Abstract
Abstract. We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras.
Introduction
Any classification of reflection groups or root systems raises the obvious question how different types are related. This work presents an attempt to answer this question for crystallographic Coxeter groups by constructing an appropriate category. The morphisms in this category are described in terms of non-crossing partitions.
Non-crossing partitions were introduced by Kreweras [29] and later generalised in the context of Coxeter groups by Brady and Watt [7, 8] , and independently by Bessis [4] ; see also Armstrong's memoir [1] . A beautiful link between non-crossing partitions and representations of quivers arises from the categorification of cluster algebras [15] via cluster categories [9, 32] ; this was pointed out by Ingalls and Thomas [25] and is the starting point of our work.
In this note we introduce a category of generalised Cartan lattices and attach to each object a sequence of combinatorial objects as follows:
generalised Cartan lattice Γ generalised Cartan matrix C(Γ)
Weyl group W (Γ) poset of non-crossing partitions NC(Γ).
A typical example of a generalised Cartan lattice arises from a hereditary artin algebra A by forming its Grothendieck group K 0 (A). A homomorphism A → A ′ of
Version from October 7, 2013. such algebras induces a morphism K 0 (A ′ ) → K 0 (A) of generalised Cartan lattices provided that restriction induces an isomorphism ′ -modules X, Y and n ≥ 0. In fact, each morphism in this category is up to isomorphism of this form (Theorem 8.2). Moreover, each symmetrisable generalised Cartan matrix [27] is of the form C(Γ) for some generalised Cartan lattice Γ (Lemma 4.1). Thus we categorify the non-crossing partitions given by crystallographic Coxeter groups.
Let us explain our main result. A generalised Cartan lattice Γ = (Γ, −, − ) is a torsion-free abelian group of finite rank together with an integral bilinear form such that its symmetrised form (x, y) = x, y + y, x yields a symmetrisable generalised Cartan matrix C(Γ) = ( e i , e i −1 (e i , e j )) i,j .
Note that Γ is essentially determined by C(Γ) together with a particular choice of a Coxeter element cox(Γ) in the corresponding Weyl group W (Γ), which is given by −, cox(Γ)(x) = − x, − . The poset of non-crossing partitions is by definition NC(Γ) = {w ∈ W (Γ) | id ≤ w ≤ cox(Γ)}, where the partial order on Weyl group elements is given by the absolute length.
The following theorem (Theorems 6.2 and 6.6) summarises the basic properties of these constructions. Theorem 1.1. The generalised Cartan lattices form a category such that all morphisms are monomorphisms.
(1) A morphism Γ ′ → Γ induces an injective homomorphism W (Γ ′ ) → W (Γ) which restricts to an order preserving map NC(Γ ′ ) → NC(Γ). (2) For an object Γ the poset of subobjects is naturally isomorphic to NC(Γ) via the assignment
This theorem is an analogue (for crystallographic Coxeter groups) of a result of Bessis [4] which describes for finite Coxeter groups the non-crossing partitions as Coxeter elements of parabolic subgroups. However, W (Γ ′ ) need not be parabolic when W (Γ) is infinite (Example 6.7). It turns out that the subgroups W (Γ ′ ) ⊆ W (Γ) arising from subobjects Γ ′ ⊆ Γ form a distinguished class of subgroups which are determined by their Coxeter elements cox(Γ ′ ) (Corollary 6.8). Our motivation for this work is to find a combinatorial model for the category H of hereditary abelian categories arising in represention theory of algebras. More precisely, the objects in H are the categories mod A of finitely generated modules over an hereditary artin algebra A. The morphisms in H are fully faithful exact functors, modulo natural isomorphisms, having an extension closed essential image.
Let C denote the category of generalised Cartan lattices. Then the functor
which takes an hereditary abelian category C to its Grothendieck group provides the link between representation theory and combinatorics we are looking for. More precisely, we have the following result (Corollary 8.4) which is a reformulation of a theorem of Ingalls and Thomas [25] . The proofs of these theorems use in an essential way that every generalised Cartan lattice arises as the Grothendieck group K 0 (A) of an hereditary artin algebra A. It is well-known that there is a close relation between roots in K 0 (A) and representations of A. For instance, Gabriel's theorem establishes for representationfinite A a bijection between the isomorphism classes of indecomposable A-modules and the positive roots in K 0 (A) [16] . We extend this bijection by setting up a correspondence between exceptional sequences in mod A and K 0 (A), exploiting the braid group action on the collection of complete exceptional sequences [12, 38] . In more detail, the morphisms in C ending in K 0 (A) are determined by real exceptional sequences, which we show are precisely those sequences arising from exceptional sequences in mod A (Proposition 5.6). Using the theory of perpendicular categories, which we recall in an appendix, we know that each such exceptional sequence X in mod A generates a subcategory C(X) equivalent to some mod B, for another hereditary artin algebra B (Proposition A.4). It follows that K 0 (C(X)) is again a generalised Cartan lattice. It is this result in particular that has no obvious combinatorial proof.
In the last section we also relate our approach to the study of Hom-free sets, which are collections of exceptional objects having pairwise no non-zero homomorphisms. In finite representation type there is an obvious bijection between the two points of view, given by sending a subcategory C closed under kernels, cokernels and extensions to its set of simple objects (Proposition 8.5). It is therefore interesting to note that this approach linking Catalan combinatorics and the representation theory of algebras was already observed in the early 1980s by Gabriel and his school [36, 17] . In [18] Gabriel and de la Peña counted the Hom-free sets of indecomposable modules for Dynkin quivers and obtained the Coxeter-Catalan numbers of ADE-type. On the other hand, Riedtmann used such sets to classify the representation-finite self-injective algebras of type A [36] .
For another intriguing correspondence between representations of hereditary algebras and Weyl group elements see [34] .
Acknowledgements. The idea for this work goes back to an Oberwolfach meeting in 2005 when the results of Ingalls and Thomas involving generalised non-crossing partitions were presented. Since then the second named author discussed this with many colleagues -too many to be listed here -and received valuable comments. He wishes to express special thanks to Christof Geiß, Lutz Hille, Claus Michael Ringel, Christian Stump, and Hugh Thomas.
Bilinear lattices and exceptional sequences
The Grothendieck group of an abelian or triangulated category is an abelian group with some additional structure given by the corresponding bilinear Euler form. In this section we provide an axiomatic treatment which is inspired by work of Lenzing on Grothendieck groups of canonical algebras [31] . In particular, the following definition of a bilinear lattice is taken from there. Then we consider exceptional sequences and the action of the braid group in this setting, modeling their properties in the Grothendieck group of an abelian or triangulated category.
Exceptional sequences were introduced in the Moscow school of vector bundles, see for instance [6, 20, 21, 39] ; later they appeared in representation theory [12, 38] . The subsequent axiomatic treatment in the context of bilinear lattices seems to be new.
Bilinear lattices. A bilinear lattice is an abelian group Γ with a non-degenerate bilinear form
Here, non-degenerate means that x, − = 0 implies x = 0, and −, y = 0 implies y = 0. Note that Γ is torsion-free. The corresponding symmetrised form is defined by (x, y) = x, y + y, x for x, y ∈ Γ.
