Prof Zoraida Plasencia Salcedo
SOMOS DIFERENTES
LetF be a finite field with|F| =q, a power of a primepand letG= GLn(q). We have seen that the
subgroup U =Un(q) = 1 ∗ ∗ . . . ∗ 0 1 ∗ . . . ∗ 0 0 1 . .. ∗ .. . ... ... . .. ∗ 0 0 0 . . . 1 ≤G (18)
is a Sylowp-subgroup of G. The normalizer of U is the Borel subgroupB appearing in the Bruhat decomposition: NG(U) =B = × ∗ ∗ . . . ∗ 0 × ∗ . . . ∗ 0 0 × . .. ∗ .. . ... ... . .. ∗ 0 0 0 . . . × ,
where the entries∗are arbitrary inF as before, and the diagonal entries×are nonzero elements ofF. We haveB = U oT, where T is the diagonal subgroup ofG, and |B| = qn(n−1)/2 ·(q−1)n. The
p-factorization of|G|is therefore
|G|=qn(n−1)/2·(q−1)n· (q
n−1)· · ·(q2−1)(q−1)
We observe that (qn−1)· · ·(q2−1)(q−1) (q−1)n = n Y k=1 (1 +q+· · ·+qk−1)≡1 mod p,
as guaranteed by Sylow’s theorem. This is the number of Sylowp-subgroups ofG, of whichU is only one. Let X be the set of all Sylow p-subgroups of G. By the Main Theorem of Group Actions, the mapping
G/B−→X, sending gB 7→gU g−1
is aG-equivariant bijection. The setX involves the complete projective geometry of the vector space
V =Fn, as I will explain.
Forn = 2, we have seen thatB is the stabilizer of the line`o =F e1 inV, so in this caseG/Bis also
identified with the setP(V)of all lines` ⊂V. Thus we have aG-equivariant bijective correspondence
P(V)3`=g·`o ↔gB ↔gU g−1 =U`∈X
between lines inV and Sylowp-subgroups of G= GL2(q). Given a line`, the subgroupU`is the set
of elements inGwhich act trivially on both` andV /`. All elements ofU` have the form
1 ∗
0 1
with respect to any basis{v1, v2}ofV withv1 ∈`, but not all elements ofU`have this form with respect to
the original basis{e1, e2}, unless` =`o.
Forn = 3we have both lines and planes inV =F3, and a given plane may or may not contain a given line. Aflagin V is a pair(`, π), where ` is a line contained in a planeπ ⊂ V. Such configurations comprise the complete projective geometry of V. Let F(V) be the set of all flags in V. The group
G = GL3(F) acts on F(V) via g ·(`, π) = (g ·`, g· π). This G-action is transitive and B is the
stabilizer of the flag (`o, πo), where `o = F e1 andπo = F e1 ⊕F e2. Thus we have a G-equivariant
bijective correspondence
F(V)3(`, π) =g·(`o, πo) ↔ gB ↔ gU g−1 =U(`,π) ∈X
between flags inV and Sylow p-subgroups ofG = GL3(q). Given a flag(`, π), the subgroup U(`,π)
is the set of elements inGwhich preserve` andπ and act trivially on`, π/`andV /π. All elements ofU(`,π) have the form
1 ∗ ∗ 0 1 ∗ 0 0 1
with respect to any basis {v1, v2, v3} of V for whichv1 ∈ ` and
v1, v2 ∈π, but not all elements ofU`,πwill have this form with respect to the original basis{e1, e2, e3},
unless`=`o andπ =πo.
For generaln≥2, aflaginV =Fnis a sequence of subspaces
f = (V0, V1, . . . , Vn−1, Vn), where {0}=V0 ⊂V1 ⊂V2 ⊂ · · · ⊂Vn−1 ⊂Vn=V,
anddimVi =ifor all0≤i≤n. Theflag varietyofGis the setF(V)of all flagsf ∈V. The action
ofG= GLn(q)onF(V)permutes the flags inV. That is, we have
ThisG-action is transitive andB is the stabilizer of the flag
fo = (0, F e1, F e1⊕F e2,· · ·, F e1⊕F e2⊕ · · · ⊕F en−1, V).
Thus we have aG-equivariant bijective correspondence
F(V)3f =g·fo ↔ gB ↔ gU g−1 =Uf ∈X
between flagsf ∈ V and Sylowp-subgroupsUf ≤ G= GLn(q). Given a flagf = (V0, V1, . . . , Vn),
the subgroupUf is given by
Uf ={g ∈G: gVi =Vi andg acts trivially on Vi/Vi−1 for all ≤i≤n}.
All elements ofUf have the form (18) with respect to any basis {v1, v2, . . . , vn}ofV for whichVi =
F v1 ⊕ · · · ⊕F vi, but not all elements of Uf have this form for the original basis {e1, . . . , en}unless
f =fo.
The total number of flags inV is
|F(V)|=|G/B|= (q
n−1)· · ·(q2−1)(q−1)
(q−1)n ,
which we have seen to be a polynomialPn(q)inqof degreen(n−1)/2.
For any fieldF there is still a complete geometry of flags inV, defined in the same way. Let us take
F =C, with G= GLn(C)andU, B defined as above for the fieldF = C. The polyomialPn(q)still
counts something aboutG/B, but not something so elementary as points, because now the flag variety
G/Bis infinite.
In fact,G/B is a complex projective variety of dimension n(n−1)/2. 4 Any smooth complex pro- jective varietyX of dimensiondhas cohomology groupsHi(X)fori = 0,1, . . . ,2dwhich are finite
dimensional complex vector spaces whose dimensions dimHi(X) are called the Betti numbers of
X. For example if X = Pd(
C) then dimX = d and the Betti numbers are dimHi(X) = 1 for i= 0,2,4, . . . ,2danddimHi(X) = 0for all otheri.
It turns out that the polynomial Pn(q) above encodes the Betti numbers of G/B: Regarding q as a
variable, we have n(n−1)/2 X i=0 dimH2i(G/B)qi =Pn(q) = (qn−1)· · ·(q2−1)(q−1) (q−1)n .
The observation that Betti numbers ofG/B over Care determined by the number of points on G/B
over finite fields is deep; it led to Weil’s conjectures, which were eventually proved by Deligne.
4A complex projective variety is a closed subset of some projective space
PN(C)defined by polynomial equations. It