ROOT SYSTEMS AND COXETER GROUPS 1

ROOT SYSTEMS AND COXETER GROUPS STEPHEN GRIFFETH Abstract. This is a brief review of root systems and Coxeter groups, intended as background material for a course on double affine Hecke algebras and Macdonald polynomials. They are still incomplete in places.

1. Root systems Much of this material can be found in [Bou]. Let V be a real vector space with a positive definite inner product h·, ·i. If α ∈ V is a non-zero vector, define the reflection sα in the hyperplane orthogonal to α by sα (x) = x − hα∨ , xiα

(1.1)

for x ∈ V ,

where α∨ =

2 α. hα, αi

A root system in V is a subset R ⊆ V such that (a) R is finite, does not contain 0, and spans V , (b) for all α, β ∈ R, sα (β) ∈ R, and (c) for all α, β ∈ R, hα∨ , βi ∈ Z. A root system R is reduced if α ∈ R =⇒ 2α ∈ / R,

(1.2)

and irreducible if there is not a non-trivial partition R = R1 ∪ R2 with hα, βi = 0 for all α ∈ R1 , β ∈ R2 .

(1.3)

Exercise 1. Show that if R is a root system, k ∈ R, α ∈ R, and kα ∈ R, then k ∈ {±1/2, ±1, ±2}. For example, let V = Rn with the standard basis consisting of the column vectors ǫi with a one in the ith position and zeros elsewhere: ǫi = (0, . . . , 1, . . . , 0)t

with

hǫi , ǫj i = δij .

For n ≥ 2 the root system of type Bn is Bn = {±ǫi ± ǫj | 1 ≤ i < j ≤ n} ∪ {±ǫi | 1 ≤ i ≤ n},

(1.4)

for n ≥ 3 the root system of type Cn is Cn = {±ǫi ± ǫj | 1 ≤ i < j ≤ n} ∪ {±2ǫi | 1 ≤ i ≤ n},

(1.5)

for n ≥ 4 the root system of type Dn is Dn = {±ǫi ± ǫj | 1 ≤ i < j ≤ n},

(1.6)

and for n ≥ 1 the root system of type BCn is (1.7)

BCn = {±ǫi ± ǫj | 1 ≤ i < j ≤ n} ∪ {±ǫi | 1 ≤ i ≤ n} ∪ {±2ǫi | 1 ≤ i ≤ n}.

For n ≥ 2 let (1.8)

An−1 = {±(ǫi − ǫj ) | 1 ≤ i 6= j ≤ n}. 1

2

STEPHEN GRIFFETH

Then An−1 is a root system in the subspace V ′ of V given by ) ( n n X X ai = 0 . ai ǫi ∈ V | V′ = x= i=1

i=1

The various restrictions on n are made in order to avoid redundancies. The classical reduced root systems are the root systems of types An , Bn , Cn , and Dn . They are all reduced and irreducible. The root system of type BCn is irreducible but not reduced, and contains all of the classical reduced root systems as subsystems. Exercise 2. Choose one of the sets An , Bn , Cn , Dn , or BCn above, and prove that the axioms for a root system are satisfied. The Weyl group of a root system R in V is the group W (R) generated by the reflections sα for α ∈ R: W (R) = hsα | α ∈ Ri.

(1.9)

It is a subgroup of the general linear group of V . It follows from the axioms that W (R) is a finite group. Exercise 3. Suppose R is a root system in V . Let 1 / R} and R1 = {α ∈ R | 2α ∈ / R} R0 = {α ∈ R | α ∈ 2 and check that R0 and R1 are root systems with R = R0 ∪ R1 and W (R) = W (R0 ) = W (R1 ). For instance, the Weyl group of type An−1 is the symmetric group Sn : (1.10)

W (An−1 ) = Sn ,

where Sn acts on V by permuting the coordinates, and on V ′ by restriction. Exercise 4. Check that W (An−1 ) = Sn , as described above, and give a description in terms of matrices (with respect to the basis ǫ1 , . . . , ǫn of V ) for the Weyl groups W (Bn ), W (Cn ), and W (Dn ). The dual of a root system R in V is the set R∨ = {α∨ | α ∈ R}.

(1.11)

Exercise 5. Prove that R∨ is a root system in V with the same Weyl group as R. Check that ∨ Cn = Bn∨ , and that A∨ n = An and Dn = Dn . A basis for a root system R is a subset B = {αi | 1 ≤ i ≤ n} ⊆ R, where n is the dimension of V , such that (a) B is a basis of V , P and (b) if α ∈ R and α = ni=1 ki αi , then k1 , . . . , kn are integers all of the same sign. A basis for An−1 is (1.12)

αi = ǫi − ǫi+1

for 1 ≤ i ≤ n − 1,

a basis for Bn is (1.13)

αi = ǫi − ǫi+1

for 1 ≤ i ≤ n − 1 and αn = ǫn ,

αi = ǫi − ǫi+1

for 1 ≤ i ≤ n − 1 and

a basis for Cn is (1.14)

αn = 2ǫn ,

a basis for Dn is (1.15)

αi = ǫi − ǫi+1

for 1 ≤ i ≤ n − 1 and

αn = ǫn−1 + ǫn ,

ROOT SYSTEMS AND COXETER GROUPS

3

and a basis for BCn is αi = ǫi − ǫi+1

(1.16)

for 1 ≤ i ≤ n − 1 and αn = ǫn .

Exercise 6. Choose one of the sets above, and prove that it is a basis for the corresponding root system. Exercise 7. Here we construct the root system E8 . It is one of the exceptional root systems. Let V = R8 with the standard basis ǫ1 , . . . , ǫ8 and let h·, ·i be the usual dot product. Define a subset L of V by ) ( 8 8 X X xi ∈ 2Z, 2xi ∈ Z, and xi − xj ∈ Z for 1 ≤ i, j ≤ 8 . xi ǫ i ∈ V | (1.17) L= i=1

i=1

Show that L is a lattice in V ; i.e., that it is a free abelian group on a basis of V . Show that hv, vi ∈ 2Z for all v ∈ L and L = {v ∈ V | hv, Li ⊆ Z}.

(1.18)

Let E8 be the set of length 2 vectors in L: E8 = {α ∈ L | hα, αi = 2}.

(1.19) Show that (1.20)

E8 = {±ǫi ± ǫj | 1 ≤ i < j ≤ 8} ∪

(

8 8 X 1X ai ∈ 2Z (−1)ai ǫi | 2 i=1

i=1

)

and that E8 is a reduced irreducible root system in V .

2. Affine root systems Let V be a real inner product space. A function f : V → R is affine linear if there exist α ∈ V and a constant m ∈ R such that f (x) = hα, xi + m

(2.1)

for all x ∈ V .

The linear function α and the constant m are uniquely determined by f , and we will write (2.2)

f = α + mc if α = Df and f (0) = m,

where c is the constant function 1 on V : c(v) = 1 for v ∈ V .

(2.3)

Let F be the set of all affine linear functions on V : F = {f : V → R | f is affine linear.}

(2.4)

Define a bilinear form on F , also denoted h·, ·i, by hf, gi = hDf, Dgi for f, g ∈ F .

(2.5) This form satisfies (2.6)

hf, f i ≥ 0,

and hf, f i = 0 exactly if f is a constant function.

