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ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS

PAOLA CELLINI AND MARIO MARIETTI

Abstract. We tackle several problems related to a finite irreducible crystallographic

root system Φ in the real vector space E. In particular, we study the combinatorial

structure of the subsets of Φ cut by affine subspaces of E and their projections. As

byproducts, we obtain easy algebraic combinatorial proofs of refinements of Oshima’s

Lemma and of a result by Kostant, a partial result towards the resolution of a problem

by Hopkins and Postnikov, and new enumerative results on root systems.

Keywords: Root system; Root polytope; Weyl group

1. Introduction

Let Φ be a finite irreducible crystallographic root system and let Π be a set of simple

roots of Φ. Given a subset S of Π and β ∈ Φ, with β =∑α∈Π

cα(β)α, we set

ΦS,β =γ =

∑

α∈Π

cα(γ)α ∈ Φ : cα(γ) = cα(β) for all α ∈ S,

ΦS,Zβ =γ =

∑

α∈Π

cα(γ)α ∈ Φ : ∃k ∈ Z with cα(γ) = kcα(β) for all α ∈ S.

If Supp(β) ∩ S = ∅, the set ΦS,β coincides with ΦS,Zβ and is the standard parabolic

subsystem generated by Πr S. So the nontrivial case happens when Supp(β) ∩ S 6= ∅.

By definition, ΦS,β is the intersection of Φ with the affine subspace that contains β

and has Span(Πr S) as directional linear subspace. The subset ΦS,Zβ is, in fact, a root

subsystem of Φ and is a union of strata each obtained by intersecting Φ with an affine

subspace parallel to Span(Πr S).

In this paper, we study several problems concerning the subsets ΦS,β and ΦS,Zβ and

their applications.

Section 2 provides the notational conventions used throughout the paper.

In Section 3, we give a uniform case-free description of the root polytope of an arbitrary

(possibly reducible) root system. The root polytope of a root system is the convex hull of

all roots and, in the irreducible case, it is studied in [2], [3], [4]. The results in this section

are needed in Section 7.

©2021. This manuscript version is made available under the CC-BY-NC-ND 4.0 license

https://creativecommons.org/licenses/by-nc-nd/4.0/

1

2 PAOLA CELLINI AND MARIO MARIETTI

In Section 4, we study the root subsystem ΦS,Zβ and find two of its simple systems that

provide in a natural way the existence of the minimum and the maximum of ΦS,β viewed

as a subposet of the root poset.

As consequences of the results in this section, we obtain easy case-free algebraic com-

binatorial proofs of a result by Kostant, of a result by Oshima, and some refinements.

Kostant’s result, which is presented in [9, Section 2], asserts the irreducibility of a certain

representation of the semisimple subalgebra gΠrS associated with ΠrS (see Remark 4.3).

Oshima’s result [10, Lemma 4.3], which is Corollary 4.7 in this paper, has received atten-

tion recently for its applications and is referred to as “Oshima’s Lemma”. The purpose

of Dyer and Lehrer in [5] is to provide an elementary self-contained proof of (a gener-

alised version of) Oshima’s Lemma. Our proof here is very elementary, self-contained,

and independent of the representation theory of semisimple complex Lie algebras (see [5]).

Oshima’s Lemma states that the action of W 〈ΠrS〉 partitions the roots in ΦS,β according

to their length, hence in at most two orbits. In the representation theory interpretation,

this means that the weights of the irreducible representation studied by Kostant are di-

vided in at most two orbits. Indeed, the fact that the roots of maximal length constitute

a single orbit follows directly from the irreducibility of this representation. We provide

general necessary and sufficient conditions for establishing whether ΦS,β contains roots

of one or two lengths (Propostion 4.12). Moreover, we provide an easy combinatorial

criterion on the pair (S, β) for ΦS,β to contain long roots (Proposition 4.14). Though this

last criterion is uniform, our proof requires some case by case check.

In Sections 5 and 6, we study in more detail the case of codimension 1, when S consists

of a single simple root α. In this case, we obtain a partition of Φ into subsets Φα,k =

Φ∩x ∈ E : (x, qωα) = k that are all intervals w.r.t. the usual partial order, except for the

0-level Φα,0, which is a possibly reducible root subsystem. In particular, we analyse the

projections of the subsets Φα,k on the hyperplane Span(Πrα). As a consequence of that

analysis, we obtain a partial result towards the resolution of a problem by Hopkins and

Postnikov. In [8], Hopkins and Postnikov show that the projection of the root polytope

of Φ fall inside the polytope formed by expanding by a factor of 2 the root polytope

of the root subsystem generated by Π r α (see Lemma 7.1 ). Their proof uses the

classification of root systems and finding a case-free proof is an open problem in [7]. In

Section 7, we give a uniform case-free argument that proves that the result of Hopkins

and Postnikov is equivalent to the fact that a certain number rα, defined by a uniform

formula (7.5.1), is smaller than 2, for each simple root α. Unfortunately, this final check

is done case-by-case.

In Section 8, we give some enumerative results on irreducible root systems.

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 1 3

Except for the last part of the proof of Proposition 4.14 (which is not needed in the

rest of the paper), its Lemma 4.13, and the final check in Section 7 discussed above, all

uniform statements are proved independently of the classification of root systems.

2. Notational conventions

Let Φ be a finite (reduced, possibly reducible) crystallographic root system in the real

vector space E = SpanRΦ endowed with the positive definite bilinear form ( · , · ). We

identify E with its dual space via its bilinear form.

We fix our notation on the root system and its Weyl group in the following list:

Φ∨ the set of coroots of Φ,

α∨ the coroot associated with the root α ∈ Φ,

Π the set of simple roots of Φ,

Π∨ the set of simple coroots of Φ,

Ω = ωα : α ∈ Π the set of fundamental weights (the dual basis of Π∨),

qΩ = qωα : α ∈ Π the set of fundamental co-weights (the dual basis of Π),

Φ+ the set of positive roots w.r.t. Π,

C the fundamental chamber associated with Π,

cα(β) the α-coordinate of β w.r.t. Π: β =∑

α∈Π cα(β)α,

Supp(α) = αi ∈ Π : ci(α) 6= 0, the support of α,

W the Weyl group of Φ,

sα the reflection with respect to α,

W 〈S〉 the subgroup of W generated by sα : α ∈ S (for S ⊆ Φ),

Φ〈S〉 the root subsystem generated by S (for S ⊆ Φ),

PΦ the convex hull of all roots in Φ, called the root polytope of Φ.

For any x ∈ E, x dominant means x ∈ C.

The root poset of Φ (w.r.t. the basis Π) is Φ+ with the standard partial order, i.e., for

all α, β ∈ Φ+ , α ≤ β if and only if β − α is a nonnegative linear combination of roots in

Φ+. The root poset is the transitive closure of the relation α⊳ β if and only if β − α is a

simple root. The root poset hence is ranked by the height function.

In the irreducible case, we denote by θ the highest root in Φ, which is the maximum of

the root poset, and by mα the coefficient cα(θ), for α ∈ Π. Furthermore, we denote by Φ

the affine root system associated to Φ and realized as follows. We extend E to a vector

space E⊕Rδ, and extend the bilinear form of E to a semidefinite positive form on E⊕Rδ

by the condition that Rδ is the kernel of the new form. Then we set:

4 PAOLA CELLINI AND MARIO MARIETTI

α0 = −θ + δ,

Π = α0 ∪Π,

Φ = β + kδ : β ∈ Φ, k ∈ Z the affine root system with simple system Π,

Φ+ = β + kδ : β ∈ Φ+, k ∈ Z≥0 ∪ −β + kδ : β ∈ Φ+, k ∈ Z>0 the set of

(real) positive roots of Φ w.r.t. Π.

For k ∈ N, we let [1, k] = 1, . . . , k.

3. Root polytopes of possibly reduced root systems

Recall that we denote by PΦ the convex hull of all roots in Φ, and we call it the root

polytope of Φ. For all irreducible Φ, a uniform explicit description of the root polytope

PΦ is given in [3]. In this section, we extend some of the results in [3] to the reducible

case.

