A Length Function for Weyl Groups of Extended Affine Root Systems

A Length Function for Weyl Groups of ExtendedAffine Root Systems of Type A1

Mohammad Nikouei

Department of MathematicsUniversity of Isfahan

September 6-8, 2012Upssala University, Upssala, Sweden

M. Nikouei (University of Isfahan) Length Function for Weyl Groups of Type A1 Upssala University 2012 1 / 15

Outline

1 IntroductionAffine Reflection SystemsWeyl Group

2 Length FunctionGeometric Structure of Root SystemCalculation of Length Function


Introduction Affine Reflection Systems

Definition

A an abelian group.(., .) : A× A −→ Q a bi-homomorphism of A (called form).(., .) symmetric & positive semi-definite.A0 := Rad(., .) := {α ∈ A | (α, α) = 0}.

Affine Reflection System (or ARS for short)I R ⊆ A, R0 = R ∩ A0 and R× = R \ R0,I R = −R,I 〈R〉 = A,I ∀α ∈ R×, β ∈ R, ∃u, d ∈ Z,

(β + Zα) ∩ R = {β − dα, . . . , β + uα}

d − u = (β, α∨).



Definition



(β + Zα) ∩ R = {β − dα, . . . , β + uα}

d − u = (β, α∨).



Definition



(β + Zα) ∩ R = {β − dα, . . . , β + uα}

d − u = (β, α∨).



Definition



(β + Zα) ∩ R = {β − dα, . . . , β + uα}

d − u = (β, α∨).



Definition



(β + Zα) ∩ R = {β − dα, . . . , β + uα}

d − u = (β, α∨).



Tame Irreducible Affine Reflection System

R is Irreducible, if R× is connected.R is Tame, if R0 ⊆ R× − R×.R is reduced, if α ∈ R× ⇒ 2α 6∈ R×.

R is locally finite root system, if A0 = {0}.Type of R is type of locally finite root system R ⊆ A := A/A0.R is called an extended affine root system (EARS for short), if A isa free abelian group. Rank of A0 is called the nullity of R andshown by ν.

Examples:Finite and affine Kac-Moody Root systems.Locally finite root systems.Locally extended affine root systems.Root systems extended by abelian groups.







































Structure of Tame Irreducible ARS of Type A1

R is a fixed affine reflection system of type A1.

R = (S + S) ∪ (R + S)

where R = {0,±ε} and S ⊆ A0 such that

0 ∈ S, S± 2S ⊆ S and 〈S〉 = A0

R = A0 ∪ (R + A0) is called toroidal root system.

A = A⊕ A0, where A := 〈R〉

sgn : A −→ Z and p : A −→ A0

α = sgn(α)ε+ p(α), ∀α ∈ R

(β, α) = 2sgn(α)sgn(β)





R = (S + S) ∪ (R + S)


0 ∈ S, S± 2S ⊆ S and 〈S〉 = A0



sgn : A −→ Z and p : A −→ A0







R = (S + S) ∪ (R + S)


0 ∈ S, S± 2S ⊆ S and 〈S〉 = A0



sgn : A −→ Z and p : A −→ A0







R = (S + S) ∪ (R + S)


0 ∈ S, S± 2S ⊆ S and 〈S〉 = A0



sgn : A −→ Z and p : A −→ A0







R = (S + S) ∪ (R + S)


0 ∈ S, S± 2S ⊆ S and 〈S〉 = A0



sgn : A −→ Z and p : A −→ A0




Introduction Weyl Group

Reflections

(β, α∨) := 2(β, α)

(α, α), α ∈ R×

(β, α∨) := 0, α ∈ R0

(β, α∨) = (β, α) = 2sgn(α)sgn(β)

wα ∈ Aut(A), α ∈ R

wα(β) = β − (β, α∨)α

= −sgn(β)ε+ p(β)− 2sgn(α)sgn(β)p(α)



Reflections

(β, α∨) := 2(β, α)

(α, α), α ∈ R×

(β, α∨) := 0, α ∈ R0

(β, α∨) = (β, α) = 2sgn(α)sgn(β)


wα(β) = β − (β, α∨)α




Reflections

(β, α∨) := 2(β, α)

(α, α), α ∈ R×

(β, α∨) := 0, α ∈ R0

(β, α∨) = (β, α) = 2sgn(α)sgn(β)


wα(β) = β − (β, α∨)α




Weyl Group of R

DefinitionW := 〈wα | α ∈ R〉 is called the Weyl group of R.