For a subset S of Γ one defines the right and left orthogonal complements
In the following Γ denotes a bilinear lattice.
Roots. An element a ∈ Γ is called a pseudo-real root, or just a root, if a, a > 0 and a,x a,a ,
x,a a,a ∈ Z for all x ∈ Γ. For such a root a one defines the following transformations:
Note that r a and l a are adjoint with respect to the bilinear from, in the sense that r a (x), y = x, l a (y) for all x, y ∈ Γ, and each s a is a reflection, so fixes a subgroup of corank one and sends a → −a. If Γ ′ is another bilinear lattice, with bilinear form −, − ′ , then an isometry φ : Γ ′ → Γ is a group homomorphism such that φ(x), φ(y) = x, y ′ for all x, y ∈ Γ ′ . We denote by Aut(Γ) the group of all bijective isometries of Γ.
Lemma 2.1. Let a ∈ Γ ′ be a root and φ : Γ ′ → Γ an isometry of bilinear lattices. Then s φ(a) φ = φs a . In particular, if b is a root, then so too is s b (a) and
Proof. Straightforward computation, where for the second statement we put Γ ′ = Γ and φ = s b . Note however that φ(a) ∈ Γ need not be a root.
For a sequence of roots E = (e 1 , . . . , e r ) we write
and denote by ZE the subgroup of Γ generated by e 1 , . . . , e r . We observe that, for each x ∈ Γ, the following all lie in ZE
Exceptional sequences. A sequence of roots E = (e 1 , . . . , e r ) in Γ is called exceptional of length r if e i , e j = 0 for all i > j. The sequence E is complete if ZE = Γ. The empty sequence E = ∅ is exceptional of length zero. An exceptional sequence of length two is also called an exceptional pair. Of particular interest are the transformations s E for exceptional sequences E. We begin with some elementary observations. Lemma 2.2. Let E = (e 1 , . . . , e r ) be an exceptional sequence. Then ZE ∩ E ⊥ = 0. In particular, the e i are linearly independent, so ZE has rank r.
Proof. Take i a i e i = ZE ∩ E ⊥ and apply e i , − for i = r, . . . , 1 in turn.
Lemma 2.3. Let E = (e 1 , . . . , e r ) be an exceptional sequence in Γ. Then the following hold:
In particular, l E is the projection from Γ onto E ⊥ along ZE.
Proof. The proofs are by induction on r. Set E ′ = (e 1 , . . . , e r−1 ). (1) Let x ∈ Γ. We have r E ′ (e r ) = e r , so e r , l E ′ l er (x) = r E ′ (e r ), l er (e r ) = e r , l er (x) = 0.
If l E (x) = 0, then l er (x) ∈ ZE ′ by induction, and therefore x = l er (x) + e r , x e r , e r e r ∈ ZE.
(3) This follows from (1) and (2) .
Proposition 2.4. Let E = (e 1 , . . . , e r ) be an exceptional sequence in Γ and x ∈ Γ. Then
Proof. For y ∈ ⊥ E use that s E (x) − x ∈ ZE. Suppose therefore that y ∈ ZE. The proof is by induction on r. Set E ′ = (e 1 , . . . , e r−1 ). If y ∈ ZE ′ , then e r ∈ ⊥ y, so
If y ∈ Ze r , then since s E ′ (z) − z ∈ ZE ′ ⊆ e ⊥ r for all z, we have y, s E ′ s er (x) = y, s er (x) , and a direct computation shows that this equals − x, y .
The radical of Γ is by definition
Corollary 2.5. Let E and F be exceptional sequences in Γ.
If e, f ∈ Γ are roots and s e = s f , then Ze = Zf .
Proof. (1) Suppose that ZE = ZF . Then s E = s F by Proposition 2.4, using that the form −, − is non-degenerate and Γ = ZE ⊕ ⊥ E. (2) Suppose that s E = s F . Given x ∈ ZF write x = x ′ + x ′′ with x ′ ∈ ZE and x ′′ ∈ ⊥ E. Using Proposition 2.4 we have for all y ∈ Γ that
Thus x ′′ ∈ rad Γ, and it follows that ZF ⊆ ZE + rad Γ. The other inclusion holds by symmetry.
(3) Using (2) we have e = αf + r with α ∈ Z and r ∈ rad Γ. Thus −e = s e (e) = s f (e) = e − 2αf and therefore r = 0. It follows that Ze ⊆ Zf . The other inclusion holds by symmetry.
As a consequence we observe that if e and αe are both roots, then α = ±1.
The Coxeter transformation. Let Γ be a bilinear lattice and suppose that Γ admits a complete exceptional sequence E = (e 1 , . . . , e n ). The Coxeter transformation of Γ is by definition
This does not depend on the choice of E by Proposition 2.4. Now identify Γ = Z n and define an n × n matrix C by x, y = x t Cy.
Proposition 2.6. The matrix C is invertible. The automorphism of Γ given by
equals cox(Γ), and satisfies −, c(x) = − x, − for all x ∈ Γ.
Proof. We have
Thus c = cox(Γ) by Proposition 2.4.
Braid group actions and non-crossing partitions
For a bilinear lattice we introduce an action of the braid group on exceptional sequences of some fixed length. Then we introduce the Weyl group with respect to a fixed complete exceptional sequence and define inside that the poset of noncrossing partitions. This leads to a correspondence between exceptional sequences and non-crossing partitions.
Braid and sign group actions. For an integer r ≥ 1 let B r be the braid group on r strands, so with generators σ 1 , . . . , σ r−1 and relations
We also consider the wreath product {±1} ≀ B r , so the semi-direct product {±1} r ⋊ B r of the braid group with the sign group, with multiplication given by Proof. We check that the relations for the braid group hold, the rest being clear. Let E = (e 1 , . . . , e r ) be an exceptional sequence. Using Lemma 2.1 we see that σ i E and σ
−1
i E are again exceptional sequences and that σ i σ
it is enough to show this when i = 1 and r = 3. In this case we have E = (e, f, g) and
Now use the identity s sg (f ) = s g s f s g from Lemma 2.1.
Note that, if (e, f ) is an exceptional pair, then s e (f ) = l e (f ) by Lemma 2.3, and dually s f (e) = r f (e), so we can express the action of the braid group in terms of the maps l and r. Lemma 3.2. Let E be an exceptional sequence of length r and σ ∈ {±1}≀B r . Then
Proof. It is clear that ZσE = ZE for each generator σ = σ i ∈ B r and each element of the sign group. Thus ZσE = ZE for all σ ∈ {±1} ≀ B r , and then s σE = σ E by Corollary 2.5.
The braid group B r acts on the set of all r element sequences in any group via
Note that the product of the group elements remains the same.
Lemma 3.3. Consider the map π sending an exceptional sequence (e 1 , . . . , e r ) in Γ to the sequence of reflections (s e1 , . . . , s er ) in Aut(Γ). Then πσ = σπ for σ ∈ B r , whereas the fibres of π are precisely the orbits of the sign group.
Proof. Recall from Lemma 2.1 that s s b (a) = s b s a s b , and also that s e = s f for two roots e, f implies that e = ±f by Corollary 2.5 (3).