Let us first recall some results from [3]. If G is a graph with set of vertices V and U is

a subset of V , we denote by G r U the subgraph of G induced by the subset of vertices

V r U .

Definition 3.1. Let Φ be irreducible and Γ be its extended Dynkin diagram. For any

subset I of Π, we say that I is Π-extremal if Γr I is connected.

We say that the root α is Π-extremal if α is.

Remark 3.2. (1) If Φ is of type An, all roots in Π are Π-extremal. For all other

types, the extremal roots are the leaves of the extendend Dynkin diagram other

than the affine vertex.

(2) Let θ be the highest root of Φ (supposed irreducible). A subset I is Π-extremal

if and only if −θ ∪Πr I is the simple system of an irreducible root subsystem

of Φ.

Let Φ be irreducible with highest root θ =∑

α∈Π mαα, and set oα = qωα/mα for all

α ∈ Π.

Given I ⊆ Π, we set

FI = Convγ ∈ Φ : (γ, oα) = 1 for all α ∈ I

Proposition 3.3, Proposition 3.5, and Theorem 5.5 of [3] imply the following result.

Theorem 3.3. Let Φ be irreducible.

(1) For each Π-extremal subset I, the set FI is a face of PΦ of dimension rk(Φ)− |I|.

(2) For each subset I of Π, there exists a Π-extremal subset I such that FI = FI .

(3) For each face F of PΦ, there exist some w ∈ W and a unique Π-extremal subset I

such that F = w(FI).

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 2 5

Following [3], we call the faces of type FI the standard parabolic faces. Note that the

standard parabolic facets are exactly the faces of type Fα where α is a Π-estremal root.

Remark 3.4. When Φ is irreducible, for each subset I of Π, there exists a root α ∈ I

such that α is not orthogonal to Π r I. If I is Π-extremal, for such a root α we have

that I r α is Π-extremal too, if nonempty. This implies that any standard parabolic

face either is a facet, or is contained in a standard parabolic face of higher dimension. In

particular, each standard parabolic face is contained in a standard parabolic facet.

Now we suppose that Φ be possibly reducible.

For any face F of PΦ, we denote byF the relative interior of F , i.e. its interior within

the affine hull of F . The relative interiorF of F consists of all strictly positive convex

linear combinations of the vertices of F . If F and F ′ are different faces, thenF ∩

F ′ = ∅.

Definition 3.5. Let F be a face of PΦ. We call F standard provided that eitherF∩C 6= ∅,

or F = ∅.

The Weyl group W acts on the faces of PΦ. We prove that the standard faces are a

set of representatives of the orbits of this action. Given a face F with vertex set V , we

denote by βF the barycenter of V , i.e. βF = 1|V |

∑v∈V v.

Theorem 3.6. Let Φ be a (possibly reducible) root system and let F be a face of PΦ.

(1) F is standard if and only if the barycenter βF is dominant. In this case, the

stabilizer StabW (F ) of F is the standard parabolic subgroup of W generated by

Π ∩ β⊥F .

(2) The orbit W · F of F under the action of W contains exactly one standard face.

Proof. (1) Given γ ∈ Φ, we denote by sγ the reflection through γ and byHγ the hyperplane

fixed by sγ.

For each w ∈ W , if w(F ) 6= F thenF ∩ w(

F ) = ∅. In particular, for each γ ∈ Φ, since

Hγ ∩F is fixed by sγ, if sγ(F ) 6= F then Hγ ∩

F = ∅, and

F lies in one of the two open

half spaces determined by Hγ . If, in addition, γ > 0 andF ∩ C 6= ∅, then (x, γ) > 0 for

all x ∈F ∩ C, and hence (x, γ) ≥ 0 for all x ∈ F .

The stabilizer StabW (F ) of F coincides with the stabilizer of the baricenter βF , hence

StabW (F ) is the parabolic subgroup of W generated by sγ : (γ, βF ) = 0. We have just

seen that, if F is standard, then (γ, βF ) > 0 for all γ 6∈ StabW (F ), hence βF ∈ C.

Conversely, if βF ∈ C, then F is standard by definition, since βF ∈F .

Furthermore, if βF ∈ C, then StabW (F ) is a standard parabolic subgroup, namely,

StabW (F ) = W 〈Π ∩ β⊥F 〉.

(2) Since C is a fundamental domain for W , there is one and only one face in W · F

with dominant barycenter, hence W · F contains exactly one standard face by (1).

6 PAOLA CELLINI AND MARIO MARIETTI

As a direct consequence of Theorem 3.6, we have the following result in the irreducible

case.

Corollary 3.7. Let Φ be irreducible. The standard faces of PΦ are exactly the standard

parabolic faces.

Proof. Since the barycenters of the standard parabolic faces are dominant (see [3, Propo-

sition 4.4]), the statement follows by Theorem 3.6.

As we see below, each dominant point belongs to the cone of some standard face.

Proposition 3.8. Let Φ be a (possibly reducible) root system and let F be a face of PΦ

such that F ∩ C 6= ∅. Then there is a standard subface F ′ of F such that F ∩ C = F ′ ∩ C.

Proof. If F is a vertex of PΦ, then F is standard. If F is not standard, then there is a

proper subface F ′ of F such that F ∩ C = F ′ ∩ C. The assertion follows by induction on

the dimension of F .

Proposition 3.8 directly implies the following result.

Corollary 3.9. Let Φ be a (possibly reducible) root system and let ∂PΦ be the boundary

of PΦ, i.e. the union of its faces. Then C ∩ ∂PΦ is contained in the union of the standard

faces. In particular, for each nonzero x ∈ C, there exist a positive real number r and a

standard face F such that rx ∈ F .

We now determine explicitly the standard faces in the case Φ is any (possibly reducible)

root system.

Let Φ1, . . . ,Φk be the irreducible components of Φ, Ei = Span(Φi), Πi = Π ∩ Φi,

Ci = C ∩ Ei, and let Pi = PΦ ∩ Ei be the root polytope of Φi. We say that x ∈ E is

Φi-dominant provided that x ∈ Ci.

Let F be a face of PΦ and

Fi = F ∩ Ei.

If F = PΦ, then Fi = Pi for all i ∈ [1, k]. Conversely, if F 6= PΦ, then Fi 6= Pi for

all ∈ [1, k]. By a proper face, we intend a possibly empty face different from the whole

polytope (this definition slightly differs from the usual one but is more convenient for our

purposes).

Proposition 3.10. Let Φ be a (possibly reducible) root system.

(1) The map sending a face F to (F1, . . . , Fk), where Fi = F∩Ei, is a bijection between

the proper faces of PΦ and the k-tuples F 1, . . . , F k, where F i is a proper face of

Pi for every i ∈ [1, k].

(2) A proper face F has dimension

dimF = k − 1 +

k∑

i=1

dimFi.

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 3 7

(3) A face F of PΦ is standard if and only if Fi is standard for all i ∈ [1, k].

Proof. (1) Let F be a proper face of PΦ, and denote by HF a supporting hyperplane

for F , i.e., a hyperplane such that F = PΦ ∩ HF and PΦ is contained in one of the

two halfspaces cut by HF . Since Φ is symmetric and spans E, the hyperplane HF does

not contain the origin, hence HF ∩ Ei is either empty, or a hyperplane in Ei. Since

Fi = F ∩ Ei = PΦ ∩ HF ∩ Ei = Pi ∩ (HF ∩ Ei), it follows that Fi is a proper (possibly

empty) face of Pi. Since F = Conv(F1 ∪ · · · ∪ Fk), the map in the statement is injective.

