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Essential p-dimension of a normalizer of a maximal torus

Mark L. MacDonald

University of British Columbia

May 2011

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential dimension

Let k0 be a field, and consider a functor F : Fields/k0 → Sets Definition The essential dimension of x ∈ F(k) is edk0(x) := min{tr.degk0(L)|x is defined over L}. The essential dimension of F is edk0(F) := sup{edk0(x)|∀x ∈ F(k)}. k L k0

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential dimension

Let k0 be a field, and consider a functor F : Fields/k0 → Sets Definition The essential dimension of x ∈ F(k) is edk0(x) := min{tr.degk0(L)|x is defined over L}. The essential dimension of F is edk0(F) := sup{edk0(x)|∀x ∈ F(k)}. k L k0 ed(G) := ed(H1(−, G)).

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential dimension

Let k0 be a field, and consider a functor F : Fields/k0 → Sets Definition The essential dimension of x ∈ F(k) is edk0(x) := min{tr.degk0(L)|x is defined over L}. The essential dimension of F is edk0(F) := sup{edk0(x)|∀x ∈ F(k)}. k L k0 ed(G) := ed(H1(−, G)). ed(SLn) = ed(GLn) = ed(Sp2n) = 0.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential p-dimension

Definition Let p be a prime. The essential p-dimension of x ∈ F(k) is ed(x; p) := min{ed(xk′)| k′/k of finite degree not divisible by p }. The essential p-dimension of F is ed(F) := sup{ed(x; p)|∀x ∈ F(k)}. k′

• L
• k

k0

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential p-dimension

Definition Let p be a prime. The essential p-dimension of x ∈ F(k) is ed(x; p) := min{ed(xk′)| k′/k of finite degree not divisible by p }. The essential p-dimension of F is ed(F) := sup{ed(x; p)|∀x ∈ F(k)}. k′

• L
• k

k0 0 ≤ ed(G; p) ≤ ed(G).

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Goal of this talk

Let G be a connected (almost) simple split linear algebraic group, with split maximal torus T. Then the normalizer fits in an exact sequence 1 → T → N → W → 1. Goal Find the exact value of ed(N; p).

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Let N be an extension of a p-group by a torus; i.e. an algebraic group such that 1 → T → N → F → 1.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Let N be an extension of a p-group by a torus; i.e. an algebraic group such that 1 → T → N → F → 1. Theorem [L¨

• tscher, M-, Meyer, Reichstein, 2010]

Assume every finite field extension of k has degree a power of p. min dim(V ) − dim N ≤ ed(N; p) ≤ min dim(W ) − dim N, where the minimums are taken over all N-representations defined

• ver k such that V is p-faithful, and W is p-generically free.

A representation is said to be p-faithful if its kernel is finite order not divisible by p. If additionally N/ ker is generically free, then the representation is said to be p-generically free.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Discrete data

To a split algebraic group we can associate the following discrete data: Root system R Character lattice ˆ T Weyl group representation W → GL( ˆ T) ˆ TSpin16

• ˆ

TSO16

• ˆ

THSpin16

• ˆ

TPSO16

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Discrete data

To a split algebraic group we can associate the following discrete data: Root system R Character lattice ˆ T Weyl group representation W → GL( ˆ T) ˆ TSpin16

• ˆ

TSL9

• ˆ

TSO16

• ˆ

THSpin16

• = ˆ

TE8 = ˆ TSL9/µ3

• ˆ

TPSO16 ˆ TPSL9

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Representations vs. subsets of the character lattice

Assume 1 → T → N → F → 1 with T split and F constant p-group. An N-representation decomposes as V = ⊕λ∈ΛVλ for some F-invariant subset Λ ⊂ ˆ T, V Λ. V is p-faithful iff Λ is p-generating (i.e. generates a sublattice of ˆ T whose index is not divisible by p).