The Weyl group. Let Γ be a bilinear lattice. Given a complete exceptional sequence E = (e 1 , . . . , e n ) we define the Weyl group of (Γ, E) to be the group
Note that the Weyl group depends only on the orbit of E under the braid group. The set of real roots (with respect to E) is
By Lemma 2.1 we see that each real root is a (pseudo-real) root; also, if a = w(e i ) ∈ Φ, then −a = ws ei (e i ) ∈ Φ and
Non-crossing partitions. Let W = (W, S) be any group generated by a set S = {s 1 , . . . , s n } of elements of order two, which we shall call the simple reflections by analogy with Coxeter groups. We define the set of all reflections to be
is the Weyl group of a bilinear lattice, then T = {s a | a ∈ Φ} is the set of reflections at the real roots. Given w ∈ W , its absolute length ℓ(w) is the minimal r ≥ 0 such that w can be written as product w = x 1 . . . x r of reflections x i ∈ T . The absolute order on W is then defined as
A Coxeter element is a product of all the generators in S, in some order. Relative to a Coxeter element c one defines the poset of non-crossing partitions
The braid group action on factorisations of the Coxeter element is transitive whenever W is a Coxeter group; we record this important result for later use. For finite Coxeter groups, a proof can be found in a letter of Deligne [13] . For the absolute length of the Coxeter element, see Dyer [14] . Theorem 3.4 (Igusa-Schiffler [26] ). Let (W, S) be a Coxeter system of rank |S| = n and let c be a Coxeter element. Then ℓ(c) = n and the braid group B n acts transitively on the set of sequences of reflections (x 1 , . . . , x n ) such that x 1 · · · x n = c.
Real exceptional sequences. Let (Γ, E) be a pair consisting of a bilinear lattice Γ and a complete exceptional sequence E = (e 1 , . . . , e n ) such that the Weyl group W (Γ, E) is a Coxeter group.
We are mostly interested in exceptional sequences consisting of real roots. Observe that the action of the wreath product {±1} ≀ B r on exceptional sequences of length r restricts to an action on sequences of real roots. This is clear from the definition of the action, using that Φ = −Φ. Proposition 3.5. Let Γ be a bilinear lattice, and let E be a complete exceptional sequence of length n such that the Weyl group W (Γ, E) is a Coxeter group. Then the wreath product {±1} ≀ B n acts transitively on the set of complete exceptional sequences of real roots.
Proof. Let F = (f 1 , . . . , f n ) be a complete exceptional sequence of real roots. Then s E = s F = cox(Γ), so by Theorem 3.4 there exists σ ∈ B n such that σ(s e1 , . . . , s en ) = (s f1 , . . . , s fn ). Consider (g 1 , . . . , g n ) := σ(e 1 , . . . , e n ). Then each g i is a real root and f i = ±g i by Lemma 3.3.
A subsequence of a sequence (f 1 , . . . , f r ) of elements of Γ is one of the form (f i1 , . . . , f is ) for 1 ≤ i 1 < i 2 < · · · < i s ≤ r; it is an initial subsequence if i j = j for all j.
A real exceptional sequence is a subsequence of a complete exceptional sequence consisting entirely of real roots. As an immediate corollary of the transitivity of the braid group action, Theorem 3.4 and Proposition 3.5, we obtain the following.
Corollary 3.6. The real exceptional sequences in Γ are precisely the initial subsequences of the sequences σE for σ ∈ {±1} ≀ B n .
It follows that the real exceptional sequences depend only on the orbit of E under the braid group.
We also have the following result concerning factorisations of non-crossing partitions.
Proposition 3.7. Let Γ be a bilinear lattice, and let E be a complete exceptional sequence such that the Weyl group W (Γ, E) is a Coxeter group. Then the map (f 1 , . . . , f r ) → (s f1 , . . . , s fr ) yields a bijection between
(1) the real exceptional sequences of length r (modulo the sign group action) and
Proof. We claim that the inverse sends (x 1 , . . . , x r ) to (f 1 , . . . , f r ), where s fi = x i . Let (x 1 , . . . , x r ) be a sequence of reflections such that x = x 1 · · · x r is a noncrossing partition of length r. Write x i = s fi for some roots f i , unique up to sign by Corollary 2.5. By Theorem 3.4 we know that (x 1 , . . . , x r ) is an initial subsequence of σ(s e1 , . . . , s en ), and Lemma 3.3 implies that, up to sign, (f 1 , . . . , f r ) is an initial subsequence of σ(e 1 , . . . , e n ), so is a real exceptional sequence.
For an exceptional sequence F , the element s F depends only on the orbit of F under the braid group. Thus we deduce that F → s F induces a surjective map from orbits of real exceptional sequences of length r to non-crossing partitions of absolute length r. We will see later in Theorem 6.2 that it is even a bijection.
Generalised Cartan lattices
In this section we introduce the category of generalised Cartan lattices.
Generalised Cartan lattices. An exceptional sequence E = (e 1 , . . . , e r ) in a bilinear lattice Γ is said to be orthogonal provided e i , e j ≤ 0 for all i = j. A generalised Cartan lattice is a pair (Γ, E) consisting of a bilinear lattice Γ and a complete exceptional sequence E = (e 1 , . . . , e n ) which is orthogonal.
If (Γ, E) is a generalised Cartan lattice, then the matrix
is a symmetrisable generalised Cartan matrix 1 and the Weyl group W = W (Γ, E) is a Coxeter group with respect to the set of simple reflections {s e1 , . . . , s en } [27, Proposition 3.13]. In particular, the Weyl group depends only on the Cartan matrix C(Γ, E). Note however that different choices of orthogonal exceptional sequences in Γ can give rise to the same Cartan matrix.
1 Following Kac [27] we call an integral square matrix C a symmetrisable generalised Cartan Proof. Let C = D −1 B be a symmetrisable generalised Cartan matrix of size n with D = diag(d i ). Take Γ = Z n with standard basis {e 1 , . . . , e n } and equip this with the bilinear form given by
Then E = (e 1 , . . . , e n ) is a complete, orthogonal exceptional sequence, (Γ, E) is a generalised Cartan lattice, and C(Γ, E) = C.
In summary we have now associated to a generalised Cartan lattice (Γ, E) with E = (e 1 , . . . , e n ) the following objects:
• the symmetrisable generalised Cartan matrix C(Γ, E).
• the Weyl group W (Γ, E) = s e1 , . . . , s en ≤ Aut(Γ).
• the Coxeter element cox(Γ) = s E .
• the set of real roots Φ.
• the poset of non-crossing partitions
For a fixed choice of E it will often be convenient to simplify the notation and write, for example, C(Γ) instead of C(Γ, E).
Morphisms of generalised Cartan lattices.
Cartan lattices is an isometry φ : Γ ′ → Γ such that φE ′ is a real exceptional sequence, so an initial subsequence of σE for some σ ∈ {±1} ≀ B n . Lemma 4.2. The generalised Cartan lattices form a category in which all morphisms are monomorphisms.
