gl(2,f_3) ... ?? aka e7 mckay group = "binary cubic group" ... ??? ....




?? maximal toruses .... ???? .....




?? split ... diagonal ...




?? unsplit .... ??? ....




gl(1,f_9) -> gl(2,f_3) .... ??? ....

?? bit about ... ??? "nice" .... ??? way for field structure to get along with basis ?? ... ??? ... ??? ..... ?? or "nice generator" ... ??? whose powers form ... ?? a basis, or some particularly nice sort of basis ??? ....

?? "necklace" .... ?? free lie algebra ... ?? lyndon .... ???? ...... ??? ....

?? more combinatorics .... ???? .....

?? apparently i once wrote about "field structures on the vector space [f_2]^[2^[n-1]] for which the frobenius automorphism is the obvious rotation operator" ... ??? .... ??? "aperiodic necklace" ??? .... ???? .... ???? .....

?? "and its multiplicative identity is the vector each of whose components is the multiplicative identity of f" ... ??? ....

???? hmmm, ramification here ??? ... ???? .....

?? hmmm, apparently there _was_ some special thing about the case of [f_2]^[2^n], but more generally i was interested in [f_2]^n .... ??? "i'll call a field structure on the vector space f^n over a finite field f "good" iff its frobenius automorphism is the obvious rotation operator and its multiplicative identity is the vector each of whose components is the multiplicative identity of f" ... ???

?? "when f = f_p with p prime we can straightforwardly identify the good field structures on f^n with the irreducible factors of x^p^[n-1] + x^p^[n-2] + ... + x^p^1 + x^p^0 - 1 over f_p that are of maximal degree and have linearly independent roots" ... ??? .....

f_3[x]/x^2=2 .... ???? .....

?? p=3, n=2

?? x^3 + x - 1

?? wild ramification and associated graded .... ????? ...... "tangent cone" ... ??? ....

?? well , let's at least get _some_ explicit unsplit maximal toruses here ... ??? ....

x^2 = 2 ... ???

(a+bx)(c+dx) = (ac+2bd)+(ad+bc)x .... ????? .....

c 2d
d c

?? generator here ???? ..... ????? ......

?? hmmm, for generator maybe _neither_ c nor d should be zero ??? ..... ???? .... ??? mystically suggestive ??? .... ???? .....

12
11

01
20

11
21

20
02

21
22

02
10

22
12

?? "cycle" in 4! ... ??? ... ?? vs split maximal torus here, which .... ??? fixes 2 points and cycles just the other 2 ... ??? ..... ??? so can unsplit maximal torus be thought of as fixing two "imaginary" points, and is this a useful idea ??? .....

?? consider real case (?? again ??) here ... ??? ... ?? gl(1,c) -> gl(2,r) as ... ??? "fixing two complex points" ??? ... ??? ??? which two ??? .... ?? obvious naive guess as ... ?? i and -i as points of riemann sphere (as cp^1) .... ???? ...... ?? seems to work ??? ..... ???? ..... ?? moebius transformation preserving equator and preserving north pole = moebius transformation preserving equator and metric structure (?? and orientation ??? ....) on it ..... ???? .... ?? one pole vs two here ... ??? borel vs cartan .... ???? ... ???? ..... ??? kaleidoscope ....... ????? ..... ????? .....

?? functoriality of complexification here ??? .... ??? ....

?? some confusion here ?? ... ?? q^2-1 vs (q-1)^2 .... ??? vs ... q^2-1 vs q^2-q ???? ..... ????? ..... ????? ....... ?? check recent numerology posts where i might have gotten into such confusion ?? ...

?? 4 "real" points ... ?? 10 "complex" .... ??? ..... 10 = 1 + 1 + 4 + ?4? ... ???? .... ?? = 1 + 1 + 8 .... ???? ....

?? so given character of unsplit maximal torus, try splitting the torus, inducing the character, then applying parabolic induction, then trying to "de-split" ...... ?????? ......

?? second term as swing vote between "not induced" first term and "induced" third term ?? ... ??? ... accusing bump of using "induced" in peculiar way ?? ..... ??? trying to systematize / make more precise such peculiar and/or non-peculiar ways .... ?? "collection of all ways of seeing rep r as induced / quasi-induced" .... ?? and / or universal such .... ???? ...

