A visualization of Quillen stratification with applications in restricted Lie ...
A visualization of Quillen stratification with applications in restricted Lie algebras Jared Warner City University of New York, Guttman Community College
Quillen Stratification
An Extraspecial Group
Let p be a prime number and fix an algebraically closed field, k, of characteristic p. To a finite group Γ we can functorially associate an affine algebraic variety, XΓ, via group cohomology: Γ 7−→ XΓ := Spec H •(Γ, k) Since the association is functorial, to every subgroup H ,→ Γ we have a map of varieties XH → XΓ. In 1971, Daniel Quillen showed that XΓ is covered by the collection of im(XE → XΓ) for E an elementary abelian p-subgroup of Γ.
Let Γ be the extraspecial group of order 128 with two elements of order 4. In other words, Γ is a central product of 5 copies of D8. The picture below contains the data necessary to use Quillen’s theorem to build XΓ (which we don’t do here!).
Rational Points and Orbits of E(r, g) The following theorem helps elucidate the structure of E(r, g) in certain cases. Theorem - (W, 2015) Let G be a reductive, connected algebraic group defined over Fp, and let g be the Lie algebra of G. For p large enough, there is a one-to-one correspondence between the orbits of E(r, g) defined over Fp and conjugacy classes of rank r elementary abelian p-subgroups of the finite group G(Fp). The dimension of an orbit is roughly the base p logarithm of the size of its corresponding conjugacy class.
Theorem (Quillen, 1971) [ XΓ = im(XE → XΓ)
We return to the example of GL3(Fp) to determine the number of orbits defined over Fp and their dimensions for the projective varieties E(1, gl3) and E(2, gl3).
E⊂Γ
E(2, gl3)
It turns out that XE is an affine space of dimension equal to the rank of E, so that this union may be thought of, roughly, as a decomposition of XΓ into affine pieces. We also note that if E and E 0 are conjugate, then their image in XΓ is equal. Thus, in our cover we can take one representative from each conjugacy class. The way that the different affine pieces glue together (and to themselves!) in XΓ is determined by the inclusions and conjugations of the various elementary abelian p-subgrops.
An Illustrating Example - GL3(Fp) Suppose p ≥ 3, and let Γ be the finite group GL3(Fp). Then we have the following exhaustive list of representatives of nontrivial conjugacy classes of elementary abelian p-subgroups (the trivial subgroup is the only elementary abelian p-subgroup of rank 0): 1 a b 1 0 a 1 a b • 0 1 0 a, b ∈ Fp , • 0 1 b a, b ∈ Fp , • 0 1 a a, b ∈ Fp 001 001 001 1 a 0 • 0 1 0 a ∈ Fp , 001
1 a 0 • 0 1 a a ∈ Fp 001
According to Quillen’s theorem, we obtain a decomposition of the cohomology space as illustrated in the pictures below.
GL3(Fp) r=2 E(1, gl3)
r=1 Dimension meter Each node represents a conjugacy class of elementary abelian 2-subgroups of Γ. Notice that XΓ has dimension 4, which is the largest rank of an elementary abelian 2-subgroup. As before, the nodes are connected based on inclusions of conjugacy classes. Here, the size of a particular conjugacy class is proportional to the wavelength of light of its representing node’s color.
The Variety of Elementary Subalgebras - E(r, g) Elementary abelian p-subgroups have an analogue in the setting of restricted Lie algebras, as detailed in what follows. • A restricted Lie algebra g is a Lie algebra with a restriction map (·)[p] : g → g, which interacts with the Lie algebra structure as a generalized pth-power map. • If a restricted Lie algebra g is abelian and satisfies x[p] = 0 for all x ∈ g, then we say that g is elementary.
2
3
4
5
The picture shows that E(2, gl3) has three orbits defined over Fp: two of dimension 2, and one of dimension 4. Similarly, the projective variety E(1, gl3) has two orbits defined over Fp: one of dimension 3 and one of dimension 5.