The following gives a more conceptual description of the morphisms.
between generalised Cartan lattices are precisely those isometries φ : Γ ′ → Γ sending real exceptional sequences in Γ ′ to real exceptional sequences in Γ.
We observe that a morphism φ :
. This functoriality will be established in §6.
Real roots for generalised Cartan lattices. We discuss the difference between pseudo-real and real roots.
Lemma 4.4. Let E = (e 1 , . . . , e n ) be a complete exceptional sequence in a bilinear lattice Γ. Then a = i α i e i is a pseudo-real root if and only if a, a > 0 and α i ei,ei a,a ∈ Z for all i. 2 We do not know a reasonable definition of a morphism which covers, for instance, morphisms
Proof. Let a = i α i e i be a pseudo-real root. We want to show that α i e i , e i ∈ a, a Z for all 1 ≤ i ≤ n. Clearly α 1 e 1 , e 1 = a, e 1 ∈ a, a Z, so the result holds for i = 1. Now let i > 1 and assume by induction that the result holds for all j < i. Since e j is a root we know that e j , e i ∈ e j , e j Z, so for each j < i we can find an integer λ j such that α j e j , e i = λ j a, a . Since a, e i ∈ a, a Z, the same holds for
Conversely, suppose a = i α i e i satisfies a, a > 0 and α i e i , e i ∈ a, a Z for all i. Fix r ∈ {1, . . . , n}. Again, since e i is a root, we have α i e i , e r , α i e r , e i ∈ a, a Z. It follows that a,er a,a , er ,a a,a ∈ Z, and hence a is a pseudo-real root.
Let (Γ, E) be a generalised Cartan lattice, where E = (e 1 , . . . , e n ), and with Cartan matrix C. We define a partial order on Γ by saying a ≥ 0 provided a = i α i e i with α i ≥ 0 for all i. We say that C (or Γ) is indecomposable if we cannot permute the rows and columns to obtain a block diagonal matrix. When C is indecomposable we say that C has In general there are pseudo-real roots which are not real. 
Thus Γ is the Grothendieck group of the path algebra of the quiver (see §5).
We take E = (e 1 , . . . , e 4 ) to be the standard basis in order. Then the element a = (1, 1, 3, 1) is a pseudo-real root but not a real root.
Grothendieck groups of hereditary algebras
The Grothendieck group of a finite dimensional hereditary algebra is an example of a generalised Cartan lattice, and in fact each generalised Cartan lattice is of this form (Lemma 5.3). In this section we concentrate on exceptional sequences of modules over hereditary algebras and discuss the braid group action.
Hereditary algebras. Let k be a field and A a finite dimensional k-algebra. We denote by mod A the category of finite dimensional A-modules and by proj A the full subcategory consisting of projective A-modules.
The Grothendieck group K 0 (A) is by definition the Grothendieck group of the exact category proj A with the bilinear form given by
This group is free of finite rank. The classes of a representative set of indecomposable projective A-modules form a basis. Now suppose that A has finite global dimension. Let K 0 (mod A) denote the Grothendieck group of the abelian category mod A and observe that the inclusion proj
We view this isometry as an identification. Finally, an algebra A is called hereditary if each A-module has a projective resolution of length at most one. This is equivalent to saying that every submodule of a projective module is again projective.
Exceptional sequences. Let A be a finite dimensional hereditary algebra. A module X ∈ mod A is called exceptional if it is indecomposable and Ext
Such a sequence is complete if r equals the rank of K 0 (A), and is orthogonal if Hom A (X i , X j ) = 0 for all i = j. For example, any exceptional sequence consisting of simples is necessarily orthogonal.
We begin by recalling the following useful lemma of Happel and Ringel. A (X, Y ) = 0, then any homomorphism Y → X is either mono or epi. In particular, if X is exceptional, then End A (X) is a division algebra.
We immediately see that if X is exceptional, then [X] ∈ K 0 (A) is a pseudo-real root. We will see later in Corollary 5.7 that it is even a real root.
Proof. We know that End A (X) is a division algebra, so all finite length modules are free. In particular this applies to the right modules Hom A (X, Y ) and Ext [X], [X] are integers, so [X] is a pseudo-real root.
It follows that each exceptional sequence (X 1 , . . . , X r ) in mod A yields an exceptional sequence ( 
Moreover, if the former is complete (respectively orthogonal), then so too is the latter.
The following lemma shows that every generalised Cartan lattice can be realised as the Grothendieck group of a finite dimensional hereditary algbra.
Lemma 5.3. The assignment A → K 0 (A) has the following properties:
(1) Let A be a finite dimensional hereditary algebra. Then there are simple A-modules S 1 , . . . , S n such that (S 1 , . . . , S n ) is a complete, orthogonal exceptional sequence in mod A. In particular, setting e i := [S i ] and E := (e 1 , . . . , e n ), we see that (K 0 (A), E) is a generalised Cartan lattice. (2) Let (Γ, E) be a generalised Cartan lattice, where E = (e 1 , . . . , e n ). Given a finite field k there exists a finite dimensional hereditary algebra A and an isometry Γ ∼ − → K 0 (A) sending each e i to the class of a simple A-module.
Proof.
(1) Each non-zero morphism between indecomposable projective A-modules is a monomorphism since A is hereditary. Thus each indecomposable projective is exceptional (since A is finite dimensional) and we can order a representative set of indecomposable projective modules as P 1 , . . . , P n such that Hom A (P i , P j ) = 0 for i < j. Let S i = P i / rad P i be the simple top of the projective P i . Then the long exact sequence for Hom A (−, S j ) yields
Since Hom A (rad P i , S j ) = 0 for all j ≤ i it follows that each S i is exceptional, and that (S 1 , . . . , S n ) is a complete, orthogonal exceptional sequence.
(2) We follow [16, §7] and [24, §5] . Let k i /k be a field extension of degree e i , e i and k ij /k a field extension of degree − e i , e j for i < j. Set k ij = 0 for i ≥ j. We regard each k ij as k j -k i -bimodule. Then A 0 = i k i is a semisimple k-algebra and A 1 = i,j k ij an A 0 -bimodule, so the tensor algebra
is a finite dimensional hereditary k-algebra. Denote by ε i the idempotent of A given by the identity of k i . Then the P i := Aε i give a representative set of indecomposable projective A-modules, with simple tops S i = k i , and ε j (rad P i / rad The braid group action. Let X = (X 1 , . . . , X r ) be an exceptional sequence in mod A. We define C(X) to be the smallest full subcategory of mod A containing each X i and closed under kernels, cokernels and extensions. Then C(X) is equivalent to the module category of a finite dimensional hereditary algebra by Proposition A.4. It follows from Corollary A.8 that for any integer 1 ≤ i < r there exist unique modules R Xi+1 (X i ) and L Xi (X i+1 ) in C(X) yielding exceptional sequences
Following [12, 38] , the braid group B r acts on exceptional sequences of length r via
In fact, we can describe the modules L X (Y ) and R Y (X) explicitly, using the five term exact sequences (A.1) and (A.2), together with Remark A.2. Let (X, Y ) be an exceptional pair. If Hom A (X, Y ) = 0, then L X (Y ) is the middle term of the universal extension
An analogous description is used for R Y (X).