?? maximal toruses in gl(3) ... trying to match up with kinds of irreps we seem to see in classification .... ??? young-diagrammatic flavor here ?? ....

?? q^3-1 ...

?? (q^2-1)(q-1) ...

?? (q-1)^3 ...

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


(q^3-1)*((q^2-1)(q-1))*((q-1)^3) = ??? .....

??? hmmm .... ??? _something_ being parameterized by "one character for "each" maximal torus" ????? ...... ????? .....

?? double life of these particular q-polynomials ... ?? well, or maybe triple, or single .... ???? .... ?? i was going to say "flag combinatorics vs ... this new (?? ...) stuff that we're doing now" ... ?? but ... maybe more unified, as to do with "factorization" of group gl(3) .... ???? ... ?? is there something about .... ???? subgroup pair of complementary index, with corresponding "decomposition" ??? ...

?? hmm, just remembering now that we didn't quite straighten out the q-polynomial for the cuspidal sector .... ???? ......

?? associated graded object of ideal power filtration of ideal

in z ..... ??? .... ???? ....

p = 5, say ....

?? then consider, say, extension z[i] .... ??? does this come along for the ride in some interesting way ??? .....

?? polynomials over z/5 .... ???? ....

-1 .... ???? .....

?? ideal power filtration of certain (??which??) ideal in gaussian integers .... ????? ......

?? adjoining square root of -1 to f_5[x] ??? .... ??? is that what happens here ??? .... ???? .... ?? how "involved" is x here ??? .... ???? ... ?? maybe not much ?? .... ?? and generally doesn't get any more involved in other cases either ?? .... ???? ... ?? hmmmm ..... ???? ramification ??? .... ?? ideal power filtration of ideal associated with 2 in gaussian integers .... ????? ..... ?? ideal power filtration associated with 5 in golden integers .... ?????? ..... ????? .....

"symbol" ... ???? ... "principal symbol" ??? .... ??? "secondary symbol" ???? ..... ?? "obstruction" ??? ... ??? ... ???? ..... ???? "hessian" ???? .... ???? ..... ??? .....

?? "y^2 = x^3" ..... ???? ......

?? "split case" =?= "parabolic induction" ??? .... ???? ....

?? extend, then induce ... ?? ....

?? unsplit case ... ???? .....

?? structure preserved by unsplit maximal torus .... ???? .....

?? when unsplit maximal torus splits, keeping track of what happens (??? under induction ??? .... ??? ....) to those irreps which corresponded to characters of it .... ???? .....

?? "algebraic representation of algebraic group" ??? .... ???? .... .... ???? ......

?? what is given maximal torus doing while particular irrep is being parameterized by character of _other_ kind of maximal torus ??? .....

?? long axises of cube as forming f_3 projective line .... ??? some (?? ...) axises of dodecahedron as forming projective line over f_5 ?? ...

?? but gl(2,f_5) has 10+8+6 = 24 irreps, whereas e8 mckay group has 8 ?? .... ???? .....

?? interpretation of paths in ade graphs under mckay correspondence as "clebsch-gordan terms" ?? ... ??? .....

?? "t-genic ag theory" .... ?? ....in mckay correspondence situation ... ??? ... ?? in situation of trying to prove nice universal property of ag theory given syntactically by quasicoherent sheaves over projective space of v .... ??? .... ???? ....

?? 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 ??? ..... ???? .....

?? seems like there really should be some sort of "kaleidoscope-folding" here because ... ?? bump says "maximal toruses" but seeming to mean conjugacy (???? ....) classes thereof ..... ???? ....

gl(2,f_3) .... ???? .....

?? maximal toruses ....

?? might help if group here has some semi-familiar secret identity .... ???? .....

?? 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" .... ??? .....

?? "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" ... ??? used to know something about something like that in lie case ... ?? ....

?? left-universal property of _fd f_q-vsp_ as braided monoidal finitely cocomplete algebroid .... ???? ..... ????? .....

?? "green substitution" .... ????? .....

[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 ....

?? "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 .... ???? ......

?? "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 .... ????? ......