A map s : V → V is affine linear if there exist λ ∈ V and a linear transformation Ds : V → V such that (2.7)

s(x) = Ds(x) + λ

for all x ∈ V .

The linear transformation Ds and vector λ are uniquely determined by this condition. In particular, we define the translation tλ by λ ∈ V via (2.8)

tλ (x) = x + λ

for x ∈ V .

4

STEPHEN GRIFFETH

Let a ∈ F be a non-constant function and define the reflection sa : V → V in the hyperplane Ha = {x ∈ V | a(x) = 0} by (2.9)

sa (x) = x − a∨ (x)Da

for x ∈ V , where a∨ =

2 a. ha, ai

The map sa is an affine linear isometry of V depending only on the hyperplaneH = Ha and not on the choice of a. The group A(V ) of invertible affine linear transformations g : V → V acts on F via (2.10)

g.f = f g−1

for f ∈ F and g ∈ A(V ).

Let α ∈ V and m ∈ R. Then one has the explicit formulas (2.11) (2.12)

tλ .(α + mc) = α + (m − hα, λi)c

for λ ∈ V and

g.(α + mc) = g(α) + mc for g ∈ O(V ),

where (2.13)

O(V ) = {g ∈ GL(V ) | hg(x), g(y)i = hx, yi

for all x, y ∈ V },

and for a non-constant function a ∈ F (2.14)

sa .f = f − ha∨ , f ia

for f ∈ F .

Exercise 8. Prove the three formulas above. An affine root system in V is a subset S ⊆ F such that (a) S consists of non-constant functions and spans F , (b) for all a ∈ S, sa .S ⊆ S, (c) for all a, b ∈ S, ha∨ , bi ∈ Z, and (d) the group W (S) generated by the reflections sa for a ∈ S acts properly on V . The Weyl group of the affine root system S is the group W (S) defined in (d) above. It is a subgroup of the group of isometries of V . We regard it as a discrete topological group, so condition (d) may be stated: for all compact subsets K, L ⊆ V , there are at most finitely many w ∈ W (S) with w.K ∩ L 6= ∅. An affine root system is reduced if 1 / S, (2.15) a ∈ S =⇒ a ∈ 2 and irreducible if there is not a non-trivial partition S = S 1 ∪ S 2 such that (2.16)

ha, bi = 0 for all a ∈ S 1 , b ∈ S 2 .

The dual of an affine root system S in V is the set (2.17)

S ∨ = {a∨ | a ∈ S}.

Exercise 9. Prove that S ∨ is an affine root system in V with the same Weyl group as S. Show that if S is an affine root system and 1 / S} and S1 = {a ∈ S | 2a ∈ / S} S0 = {a ∈ S | a ∈ 2 then S = S0 ∪ S1 and S0 and S1 are affine root systems in V with W (S) = W (S0 ) = W (S1 ). Let V be a real inner product space and let R ⊆ V be a reduced root system in V . The affine root system S(R) associated to R is (2.18)

S(R) = {α + mc | α ∈ R and m ∈ Z},

ROOT SYSTEMS AND COXETER GROUPS

5

where α + mc is regarded as an affine linear function on V via (α + mc)(x) = hα, xi + m.

(2.19)

It follows from (2.9) that the reflection sα+mc : V → V is given by the formula sα+mc (x) = x − hα∨ , xiα − mα∨ = t−mα∨ sα (x)

(2.20)

for x ∈ V ,

where for λ ∈ V the translation tλ : V → V is defined by (2.21)

tλ (x) = x + λ

for x ∈ V .

By (2.20) one has (2.22)

W (S(R)) = Q∨ ⋊ W (R),

with wtλ = tw.λ w for λ ∈ Q∨ and w ∈ W (R),

where Q∨ is the group of translations generated by all tα∨ for α ∈ R. The root lattice Q and coroot lattice Q∨ are the subgroups of V generated by R and R∨ , respectively: (2.23)

Q = Z-span{α | α ∈ R}

and Q∨ = Z-span{α∨ | α ∈ R}.

Exercise 10. Let R be a root system in V . Check that Q and Q∨ are lattices in V (i.e., check that they span V and have no limit points in V ). Prove that S(R) satisfies the axioms for an affine root system, which is always reduced, and is irreducible if R is. Our main example of a non-reduced affine root system is the affine root system (Cn∨ , Cn ). Using the notation from Section 1, (Cn∨ , Cn ) = {±ǫi ± ǫj + mc | 1 ≤ i < j ≤ n and m ∈ Z} 1 ∪ {±ǫi + mc | 1 ≤ i ≤ n and m ∈ Z} 2 ∪ {±2ǫi + mc | 1 ≤ i ≤ n and m ∈ Z}. We will always use the basis 1 a0 = −ǫ1 + c, a1 = ǫ1 − ǫ2 , . . . , an−1 = ǫn−1 − ǫn , an = ǫn , 2 which, as we will see in the following sections, corresponds to the “fundamental alcove” 1 (2.25) A = {(x1 , x2 , . . . , xn | > x1 > x2 > · · · > xn > 0}. 2 We have (2.24)

(Cn∨ , Cn ) = S(Cn )∨ ∪ S(Cn ),

(2.26) whence the notation.

3. Groups generated by Euclidean reflections Let V be a real inner product space. A hyperplane H in V is a set of the form (3.1)

H = {x ∈ V | f (x) = 0},

where f is a non-constant affine linear function on V .

We write H = Hf if H is the zero set of f as above; f is then determined by H up to multiplication by non-zero scalars. The reflection with respect to H is the function sH : V → V defined by (3.2)

sH (x) = x − f ∨ (x)Df

for x ∈ V , where H = Hf is the zero set of f , as above.

The function sH depends only on H, not on the choice of f . Throughout this section we assume that A is a collection of hyperplanes in V satisfying (a) For all H, H ′ ∈ A, sH .H ′ ∈ A, and (b) the group W (A) generated by the reflections sH for H ∈ A acts properly on V .

6

STEPHEN GRIFFETH

As in Section 2 we regard W (A) as a discrete topological group, so condition (b) means that for all compact sets K, L ⊆ V , there are finitely many w ∈ W (A) such that w.K ∩ L 6= ∅. Note that if R is a root system and S is an affine root system, then the sets of hyperplanes {Hα | α ∈ R} and {Ha | a ∈ S} satisfy these conditions. Thus results proved about A and W (A) apply to the hyperplane arrangements and Weyl groups of roots systems and affine root systems. By condition (b) on A, each point x ∈ V has a neighborhood that meets only finitely many of the hyperplanes in A (i.e., the set A is locally finite). The chambers of A are the connected components of the topological space [ (3.3) V◦ =V − H. H∈A

◦

By the local finiteness of A, the subset V ⊆ V is open. Define an equivalence relation ∼ on V by

x ∼ y ⇐⇒ {H ∈ A | x ∈ H} = {H ∈ A | y ∈ H} and x and y are on the same side of H for all H ∈ A with x, y ∈ / H. The faces of A in V are the equivalence classes for this relation. The linear span of a face is the smallest affine linear subspace of V containing it. The dimension of a face is the dimension of its linear span. The chambers are the faces of maximal dimension (equal to the dimension of V ) and the closure (in V ) F of a face F is the union of F and faces of strictly smaller dimension. Exercise 11. Draw pictures of the set of faces for the root system of type B2 and for the affine root system S(B2 ), and prove that the chambers are indeed the faces of maximal dimension and that the closure of a face F is the union of F and faces of strictly smaller dimension, as claimed above. A facet of a chamber C is a face contained in C whose linear span is a hyperplane in V . A wall of a chamber C is a hyperplane H that contains a facet of C. Each hyperplane H = Hf ⊆ V determines two open half-spaces Vf+ and Vf− by (3.4)

Vf+ = {x ∈ V | f (x) > 0}

and Vf− = {x ∈ V | f (x) < 0}.