We observe that this definition appears more natural once one passes to the derived category D b (mod A), where functors L E and R E are defined with respect to any exceptional object E; see [6] . Then L X (Y ) and R Y (X) coincide up to translation with the objects defined in D b (mod A) via the functors L X and R Y . The braid group actions on exceptional sequences in mod A and K 0 (A) are related as follows.
Lemma 5.4. Let (X 1 , . . . , X r ) be an exceptional sequence and σ ∈ B r . Then
for some ε ∈ {±1} r .
Proof. Let (X, Y ) be an exceptional pair in mod A. By Lemma 5.1 any nonzero homomorphism X → Y is either mono or epi, so Hom A (X, Y ) = 0 implies Ext
) (in fact, we get '−' in the first case and '+' in the second and third cases), and similarly for R Y (X). This proves the result when σ = σ ±1 i ; the general case follows immediately.
Transitivity of the braid group action. We have seen in Theorem 3.4 that the braid group acts transitively on factorisations of a Coxeter element. The following theorem is the analogue for exceptional sequences in module categories.
We say that two exceptional sequences (X 1 , . . . , X r ) and (
Theorem 5.5 (Crawley-Boevey [12] , Ringel [38] ). Let A be a finite dimensional hereditary algebra, and set n to be the rank of K 0 (A). Then the braid group B n acts transitively on the isomorphism classes of complete exceptional sequences in mod A.
The close relation between Theorems 3.4 and 5.5 is clarified in Remark 5.14.
Connection to real exceptional sequences. We now want to compare exceptional sequences in mod A with real exceptional sequences in K 0 (A), where A is a finite dimensional hereditary algebra. Proof. As in Lemma 5.3 let S = (S 1 , . . . , S n ) be a complete, othogonal exceptional sequence in mod A consisting of simple modules. Set e i := [S i ], so that E := (e 1 , . . . , e n ) is a complete, orthogonal exceptional sequence in K 0 (A). Now, by Corollary 3.6, a real exceptional sequence in K 0 (A) is an initial sequence ofσE someσ ∈ {±1}≀B n . On the other hand, every exceptional sequence in mod A can be extended to a complete exceptional sequence (Corollary A.8), so is an initial sequence of σS for some σ ∈ B n by Theorem 5.5. The result now follows from Lemma 5.4.
The real roots of the form ±[X] for an exceptional object X are called real Schur roots. We can characterise them amongst all real roots using non-crossing partitions.
Corollary 5.8. Let a ∈ K 0 (A) be a real root. Then a is a real Schur root if and only if s a ∈ NC(K 0 (A)).
For an algebra A of finite representation type (so the corresponding Weyl goup W (K 0 (A)) is finite) all real roots are real Schur roots. The following example provides a real root which is not a real Schur root. The module X is not exceptional, but [X] is a real root.
) yields a bijection between the orbits of the braid group B r on exceptional sequences of length r in mod A, and the orbits of the wreath product {±1}≀B r on real exceptional sequences of length r in K 0 (A).
Combining with Proposition 3.7 we get the following.
, . . . , s [Xr] ) yields a bijection between (1) isomorphism classes of exceptional sequences of length r in mod A, and (2) sequences of reflections (x 1 , . . . , x r ) in the Weyl group W (K 0 (A)) such that x = x 1 · · · x r belongs to NC(K 0 (A)) and ℓ(x) = r.
This result can be reformulated as saying that exceptional sequences in mod A correspond to paths in the Hasse diagram of NC(K 0 (A)). For, sequences of reflections (x 1 , . . . , x r ) in W (K 0 (A)) with x = x 1 · · · x r in NC(K 0 (A)) and ℓ(x) = r are represented by paths of length r which start at the unique minimal element; see also [25, p. 1534] . The number of complete exceptional sequences for algebras of finite representation type is computed in [33] ; we refer to their paper for further historical comments.
It seems to be a hard problem to distinguish the real exceptional sequences inside the set of all exceptional sequences. Every complete exceptional sequence of real roots is necessarily a real exceptional sequence, and so (modulo signs) can be lifted to an exceptional sequence in mod A. In general it is enough to check this pairwise (see also Proposition 7.3). Here one might naively hope that if x, y are real Schur roots and y, x = 0, then (x, y) is a real exceptional pair. This is known to fail, as the following counter-example by Schofield [41] shows. Proof. By Proposition A. 4 we know that C(X) is equivalent to the module category of a finite dimensional hereditary algebra, so K 0 (C(X)) is again a generalised Cartan lattice. Moreover, since C(X) is a full subcategory of mod A closed under extensions, the inclusion functor induces an isometry of generalised Cartan lattices. Finally, by Corollary A.8, there is a complete exceptional sequence containing X as an initial subsequence, so φ sends ([X 1 ], . . . , [X n ]) to a real exceptional sequence by Theorem 5.5.
Remark 5.14. We finish this section by noting that Theorem 5.5 is an immediate consequence of Proposition 3.5 once one knows that the class of an exceptional A-module is a real root in K 0 (A). 
The subobjects of generalised Cartan lattices
In this section we establish the correspondence between subobjects of generalised Cartan lattices and non-crossing partitions. Later on we illustrate this by looking at representations of hereditary algebras. In fact, the proof of our main result uses representations of hereditary algebras in an essential way.
Subobjects. Fix a category and an object X in this category. Two monomorphisms φ : U → X and φ ′ : U ′ → X are equivalent if φ factors through φ ′ and φ ′ factors through φ. The equivalence class of a monomorphism φ ending in X is denoted by [φ]; they form the subobjects of X and Sub(X) denotes the set of these subobjects. Defining [φ] ≤ [φ ′ ] if φ factors through φ ′ gives a partial order on Sub(X).
The following lemma provides a crucial step in our proof of the main theorem. It would be interesting to have a purely combinatorial proof which avoids the use of hereditary algebras.
Lemma 6.1. Let (Γ, E) be a generalised Cartan lattice. Given a real exceptional sequence E ′ of length r, there exists τ ∈ {±1} ≀ B r such that τ E ′ is orthogonal.
Proof. Let E = (e 1 , . . . , e n ). By Lemma 5.3 we can find a finite dimensional hereditary k-algebra A and simple A-modules S i such that [S i ] = e i . We use the braid group action on the set of complete exceptional sequences in mod A. Fix σ ∈ {±1} ≀ B n such that E ′ is an initial subsequence of σE; see Corollary 3.6. Let σ ′ ∈ B n be the projection of σ and write (X 1 , . . . , X n ) := σ ′ (S 1 , . . . , S n ). Set X := (X 1 , . . . , X r ) and consider the subcategory C(X) of mod A. Then C(X) ∼ = mod A ′ for some finite dimensional hereditary k-algebra A ′ by Proposition A.
Under this equivalence E
′ is sent to a complete exceptional sequence in the generalised Cartan lattice K 0 (A ′ ), so by Proposition 3.5 there exists some τ ∈ {±1} ≀ B r such that the image of τ E ′ is orthogonal in K 0 (A ′ ), and hence τ E ′ is orthogonal in Γ.