Conversely, for any choice of proper faces F 1, . . . , F k of P1, . . . ,Pk, respectively, consider

the convex hull F of F 1 ∪ · · · ∪ F k. Clearly, F ∩ Ei = F i, for every i ∈ [1, k]. We need

to show that F is a proper face of PΦ. Indeed, since the supporting hyperplanes of the

F i do not contain the origin, there exist yi ∈ Ei such that (yi, x) ≤ 1 for all x ∈ Pi, and

F i = x ∈ Pi : (x, yi) = 1, for every i ∈ [1, k]. If we define y = y1 + · · ·+ yk, we obtain

(y, x) ≤ 1 for all x ∈ PΦ and (y, x) = 1 if and only if x ∈ F .

(2) Since a face of a polytope is the convex hull of the vertices of the polytope that belong

to the face, F is the convex hull of F ∩Φ, and Fi is the convex hull of F ∩Φi. If F is proper

and nonempty, then dimF = dim(Span(F ∩Φ))− 1 and dimFi = dim(Span(F ∩Φi))− 1

The dimension formula follows by the preceding formulas (which hold also for the empty

face, since it has dimension −1 by definition).

(3) The set of vertices of F is the union of the sets of vertices of the subfaces Fi.

Hence, the barycenter βF of F is a positive linear combination of the barycenters βFiof

the nonempty faces Fi, and βF is dominant if and only if all these barycenters βFiare

Φi-dominant. This implies the claim by Theorem 3.6.

Now let F be standard. For i ∈ [1, k], denote by θi the highest root of Φi. For α ∈ Πi,

define mα by the condition θi =∑

α∈Πimαα, and set oα = qωα/mα. By Theorem 3.3,

Corollary 3.7, and Proposition 3.10, for every nonempty Fi = F ∩ Ei, there exists a

unique Πi-extremal subset Πi(F ) of Πi such that:

Fi = x ∈ Pi : (x, oα) = 1 for all α ∈ Πi(F ).

We let

oi(F ) =1

|Πi(F )|

∑

α∈Πi(F )

oα.

For Fi = ∅, we set oi(F ) = 0. Since (x, oα) ≤ 1 for all α ∈ Πi and x ∈ Pi, we have

(3.10.1) (x, oi(F )) ≤ 1, for all x ∈ Pi, and Fi = x ∈ Pi : (x, oi(F )) = 1.

The following result holds.

8 PAOLA CELLINI AND MARIO MARIETTI

Proposition 3.11. Let Φ be a (possibly reducible) root system. If F is a standard face

of PΦ, then

(x,

k∑i=1

oi(F )

)≤ 1 for all x ∈ PΦ and

F =

x ∈ PΦ :

(x,

k∑

i=1

oi(F )

)= 1

.

Proof. Let x ∈ PΦ. From any expression of x as a convex linear combination of the roots

in Φ, we easily obtain a subset J of [1, k] and nonzero elements xi ∈ Pi, for all i ∈ J , such

that x is a strictly positive convex linear combination of such elements xi. By (3.10.1)

above, (xi, oi(F )) ≤ 1 for all i ∈ J , hence

(x,

k∑i=1

oi(F )

)=

(x,∑i∈J

oi(F )

)≤ 1. Moreover,

(x,∑i∈J

oi(F )

)= 1 if and only if (xi, oi(F )) = 1 for all i ∈ J , hence if and only if xi ∈ Fi

for all i ∈ J : this is equivalent to x ∈ F .

In the following corollary, we isolate the special case of facets. The proof is a direct

application of the preceding results.

Corollary 3.12. Let Φ be a (possibly reducible) root system and let F be a facet of PΦ.

Then Fi 6= ∅ and dimFi = dimEi − 1, for all i ∈ [1, k].

In particular, F is standard if and only Fi is a standard facet of Pi. In this case, for

all i ∈ [1, k], the set Πi(F ) is a singleton containing a Πi-extremal root αi ∈ Πi, hence

oi(F ) = oαiand the supporting hyperplane of F is

x ∈ E : (x, oα1 + · · ·+ oαk) = 1.

Conversely, for each k-tuple (α1, . . . , αk) ∈ Π1 × · · · × Πk such that αi is Πi-extremal for

all i ∈ [1, k], the hyperplane

x ∈ E : (x, oα1 + · · ·+ oαk) = 1

supports a standard facet of PΦ.

Corollary 3.12 and Remark 3.4 together imply that each standard face is a subface of

a standard facet. Hence, by Corollary 3.9, we obtain the following result.

Corollary 3.13. Let x be a dominant element in E. Then Conv(W ·x) ⊆ PΦ if and only

if

(x, oα1 + · · ·+ oαk) ≤ 1,

for each k-tuple (α1, . . . , αk) ∈ Π1 × · · · ×Πk such that αi is Πi-extremal for all i ∈ [1, k].

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 4 9

4. Root systems and affine subspaces

In this section, we study the intersections of an irreducible root system with certain

affine subspaces.

Let Φ be irreducible.

Given a subset S of Π and β ∈ Φ, with β =∑α∈Π

cα(β)α, we set

ΦS,β =γ =

∑

α∈Π

cα(γ)α ∈ Φ : cα(γ) = cα(β) for all α ∈ S,

ΦS,Zβ =γ =

∑

α∈Π

cα(γ)α ∈ Φ : ∃k ∈ Z with cα(γ) = kcα(β) for all α ∈ S.

Remark 4.1. Let β ∈ Φ. If Φ were reducible and Ψ were the irreducible component of

Φ containing β, then ΦS,β = ΨS,β and ΦS,Zβ = ΨS,Zβ for all nonempty S. Thus, there is

no loss of generality in our assumption that Φ be irrreducible.

Recall that δ denotes the indivisible imaginary root of the affine root system Φ (see

Section 2).

Proposition 4.2. Let S ⊆ Π and β ∈ Φ+ with Supp(β) ∩ S 6= ∅.

(1) ΦS,Zβ is a root subsystem of Φ.

(2) ΦS,β is an interval of the root poset, i.e. it has minimum and maximum and

contains every root in between. Moreover, both

minΦS,β ∪ (Πr S) and −maxΦS,β ∪ (Πr S)

are simple systems of ΦS,Zβ; in particular, minΦS,β and maxΦS,β have the same

length.

(3) Let Ψ = ΦS,Zβ and choose γ ∪ (Πr S) as a set of simple roots of Ψ, with either

γ = minΦS,β or γ = −maxΦS,β. Then either

ΦS,β = Ψγ,γ or ΦS,β = −Ψγ,γ ,

respectively.

Proof. (1) The assertion is clear.

(2) Let U = Span ((Πr S) ∪ β + δ) and ΦU = Φ ∩ U . Then ΦS,β + δ ⊆ ΦU .

Since Supp(β) ∩ S is nonempty, Rδ ∩ U = 0, hence the restriction of (· , ·) to U

is positive definite and ΦU is a finite dimensional crystallographic root system, natu-

rally isomorphic to ΦS,Zβ through the natural projection E + Rδ → E. We can choose

ΦU ∩ Φ+ as a positive system for ΦU . The simple system determined by this choice is

(Πr S) ∪ γ + δ, for a certain γ ∈ ΦS,β with γ ≤ β. Indeed, since all roots in ΦS,β + δ

must be non-negative linear combination of the basis, γ = minΦS,β.

10 PAOLA CELLINI AND MARIO MARIETTI

If we make the above construction with −β+ δ in place of β+ δ, we obtain, as a simple

system for ΦU , (Πr S) ∪ −γ + δ, with γ = maxΦS,β.

By definition of ΦS,β, every root between minΦS,β and maxΦS,β in the root poset

belongs to ΦS,β.

Finally, it is clear that, if (Π r S) ∪ γ + δ is a basis of ΦU , then (Π r S) ∪ γ is a

basis of ΦS,Zβ.

(3) The assertion follows by (1) and (2).