Two subsets X and Y of V are on the same side of H if X, Y ⊆ Vf+ or X, Y ⊆ Vf− . Exercise 12. Prove that for any face F of A in V , any face F ′ ⊆ F − F is contained in the closure of a face F ′′ ⊆ F − F with dim(F ′′ ) = dim(F ) − 1. Prove that if C is a chamber then \ C= {x ∈ V | x is on the same side of H as C}. H a wall of C Exercise 13. Let g ∈ O(V ), λ ∈ V , and a ∈ F with Da 6= 0. Show that (tλ g)sa (tλ g)−1 = stλ g.a . Show that if H is a hyperplane in V and H ′ = tλ g(H) then sH ′ = (tλ g)sH (tλ g)−1 Proposition 3.1. Let C be a chamber of A, and let S be the set of reflections with respect to the walls of C. Then W (A) is generated by S and acts transitively on the set of chambers of A. Proof. Let W be the group generated by S, and fix a point x ∈ C. Choose a point y ∈ V ◦ . We will first show that w(y) ∈ C for some w ∈ W . Since W ⊆ W (A) acts properly on V any closed ball about x meets only finitely many points in the orbit W.y; thus there exists z = w(y) ∈ W.y such that (3.5)

d(x, z) ≤ d(x, z ′ ) for all z ′ ∈ W.y.

ROOT SYSTEMS AND COXETER GROUPS

7

Suppose H is a wall of C. We will show that z is on the same side of H as C. By moving the origin if necessary, we may assume that H passes through the origin of V so that sH is linear isometry of V , and there is a non-zero α ∈ V with H = Hα and sH (p) = p − hα∨ , piα

(3.6)

for p ∈ V .

Then (3.5) gives hx − z, x − zi ≤ hx − sH (z), x − sH (z)i, or upon canceling the terms hz, zi = hsH (z), sH (z)i and hx, xi from both sides and dividing by 2, −hx, zi ≤ −hx, sH (z)i = −hx, z − hα∨ , ziαi. Finally, canceling the term −hx, zi from both sides gives 0 ≤ hα∨ , zihα, xi, so that x and z are on the same side of H. Since H was arbitrary, w(y) = z ∈ C by exercise 12. It follows that for any chamber C ′ of A, there is some w ∈ W with w.C ′ = C. Now let H ′ be a hyperplane in A and choose a chamber C ′ that has H ′ for a wall and w ∈ W with w.C ′ = C. Then there is a wall H of C with w.H ′ = H, whence sH ′ = w−1 sH w ∈ W.

(3.7)

Since W (A) is generated by the reflections sH for H ∈ A the proposition is proved.



Lemma 3.2. Let H 6= H ′ be walls of C defined by affine linear functions f and f ′ , respectively. If f and f ′ are positive on C then hDf, Df ′ i ≤ 0. Proof. Let W ′ be the subgroup of W generated by sH and sH ′ , let V ′ be the span of Df and Df ′ and let W ′ act on V ′ by restriction. Let A′ = W ′ .{H ∩ V ′ , H ′ ∩ V ′ } and C ′ = {x ∈ V ′ | f (x), f ′ (x) > 0}. If some element of A′ meets C ′ , then it would meet C as well. We are thus reduced to the case of a vector space of dimension two (if Df and Df ′ are linearly independent; that is, if H ∩ H ′ 6= ∅) or one (if Df and Df ′ are parallel; that is, H and H ′ are parallel). In each case the result is easy to check (especially by drawing a picture).  From now on we suppose that A is the set of hyperplanes for a root system R or an affine root system S, and fix a chamber C of A. The set of positive roots with respect to C is (3.8)

R+ = {α ∈ R | hα, xi > 0 for all x ∈ C}

or

S + = {a ∈ S | ha, xi > 0 for all x ∈ C}

and the set of negative roots with respect to C is (3.9) R− = {α ∈ R | hα, xi < 0 for all x ∈ C.} or

S − = {a ∈ S | hα, xi < 0 for all x ∈ C.}

One has (3.10)

R = R+ ∪ R−

or S = S + ∪ S −

and

R− = −R+

or

S − = −S + .

We define sets R1 and S1 of indivisble roots by 1 1 / R} and S1 = {a ∈ S | a ∈ / S} (3.11) R1 = {α ∈ R | α ∈ 2 2 The sets of indivisible positive roots R1+ and S1+ are defined similarly (and coincide with the sets of positive roots in the corresponding root system R1 or S1 ). The basis of R with respect to C is the set (3.12)

B = {α ∈ R1+ | Hα is a wall of C}

or

B = {a ∈ S1+ | Ha is a wall of C}

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STEPHEN GRIFFETH

We fix index sets I0 and I with (3.13)

B = {αi | i ∈ I0 }

or for an affine root system

B = {ai | i ∈ I}.

We will prove that B is actually a basis of V (or F ) with some especially nice properties. The following corollary is a first step. Corollary 3.3. We have R ⊆ Z-span(B)

or

S ⊆ Z-span(B).

Proof. The proof for the two cases (R or S) is exactly the same, so we will give the proof for S (or, the R case follows from the S case by using S(R)). It suffices to prove that an indivisible root a ∈ S1+ can be written as an integer linear combination of ai ’s. Since W (S) acts transitively on the chambers by Proposition 3.1, there is some w ∈ W (S) with w.Ha a wall of C. Since a is indivisible, so is w(a), and it follows that w(a) = ±ai

(3.14)

for some i ∈ I.

On the other hand, it follows from Proposition 3.1 and the formula si (b) = b − ha∨ i , biai

(3.15)

that every element of the set W.{a0 , a1 , . . . , an } is an integer linear combination of a0 , a1 , . . . , an .  Lemma 3.4. (a) Let R be a finite root system in V with chamber C and corresponding basis αi for i ∈ I0 . If there is a proper non-empty subset J ⊆ I0 with hαi , αj i = 0

for i ∈ / J and j ∈ J

then R is not irreducible. (b) Let S be an affine root system in V with chamber C and corresponding basis ai for i ∈ I. If there is a proper non-empty subset J ⊆ I with hai , aj i = 0

for j ∈ J and i ∈ /J

then S is not irreducible. Proof. The two parts (a) and (b) are proved in the same way; we will prove (b). Let (3.16)

S0 = Z-span{aj | j ∈ J} ∩ S

and S1 = Z-span{ai | i ∈ / J} ∩ S.

By assumption S0 ⊥ S1 . Furthermore, each of S0 and S1 is W -stable: if i ∈ / J and j ∈ J then sai .aj = aj and if i, j ∈ J then sai .aj = ai − ha∨ , a ia ∈ S , proving that S0 is W -stable; j i 0 i symmetrically, S1 is also W -stable. If a ∈ S there is some w ∈ W and i ∈ J or i ∈ / J with a = w.aj ; hence a is in either S0 or S1 and it follows that (3.17)

S = S0 ∪ S1

is not irreducible. 