Exercise The picture below shows the lattice of conjugacy classes of elementary abelian p-subgroups of GL4(Fp). Using the dimension meter provided, we’ve colored each node according to the base p logarithm of the size of its corresponding class.
GL4(Fp), p > 5
• For a fixed positive integer r and a fixed restricted Lie algebra g, define E(r, g) to be the subset of Grass(r, g) consisting of all elementary subalgebras of g of dimension r.
r=2
Since the bracket and restriction are polynomial in nature, we have the following theorem.
r=1 r=0
Theorem (Carlson-Friedlander-Pevtsova, 2012)
XGL3(Fp)
E ⊂ GL3(Fp))
Notice that the dimensions of the different pieces in XGL3(Fp) are equal to the ranks of the corresponding subgroups E, and that the gluing is determined by the inclusion of subgroups. For example, we have the following inclusions of conjugacy classes: •⊂•
•⊂•
•⊂•
•⊂•
These inclusions determine which colored planes contain a particular line. There are further identifications not pictured which are due to conjugation (in other words, the map XE → XΓ is not necessarily injective).
E(r, g) is a projective subvariety of Grass(r, g). If g = Lie(G) for some algebraic group G, then E(r, g) is a G-variety via the adjoint action of G on g.
Dimension meter
A helpful example to keep in mind is that of g = gln, where the bracket is given by the commutator of two matrices, the restriction map is given by the pth power of a matrix, and the adjoint action is given by conjugation.
3 4 5 6 7 8 9 10 11
[A, B] := AB − BA,
A[p] := Ap,
AdA(B) = ABA−1
Then, E(r, gln) ⊂ Grass(r, gln) is the set of all r-dimensional vector subspaces of gln consisting of p-nilpotent, pairwise commuting matrices.
• What is the dimension of the affine variety XGL4(Fp)? • In Quillen’s cover of XGL4(Fp), how many three-dimensional constituents are contained in no four-dimensional one? • How many orbits of E(2, gl4) are defined over Fp? What are their dimensions?
Quillen Stratification
An Extraspecial Group
Let p be a prime number and fix an algebraically closed field, k, of characteristic p. To a finite group Γ we can functorially associate an affine algebraic variety, XΓ, via group cohomology: Γ 7−→ XΓ := Spec H •(Γ, k) Since the association is functorial, to every subgroup H ,→ Γ we have a map of varieties XH → XΓ. In 1971, Daniel Quillen showed that XΓ is covered by the collection of im(XE → XΓ) for E an elementary abelian p-subgroup of Γ.
Let Γ be the extraspecial group of order 128 with two elements of order 4. In other words, Γ is a central product of 5 copies of D8. The picture below contains the data necessary to use Quillen’s theorem to build XΓ (which we don’t do here!).
Rational Points and Orbits of E(r, g) The following theorem helps elucidate the structure of E(r, g) in certain cases. Theorem - (W, 2015) Let G be a reductive, connected algebraic group defined over Fp, and let g be the Lie algebra of G. For p large enough, there is a one-to-one correspondence between the orbits of E(r, g) defined over Fp and conjugacy classes of rank r elementary abelian p-subgroups of the finite group G(Fp). The dimension of an orbit is roughly the base p logarithm of the size of its corresponding conjugacy class.
Theorem (Quillen, 1971) [ XΓ = im(XE → XΓ)
We return to the example of GL3(Fp) to determine the number of orbits defined over Fp and their dimensions for the projective varieties E(1, gl3) and E(2, gl3).
E⊂Γ
E(2, gl3)
It turns out that XE is an affine space of dimension equal to the rank of E, so that this union may be thought of, roughly, as a decomposition of XΓ into affine pieces. We also note that if E and E 0 are conjugate, then their image in XΓ is equal. Thus, in our cover we can take one representative from each conjugacy class. The way that the different affine pieces glue together (and to themselves!) in XΓ is determined by the inclusions and conjugations of the various elementary abelian p-subgrops.