For w = wα1 · · ·wαn ∈ W, define

ε(w) = (−1)n and T(w) = ε(w)

n∑i=1

(−1)isgn(αi)p(αi).

ε :W −→ {−1, 1} is a homomorphism. SetW0 := Kerε.T :W −→ A0 is a map and T(w1w2) = ε(w2)T(w1) + T(w2).w(α) = ε(w)sgn(α)α+ p(α)− 2sgn(α)T(w) andW0 ∼= A0 with T|W0 .



Weyl Group of R



ε(w) = (−1)n and T(w) = ε(w)

n∑i=1





Weyl Group of R



ε(w) = (−1)n and T(w) = ε(w)

n∑i=1





Weyl Group of R



ε(w) = (−1)n and T(w) = ε(w)

n∑i=1





Weyl Group of R



ε(w) = (−1)n and T(w) = ε(w)

n∑i=1





Structure of the Weyl Group

TheoremWR =WR, for any ARS R in A.

TheoremEach element w inW has a unique expression in the form

w = wδε(w),1ε wε+T(w),

where δ is the Kronecker delta. In particular, if W := 〈wε〉 is the finite Weylgroup of type A1, we have

W = W nW0 ∼= W n A0.

W =W0 ∪W1, whereW1 = {wα | α = ε+ σ, σ ∈ R0}.








W = W nW0 ∼= W n A0.









W = W nW0 ∼= W n A0.



Length Function Geometric Structure of Root System

Geometry of Root System

A is a free abelian group of finite rank ν.R is a fixed extended affine root system in A.B0 = {σ1, . . . , σν} is a Z-basis for A0.Let (·, ·)E : A0 × A0 −→ Z be a form on A0 defined by

(

ν∑i=1

miσi,

ν∑i=1

niσi)E :=

ν∑i=1

mini,

where mi, ni ∈ Z, for 1 ≤ i ≤ ν.If one considers the ν dimensional vector space V0 := R⊗Q A0 asan Euclidean space, then (·, ·)E is the restriction of the Euclideaninner product on V0 to A0.R = {kε+ σ | k ∈ {0,±1}, σ ∈ A0}.





(

ν∑i=1

miσi,

ν∑i=1

niσi)E :=

ν∑i=1

mini,






(

ν∑i=1

miσi,

ν∑i=1

niσi)E :=

ν∑i=1

mini,






(

ν∑i=1

miσi,

ν∑i=1

niσi)E :=

ν∑i=1

mini,






(

ν∑i=1

miσi,

ν∑i=1

niσi)E :=

ν∑i=1

mini,






(

ν∑i=1

miσi,

ν∑i=1

niσi)E :=

ν∑i=1

mini,




Positive and Negative Roos

DefinitionLet α ∈ R \ {0}.(i) We call α positive, if

sgn(α) 6= −1 and (p(α),

ν∑i=1

σi)E ≥ 0

or

sgn(α) = −1 and (p(α),

ν∑i=1

σi)E > 0.

We denote the set of positive roots by R+.(ii) We call a non-zero root α negative, if α ∈ R \ R+. We denote the set ofnegative roots by R−.

We have R = R+ ∪ {0} ∪ R−.



Positive and Negative Roos

DefinitionLet α ∈ R \ {0}.(i) We call α positive, if

sgn(α) 6= −1 and (p(α),

ν∑i=1

σi)E ≥ 0

or

sgn(α) = −1 and (p(α),

ν∑i=1

σi)E > 0.

We denote the set of positive roots by R+.(ii) We call a non-zero root α negative, if α ∈ R \ R+. We denote the set ofnegative roots by R−.

We have R = R+ ∪ {0} ∪ R−.



Height Function

For σ =∑ν

i=1 miσi, let m+ :=∑ν

i=1mi>0

mi and m− :=∑ν

i=1mi<0

mi.