Theorem 3.5. (a) Let R be a root system in V . Then the vectors {αi }i∈I0 are linearly independent. (b) Let S be an irreducible affine root system in V . Then the vectors ai are linearly independent and there is a unique relation of the form X (3.18) mi ai = c0 , i∈I

where mi are positive integers without a common factor and c0 is a constant function on V .

ROOT SYSTEMS AND COXETER GROUPS

9

Proof. We first prove (a). Suppose there is a relation of the form X ki αi . i∈I0

Since hαi , αj i ≤ 0 for i 6= j we have X X X X |ki kj |hαi , αj i ≤ ki kj hαi , αj i = 0, |ki |αi i = |ki |αi , (3.19) 0≤h i∈I0

so

X

i,j∈I

i,j∈I0

i∈I0

|ki |αi = 0 and since αi (x) > 0 for x ∈ C we obtain ki = 0.

i∈I

The proof of (b) is similar. Suppose there is a relation of the form X (3.20) ki ai = c0 with c0 a constant function and ki not all zero. i∈I

Since hai , aj i ≤ 0 for i 6= j we have X X X X (3.21) 0≤h |ki |ai , |ki |ai i = |ki kj |hai , aj i ≤ ki kj hai , aj i = 0, i∈I

so

i∈I

X

i,j∈I

|ki |ai = c1

i,j∈I

for some constant c1 .

i∈I

We next show that |ki | > 0 for i ∈ I. Let J ⊆ I be the set of indices j for which |kj | > 0. If i∈ / J then the equation X X (3.22) 0= |kj | hai , aj i = |kj |hai , aj i j∈I

j∈J

and the inequalities hai , aj i ≤ 0 and |kj | > 0 imply that hai , aj i = 0. Since this holds for arbitrary j ∈ J and i ∈ / J, we must have J = I by irreducibility of S. In particular, if X ki ai = 0 i∈I

then there is a constant c0 with X X |ki |ai = c0 =⇒ (ki + |ki |)ai = c0 =⇒ ki > 0 for i ∈ I. i∈I

i∈I

But ai (x) > 0 for x ∈ C, a contradiction. Thus {ai }i∈I is a basis of F . From now on we assume I = {0, 1, . . . , n} where n = dim(V ). For the last assertion of the theorem, note that the map Rn+1 −→ P F n (m0 , m1 , . . . , mn ) 7−→ i=0 mi ai

is an isomorphism, and that the set of tuples (m0 , m1 , . . . , mn ) mapping to constant functions is therefore 1-dimensional, and by what was proved above it contains vectors with all positive entries. On the other hand, (m0 , m1 , . . . , mn ) maps to a constant function exactly if it is in the kernel of the matrix ha∨ i , aj ii,j∈I ; by the definition of an affine root system, this matrix has integer entries, whence there is a vector (m0 , m1 , . . . , mn ) with rational coordinates mapping to a constant function. By scaling, we may assume they are positive integers without common factors.  If R is a root system in V and α1 , . . . , αn is the basis of R corresponding to the chamber C, then the fundamental coweights are the dual basis ω1∨ , . . . , ωn∨ to α1 , . . . , αn : (3.23)

hαi , ωj∨ i = δij .

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STEPHEN GRIFFETH

Corollary 3.6. (a) Suppose that R is a root system in V and fix a chamber C with corresponding basis α1 , . . . , αn and fundamental coweights ω1∨ , . . . , ωn∨ . Then ) ( n X ∨ ti ωi | ti > 0 for 1 ≤ i ≤ n , C= i=1

and the set of elements of V that are non-negative on C is ) ( n X ki αi | ki ∈ R≥0 for 1 ≤ i ≤ n . i=1

In particular,

R+ =

( n X

)

ki αi ∈ R | ki ∈ Z≥0 for 1 ≤ i ≤ n .

i=1

(b) Suppose that S is an irreducible affine root system in V and fix a chamber C for S with corresponding basis a0 , . . . , an . There exist points xi ∈ V with ai (xj ) = δij c0 /mi , and ) ( n n X X ti = 1 and ti > 0 for 0 ≤ i ≤ n . t i xi | C= i=0

i=0

The set of elements of F that are non-negative on C is ) ( n X ki ai | ki ≥ 0 for 0 ≤ i ≤ n, i=0

and in particular

S+ =

(

n X

)

ki ai ∈ S | ki ≥ 0

for 0 ≤ i ≤ n .

i=0

Proof. The assertions in (a) are straightforward. We prove (b); use the notation from Theorem 3.5. By Theorem 3.5, for any proper subset J of {0, 1, . . . , n}, the elements Daj for j ∈ J are linearly independent, so there exist xi ∈ V with for j 6= i and hence ai (xi ) = c0 /mi . P Now for any affine linear function f on V , yi ∈ V , and ti ∈ R with ni=0 ti = 1, we have (3.24)

(3.25)

aj (xi ) = 0

n n X X ti f (yi ), ti y i ) = f( i=0

so each x ∈ V may be written uniquely as n X t i xi (3.26) x= i=0

i=0

with

Pn

i=0 ti

= 1,

and appying ai to this formula for x shows that x ∈ C exactly if ti > 0 for 0 ≤ i ≤ n. The remaining assertions of (b) are now straightforward.  Exercise 14. Show that the closure of C is C = {x ∈ V | hαi , xi ≥ 0 for 1 ≤ i ≤ n, or C = {x ∈ V | ai (x) ≥ 0 for 0 ≤ i ≤ n, as appropriate.

ROOT SYSTEMS AND COXETER GROUPS

11

4. The relationship between R and S(R) Let R be a root system in V with fundamental chamber C and basis B(C) = {α1 , α2 , . . . , αn }.

(4.1)

Let R+ be the set of positive roots with respect to C. The root lattice and coroot lattice of R are (4.2)

Q = Z-span{α ∈ R} and Q∨ = Z-span{α∨ ∈ R∨ },

and the weight lattice and coweight lattice of R are (4.3) P = {λ ∈ V | hα∨ , λi ∈ Z for all α ∈ R} and

P ∨ = {λ ∈ V | hα, λi ∈ Z

for all α ∈ R}.

The dominance order on V is the partial order ≤ defined by n X ′ ′ ki αi with ki ∈ Z≥0 for 1 ≤ i ≤ n. (4.4) v≤v ⇐⇒ v −v = i=1

A highest root for R is a maximal element of R with respect to dominance order. Lemma 4.1. Let α, β ∈ R be non-proportional roots and suppose hα∨ , βi > 0. Then for all 0 ≤ k ≤ hα∨ , βi, we have β − kα ∈ R. Proof. First observe that by the Cauchy-Schwartz inequality hα∨ , βihβ ∨ , αi = 4

(4.5)

hα, βi2 ≤ 4, hα, αihβ, βi

with equality exactly if α and β are proportional. Thus for non-proportional α, β with hα∨ , βi > 0 we must have hα∨ , βi = 1 or hβ ∨ , αi = 1. If hα∨ , βi = 1 there is nothing to prove. So we may assume hβ ∨ , αi = 1, whence (4.6)

sβ (α) = α − hβ ∨ , αiβ = α − β ∈ R

hα∨ , βi

If = 2 we are done; the other possibility is ∨ hα , βi = 1 by replacing β by β − α.