Remark 4.3. Proposition 4.2 implies a representation theoretical result due to Kostant,

which is presented in [9, Section 2]. We briefly recall this result. Let g be a simple

Lie algebra with root system Φ, and pΠrS be the standard parabolic subalgebra of g

corresponding to Πr S. Then

pΠrS = gΠrS ⊕ h(ΠrS)⊥ ⊕ n+ΦrΦ(ΠrS),

where gΠrS is the semisimple subalgebra generated by the root spaces gα with α ∈ ΠrS,

h(ΠrS)⊥ is defined as h ∈ h : α(h) = 0, ∀α ∈ Π r S and is the center of the reductive

subalgebra gΠrS⊕h(ΠrS)⊥ , and n+ΦrΦ(ΠrS) =⊕

γ∈Φ+rΦ(ΠrS)

gγ . The adjoint representation of

g restricts to a representation of gΠrS⊕h(ΠrS)⊥ on the nilpotent summand n+ΦrΦ(ΠrS). For

any β ∈ Φ+, if β =∑α∈Π

cαα, the action of h(ΠrS)⊥ on gβ depends only on∑α∈S

cαα, hence,⊕

γ∈ΦS,β

gγ , which we denote by nΦS,β, is a h(ΠrS)⊥-isotypic component. Moreover, nΦS,β

is

stable under the action of gΠrS. Kostant’s results is that, whenever Supp(β) ∩ S 6= ∅,

the subrepresentation of gΠrS on nΦS,βis irreducible. The whole set of weights of this

irreducible representation is ΦS,β, hence its unique highest weight is the maximum of

ΦS,β. We show that the set of weights ΦS,β is partioned into one or two W 〈Πr S〉-orbits

according to Propositions 4.11, 4.12, and 4.14 (see also Remark 4.9).

We also note that, by a general result of representation theory (see [6, p. 204]), the set

of weights ΦS,β is contained in the convex hull of the orbit of the highest weight under

the action of the Weyl group of gΠrS, hence

Conv(ΦS,β) = Conv(W 〈Πr S〉 ·maxΦS,β

).

Remark 4.4. Let α ∈ Π and S = α. Then ΦS,Zα = Φ and minΦS,α ∪ (Πr S) = Π.

By Proposition 4.2, also −maxΦS,α ∪ (Πr S) is a simple system of Φ. When mα = 1,

maxΦS,α = θ and, since the affine simple root α0 satisfies (α0, x) = (−θ, x) for all x ∈ E,

we reobtain the following well-known fact:

the coefficient mα is equal to 1 if and only if the Dynkin diagram of Π is equal to the

subdiagram Π ∪ α0r α of the extended Dynkin diagram of Φ.

Recall that qωα, for each α ∈ Π, denotes the fundamental co-weight corresponding to

α, i.e. the element of E defined by the conditions (qωα, α) = 1 and (qωα, α′) = 0 for

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 5 11

α′ ∈ Πr α. Let

A = Span(Πr S), so that A⊥ = Span(qωα : α ∈ S),

and denote by πA and πA⊥ the orthogonal projections from E onto A and A⊥, respectively.

Lemma 4.5. For all γ ∈ ΦS,β and w ∈ W 〈Πr S〉,

(1) πA⊥(γ) = πA⊥(β), hence πA(γ) = γ − πA⊥(β),

(2) πA w(γ) = w πA(γ).

Proof. The equalities in (1) follow directly from the definitions. Equality (2) follows easily

from (1), since W 〈Πr S〉 stabilizes A and fixes A⊥ pointwise.

Given a subset T of Π, we say that v ∈ E is T -dominant provided that (v, α) ≥ 0 for

all α ∈ T . We note that, for all v ∈ E, v is (ΠrS)-dominant if and only if πA(v) belongs

to the fundamental chamber of the possibly reducible root system Φ〈Πr S〉.

Proposition 4.6. Let S ⊆ Π and β ∈ Φ+, with Supp(β) ∩ S 6= ∅. Then

(1) maxΦS,β and −minΦS,β are both (Πr S)-dominant;

(2) if ΦS,β contains some long roots, then maxΦS,β is the unique long (ΠrS)-dominant

root; otherwise, maxΦS,β is the unique (short) (Πr S)-dominant root.

Proof. Assertion (1) is implicit in Proposition 4.2 (2), and may also be seen directly since,

for each γ ∈ Φ, if (γ, α) < 0 for some α ∈ Πr S, then γ + α ∈ Φ.

Let us prove (2). We first show that, for each γ ∈ ΦS,β, there exists a (ΠrS)-dominant

root of the same length as γ. The parabolic subgroup W 〈Πr S〉 acts on the set of long

(short) roots in ΦS,β. By Lemma 4.5, the orthogonal projection πA : E → A restricts

to a bijection from the W 〈Π r S〉-orbit of γ to the W 〈Π r S〉-orbit of πA(γ). Since

W 〈Π r S〉 · πA(γ) contains a unique element belonging to the fundamental chamber of

Φ(Πr S), the orbit W 〈Πr S〉 · γ contains a unique (Πr S)-dominant root.

Next, we prove that, if γ is a (Πr S)-dominant root in ΦS,β other than maxΦS,β, then

maxΦS,β is a long root and γ is short. Since γ < maxΦS,β, we have maxΦS,β − γ ∈

N(Π r S), and hence (maxΦS,β, γ∨) ≥ (γ, γ∨) = 2, since (α, γ∨) ≥ 0 for all α ∈ Π r S.

This implies that maxΦS,β is long and γ is short.

Hence, if ΦS,β contains either only long roots or both long and short roots, then maxΦS,β

is the unique long (Π r S)-dominant root. Moreover, if maxΦS,β is short, then it is the

unique short (Πr S)-dominant root in ΦS,β, and there is no long root in ΦS,β.

As a consequence, we obtain an algebraic combinatorial short proof of Oshima’s Lemma

(see [10]), which is case-free and independent of the representation theory of semisimple

complex Lie algebras (see [5]).

Corollary 4.7 (Oshima’s Lemma). Let S ⊆ Π and β ∈ Φ+, with Supp(β)∩S 6= ∅. Then

12 PAOLA CELLINI AND MARIO MARIETTI

(1) if ΦS,β contains a long (short) root, then it contains exactly one (ΠrS)-dominant

long (short) root;

(2) W 〈Πr S〉 is transitive on the set of long (short) roots in ΦS,β.

Proof. Let us prove (1). By Proposition 4.6(2), if ΦS,β contains some long roots, then it

contains exactly one (Πr S)-dominant long root.

It remains to show that, if ΦS,β contains some short roots, then it contains exactly one

(Πr S)-dominant short root. We prove it by duality.

For a subset T of Φ, let T∨ = α∨ : α ∈ T. Let βs be any short root in ΦS,β. Then γ

is a short root in ΦS,β if and only if γ∨ is a long root in Φ∨S∨,β∨

s, and γ is (ΠrS)-dominant

root if and only if γ∨ is (Π∨ r S∨)-dominant. Hence, we are done.

Part (2) is, in fact, equivalent to Part (1) since, as already noticed, Lemma 4.5 implies

that the orthogonal projection πA : E → A restricts to a bijection from theW 〈ΠrS〉-orbit

of each γ ∈ ΦS,β to the W 〈Πr S〉-orbit of πA(γ).

Remark 4.8. If ΦS,β contains some short roots, then the unique short (ΠrS)-dominant

root is also the unique maximal short root in ΦS,β. Indeed, if γ ∈ ΦS,β is not (Π r S)-

dominant, for α ∈ ΠrS with (γ, α) < 0, we have that sα(γ) > γ, so γ cannot be maximal

among the short roots.

Remark 4.9. In Remark 4.3, we have seen that maxΦS,β is the highest weight of an

irreducible representation of the semisimple Lie algebra associated to Φ〈Π r S〉. Corol-

lary 4.7 shows that the action of W 〈Πr S〉 on the weights of this representation has one

orbit for each length of the roots ΦS,β, hence at most two orbits. In particular, in the

simply laced case there is a single orbit action. By Proposition 4.6, there is a single orbit

action also when maxΦS,β is short.

In the rest of the section, we study when ΦS,β contains roots of one or two lengths.

Recall that the lacing number of Φ is the maximal number of edges connecting two

vertices of the Dynkin diagram of Φ, i.e. max(α, α)/(β, β) : α, β ∈ Φ. As usual, in

simply laced root system, we consider all roots as long. The following result is an exercise

of Bourbaki’s [1]. We include a proof for completeness.