Let Exc(Γ) be the set of equivalence classes of real exceptional sequences in Γ, where two such sequences of length r are equivalent if they determine the same orbit under the action of {±1} ≀ B r . We make this into a poset by saying that
′′ is an initial subsequence of some τ E ′ .
Theorem 6.2. Let (Γ, E) be a generalised Cartan lattice. There are canonical isomorphisms
Proof. We know from Proposition 3.7 that the map Exc(Γ) → NC(Γ), E ′ → s E ′ , is well-defined and surjective. To see that it is injective, take x ∈ NC(Γ) of length r, and set y = x −1 c. Then y is again a non-crossing partition (we have y(c −1 xc) = c), say of length n − r. By Proposition 3.7 we can find real exceptional sequences X = (f 1 , . . . , f r ) and Y = (f r+1 , . . . , f n ) such that s X = x and s Y = y, and hence (f 1 , . . . , f n ) is a complete exceptional sequence. Moreover by Lemma 6.1 we may further assume that X is orthogonal, so that
be any other real exceptional sequence satisfying s X ′ = x. Applying Proposition 3.7 again, we have r ′ = r and (f ′ 1 , . . . , f ′ r , f r+1 , . . . , f n ) is also a complete exceptional sequence, so ZX ′ = Y ⊥ and by Proposition 3.5 we have X ′ = σX for some σ ∈ {±1} ≀ B r . To check that this is an isomorphism of posets, note that if E ′ is an initial subsequence of a real exceptional sequence E ′′ , then clearly s E ′ ≤ s E ′′ . Conversely, let x ≤ y be non-crossing partitions. Then x −1 y is also a non-crossing partition, and so by Proposition 3.7 we can find real exceptional sequences X = (f 1 , . . . , f r ) and X ′ = (f r+1 , . . . , f s ) such that s X = x and s X ′ = x −1 y. Setting Y = (f 1 , . . . , f s ) we have s Y = y, so Y is also a real exceptional sequence, containing X as an initial subsequence. 
. It follows that E ′ → φ induces a poset homomorphism Exc(Γ) → Sub(Γ), and that this is inverse to the map described above.
We write cox for the map Exc(Γ) → NC(Γ), E ′ → s E ′ , as well as for the composition Sub(Γ) → NC(Γ). 
It follows that Γ → NC(Γ) determines a functor from the category of generalised Cartan lattices to the category of posets.
Proof. We know that φ sends real exceptional sequences in Γ ′ to real exceptional sequences in Γ. This preserves the action of the wreath product and the notion of being an initial subsequence, so induces a morphism of posets Exc(
The Weyl group. We show that the assignment Γ → W (Γ) is also functorial, and restricts to the functor Γ → NC(Γ) constructed above. The proof requires several steps.
Lemma 6.4. Let (Γ, E) be a generalised Cartan lattice of rank n + 1 and let E ′ = {f 1 , . . . , f n } be a linearly independent set of real roots. Set Γ ′ = ZE ′ and write W ′ for the subgroup of W (Γ) generated by s f1 , . . . , s fn . Then restriction to
Proof. We need to show that if w ∈ W ′ fixes Γ ′ pointwise, then it is the identity. For simplicity we assume that Γ is connected.
Write E = (e 0 , . . . , e n ) and take i such that e i ∈ Γ ′ . Set x = w(e i ) − e i . Since W ′ is generated by the s fj we know that x ∈ Γ ′ . Also, the symmetric bilinear form is W -invariant, so (y, y) = (w(y), w(y)) for all y ∈ Γ. Applying this to y = e i gives 2(e i , x) + (x, x) = 0, whereas applying it to y = e i + a for a ∈ Γ ′ and using that w(e i + a) = e i + a + x gives 2(e i + a, x) + (x, x) = 0. It follows that (a, x) = 0 for all a ∈ Γ ′ ; in particular (x, x) = 0, so (e i , x) = 0 and hence x ∈ rad Γ. Now write x = x + − x − , where x + , x − ≥ 0 have disjoint support. Since x is radical we must have (x + , e j ) = (x − , e j ) for all j. If j ∈ supp x + then the left-hand side is non-positive, whereas if j ∈ supp x − then the right-hand side is non-positive. Since the supports are disjoint we deduce that (x + , e j ) = (x − , e j ) ≤ 0 for all j.
Next observe that w(e i ) = e i + x is a real root, so is either positive or negative when expressed as a linear combination of the e j (see for example [27, Theorem 1.2]). If x + > 0, then e i + x must be a positive root, whence 0 ≤ x − ≤ e i . Since (e i , x − ) ≤ 0 we must have x − = 0, so x = x + > 0. On the other hand, if x + = 0, then x = x − ≤ 0.
In either case we see that, if w = id, then x ∈ rad Γ is either positive or negative. We can now apply the classification of symmetrisable generalised Cartan matrices ( [27, Corollary 4.3] or [11, Corollary 15.11] ) to conclude that Γ is of affine type and x = mδ is a multiple of the minimal positive imaginary root δ.
We have thus reduced to the case when Γ is of affine type. In this case we may assume that e 0 is an extending vertex 
When Γ has type A
2l we set
(In type A Proposition 6.5. Let (Γ, E) be a generalised Cartan lattice and E ′ = (e 1 , . . . , e r ) a real exceptional sequence. Set Γ ′ = ZE ′ . Then the assignment w → w| Γ ′ induces an isomorphism
Proof. Let w ∈ s e1 , . . . , s er fix Γ ′ pointwise. We need to show that w = id. Extend E ′ to a complete real exceptional sequence (e 1 , . . . , e n ) for Γ. Given r < i ≤ n set E i = (e 1 , . . . , e r , e i ) and Γ i = ZE i . By Lemma 6.1 there exists τ ∈ {±1} ≀ B r+1 such that (Γ i , τ E i ) is a generalised Cartan lattice, so we can apply Lemma 6.4 to deduce that w is the identity on Γ i . This holds for each such i, so w is the identity on the whole of Γ. Theorem 6.6. Let φ : Γ ′ → Γ be a morphism of generalised Cartan lattices, so yielding an injection on the set of real roots. Then the map s a → s φ(a) yields an injective group homomorphism φ * : W (Γ ′ ) → W (Γ), and this restricts to the poset homomorphism NC(Γ ′ ) → NC(Γ) constructed previously. In particular, the map φ * identifies NC(Γ ′ ) with {x ∈ NC(Γ) | x ≤ φ * (cox(Γ ′ ))}.
Proof. The map φ * is just the inverse of the isomorphism (6.1) construced above.
For the second statement we just need to observe that, if E ′ is a real exceptional sequence in Γ ′ , then φ * (s E ′ ) = s φ(E ′ ) .
Pointed Coxeter groups. For a Coxeter group W = W (Γ) we give a group theoretic description of the subgroups of W which are of the form W (Γ ′ ) for some subobject Γ ′ ⊆ Γ. If W is finite, then these subgroups are known to be parabolic [4, Lemma 1.4.3] . This is no longer true when W is infinite. Moreover, the Coxeter diagram of such a subgroup is not necessarily obtained by removing vertices from the diagram of W . The authors are grateful to Christian Stump for suggesting the following example.