β − α ∈ R.

=⇒

hα∨ , βi

= 3, which we reduce to the case 

Proposition 4.2. Let R be an irreducible root system in V . Then R has a unique highest root φ. Furthermore, (a) φ ∈ C, (b) hφ, φi ≥ hα, αi for all α ∈ R, and (c) for every α ∈ R+ not proportional to φ, hφ∨ , αi ∈ {0, 1}. Proof. Suppose that φ is a maximal element of R with respect to dominance order. Evidently ∨ φ ∈ R+ . Also observe that hα∨ i , φi ≥ 0 for 1 ≤ i ≤ n: if hαi , φi < 0, then φ < φ − hα∨ i , φiαi = si (φ) ∈ R is a contradiction. Write (4.7)

φ=

n X

mi αi

with mi ∈ Z≥0 for 1 ≤ i ≤ n.

i=1

Let I be the set of 1 ≤ i ≤ n such that mi 6= 0. If i ∈ / I, then X (4.8) hα∨ mj hα∨ i , φi = i , αj i ≤ 0 j∈I

whence hα∨ i , αj i = 0 for all j ∈ I, contradicting Now suppose φ′ 6= φ is another maximal element

(4.9)

hφ′ , φi =

n X i=1

irreducibility of R. Thus mi > 0 for 1 ≤ i ≤ n. of R. We have

mi hφ′ , αi i > 0 since hφ′ , αi i > 0 for some i.

12

STEPHEN GRIFFETH

It follows from Lemma 4.1 that φ − φ′ ∈ R and φ′ − φ ∈ R, contradicting the maximality of φ and φ′ . We have proved the uniqueness assertion and part (a). For (b), assume α ∈ R. We have w(α) ∈ C for some w ∈ W , so we may assume α ∈ C. Now by (a) and maximality of φ we have hα, φ − αi ≥ 0

and hφ, φ − αi ≥ 0

and therefore (4.10)

hφ, φi ≥ hφ, φi − hφ, φ − αi = hφ, αi ≥ hφ, αi − hφ − α, αi = hα, αi,

proving (b). Now (c) follows from (a), (b), and Proposition ??.



From now on, we will refer to the chambers for an affine root system S in V as alcoves, and reserve the word chamber for the chambers of a finite root system R. Suppose R is an irreducible reduced root system in V , fix a chamber C of R, and let φ be the highest root of R with respect to C. Define the fundamental alcove of S(R) by (4.11)

A = {x ∈ V | hx, φi < 1 and hx, αi i > 0 for 1 ≤ i ≤ n.}

Proposition 4.3. The set A is an alcove of S(R) in V , the set of positive roots of S(R) with respect to A is S + = {α + mc | α ∈ R, m ∈ Z≥0 , and m ≥ 1 if α ∈ R− }, and the basis of S(R) with respect to A is B = {−φ + 1, α1 , α2 , . . . , αn }. Proof. We first show that no hyperplane Ha for a ∈ S(R) meets A. Suppose x ∈ A and (4.12)

α(x) = m

for some α ∈ R and m ∈ Z.

We may then α(x) < 0, and this is impossible. Thus α ∈ R+ . If Pn assume m ≥ 0. If α ∈ φ = i=1 mi αi is the highest root of R, we have by Proposition ?? R− ,

(4.13)

0 ≤ m = α(x) =

n X i=1

ki αi (x) ≤

n X

mi αi (x) = φ(x) < 1.

i=1

Thus m = 0, a contradiction since α(x) > 0. On the other hand, it follows from its definition that A is a union of the equivalence classes from (??), and since it is open, connected, and contained in V ◦ it must be an alcove. By Theorem ??, the set −φ + 1, α1 , . . . , αn is the basis of S(R) corresponding to A and the set of positive roots is as claimed.  As an example of these constructions for the root system of type Bn , we take the set C = {(x1 , . . . , xn ) ∈ Rn | x1 > x2 > · · · > xn > 0}

(4.14)

as a fundamental chamber. The basis with respect to this chamber is (4.15)

α1 = ǫ1 − ǫ2 , α2 = ǫ2 − ǫ3 , . . . , αn−1 = ǫn−1 − ǫn , αn = ǫn ,

the set of positive roots is R+ = {ǫi ± ǫj | 1 ≤ i < j ≤ n} ∪ {ǫi | 1 ≤ i ≤ n},

(4.16) and the highest root is (4.17)

φ = α1 + 2α2 + · · · + 2αn = ǫ1 + ǫ2 .

The fundamental alcove for S(Bn ) corresponding to C is therefore (4.18)

A = {(x1 , x2 , . . . , xn ) | x1 + x2 < 1 and x1 > x2 > · · · > xn > 0}

ROOT SYSTEMS AND COXETER GROUPS

13

and the basis of S(Bn ) is (4.19)

a0 = −ǫ1 − ǫ2 + 1, a1 = ǫ1 − ǫ2 , . . . , an−1 = ǫn−1 − ǫn , an = ǫn .

The simple reflections are the functions (4.20)

s0 (x1 , x2 , . . . , xn ) = (−x2 + 1, −x1 + 1, x3 , . . . , xn ),

(4.21)

si (. . . , xi , xi+1 , . . . ) = (. . . , xi+1 , xi , . . . ) for 1 ≤ i ≤ n − 1,

and sn (x1 , x2 , . . . , xn ) = (x1 , x2 , . . . , −xn ).

(4.22) The affine Weyl group is (4.23)

W (S(Bn )) = {tλ w | λ = (λ1 , . . . , λn ) ∈ Zn with λ1 + · · · + λn ∈ 2Z and w ∈ W (Bn )},

and the extended affine Weyl group (see Section 5) is c = {tλ w | λ ∈ Zn and w ∈ W (Bn )}. W

(4.24)

5. The extended affine Weyl group As in the previous section, we let R be an irreducible root system in V with coweight lattice P ∨ = {λ ∈ V | hαi , λi ∈ Z for 1 ≤ i ≤ n,} and fix a fundamental chamber C for R. Write φ for the corresponding highest root and α1 , . . . , αn positive roots defining the walls of C. Let S(R) be the affine root system in V associated to R, let A = {x ∈ V | hαi , xi > 0

and hφ, xi < 1}

be the fundamental alcove for S(R) in V , and let a0 , a1 , . . . , an and s0 , s1 , . . . , sn be the corresponding sets of simple roots and simple reflections, respectively. Let S + be the set of positive roots. Let L be a lattice in V with Q∨ ≤ L ≤ P ∨ . The extended affine Weyl group of R and L is the group (5.1)

W = L ⋊ W (R).

The affine Weyl group W (S) is a subgroup of W with quotient W/W (S) ≃ L/Q∨ .

(5.2) The inversion set of w ∈ W is

S(w) = {α ∈ S + | w(α) ∈ S − },

(5.3)

and the length of w ∈ W (S) is the minimum number p occurring in an expression for w as a product of simple reflections: (5.4)

l(w) = min{p ∈ Z≥0 | w = si1 · · · sip } for w ∈ W (S).

The subgroup Ω of W is (5.5)

Ω = {w ∈ W | |S(w)| = 0}.

Lemma 5.1. For v, w ∈ W , one has S(vw) ⊆ w−1 .S(v) ∪ S(w)

with equality exactly if w−1 .S(v) ⊆ S + .