Lemma 4.10. [1, Exercise 20, Chapter VI, §1] Let γ =∑

α∈Π cαα ∈ Φ. Then, γ is long

if and only if, for all short α in Π, the coefficient cα is a multiple of the lacing number of

Φ.

Proof. Let γ be a long root. The coroot γ∨ satisfies

γ∨ =2γ

(γ, γ)=∑

α∈Πℓ

2

(γ, γ)cαα +

∑

α∈Πs

2

(γ, γ)cαα =

∑

α∈Πℓ

cαα∨ +

∑

α∈Πs

(α, α)

(γ, γ)cαα

∨,

where Πℓ and Πs denote, respectively, the set of simple long and short roots. Now, for

all short root α, the ratio (α, α)/(γ, γ) coincides with the inverse of the lacing number of

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 6 13

Φ; since the coefficients of γ∨ must be integers, cα is a multiple of the lacing number, for

each short α ∈ Π.

Conversely, let γ =∑

α∈Π dαα be a root such that dα is a multiple of the lacing number

of Φ, for all short α ∈ Π. We prove that γ is long by contradiction. Suppose that it is

short. The coroot γ∨ satisfies

γ∨ =2γ

(γ, γ)=∑

α∈Πℓ

2

(γ, γ)dαα +

∑

α∈Πs

2

(γ, γ)dαα =

∑

α∈Πℓ

(α, α)

(γ, γ)dαα

∨ +∑

α∈Πs

dαα∨.

Since (α, α)/(γ, γ) coincides with the lacing number of Φ for each long root α, all coeffi-

cients of γ∨ are multiples of the lacing number. This is a contradiction since a (co)root

cannot be a multiple of an element in the (co)root lattice, since any (co)root can be a

basis element of the (co)root lattice.

A direct consequence of Lemma 4.10 is the following result.

Proposition 4.11. If all short simple roots belong to S, then all roots in ΦS,β share the

same length.

Lemma 4.10, together with Proposition 4.2(3), yields a further proof of the fact that,

if minΦS,β is short, then all roots in ΦS,β are short (see Proposition 4.6(2). Indeed, by

Proposition 4.2(3) ΦS,β consists of all the roots in ΦS,Zβ that, with respect to the basis

(Πr S) ∪ minΦS,β, have the minΦS,β-coordinate equal to 1.

Similarly, we may obtain a criterion for ΦS,β to contain short roots in case minΦS,β is

long.

Proposition 4.12. Let S ∩ Supp(β) 6= ∅.

(1) The set ΦS,β contains only short roots if and only if minΦS,β is short.

(2) The set ΦS,β contains only long roots if and only if minΦS,β is long and the con-

nected component of minΦS,β in the Dynkin diagram of (Π r S) ∪ minΦS,β

contains only long simple roots.

(3) The set ΦS,β contains both long and short roots if and only if minΦS,β is long and

the connected component of minΦS,β in the Dynkin diagram of (ΠrS)∪minΦS,β

contains some short simple root.

Proof. Part (1) has already been proved.

Parts (2) and (3) follow from the following argument.

The roots in ΦS,β belong to the irreducible component of minΦS,β in ΦS,Zβ. Hence, if

all simple roots in such connected component are long, then all roots in ΦS,β are long.

On the contrary, if such connected component contains some simple short root, then the

sum of all simple roots in such connected component is a root (it is a general well-known

fact that the sum of the simple roots of an irreducible root system is a root), belongs to

ΦS,β, and moreover, by Lemma 4.10, is a short root.

14 PAOLA CELLINI AND MARIO MARIETTI

Lemma 4.10 provides a necessary condition on β in order that ΦS,β contain some long

roots, namely, that for any short simple root α ∈ S, the coefficient cα(β) should be a

multiple of the lacing number. In Proposition 4.14, we see that this is also a sufficient

condition, provided S ∩Supp(β) 6= ∅. The proofs of Proposition 4.14 and its Lemma 4.13

require the classification of the irreducible root systems.

We note that in case S ∩ Supp(β) = ∅, the above condition on β is trivially satisfied,

but ΦS,β is the root subsystem generated by ΠrS, and hence, if S contains all long simple

roots, then ΦS,β only contains short roots.

Lemma 4.13. If µ ∈ Φ+ is a short root and satisfies Supp(µ) = Π, then there exists a

short simple root γ such that µ+γ ∈ Φ+ and cγ(µ) is not a multiple of the lacing number.

Proof. We prove the assertion by a a case-by-case analysis. For each type, we list all short

roots µ satisfying Supp(µ) = Π and, for each such µ, we provide a short simple root γ

with the required properties. The simple roots are denoted according to the Tables in [1].

If Φ is of type Bl, then µ =∑l

i=1 αi and γ = αl.

If Φ is of type Cl, then µ =∑j−1

i=1 αi + 2∑l−1

i=j αi + αl, with 1 < j ≤ l, and γ = αj−1.

If Φ is of type F4, then either µ = α1+α2+α3+α4 and γ = α3, or µ = α1+α2+2α3+α4

and γ = α4, or µ = α1 + 2α2 + 2α3 + α4 and γ = α4, or µ = α1 + 2α2 + 3α3 + α4 and

γ = α4, or µ = α1 + 2α2 + 3α3 + 2α4 and γ = α3.

If Φ is of type G2, then either µ = α1 + α2, or µ = 2α1 + α2, and, in both cases,

γ = α1.

Proposition 4.14. Let S ∩ Supp(β) 6= ∅. The set ΦS,β contains long roots if and only if,

for every short root α in S, the coefficient cα(β) is a multiple of the lacing number.

Proof. Let γ =∑α∈Π

cα(γ)α be a long root in ΦS,β. By Lemma 4.10, the coefficient cα(γ)

is a multiple of the lacing number for every short root α in Π and so, in particular, for

every short root α in S. By the definition of ΦS,β, we have cα(β) = cα(γ), for all α ∈ S.

Conversely, suppose that cα(β) is a multiple of the lacing number for each short root

α in S. If β is long, then we are done. So we suppose that Φ is not simply laced and β

is short. By symmetry, we may also suppose β ∈ Φ+. Let µ∗ be a maximal root in the

set µ ∈ ΦS,β : µ is short and Supp(µ) ⊆ Supp(β), which is not empty since it contains

β. It is well known that the support of a root is connected and hence Φ〈Supp(β)〉 is

a standard parabolic irreducible subsystem of Φ. Thus, if Supp(β) contains only short

roots, then, by the classification of the irreducible root systems, Supp(β) is a simple root

system of type A. This is impossible since cα(β) > 1 for α ∈ S∩Supp(β). Hence, Supp(β)

is a simple system of an irreducible not simply laced root system. By applying Lemma

4.13 with µ∗ and Supp(β) in place of µ and Π, we find a short simple root γ ∈ Supp(β)

such that µ∗ + γ is short and cγ(µ∗) is not a multiple of the lacing number. In particular,

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 7 15

γ 6∈ S and hence µ∗ + γ still belongs to ΦS,β. By the maximality of µ∗, the root µ∗ + γ is

long.

5. Root systems and affine hyperplanes

In this section, we analyse in more details the case of codimension 1 of the intersections

studied in Section 4.

Let Φ be an irreducible root system (see Remark 4.1). Throughout this section, we fix a

simple root α ∈ Π. According to its coordinate cα(β), each root β =∑γ∈Π

cγ(β)γ lies in one

of the 2mα + 1 parallel affine hyperplanes x ∈ E : (x, qωα) = cα, with −mα ≤ cα ≤ mα.