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Representations vs. subsets of the character lattice

Assume 1 → T → N → F → 1 with T split and F constant p-group. An N-representation decomposes as V = ⊕λ∈ΛVλ for some F-invariant subset Λ ⊂ ˆ T, V Λ. V is p-faithful iff Λ is p-generating (i.e. generates a sublattice of ˆ T whose index is not divisible by p). Conversely: If N = NG(T) for a split maximal torus in a simple algebraic group, then we can construct an N-representation from an F-invariant subset of ˆ T. Λ VΛ.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Symmetric p-rank

Definition The symmetric p-rank of an (integral) representation W → GL(L), is the minimal size of an F-invariant p-generating subset of L, where F ⊂ W is a Sylow p-subgroup.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Symmetric p-rank

Definition The symmetric p-rank of an (integral) representation W → GL(L), is the minimal size of an F-invariant p-generating subset of L, where F ⊂ W is a Sylow p-subgroup. For example, L = Z, and W = Z/2 acting by negation. Then the symmetric 2-rank equals 2, by taking Λ = {−1, 1}, or Λ = {−3, 3}, etc.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Example: E8, p = 5

F = W (E8)(5) ∼ = W (A4)(5) × W (A4)(5). α2 α1 α3 α4 α5 α6 α7 α8 α0

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Example: E8, p = 5

F = W (E8)(5) ∼ = W (A4)(5) × W (A4)(5). α2

• α1
• α3
• α4
• α5

α6 α7 α8 α0 1) Consider elements of ˆ TE8 in the Z-basis {α1, · · · , α8}. If Λ ⊂ ˆ TE8 is 5-generating, it must contain an element whose α5 coefficient is not divisible by 5. 2) Such an element has F-orbit of size 25. 3) Therefore ed(NE8(T); 5) = 25 − 8 = 17.

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Other exceptional cases

φ G p SymRank(φ; p) (K)

2A2

G2 2 4 n

1A2

G2 3 3 n

2D4

F4 2 16 y

3D4

F4 3 9 y

1E6

E6 2 16 y

2E6 2E6

2 32 y

1E7

E7 2 64 y

1E +2 7

2E7 2 40 y

1E8

E8 2 128 y

1E6

E6 3 27 y

1E +3 6

3E6 3 27 y

1E7

E7 3 27 y

1E8

E8 3 81 y

1E8

E8 5 25 y

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential p-dimension of Weyl groups W (R)

H1(k, N) ։ H1(k, G) ⇒ ed(G; p) ≤ ed(N; p) H1(k, N) ։ H1(k, W ) ⇒ ed(W ; p) ≤ ed(N; p)

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Essential p-dimension of Weyl groups W (R)

H1(k, N) ։ H1(k, G) ⇒ ed(G; p) ≤ ed(N; p) H1(k, N) ։ H1(k, W ) ⇒ ed(W ; p) ≤ ed(N; p) ed(W (R); p) p = 0 p = 2 p = 3 p = 5 p = 7 p ≥ 11 An ?? ⌊ n+1

2 ⌋

⌊(n + 1)/p⌋ Bn n n ⌊n/p⌋ Cn n n ⌊n/p⌋ Dn (n odd) n − 1 n − 1 ⌊n/p⌋ Dn (n even) n n ⌊n/p⌋ E6 ?? 4 3 1 E7 7 7 3 1 1 E8 8 8 4 2 1 F4 4 4 2 G2 2 2 1

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Summary

Convert the essential dimension problem ed(N; p) into a problem about Weyl group actions on character lattices. Compute the symmetric p-rank by considering each connected simple split group separately. In “most” cases, the bounds match: min dim V − dim N ≤ ed(N; p) ≤ min dim W − dim N, where V is p-faithful, and W is p-gen. free. All known cases are consistent with the conjecture that the upper bound is an equality

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus

Appendix: Dn lattices

φ G p Conditions SymRank(φ; p) (K)

1Dn

PSO2n 2 n = 2rs, s > 1 22r+2(s − 1) y n = 2r ≥ 4 22r y

2Dn

PSp2n 2 n = 2rs, s > 1 22r+2(s − 1) y n = 2r ≥ 4 22r y

1D+2 n

HSpin2n 2 n ≥ 6 2n−1 y

1In

SO2n 2 n ≥ 4 2n n

2In

Sp2n, SO2n+1 2 n ≥ 2 2n n

1D+4 n

Spin2n 2 n ≥ 5 odd 2n−1 y n ≥ 4 even 2n−1 + 2r+1 y

2D+4 n

Spin2n+1 2 n ≥ 2 2n y

3D4

F4 3 9 y

Mark L. MacDonald Essential p-dimension of a normalizer of a maximal torus