DefinitionFor α = kε+ σ ∈ R, we define

ht(α) :=

{k − htB0(σ), k = −1, m+ = |m−|,k + htB0(σ), otherwise,

where

htB0(σ) :=

{2m+, m+ ≥ |m−|,2m−, m+ < |m−|.

If α ∈ R is a nonzero root, then ht(α) 6= 0.

LemmaLet α ∈ R. We have α ∈ R+ (resp. α ∈ R−) if and only if ht(α) > 0 (reps.ht(α) < 0).



Height Function

For σ =∑ν


i=1mi>0

mi and m− :=∑ν

i=1mi<0

mi.


ht(α) :=


where

htB0(σ) :=

{2m+, m+ ≥ |m−|,2m−, m+ < |m−|.





Height Function

For σ =∑ν


i=1mi>0

mi and m− :=∑ν

i=1mi<0

mi.


ht(α) :=


where

htB0(σ) :=

{2m+, m+ ≥ |m−|,2m−, m+ < |m−|.





Root Bases

We call 0 6= σ =∑ν

i=1 miσi ∈ Λ a strictly positive isotropic root, ifmi ≥ 0, for 1 ≤ i ≤ ν.

DefinitionConsider a Z-basis Π = {β0, . . . , βν} for A. We call Π a root base, if eachstrictly positive isotropic root σ can be written in the form σ =

∑νi=0 niβi,

where ni ≥ 0.

Π0 = {α0 := ε, α1 := −ε+ σ1, . . . , αν := −ε+ σν} is a root base for R.

Proposition

Let α =∑ν

i=0 niαi ∈ R. We have

|ht(α)| =ν∑

i=0

|ni|.

In particular∑ν

i=0 |ni| is odd, if α ∈ R× and∑ν

i=0 |ni| is even, if α ∈ R0.



Root Bases




∑νi=0 niβi,

where ni ≥ 0.


Proposition

Let α =∑ν


|ht(α)| =ν∑

i=0

|ni|.

In particular∑ν





Root Bases




∑νi=0 niβi,

where ni ≥ 0.


Proposition

Let α =∑ν


|ht(α)| =ν∑

i=0

|ni|.

In particular∑ν





Root Bases




∑νi=0 niβi,

where ni ≥ 0.


Proposition

Let α =∑ν


|ht(α)| =ν∑

i=0

|ni|.

In particular∑ν




Length Function Calculation of Length Function

Calculation of Length

In a group G, we defineThe length of g ∈ G with respect to an index set of generators Π,which is denoted by `Π(g), is the smallest length of any expressionof g. By convention, `Π(1) = 0.Any expression of g ∈ G with length `Π(g) is called a reducedexpression of g with respect to Π.

TheoremFor w ∈ W , we have

`Π0(w) = |ht(ε+ T(w))| − sgn(ht(ε+ T(w)))t,

where t = δε(w),1, and sgn(ht(ε+ T(w))) is the sign of ht(ε+ T(w)) as aninteger.
























Explanation

Theorem(i) For α ∈ R×, we have

`Π0(wα) = |ht(α)|.

In particular, if α =∑ν

i=0 niαi, we have `Π0(wα) =∑ν

i=0 |ni|.(ii) Let w ∈ W0. We have

`Π0(w) = |ht(T(w))|.

In particular, if T(w) =∑ν

i=0 niαi then `Π0(w) =∑ν

i=0 |ni|.

Since every element ofW is either a reflection or an element ofW0,according to the above theorem, for each w ∈ W, there exists a rootα ∈ R such that `Π0(w) = |ht(α)|.



Explanation

Theorem(i) For α ∈ R×, we have

`Π0(wα) = |ht(α)|.

In particular, if α =∑ν

i=0 niαi, we have `Π0(wα) =∑ν

i=0 |ni|.(ii) Let w ∈ W0. We have

`Π0(w) = |ht(T(w))|.

In particular, if T(w) =∑ν

i=0 niαi then `Π0(w) =∑ν

i=0 |ni|.

Since every element ofW is either a reflection or an element ofW0,according to the above theorem, for each w ∈ W, there exists a rootα ∈ R such that `Π0(w) = |ht(α)|.


Thank you for your attention!


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A Length Function for Weyl Groups of Extended Affine Root Systems