To lighten the notation, we set:

• Φα,k = Φ ∩ x ∈ E : (x, qωα) = k

• Φα,Zk = Φ ∩ x ∈ E : k divides (x, qωα)

• Φα = Φ〈Πr α〉

• Wα = W 〈Πr α〉

• µα,k = maxΦα,k, for k ∈ [1, mα] (see Proposition 4.2)

In the notation of Section 4, Φα,k = Φα,β and Φα,Zk = Φα,Zβ , where β =∑γ∈Π

cγ(β)γ is

any root with cα(β) = k. Moreover, Φα = Φα,0, Φα,Zk ⊇ Φα,Zk′ whenever k divides k′, and

Φα,Zk ∩ Φα,Zh = Φα,Zl where l is the least common multiple of k and h. For k = 1, we

have Φα,Zk = Φ.

Notice that, by Proposition 4.2,(2), the Φα,k, with k 6= 0, provide a partition of Φ \Φα

into intervals.

By Proposition 4.6 and Corollary 4.7, we have the two following possibilities:

• if all roots in Φα,k have the same length, then

Φα,k = Wα · µα,k,

• if Φα,k contains both short and long roots, then µα,k is long, the set β ∈ Φα,k :

β is short has a maximum, denoted σα,k, and

Φα,k = Wα · µα,k ∪ Wα · σα,k.

By the last formula in Remark 4.3, however, in both cases

Conv(Φα,k) = Conv(Wα · µα,k).(5.0.1)

Recall that the subset Φα,Zk, with k ≤ mα, is a root subsystem of Φ, by Proposi-

tion 4.2,(1). For instance, if Φ has type G2, then the root subsystem Ψα2,k has type A1,

G2, A1 × A1, or A2 depending on whether k has the value 0, 1, 2, or 3.

16 PAOLA CELLINI AND MARIO MARIETTI

Let A = Span(Π r α). For x ∈ E and X ⊆ E, we let x = πA(x) and X = πA(X).

For all x ∈ E, we have x = x− (x,qωα)(qωα,qωα)

qωα. In particular,

γ = γ −k

(qωα, qωα)qωα,

for all γ ∈ Φα,k.

Let γ ∈ Φα,k be a root of the same length as µα,k. The transitivity of Wα on the long

roots in Φα,k and (5.0.1) imply

Conv(Φα,k) = Conv(Φα,k) = Conv(Wα · γ).(5.0.2)

Thus, since γ is a weight of Φα, we have that Conv(Φα,k) is the weight polytope of γ,

relatively to Φα in the space A. Clearly, γ = 0 if and only if γ ∈ A⊥.

Proposition 5.1. Fix α ∈ Π and k ∈ [1, mα]. Let γ ∈ Φα,k be a root of the same length as

µα,k. Enumerate the irreducible components Ψ1, . . . ,Ψt of Φα so that there exists h ∈ [1, t]

satisfying γ 6⊥ Ψi for 1 ≤ i ≤ h and γ ⊥ Ψi for h < i ≤ t. Then Φα,k ⊆ Span(Ψ1∪· · ·∪Ψh)

and

dim(Conv(Φα,k)) = dim(Conv(Φα,k)) = rk(Ψ1 ∪ · · · ∪Ψh).

Proof. The first equality follows from the fact that Conv(Φα,k) belongs to the affine hyper-

plane A+ k(qωα,qωα)

qωα, and the restriction of πA to this affine hyperplane is the translation

by − k(qωα,qωα)

qωα.

Since γ ⊥ Ψi for h < i ≤ t, we have Wα · γ ⊆ Span(Ψ1 ∪ · · · ∪ Ψh) + γ. By (5.0.2), in

order to prove the second equality, it suffices to prove that, for all β ∈ Π∩ (Ψ1∪· · ·∪Ψh),

there exist γ1, γ2 ∈ Wα · γ such that γ1 − γ2 = cβ, for some c 6= 0.

Let β ∈ Π ∩ Ψi, for some i ∈ [1, h]. Since γ 6⊥ Ψi, there exist l (possibly l = 1) and a

path (β1, . . . , βl) in the Dynkin diagram of Ψi such that β1 = β, βs ⊥ γ for s ∈ [1, l − 1],

and βl 6⊥ γ. Hence, for all j ∈ [1, l − 1],

sβj. . . sβl

(γ) = cjβj + · · ·+ clβl + γ

for certain nonzero real cj , . . . , cl. We can take the elements sβ1 . . . sβl(γ) and sβ2 . . . sβl

(γ)

as γ1 and γ2, respectively.

We state the following lemma for later reference.

Lemma 5.2. Let α ∈ Π. The roots α and µα,1 have the same length (the maximal length).

Proof. The equality of the lengths follows from Proposition 4.2,(2), since α = minΦα,1.

The length is maximal by Proposition 4.6,(2).

As a direct consequence of Proposition 5.1 and Lemma 5.2, we obtain the following

result.

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 8 17

Corollary 5.3. Let α ∈ Π and let Ψ be any irreducible component of Φα. Let k ∈ [1, mα]

and denote by Φα,k the set of roots of Φα,k of the same length as µα,k.

(1) For every γ, γ′ ∈ Φα,k, we have γ ⊥ Ψ if and only if γ′ ⊥ Ψ.

(2) For every γ ∈ Φα,1, we have

γ 6⊥ Ψ.

In particular, dim(Conv(Φα,1)) = rk(Φ)− 1.

Proof. (1) The proof is straightforward from Proposition 5.1.

(2) Being Φ irreducible, α 6⊥ Ψ. Since α belongs to Φα,1 and has the same length as

µα,1 by Lemma 5.2, γ 6⊥ Ψ follows by (1). The assertion on the dimension follows by

Proposition 5.1.

6. Projections

Let Φ be an irreducible root system. Fix an arbitrary simple root α ∈ Π. In this

section, we study the inclusion of the projected polytopes Conv(Φα,k) = Conv(Φα,k) in

the root polytope PΦα= Conv(Φα) and, more generally, the scale factors r for which

Conv(Φα,k) ⊆ r · PΦα.

Lemma 6.1. Let k ∈ [1, mα] and r ∈ R.

(1) The inclusion Conv(Φα,k) ⊆ r · PΦαholds if and only if µ ∈ r · PΦα

for some

µ ∈ Φα,k of maximal length in Φα,k.

(2) We have Φα,k ⊆ r · PΦαif and only if Φα,−k ⊆ r · PΦα

.

Proof. (1) By Remark 4.3, Proposition 4.6 and Corollary 4.7, the set of vertices of

Conv(Φα,k) is the set of all elements of maximal length in Φα,k, and this set is a sin-

gle Wα-orbit. Since Wα and the projection πA commute by Lemma 4.5, and since Φα is

Wα-stable, if a single vertex of Conv(Φα,k) belongs to r · PΦα, then all vertices do. The

claim follows.

(2) We have Φα,−k = −Φα,k, hence Φα,−k = −(Φα,k), so the claim holds since−Φα = Φα.

By Lemma 6.1,(1), we obtain a simple criterion for checking the inclusion of Conv(Φα,1)

in the dilations of PΦα.

Proposition 6.2. Let r ∈ R. We have Conv(Φα,1) ⊆ r · PΦαif and only if α ∈ r · PΦα

.

Proof. By Lemma 5.2, the root α has maximal length in Φα,1. The assertion follows by

Lemma 6.1,(1).

Remark 6.3. Let 1 < k ≤ mα, γ = minΦα,k, and Ψ = Φα,Zk. By Proposition 4.2(2)

and (3), the roots γ ∪ Πr α form a simple system for the root subsystem Ψ and

Φα,0 = Ψγ,0, while Φα,k = Ψγ,1.

18 PAOLA CELLINI AND MARIO MARIETTI

The root system Ψ may be reducible; however

rΨγ,1 ⊆ Conv(Ψγ,0) = PΨγif and only if rΨ∗

γ,1 ⊆ Conv(Ψ∗γ,0) = PΨ∗

γ,

where Ψ∗ denotes the component of γ in Ψ.

Recall that, for each ε ∈ Π, we denote by ωε the fundamental weight corresponding

to ε, i.e. the element of E defined by the conditions (ωε, ε∨) = 1 and (ωα, ε

′∨) = 0 for

ε′ ∈ Π r ε. Note that, for ε ∈ Π r α, the projection ωε is the fundamental weight

corresponding to ε in the subsystem Φα. We let N [α] denote the set ε ∈ Π r α :

ε 6⊥ α of the neighbours of α in the Dynkin diagram of Φ. By the following result, α

is a negative linear combination of the fundamental weights associated with N [α] in the

standard parabolic subsystem Φα.