Example 6.7. Consider a Coxeter group W = s, t, u of affine type A 2 with Coxeter element c = stu. The factorisation c = s(utu)(ututu) yields a non-crossing partition s(utu) and the corresponding subgroup W ′ = s, utu is affine of type A 1 . Thus W ′ is not a parabolic subgroup.
Let us define a pointed Coxeter group as a pair (W, c) consisting of a Coxeter group W and a Coxeter element c.
Note that any such subgroup is determined by its Coxeter element since it is generated by its non-crossing partitions.
The following is an immediate consequence of Theorem 6.6.
Corollary 6.8. Let Γ be a generalised Cartan lattice. Then (W (Γ), cox(Γ)) is a pointed Coxeter group and the map Γ ′ → (W (Γ ′ ), cox(Γ ′ )) induces an order preserving bijection between the subobjects of Γ and the subgroups of (W (Γ), cox(Γ)).
Non-crossing partitions revisited
In this section we collect some properties of non-crossing partitions which are inspired by our categorification.
The Kreweras complement. Let Γ = (Γ, E) be a generalised Cartan lattice and
the corresponding poset of non-crossing partitions. Given a sublattice ∆ ⊆ Γ, we can form its orthogonal complements ⊥ ∆ and ∆ ⊥ with respect to −, − . These induce two operations on Sub(Γ) which correspond to the formation of Kreweras complements in NC(Γ) via the isomorphism cox: Sub(Γ)
Proposition 7.1. Taking ∆ ⊆ Γ to ⊥ ∆ and ∆ ⊥ induces two order reversing bijections Sub(Γ) → Sub(Γ) which are mutually inverse. Moreover,
Proof. Let (∆, E ′ ) be in Sub(Γ). Thus E ′ = (e 1 , . . . , e r ) is a subsequence of σE = (e 1 , . . . , e n ) for some σ ∈ {±1} ≀ B n . Set E ′′ = (e r+1 , . . . , e n ). Then
The argument for ∆ ⊥ is similar. Real Schur roots. Let Γ = (Γ, E) be a generalised Cartan lattice. Recall from Corollary 5.8 that x ∈ Γ is a real Schur root if it is a real root and the reflection s x belongs to NC(Γ). If Γ = K 0 (A) for some finite dimensional hereditary algebra A, then the real Schur roots correspond to the exceptional A-modules.
In [38, §3] , Ringel describes an algorithm for obtaining all exceptional modules for a given finite dimensional hereditary algebra A. The idea is to construct them inductively from the simple modules by applying in each step the braid group operations on exceptional pairs consisting of modules which are already constructed.
Non-crossing partitions from exceptional pairs. For the Coxeter group W (Γ) given by a generalised Cartan lattice Γ, one can ask for a procedure to construct its non-crossing partitions. The following result says that it suffices to know the non-crossing partitions of absolute length one and two; see [2, §2] for the case that W (Γ) is finite. Note that the non-crossing partitions of absolute length one are precisely the reflections corresponding to real Schur roots in Γ. (1) x ∈ NC(W, c) and ℓ(x) = r. (2) x i x j ∈ NC(W, c) and ℓ(x i x j ) = 2 for all i < j.
Proof. Fix a generalised Cartan lattice Γ such that W = W (Γ). In fact, it is convenient to choose Γ = K 0 (A) for some finite dimensional hereditary algebra A. Then the assertion follows from the bijection between non-crossing partitions in W (Γ) and exceptional sequences in mod A; see Corollary 5.11.
The proposition yields a procedure to construct elements of some fixed absolute length in a Coxeter group. For another description of this length we refer to [14] .
Hereditary categories
We consider the category H of hereditary abelian categories arising in the represention theory of algebras. More precisely, the objects in H are the categories mod A of finitely generated modules over an hereditary artin algebra A.
5 The morphisms in H are fully faithful exact functors, modulo natural isomorphisms, having an extension closed essential image.
Homological epimorphisms. Following [19] , a ring homomorphism A → B is called a homological epimorphism if restriction of scalars induces an isomorphism Conversely, every such subcategory of mod A arises in this way.
Proof. Since f is finite, the restriction of scalars functor goes between the categories of finitely generated modules. Moreover, this functor is always exact and has extension of scalars as a left adjoint. Finally, since f is a homological epimorphism, the functor is fully faithful and the essential image is closed under extensions. Conversely, let C ⊆ mod A be a full subcategory closed under kernels, cokernels and extensions. Then Ext
If C is also a reflective subcategory, so the inclusion has a left adjoint L, then LA is a projective generator for C. Thus C is equivalent to mod End A (LA), and the map End A (A) → End A (LA) induced by the left adjoint yields a homological epimorphism f : A → End A (LA). Finally, the isomorphism Hom A (A, LA) ∼ = End A (LA) is now an isomorphism of right A-modules, showing that f is finite.
In fact, using the isomorphisms LA ∼ = Hom A (A, LA) ∼ = End A (LA), we can endow LA with the structure of an algebra, in which case the algebra homomorphism f : A → LA corresponds to the identity in End A (LA). Theorem 8.2. Every generalised Cartan lattice is isomorphic to K 0 (A) for some hereditary artin algebra A.
If f : A → A ′ is a finite homological epimorphism between hereditary artin algebras, then restriction of scalars induces a morphism f
Conversely, every morphism of generalised Cartan lattices ending in K 0 (A) is up to isomorphism of the form f 
, since each A-module having no self-extensions is determined by its class in K 0 (A) by [28, Lemma 8.2] . This
It is clear that the functor H → C is not full. First observe that any functor φ : mod A ′ → mod A representing a morphism in H is an equivalence if K 0 (φ) is an isomorphism in C. Now fix two fields k and k ′ . We have
but mod A ′ and mod A need not to be equivalent. Corollary 8.4. Let C = mod A be the category of finitely generated modules over an hereditary artin algebra A. Then there is a natural isomorphism of posets
Proof. The first isomorphism follows from Theorem 8.2 combined with Lemma 8.1; the second one follows from Theorem 6.2.
Non-crossing partitions and Hom-free sets. Let A be an hereditary artin algebra. By a Hom-free set in mod A we will mean a set {S 1 , . . . , S r } of exceptional modules satisfying Hom A (S i , S j ) = 0 for all i = j. Such Hom-free sets were studied by various authors [10, 18, 35, 36, 42] . Proposition 8.5. Let X be an exceptional sequence in mod A. Then the map sending C(X) to the set of simple objects in C(X) induces an injective map from NC(K 0 (A)) to the Hom-free sets up to isomorphism. If A is representation-finite, then this map is a bijection.
Proof. If X is an exceptional sequence, then C(X) ∼ = mod B for some hereditary artin algebra B, by Proposition A.4, and hence the simple objects in C(X) form a Hom-free set. Conversely, C(X) is uniquely determined by its set of simple objects. Using Corollary 8.4 we know that the non-crossing partitions are in bijection with the subcategories of the form C(X) for X exceptional. Thus the map sending C(X) to its set of simple objects induces an injective map from the set of non-crossing partitions to the Hom-free sets up to isomorphism.