An Illustrating Example - GL3(Fp) Suppose p ≥ 3, and let Γ be the finite group GL3(Fp). Then we have the following exhaustive list of representatives of nontrivial conjugacy classes of elementary abelian p-subgroups (the trivial subgroup is the only elementary abelian p-subgroup of rank 0): 1 a b 1 0 a 1 a b • 0 1 0 a, b ∈ Fp , • 0 1 b a, b ∈ Fp , • 0 1 a a, b ∈ Fp 001 001 001 1 a 0 • 0 1 0 a ∈ Fp , 001
1 a 0 • 0 1 a a ∈ Fp 001
According to Quillen’s theorem, we obtain a decomposition of the cohomology space as illustrated in the pictures below.
GL3(Fp) r=2 E(1, gl3)
r=1 Dimension meter Each node represents a conjugacy class of elementary abelian 2-subgroups of Γ. Notice that XΓ has dimension 4, which is the largest rank of an elementary abelian 2-subgroup. As before, the nodes are connected based on inclusions of conjugacy classes. Here, the size of a particular conjugacy class is proportional to the wavelength of light of its representing node’s color.
The Variety of Elementary Subalgebras - E(r, g) Elementary abelian p-subgroups have an analogue in the setting of restricted Lie algebras, as detailed in what follows. • A restricted Lie algebra g is a Lie algebra with a restriction map (·)[p] : g → g, which interacts with the Lie algebra structure as a generalized pth-power map. • If a restricted Lie algebra g is abelian and satisfies x[p] = 0 for all x ∈ g, then we say that g is elementary.
2
3
4
5
The picture shows that E(2, gl3) has three orbits defined over Fp: two of dimension 2, and one of dimension 4. Similarly, the projective variety E(1, gl3) has two orbits defined over Fp: one of dimension 3 and one of dimension 5.
Exercise The picture below shows the lattice of conjugacy classes of elementary abelian p-subgroups of GL4(Fp). Using the dimension meter provided, we’ve colored each node according to the base p logarithm of the size of its corresponding class.
GL4(Fp), p > 5
• For a fixed positive integer r and a fixed restricted Lie algebra g, define E(r, g) to be the subset of Grass(r, g) consisting of all elementary subalgebras of g of dimension r.
r=2
Since the bracket and restriction are polynomial in nature, we have the following theorem.
r=1 r=0
Theorem (Carlson-Friedlander-Pevtsova, 2012)
XGL3(Fp)
E ⊂ GL3(Fp))
Notice that the dimensions of the different pieces in XGL3(Fp) are equal to the ranks of the corresponding subgroups E, and that the gluing is determined by the inclusion of subgroups. For example, we have the following inclusions of conjugacy classes: •⊂•
•⊂•
•⊂•
•⊂•
These inclusions determine which colored planes contain a particular line. There are further identifications not pictured which are due to conjugation (in other words, the map XE → XΓ is not necessarily injective).
E(r, g) is a projective subvariety of Grass(r, g). If g = Lie(G) for some algebraic group G, then E(r, g) is a G-variety via the adjoint action of G on g.
Dimension meter
A helpful example to keep in mind is that of g = gln, where the bracket is given by the commutator of two matrices, the restriction map is given by the pth power of a matrix, and the adjoint action is given by conjugation.
3 4 5 6 7 8 9 10 11
[A, B] := AB − BA,
A[p] := Ap,
AdA(B) = ABA−1
Then, E(r, gln) ⊂ Grass(r, gln) is the set of all r-dimensional vector subspaces of gln consisting of p-nilpotent, pairwise commuting matrices.
• What is the dimension of the affine variety XGL4(Fp)? • In Quillen’s cover of XGL4(Fp), how many three-dimensional constituents are contained in no four-dimensional one? • How many orbits of E(2, gl4) are defined over Fp? What are their dimensions?