14

STEPHEN GRIFFETH

Proof. First suppose a ∈ S(vw) but a ∈ / S(w). Then w(a) ∈ S + and vw(a) ∈ S − implies w(a) ∈ −1 S(v) and hence a ∈ w .S(v). This proves the first assertion. The “only if” part of the second assertion is clear. Conversely, suppose that w−1 .S(v) ⊆ S + . If a ∈ S(w) and vw(a) ∈ S + , then −w(a) ∈ S(v) whence −a ∈ w−1 .S(v), contradiction. Hence S(w) ⊆ S(vw). If a ∈ w−1 .S(v) then a ∈ S + and vw(a) ∈ S − , so w−1 .S(v) ⊆ S(vw).  Theorem 5.2. (a) For 0 ≤ i ≤ n, one has S(si ) = {ai }, (b) if u ∈ Ω, 0 ≤ i, i1 , . . . , ip , and usi1 · · · sip (ai ) ∈ S − , then usi1 · · · sip si = usi1 · · · sc ij · · · sip

for some 1 ≤ j ≤ p,

(c) for w ∈ W (S) and i ∈ {0, 1, . . . , n},

l(wsi ) < l(w)

⇐⇒

w(ai ) ∈ S − ,

(d) for any expression w = si1 · · · sip with p minimal, one has S(w) = {aip , sip (aip−1 ), sip sip−1 (aip−2 ), . . . , sip · · · si2 (ai1 )}. and hence l(w) = |S(w)|. (e) we have Ω ∩ W (S) = {1} and W = ΩW (S), and hence Ω ≃ L/Q∨ . P Proof. Let a = ni=0 ki ai ∈ S + , and suppose si (a) ∈ S − . Then X − (5.6) si (a) = ki ai + (ki − ha∨ i , ai)ai ∈ S j6=i

implies that kj = 0 for j 6= i. Since S(R) is reduced, a = ai , proving (a). If usi1 · · · sip (ai ) ∈ S − , there exists 1 ≤ j ≤ p with sij · · · sip (ai ) ∈ S −

(5.7)

and

sij+1 · · · sip (ai ) ∈ S + .

Thus (5.8)

aij = sij+1 · · · sip (ai ) and therefore sij = (sij+1 · · · sip )si (sij+1 · · · sip )−1 .

Rewriting the last equation proves (b). For (c), first assume that w.ai ∈ S − . Then by (b) if w = si1 · · · sip with p minimal we have

and therefore

wsi = si1 · · · sc ij · · · sip

for some 1 ≤ j ≤ p,

l(wsi ) < l(w).

Replacing w by wsi proves the converse. We prove (d) by induction on l(w). For l(w) = 0 it is trivial and for l(w) = 1 it is part (a). If w ∈ W (S) and l(wsi ) > l(w), then the inductive hypothesis and Lemma 5.1 imply S(wsi ) = {aip } ∪ sip S(w) = {aip , sip aip−1 , . . . , sip sip−1 · · · si2 ai1 } as claimed. For (e), we first note that the equality Ω ∩ W (S) = {1} follows from part (d). Finally, since W (S) acts transitively on the set of alcoves, for any w ∈ W we can find v ∈ W (S) with w.A = v.A, and it follows that v −1 w ∈ Ω, proving (e). 

ROOT SYSTEMS AND COXETER GROUPS

15

For an arbitrary element w ∈ W we define the length of w to be l(w) = |S(w)|.

(5.9) By the preceding theorem one has (5.10)

l(w) = min{p ∈ Z≥0 | there are u ∈ Ω and i1 , . . . , ip with w = usi1 · · · sip },

and if w = usi1 · · · sip with p minimal and u ∈ Ω, then (5.11)

S(w) = {aip , sip aip−1 , . . . , sip · · · si2 ai1 }.

Part (c) of Theorem 5.2 is sometimes referred to as the “weak exchange condition” for W . The next corollary strengthens this result, and is sometimes know as the “strong exchange condition”. Corollary 5.3. Let a ∈ S + and w ∈ W . Write w = usi1 · · · sip for some u ∈ Ω and 0 ≤ i1 , . . . , ip ≤ n. If w(a) ∈ S − , then wsa = usi1 · · · sc for some 1 ≤ j ≤ p. ij · · · sip

In particular, for w ∈ W and a ∈ S + ,

l(wsa ) < l(w)

⇐⇒

w(a) ∈ S − .

Proof. By Theorem 5.2, every element w ∈ W may be written in the indicated form. As in the proof of part (c) of Theorem 5.2, there is a 1 ≤ j ≤ p with sij sij+1 · · · sip (a) ∈ S −

and

sij+1 · · · sip (a) ∈ S + .

It follows that aij = sij+1 · · · sip (a)

and hence sij = (sij+1 · · · sip )sa (sij+1 · · · sip )−1 .

Rewriting this equation gives the first claim. Now observe that by part (d) of Theorem 5.2 and what was just proved, l(wsa ) < l(w) if w(a) ∈ S − . If w(a) ∈ S + then wsa (a) ∈ S − , and hence l(w) = l(wsa sa ) < l(wsa ).  A reduced word for w ∈ W is an expression w = usi1 · · · sip with p minimal, as in part (d) of Theorem ??. Our next result implies that W (S) is a Coxeter group. Part (a) will be fundamental in our study of the braid group of S(R). Theorem 5.4. (a) Let M be a monoid, and let f : {s0 , . . . , sn } → M be a function such that f (si )f (sj )f (si ) · · · = f (sj )f (si )f (sj ) · · · {z } | {z } | o(si sj ) factors o(si sj ) factors

for all 0 ≤ i < j ≤ n. For each w ∈ W (S) and each reduced word w = si1 . . . sip for w, define f (si1 , si2 , . . . , sip ) = f (si1 )f (si2 ) · · · f (sip ). Then this extension of f to the set of reduced words is constant. (b) Let G be a group and suppose f : {s0 , s1 , . . . , sn } → G is a map such that (f (si )f (sj ))o(si sj ) = 1

for all 0 ≤ i, j ≤ n.

Then there is a unique extension f : W (S) → G of f to a group homomorphism.

16

STEPHEN GRIFFETH

Proof. We prove (a) by induction on l(w). For l(w) = 1 it is clear. Suppose l(w) > 1 and that si1 si2 · · · sip

(5.12)

and sj1 sj2 · · · sjp

are two reduced decompositions of w. If i1 = j1 or ip = jp , then by induction we have f (si2 )f (si3 ) · · · f( sip ) = f (sj2 )f (sj3 ) · · · f( sjp ) or

f (si1 )f (si2 ) · · · f( sip−1 ) = f (sj1 )f (sj2 ) · · · f( sjp−1 )

and the result follows from either of these equations. If ip 6= jp then by part (c) of Theorem ?? so that

si1 si2 · · · sip−1 = sj1 sj2 · · · sc jk · · · sjp

for some 1 ≤ k < p

si1 si2 · · · sip−1 sip = sj1 sj2 · · · sc jk · · · sjp sip are both reduced words for w. If k 6= 1, then by the above reasoning we’re done, since the right hand side has the same first letter as sj1 · · · sjp and the same last letter as si1 · · · sip . Thus we may assume k = 1 so that w = sj2 sj3 · · · sjp sip

and symmetrically

w = si2 si3 · · · sip sjp .