If now A is representation-finite, then any Hom-free set can be arranged to form an exceptional sequence. For, the (isomorphism classes of) indecomposable Amodules are partially ordered using the Auslander-Reiten quiver, so Ext 1 A (M, N ) = 0 implies N ≺ M . It follows that every Hom-free set arises as the set of simple objects in C(X) for some exceptional sequence X.
When A has finite representation type, this observation can be used to enumerate the non-crossing partitions. This was already done by Gabriel and de la Peña in [18, § §2.6, 2.7] and from 'painful' calculations they obtained the Coxeter-Catalan numbers of ADE-type. 6 We observe that the map is not bijective in general. form a Hom-free set, but the subcategory C(X, Y ) is equivalent to a tube of period two, so to the category of nilpotent modules over an oriented cycle of length two, and as such does not contain a relative projective generator.
Let (Γ, E) be a generalised Cartan lattice, and let F = (f 1 , . . . , f r ) be a real exceptional sequence in Γ. Writing f i = j λ ij e j in terms of E we may define the height of F to be ht(F ) := ij |λ ij |.
Lemma 8.7. Let (Γ, E) be a generalised Cartan lattice, and F a real exceptional sequence in Γ. If F ′ = σF has minimal height, then the roots in F ′ are unique up to sign.
Proof. By Lemma 5.3 we can realise Γ as K 0 (A) for an hereditary artin algebra A, and by Proposition 5.6 we can lift F (up to sign) to an exceptional sequence X = (X 1 , . . . , X r ) in mod A. Moreover, ht(F ) equals the length ℓ A (X) := ℓ A (X 1 ⊕ · · · ⊕ X r ). Thus if F ′ has minimal height, then the roots in F ′ are ±[S i ], where {S 1 , . . . , S r } is the set of simple objects in C(X).
Appendix A. Perpendicular calculus
In this appendix we collect some basic facts about perpendicular categories, following Geigle-Lenzing [19, §3] , Schofield [40, §2] , and Crawley-Boevey [12] .
Let A be an hereditary artin algebra. We consider the category mod A of finitely generated A-modules.
For a subset X ⊆ mod A we define the right and left perpendicular categories to be the full subcategories
6 Note however that their form for the Coxeter-Catalan number of type D has not been simplified, and although they have the correct number for E 8 , their numbers for E 6 and E 7 are slightly wrong.
These are clearly closed under kernels, cokernels and extensions. We also consider C(X), the smallest full subcategory of mod A closed under kernels, cokernels and extensions and containing each X i ∈ X.
The following proposition describes simultaneously adjoints of the inclusions C(X) → mod A and C(X) ⊥ → mod A via a five term exact sequence; see also [19 Proposition A.1. Let C ⊆ mod A be a full subcategory closed under kernels, cokernels and extensions. Assume that X ∈ C is a relative projective generator.
(1) Each A-module M fits into a functorial exact sequence
Proof. We first note that C = C(X) and
is surjective for all X ′ ∈ add(X). We also set
is surjective for all X ′ ∈ add(X). Let N be the kernel of X M → M , and consider the push-out diagram
Clearly M C ∈ C, and applying Hom A (X, −) to the first column yieldsM
Thus we can form the push-out of ε to obtain a universal extension
Consider a right approximation X F → F , and let G by the image of the composition
We then obtain a commutative diagram 0
Putting this together yields the five term exact sequence Remark A.2. We observe that if X ∈ mod A is an exceptional module, then C = C(X) = add(X), so has X itself as a relative projective generator. In this case, given M , we can take a minimal right approximation
and the kernel already lies in X ⊥ . Similarly we can take a minimal universal extension using X Proposition A.3. Let C ⊆ mod A be a full subcategory closed under kernels, cokernels and extensions. Assume that X ∈ C is a relative injective cogenerator.
(1) Each A-module M fits into a functorial exact sequence Proposition A.4. Let A be a hereditary artin algebra and C ⊆ mod A a full subcategory. Then the following are equivalent.
(1) The inclusion C → mod A admits a left adjoint and C is closed under kernels, cokernels, and extensions. (1 ′ ) The inclusion C → mod A admits a right adjoint and C is closed under kernels, cokernels, and extensions. . . . , S r ) in mod B consisting of simple B-modules, and clearly mod B = C(S). Setting X i ∈ C to be the image of S i , we deduce that X = (X 1 , . . . , X r ) is an exceptional sequence and C(X) = C. . . . , X r ) and set X ′ = (X 1 , . . . , X r−1 ). We first claim that C(X) = ⊥ (X ⊥ ). As in the proof of (4) ⇒ (1) we know that C(X r ) = add(X r ) and X ) and therefore in C(X ′ ). Thus M ∈ C(X). Finally, the implication (4) ⇒ (3) tells us that X ⊥ = C(Z) for some exceptional sequence Z, and so C(X) = ⊥ (X ⊥ ) = ⊥ Z. Thus (4 ′ ) holds. The implication (3) ⇒ (4) is dual.
Remark A.5. Let C ⊆ mod A be a full subcategory closed under kernels, cokernels, and extensions. Then the inclusion C → mod A admits left and right adjoints iff the Loewy lengths of the objects in C are bounded [16, 8.2] . In particular, there are adjoints if A is representation-finite.
We now list some useful consequences of this result. Corollary A.6. Let X be an exceptional sequence in mod A. Then C(X) contains both a relative projective generator and a relative injective cogenerator, so Propositions A.1 and A.3 hold for C(X).
Proof. We know that C(X) ∼ = mod B for some hereditary artin algebra B.
Recall that an exceptional sequence X = (X 1 , . . . , X r ) in mod A is called complete provided that r = rk K 0 (A).
Corollary A.7. Let X be an exceptional sequence in mod A. Then mod A = C(X, X ⊥ ). In particular,
X is complete ⇐⇒ C(X) = mod A ⇐⇒ X ⊥ = 0.
Proof. Consider the five term exact sequence given in Proposition A.1 with respect to the subcategory C(X). It follows that mod A = C(X, X ⊥ ) and also that C(X) = mod A if and only if X ⊥ = 0. On the other hand, we know from Lemmas 2.2 and 2.3 that the rank of K 0 (C(X)) equals the length of X and that K 0 (mod A) = K 0 (C(X)) ⊕ K 0 (X ⊥ ). Therefore X is complete if and only if X ⊥ = 0.
Corollary A.8 ([12, Lemma 1]). Let (X 1 , . . . , X r ) be an exceptional sequence in mod A which is not complete. Then for each 0 ≤ i ≤ r there exists an A-module Y such that (X 1 , . . . , X i , Y, X i+1 , . . . , X r ) is exceptional. Moreover, if this is complete, then Y is unique up to isomorphism.
Proof. We have that ⊥ (X 1 ⊕· · ·⊕X i )∩(X i+1 ⊕· · ·⊕X r ) ⊥ is equivalent to mod B for some hereditary artin algebra B of rank rk K 0 (A) − r, so let Y be any exceptional B-module. If this is complete, then rk K 0 (A) = r + 1, so rk K 0 (B) = 1 and Y is unique up to isomorphism.

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