11]
3
[3] [21] [111]
4
[4] [31] [22] [211] [1111]
5
[5] [41] [32] [311] [221] [2111] [11111]
??? gl(2) ....
?? cuspidal ??
?? green convolution of 2 different gl(1) cuspidals ...
?? apply 2-box young diagram (?? either [2] or [11] ?? ...) to gl(1) cuspidal ... ???? ....
c2 ....
c1 X c1 - c1 ... ????
c1 X 2 ... ???? ....
??? hmmmm .... q=2 .... ???? cj = 1 so c1 X c1 - c1 = 0 ??? .... ??? ...
6 = 1^2 + 2^2 + 1^2 .....
48 = 1^2 + 3^2 + 1^2 + 3^2 + 4^2 + 12 .... ???? 12 = (2^2)*3 ??? ....
15*12 = (1^2 + 4^2)*3 + (5^2)*3 + 54 ..... ???? 54 = (3^2)*6 ??? ....
24*20 = (1^2 + 5^2)*4 + (6^2)*6 + 160 .... ???? 160 = (4^2)*10 ??? ....
48*42 = (1^2 + 7^2)*6 + (8^2)*15 + (6^2)*21 ?????? .....
2016 = 300 + 960 + 756
?? so naive guesses seem to be checking out here .... ??? ....
7*6*4 = 168
26*24*18 = 11232
63*60*48 = 181440
?? "monocuspidal" :
gl(3) cuspidal .... ??? ....
gl(1) cuspidal with 3-box young diagram applied ... ??? ...
?? "bicuspidal" :
gl(2) cuspidal green-convolve gl(1) cuspidal
(gl(1) cuspidal with 2-box young diagram applied) green-convolve gl(1) cuspidal
?? "tricuspidal" :
gl(1) cuspidal green-convolve gl(1) cuspidal green-convolve gl(1) cuspidal
q=2 ... ??? ...
168 = 1^2 + 1^2 + ??? ....
?? nothing, point, flag .... ???? .....
?? induced vs irrep ... categorified gram-schmidt .... ???? ....
?? nothing, point - nothing, flag - point*2 + nothing
1, q^2+q+1, (q^2+q+1)*(q+1)
1, q^2+q, (q^2+q+1)*(q-1)+1 = q^3 ??? ....
q=2 ...
1, 6, 8 ???? ....
168 = (1^2) + (1^2 + 6^2 + 8^2) + (7^2)
1 + (1 + 36 + 64) + 49
?? could it be that there are 2 3d gl(3) cuspidals here ???? ..... ????
168 = (3^2 + 3^2) + (1^2 + 6^2 + 8^2) + (7^2)
11232 = (?^2)*?? + (1^2 + 12^2 + 27^2)*2 + ((13*2)^2)*3*2 + (13^2 + 39^2)*1
11232 - (1748 + 4056 + 3380) = 3738 = 2*3*7*89 ?? .... must be arithmetic mistake somewhere .... ???? ....
?? hmmm, so maybe the mistake was ... to neglect that for the case where you use a distinct pair of gl(1) cuspidals, the pair are distinguishable ...... ??? (maybe figure out what i mean by difference between "distinct" and "distinguishable" here sometime ??? ...) ... so that 1690 should be doubled to 3380 ... ?? and now the arithmetic seems a bit more encouraging ??? .... 2048 left over ... = 2^11 .... ??? .....
??? so that could fit with 2 32d gl(3,f_3) cuspidals ?? ... ?? or 8 16d ones ??? ... ?? ... or 32 8d ..... ???? or 128 4d, or 512 2d, or 2048 1d .... ???? .... ?? which seems more plausible offhand ??? ....
2 1d gl(1,f_3) cuspidals ....
3 2d gl(2,f_3) cuspidals ...
8 16d gl(3,f_3) cuspidals ... ???? .....
...
1 1d gl(1,f_2) cuspidal ....
1 1d gl(2,f_2) cuspidal ....
2 3d gl(3,f_2) cuspidals .... ???? ....
...
q-1 1d gl(1,f_q) cuspidals ....
q(q-1)/2 [q-1]d gl(2,f_q) cuspidals .... ????? ...
.... ???? .....
181440 = (?^2)*?? + (1^2 + 20^2 + 64^2)*3 + ((21*3)^2)*6*3 + (21^2 + (21*4)^2)*6 + ((21*5)^2)*1
181440 - 13491 - 71442 - 44982 - 11025 = 40500 = 2^2 * 3^4 * 5^3
?? so .... ???? gl(3,f_4) cuspidals ... ??? maybe 5 90d ??? or 20 45d ?? .... ?? or 45 30d ?? .... or 125 18d ?? .... or 180 15d ?? .... or 405 10d ... or 500 9d ... or 1125 6d ... or 1620 5d .... or 4500 3d ... or 10125 2d ... or 40500 1d ... ???? ......
124*120*100 = (?^2)*?? + (1^2 + 30^2 + 125^2)*4 + ((31*4)^2)*10*4 + (31^2 + (31*5)^2)*12 + ((31*6)^2)*4
1488000 - 66104 - 615040 - 299832 - 138384 = 368640 = 2^13 * 3^2 * 5
342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =
33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7
?? some sort of categorified gram-schmidt here ?? .... ???? .... ?? try to get q-polynomials for various sectors here ... ??? ... ?? ...
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