It suffices to prove that f takes the same value on these two words. Continuing in this fashion, we see that it suffices to prove that f takes the same value on reduced words of the form w = · · · sjp sip sjp

and w = · · · sip sjp sip ,

with p factors. It follows that o(sip sjp ) divides p, but since the word is reduced, we must actually have p = o(sip sjp ). The result in (a) follows. The result from (a) implies that in the situation of part (b), we may define an extension of f to W (S) by the formula f (w) = f (si1 ) · · · f (sip ) for any reduced word w = si1 · · · sip for w. We must prove that this map is a group homomorphism. It suffices to show that f (wsi ) = f (w)f (si ) for all w ∈ W (S) and 0 ≤ i ≤ n. If l(wsi ) > l(w), the preceding equation is true by definition of f . If l(wsi ) < l(w), then by the case just treated, f (w) = f (wsi si ) = f (wsi )f (si ) and since f (si )2 = 1 by hypothesis, f (w)f (si ) = f (wsi ). The uniqueness of the extension follows from the fact that s0 , s1 , . . . , sn generate W (S).



Remark 5.5. Part (b) of the theorem may be paraphrased as “The group W (S) has a presentation by generators s0 , s1 , . . . , sn and relations (si sj )o(si sj ) = 1”. Thus (W (S), {s0 , . . . , sn }) is a Coxeter system. 6. Bruhat order Let R be an irreducible reduced root system and Q∨ ≤ L ≤ P ∨ a lattice as in the previous section and let W be the extended affine Weyl group of R and L. Let S = S(R) be the affine root system associated to R and suppose that S + is its set of positive roots with respect to a fixed fundamental alcove A. Let b ∈ S + , let sb be the corresponding reflection, and let w ∈ W . We write b

w → wsb

(6.1)

if l(wsb ) = l(w) + 1.

The Bruhat order is the partial order ≤ on W defined by (6.2)

v≤w

if there is a chain

b

b

bp

1 2 vsb1 → v→ · · · → vsb1 sb2 · · · sbp = w.

Since wsb = sw(b) w the definition is left-right symmetric. Lemma 6.1 (“Property Z” of Bruhat order). Let v, w ∈ W and suppose 0 ≤ i ≤ n is such that vsi < v and wsi < w. The following are equivalent:

ROOT SYSTEMS AND COXETER GROUPS

17

(a) v ≤ w, (b) vsi ≤ w, and (c) vsi ≤ wsi . Proof. By induction on l(w). For l(w) = 1 the three conditions all amount to v = si . That (a) implies (b) follows from vsi < v ≤ w. Assume (b) holds. Then there is b ∈ S + with vsi ≤ sb w and l(sb w) = l(w) − 1. If w−1 (b) = −ai then (c) holds, so we may assume w−1 (b) 6= −ai . We have (wsi )−1 (b) = si (w−1 (b)) ∈ S − and so by Corollary 5.3 we have l(sb wsi ) < l(wsi ) = l(w) − 1. On the other hand, l(sb wsi ) = l(sb w) ± 1 = l(w) − 1 ± 1, whence l(sb wsi ) = l(wsi ) − 1

and therefore

sb wsi < wsi and sb wsi < sb w.

By induction vsi ≤ sb wsi < wsi , so (c) holds. Finally, assuming (c) holds there is b ∈ S + with vsi ≤ sb wsi and l(sb wsi ) = l(wsi ) − 1. If sb wsi < sb w then by induction v ≤ sb w ≤ w proving (a) in this case. If sb w < sb wsi then again by induction v ≤ sb wsi ≤ wsi ≤ w, and we are finished.  Theorem 6.2. Let v, w ∈ W , and let w = usi1 · · · sip , where u ∈ Ω, be a reduced decomposition for w. The following are equivalent: (a) v ≤ w, (b) there exists a subsequence j1 , j2 , . . . , jq of 1, 2, . . . , p such that v = usij1 sij2 · · · sijq

is a reduced word for v,

(c) there exists a subsequence j1 , j2 , . . . , jq of 1, 2, . . . , p such that v = usij1 sij2 · · · sijq . Proof. Assume v ≤ w. We prove that condition (b) holds by induction on l(w). If l(w) = 0 or if v = w the result is clear. Otherwise, by definition of Bruhat order there is a ∈ S + with l(wsa ) = l(w) − 1

and v ≤ wsa ,

and there is some 1 ≤ j ≤ p such that wsa = usi1 · · · sc ij · · · sip

is a reduced word for wsa .

Part (b) follows from this by the inductive hypothesis. It is clear that (b) implies (c). We show that (c) implies (a). Again we proceed by induction on l(w). If l(w) = 0 we are done. If jq 6= p then using the inductive hypothesis and the definition of Bruhat order we have v ≤ usi1 · · · sip−1 ≤ w. If jq = p, then by induction vsip ≤ wsip and by Lemma 6.1 v ≤ w.



Corollary 6.3. If b ∈ S + and w ∈ W , then wsb < w exactly if l(wsb ) < l(w). Proof. If wsb < w then it follows from the definition of Bruhat order that l(wsb ) < l(w). The converse implication follows from Corollary ?? and the characterization of Bruhat order in part (c) of Theorem 6.2.  The Bruhat order on W0 = W (R) is obtained by restriction. Exercise 15. Draw the Hasse diagrams of the Bruhat orders on W (A1 ), W (A2 ), and the extended affine Weyl group of type A1 .

18

STEPHEN GRIFFETH

7. Length computations and coset representatives Let R be a reduced and irreducible root system in the real inner product space V , let W0 = W (R) be the Weyl group of R, fix a lattice Q∨ ≤ L ≤ P ∨ , and let W = L ⋊ W0 be the extended affine Weyl group of R and L. We fix a fundamental chamber C for R and let α1 , . . . , αn , s1 , . . . , sn , and R+ be the corresponding sets of simple roots, simple reflections, and positive roots. Let S(R) be the affine root system associated to R and let A be the fundamental alcove for S(R) associated to C. Let a0 = −φ + 1 and a1 = α1 , . . . , an = αn be the corresponding set of simple roots for S(R). Let S + be the set of positive roots for S with respect to A. Define the function χ : R → {0, 1} by ( 1 if α ∈ R− , and (7.1) χ(α) = 0 if α ∈ R+ . Proposition 7.1. Let λ ∈ L and let w ∈ W0 . Then S(wtλ ) = {α + kc ∈ S | χ(α) ≤ k ≤ hλ, αi + χ(w.α) − 1}, S(tλ w) = {α + kc ∈ S | χ(α) ≤ k ≤ hλ, w.αi + χ(w.α) − 1}, X X l(wtλ ) = |hλ, αi + χ(w(α))|, and l(tλ w) = |hλ, αi − χ(w−1 (α))| α∈R+

α∈R+

Proof. Let α ∈ R and k ∈ Z. We have tλ w.(α + kc) = tλ .(w.α + kc) = w.α + (k − hλ, w.αi)c and hence S(tλ w) = {α + kc ∈ S | χ(α) ≤ k ≤ hλ, w.αi + χ(w.α) − 1}. Similarly, wtλ .(α + kc) = w(α) + (k − hλ, αi)c, and it follows that S(wtλ ) = {α + kc ∈ S | χ(α) ≤ k ≤ hλ, αi + χ(w(α)) − 1}. Thus l(wtλ ) = |S(wtλ )| = =

X

X

max{hλ, αi + χ(w(α)) − χ(α), 0}

α∈R

max{hλ, αi + χ(w(α)) − χ(α), 0} + max{hλ, −αi + χ(−w(α)) − χ(−α), 0}

α∈R+

=

X

max{hλ, αi + χ(w(α)) − χ(α), 0} + max{−(hλ, αi + χ(w(α)) − χ(α)), 0}

α∈R+

=

X

|hλ, αi + χ(w(α))|,

α∈R+

proving the first equality. The second follows from X X l(tλ w) = l(w−1 t−λ ) = |h−λ, αi + χ(w−1 (α))| = |hλ, αi − χ(w−1 (α))|. α∈R+

α∈R+



Proposition 7.2. Let λ ∈ L and let λ− be an element of W0 .λ satisfying hλ− , αi ≤ 0 for all α ∈ R+ . Such a λ− always exists since W0 acts transitively on the chambers C of R.