Proposition 6.4. The projection α satisfies

α =∑

ε∈N [α]

(α, ε∨)ωε.

Proof. The set ωε : ε ∈ Π r α is the dual basis of (Π r α)∨ in the hyperplane A.

Hence,∑

ε∈N [α]

(α, ε∨)ωε has the same scalar products as α with the basis (Πr α)∨ of A.

This implies∑

ε∈N [α]

(α, ε∨)ωε = α.

Remark 6.5. If α is a leaf in the Dynkin diagram, the sum in Proposition 6.4 consists

of merely one addend, say (α, ε∨)ωε. Moreover, in such cases, the coefficient (α, ε∨) is

always equal to −1 (and hence α projects to −ωε) except:

• in Cn, when α = αn, and in this case αn = −2ωn−1,

• in G2, when α = α2, and in this case α2 = −3ω1.

Also notice that N [α] consists of at most three elements, hence∑

ε∈N [α]

(α, ε∨)ωε consists of

at most three summands. When k = 1, we have α = minΦα,k and Φα,Zk = Φ. For k > 1,

by Proposition 4.2, Φα,Zk is a root system with simple system minΦα,k∪Πrα, hence

also the projection minΦα,k can be written as a linear combination of three or less elements

in ωε : ε ∈ Π r α. In fact, by a direct check, we see that, if k > 1, then minΦα,k

can be written as a linear combination of two or less elements in ωε : ε ∈ Π r α.

Otherwise, Π and minΦα,k ∪ Π r α would be two non isomorphic simple systems

that produce isomorphic root subsystem after removing their trivalent vertex. This is

impossible.

7. On a problem by Hopkins and Postnikov

In this section, we use the results in the preceding sections to tackle a problem by

Hopkins and Postnikov.

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 9 19

In [8], Hopkins and Postnikov find certain applications of the root polytope of an irre-

ducible root system Φ. In particular, they need the following lemma concerning dilations

of projections of the root polytope (see [8, Lemma 2.10]).

Lemma 7.1 (Hopkins-Postnikov). Let U be a nonzero subspace of E spanned by a subset

of Φ. Let ΦU be the root subsystem Φ ∩ U of Φ. Then there exists some r ∈ R, with

1 ≤ r < 2, such that the projection on U of the root polytope PΦ of Φ is contained in the

dilation r · PΦUof the root polytope of ΦU .

Hopkins and Postnikov prove Lemma 7.1 by using Theorem 3.3 in a case-by-case check,

and ask for a uniform proof that does not require the classification of root systems. In

this section, we give a shorter proof, which unfortunately, in its last part, requires the

classification (see Remark 7.6). Towards this goal, consider the following lemma.

Lemma 7.2. There exists some positive real number r, with r < 2, such that

α ∈ r · PΦα,

for all α ∈ Π.

Lemma 7.2 is in fact equivalent to Lemma 7.1.

Proposition 7.3. Lemma 7.1 holds true for all root systems if and only if Lemma 7.2

holds true for all root systems.

Proof. Clearly, Lemma 7.1 implies Lemma 7.2.

Conversely, suppose that Lemma 7.2 holds true for all root systems. Let us prove

Lemma 7.1.

As explained in [8, Appendix A], we may reduce to the case when U is a maximal

standard parabolic subspace, i.e., U is the hyperplane generated by a codimension 1

standard parabolic subsystem of Φ. Hence the proof of Lemma 7.1 amounts to show that

there exists some r ∈ R, with 1 ≤ r < 2, such that, for all α ∈ Π,

Φα,k ⊆ r · PΦα

for all k ∈ [1, mα]. Furthermore, Remark 6.3 implies that it suffices to show that there

exists some r ∈ R, with 1 ≤ r < 2, such that, for all α ∈ Π,

Φα,1 ⊆ r · PΦα.

Hence, the result follows by Proposition 6.2.

For each ε ∈ N [α], let Φε be the component of ε in Φα and Πε = Π∩Φε. Note that, since

Φ is irreducible, for ε varying in N [α], the Φε exhaust the components of Φα. Moreover,

they are distinct and at most 3. Also, let θε be the highest root of Φε and, for all η ∈ Πε,

let mη be defined by the condition θε =∑

η∈Πε mηη, and let oη = qωη/mη.

20 PAOLA CELLINI AND MARIO MARIETTI

Proposition 7.4. Let Φ be irreducible and α ∈ Π.

minr : Φα,1 ⊆ r · PΦα = minr : α ∈ r · PΦα

=∑

ε∈N [α]

−(α, ε∨)cε

where

cε = max(ωε, oη) : η is a Πε-extremal root in Πε.

Proof. The first equality follows by Proposition 6.2. Let us prove the second equality.

Since PΦα= −PΦα

, we have α ∈ r ·PΦαif and only if −α ∈ r ·PΦα

. By Proposition 6.4,

−α =∑

ε∈N [α]

−(α, ε∨)ωε,

is dominant relatively to Φα and hence, in order to check its inclusion in r · PΦα, we

may use Corollary 3.13. If we number the elements in N [α], say N [α] = ε1, . . . , εk, by

Corollary 3.13, we obtain:

minr : Φα,1 ⊆ r · PΦα = max(−α, oη1 + · · ·+ oηk)

= max−(α, ε∨1 )(ωε1 , oη1)− · · · − (α, ε∨k )(ωεk , oηk)

where the maximum is taken over all k-tuples (η1, . . . , ηk) ∈ Πε1 × · · · × Πεk such that ηi

is Πεi-extremal. Hence, the assertion follows.

Note that (α, ε∨) = −1 except when (α, α) > (ε, ε), in which case (α, ε∨) = −(α, α)/(ε, ε).

In order to prove Lemma 7.2, we show∑

ε∈N [α]

−(α, ε∨)cε < 2, by using the classification

of root systems.

Let Φ be of type Xn, with the simple roots α1, . . . , αn numbered according to the

Tables in [1]. Let

ci[Xn] = max(ωi, oj) : αj is Π-extremal.

By definition, (ωi, oj) = (ωi, qωj)/mi. Furthermore, (ωi, qωj), for j ∈ [1, n], are the coordi-

nates of ωi with respect to α1, . . . , αn, and thus are explicit in the tables of [1]. Notice

also that((ωi, qωj)

)i,j

is the inverse of the Cartan matrix((αi, α

∨j ))i,j.

For X = A, all simple roots are Π-extremal and all mj are 1, and hence ci[An] =

max1≤j≤n

(ωi, qωj). In all other cases, the Π-extremal simple roots are very few. In the

following lemma, we list the values of ci[Xn] needed in the proof of Lemma 7.2 (ci[An] is

actually needed only for i = 1, 2, 3).

Lemma 7.5. The following equalities hold:

• ci[An] = i(n+ 1− i)/(n+ 1),

• c1[Bn] = c1[Dn] = 1,

• c1[Cn] = 1/2,

• cn[Bn] = cn[Dn] = n/4,

• cn[Cn] = n/2,

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 10 21

• c6[E6] = 4/3,

• c7[E7] = 3/2.

Proof of Lemma 7.2. Let α ∈ Π and let

rα =∑

ε∈N [α]

−(α, ε∨)cε(7.5.1)

By Proposition 7.4, it suffices to prove rα < 2. The proof consists in a case-by-case

computation.

Henceforward, α1, . . . , αn is the simple system of Φ with the numbering of [1], and

α = αi for some i ∈ [1, n].

Let Φ ∼= An. Then Φα∼= Ai−1 × An−i, where we intend A0 as the empty component.

For each ε ∈ N [α], we can view ωε as the first fundamental weight for the component of

ε in Φα, hence rα = c1[Ai−1] + c1[An−i].

