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