ROOT SYSTEMS AND COXETER GROUPS

19

(a) There is a unique minimal length element vλ ∈ W0 = W (R) such that vλ .λ = λ− . Furthermore, R(vλ ) = {α ∈ R+ | hλ, αi > 0}. (b) The element λ− is the unique element of the orbit W0 .λ satisfying hλ− , αi ≤ 0 for all α ∈ R+ . (c) The element uλ = tλ vλ−1 is the unique element of minimal length in W such that uλ .0 = λ, and S(uλ ) = {α + kc | α ∈ R− and 1 ≤ k < hλ, vλ−1 .αi + χ(vλ−1 .α)}. (d) We have l(uλ ) + l(vλ ) = l(tλ ). Proof. For (a), let v ∈ W0 have minimal length subject to v.λ = λ− . We will prove that the inversion set of v is R(v) = {α ∈ R+ | hλ, αi > 0}.

(7.2)

Suppose α ∈ R+ and hλ, αi > 0. Then 0 < hλ, αi = hv.λ, v.αi = hλ− , v.αi =⇒ v.α ∈ R− .

(7.3)

For the reverse containment, let v = si1 · · · sip be a reduced word so that R(v) = {αip , sip .αip−1 , . . . , sip · · · si2 αi1 }.

(7.4) Observe that (7.5)

hλ, sip . . . sij+1 αij i = hv.λ, si1 · · · sij .αij i = hλ− , −si1 · · · sij−1 αij i ≥ 0.

On the other hand, if hλ, sip · · · sij+1 αij i = 0 we obtain a contradiction with the minimality of l(v). This proves (a). Part (b) follows from (a) and the fact that the inversion set of w ∈ W0 determines w. For (c) we observe that by Proposition 7.1 l(tλ w) is minimized if χ(w−1 α) = 1

⇐⇒

hλ, αi > 0,

that is, exactly if w = v(λ)−1 . Since the set of w ∈ W with w.0 = λ is the coset tλ W0 , this proves the first assertion of (c). The second follows from the description of S(tλ w) given in Proposition 7.1. Finally, we observe that by (c) and Proposition 7.1 X X
l(uλ ) = hλ− , αi + χ(vλ−1 .α) − 1 = l(tλ− ) − χ(vλ−1 .α) = l(tλ ) − l(vλ−1 ), α∈R+

α∈R−

and using l(vλ ) = l(vλ−1 ) proves (d).



We can use Proposition 7.2 to obtain an explicit description of the group Ω = {w ∈ W | l(w) = 0}. A coweight λ ∈ L is cominiscule if (7.6)

hλ, αi ∈ {0, 1}

for all α ∈ R+ .

The fundamental coweights are the elements πi ∈ P ∨ defined for 1 ≤ i ≤ n by (7.7)

hπi , αj i = δij

for 1 ≤ i, j ≤ n,

and (7.8)

π0 = 0.

Exercise 16. Let λ be a cominiscule coweight. Show that λ = πj for some 0 ≤ j ≤ n. If φ is the highest root of R, show that hλ, φi = 1.

20

STEPHEN GRIFFETH

Let J ⊆ {0, 1, 2, . . . , n} be the set given by J = {0 ≤ i ≤ n | πi ∈ L and πi is cominiscule},

(7.9) and for j ∈ J write (7.10)

uj = uπ j

and vj = vπj .

Corollary 7.3. We have Ω = {uj | j ∈ J}, and the map L/Q∨ −→ Ω πj 7−→ uj is an isomorphism of groups. In particular, Ω is a finite abelian group. Proof. Let u ∈ Ω. Evidently u is the minimal length element of the coset uW0 , and it follows from Proposition 7.2 that u = uλ where λ = u.0. On the other hand, by parts (a) and (d) of Proposition 7.2 it follows that X (7.11) l(uλ ) = l(tλ ) − l(vλ ) = |hλ, αi| − χ(vλ .α), α∈R+

whence

l(uλ ) = 0

⇐⇒

for α ∈ R+ , |hλ, αi| =

(

if hλ, αi > 0, and otherwise.

1 0

⇐⇒ λ is cominiscule.

This proves the first assertion. Now observe that modulo W (S), we have t π j = uj

(7.12)

and tλ tµ = tν

if λ + µ = ν mod Q∨ ;

hence (7.13)

uj uk = tπj vj−1 tπk vk−1 = tπl = ul mod W (S) if πj + πk = πl mod Q∨ .

The natural map Ω → W/W (S) is an isomorphism by Theorem ?? part (e), so we are finished.  Exercise 17. For the root systems of type An and Bn and a fixed choice of fundamental chamber, describe set of cominiscule coweights and the group Ω as a group of affine linear symmetries of V . Define α0 by α0 = a0 − c,

(7.14) so that α0 = −φ is the lowest root of R.

Corollary 7.4. Let j ∈ J. Then uj permutes the set {a0 , a1 , . . . , an } of simple roots, and we have the formulas (7.15)

u−1 j .aj = a0

and (7.16)

vj (αi ) = αk

if u−1 j .ai = ak .

Proof. Since l(uj ) = 0 it fixes the alcove A and hence permutes {a0 , a1 , . . . , an }. If j = 0 the first formula is trivial since u0 = 1. Assuming j 6= 0 and evaluating u−1 j .aj at 0 gives (7.17)

(u−1 j .aj )(0) = aj (πj ) = hαj , πj i = 1

=⇒

u−1 j .aj = a0 ,

since a0 is the unique simple root with a0 (0) = 1. The last assertion of the corollary is true if j = 0 since v0 = u0 = 1. Now assume j ∈ J and i 6= j, 0. Then (7.18)

−1 −1 vj (αi ) = u−1 j tπj .αi = uj .αi = uj .ai = ak

ROOT SYSTEMS AND COXETER GROUPS

21

This proves the last assertion of the corollary in this case. If i = j we have −1 vj (αj ) = u−1 j tπj .αj = uj (αj − c) = a0 − c = α0 .

(7.19) Finally if i = 0 we have

−1 −1 vj (α0 ) = u−1 j tπj .(−φ) = uj .(−φ + c) = uj .a0 ,

(7.20) and we are finished.



An consequence of the corollary is the formula (7.21)

vj vk = vl

if uj uk = ul ,

since acting by the left hand side on αi gives the same result as acting by the right hand side, and α1 , . . . , αn span V . References [Bou]

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6.

Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, E-mail address: [email protected]