Let Φ ∼= Xn, with X = B,C,D. Reasoning as in case An, when either X = B,C and

i ∈ [1, n− 2], or X = D and i ∈ [1, n− 4], we find

rα = c1[Ai−1] + c1[Xn−i].

For X = B,C and i = n− 1, we have

rα = c1[An−2] + tc1[A1] where t =

2 if X = B,

1 if X = C.

For X = B,C and i = n, we have

rα = tc1[An−1] where t =

1 if X = B,

2 if X = C.

For X = D, we have:

rα =

c1[An−4] + c2[A3] if i = n− 3,

c1[An−3] + c1[A1] + c1[A1] if i = n− 2,

c2[An−1] if i = n− 1, n.

Substituting the values of ci[X] listed in Lemma 7.5, we find rα < 2 in all cases.

22 PAOLA CELLINI AND MARIO MARIETTI

If Φ ∼= En (n = 6, 7, 8), then

rα =

cn−1[Dn−1] ≤74

if i = 1,

c3[An−1] ≤158

if i = 2,

c1[A1] + c2[An−2] ≤2714

if i = 3,

c1[A1] + c1[A2] + c1[An−4] ≤5930

if i = 4,

c2[A4] + c1[An−5] ≤3920

if i = 5,

c5[D5] + c1[An−6] ≤2312

if i = 6,

c6[E6] + c1[An−7] ≤116

if i = 7 ≤ n,

c7[E7] =32

if i = 8 = n.

If Φ ∼= F4, then

rα =

c3[C3] =32

if i = 1,

c1[A1] + 2c1[A2] =116

if i = 2,

c1[A1] + c1[A2] =76

if i = 3,

c3[B3] =34

if i = 4

If Φ ∼= G2, then

rα =

c1[A1] =

12

if i = 1,

3c1[A1] =32

if i = 2.

Remark 7.6. In [8], Hopkins and Postnikov show, by a uniform argument, that Lemma 7.1

can be reduced to check that, for each α ∈ Π and each k ∈ [1, mα], the projections of

the (one or two) Φα-dominant roots belonging Φα,k fall inside 2 ·PΦα

. This final check

is done case-by-case. Here, our case-by-case analysis amounts to just one check for each

α ∈ Π, i.e. that the uniform expression∑

ε∈N [α]

−(α, ε∨)cε of Proposition 7.4 specialises to

a number smaller than 2.

8. Enumerative formulas

In this section, we use the results in the previous sections to give some enumerative

results.

For each subset X of Π, we use the shorthand notation WX for W 〈ΠrX〉. Let α ∈ Π,

Wα = Wα, A = Span(Πr α), πA be the projection on A, and, for x ∈ E, x = πA(x).

We recall that N [α] denote the set ε ∈ Πr α : β 6⊥ α of the neighbours of α in the

Dynkin diagram of Π. We also set

N [α] = N [α] ∪ α.

Then N(α) is the set of all simple roots that are not orthogonal to α.

ROOT SYSTEMS, AFFINE SUBSPACES, AND PROJECTIONS 11 23

Lemma 8.1. Let Φ be an irreducible root system and α ∈ Π. The number of the roots in

Φα,1 having the same length as α is

|Wα|

|WN [α]|

.

Proof. By Corollary 4.7, the parabolic subgroup Wα acts transitively both on the set long

roots of Φα,1 and on the set short roots of Φα,1.

Recall the well known fact that the stabilizer in a Weyl group W of a vector v ∈ C

is the standard parabolic subgroup generated by the reflections through the simple roots

orthogonal to v. By Proposition 6.4, α is a negative linear combination of the weigths

of Φα corresponding to the simple roots in N [α]. Hence, the stabilizer in Wα of α, and

hence of α, is the parabolic subgroup WN [α]. The assertion follows.

The proof of the following result uses the classification of root systems.

Proposition 8.2. Let Φ be a classical irreducible root system and α ∈ Π be a leaf in the

Dynkin diagram. Then, the number of the roots in Φ+ r Φα having the same length as α

is|Wα|

|WN(α)|

.

Proof. If mα = 1, we have Φ+ r Φα = Φα,1 and we conclude by Lemma 8.1. For the

classical types, this happens exactly when the leaf α is long. If α is short, we obtain the

claim by duality, since W 〈S∨〉 = W 〈S〉, for each S ⊆ Φ+, and the coroots corresponding

to the short roots in Φ+ whose supports contain α are exactly the long roots in (Φ∨)+

whose supports contain α∨.

It is well known that the number of roots in the irreducible root system Φ equals the

product of the rank of Φ times its Coxeter number (Kostant formula). Let hΦ be the

Coxeter number of Φ. The following result is straightforward by Proposition 8.2.

Corollary 8.3. Let Φ be a root system of type An or Dn. If α ∈ Π is a leaf of the Dynkin

diagram of Φ, then

hΦn− hΦα(n− 1) = 2

|Wα|

|WN [α]|.

Let Φ be an irreducible root system and let ε1, ε2, . . . , εr be the set of all simple roots

of a fixed length t, ordered in such a way that Φ〈Πr ε1, . . . , εi〉 is irreducible for each

i ∈ [1, r]. Such an ordering exists, is unique in the multiple laced case, and is not unique

in the simply laced case. For each i ∈ [1, r], let also

Π(i) = Πr εj : j < i, W (i) = W 〈Π(i)〉,

24 PAOLA CELLINI AND MARIO MARIETTI

(so Π(1) = Π). By definition, εi is a leaf of the Dinkyn diagram of Π(i), for i ∈ [1, r]: we

denote by ε′i the unique root in Π(i) such that

N [εi] ∩Π(i) = εi, ε′i.

Adapting the previous notation, let

W (i)εi

= W 〈Π(i)r εi〉, W

(i)εi,ε

′

i= W 〈Π(i)

r εi, ε′i〉.

Corollary 8.4. Let Φ be a classical root system, Φ+t be the set of all positive roots of

length t in Φ, and ε1, ε2, . . . , εr be the set of all simple roots of length t, ordered in such

a way that Φ〈Πr ε1, . . . , εi〉 is irreducible for each i ∈ [1, r]. Then

|Φ+t | =

r∑

i=1

|W(i)εi |

|W(i)εi,ε

′

i|.

Proof. By definition, Π(i+1) = Π(i)rεi, hence, by Proposition 8.2, |W (i)εi | / |W

(i)εi,ε

′

i| is the

number of positive roots of length t in Φ〈Π(i)〉 r Φ〈Π(i+1)〉. The claim follows directly,

since the roots of length t are contained in Φr Φ〈Π(r)〉.

Remark 8.5. Corollary 8.4 implies, in particular, that the numberr∑

i=1

|W(i)εi

|

|W(i)

εi,ε′i

|does not

depend on the particular ordering of ε1, . . . , εr, provided that each εi is a leaf of the

Dynkin diagram of Π r εj : j < i. This is trivial when the vertices of the Dynkin

diagram have degree at most two. It is not trivial for type D. For type E, it is false.

Indeed, consider type E6. If the sequence starts with the leaf (denoted α2 in [1]) adjacent

to the trivalent root, then

r∑

i=1

|W (i)εi |

|W(i)εi,ε

′

i|=

|W [A5]|

|W [A2]| × |W [A2]|+ (number of positive roots in A5) = 35,

where W [Xn] is the Weyl group of the root system of type Xn. On the other hand, if the

sequence starts with another leaf, then

r∑

i=1

|W(i)εi |

|W(i)εi,ε

′

i|=

|W [D5]|

|W [A4]|+ (number of positive roots in D5) = 36.

Acknowledgments

We thank the referee for carefully reading our manuscript and for giving helpful com-

ments for improving the article.

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Dipartimento di Ingegneria e Geologia, Universita di Chieti – Pescara, Viale Pindaro

42, 65127 Pescara, Italy

Email address : pcellini@unich.it

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universita Politecnica

delle Marche, Via Brecce Bianche, 60131 Ancona, Italy

Email address : m.marietti@staff.univpm.it