# HALL POLYNOMIALS FOR AFFINE QUIVERS In - American

код для вставкиREPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 14, Pages 355вЂ“378 (April 30, 2010) S 1088-4165(10)00374-2 HALL POLYNOMIALS FOR AFFINE QUIVERS ANDREW HUBERY Abstract. We use GreenвЂ™s comultiplication formula to prove that Hall polynomials exist for all Dynkin and aп¬ѓne quivers. For Dynkin and cyclic quivers this approach provides a new and simple proof of the existence of Hall polynomials. For non-cyclic aп¬ѓne quivers these polynomials are deп¬Ѓned with respect to the decomposition classes of Bongartz and Dudek, a generalisation of the Segre classes for square matrices. In [19], Ringel showed how to construct an associative algebra from the category of п¬Ѓnite modules over a п¬Ѓnitary ring, and whose multiplication encodes the possible extensions of modules. More precisely, one forms the free abelian group with basis the isomorphism classes of modules and deп¬Ѓnes a multiplication by taking as structure constants the Hall numbers F X := |{U в‰¤ X : U в€ј = N, X/U в€ј = M }|. MN X X Alternatively, we can write FM N = PM N /aM aN , where f g X PM в†’Xв€’ в†’ M в†’ 0 exact}| and N := |{(f, g) : 0 в†’ N в€’ aM := | Aut(M )|. In the special case of п¬Ѓnite length modules over a discrete valuation ring with п¬Ѓnite residue п¬Ѓeld, one recovers the classical Hall algebra [16]. Interesting examples of such Hall algebras arise when one considers representations of a quiver over a п¬Ѓnite п¬Ѓeld. Green showed in [12] that the subalgebra generated by the simple modules, the so-called composition algebra, is a specialisation of LusztigвЂ™s form for the associated quantum group. (More precisely, one must п¬Ѓrst twist the multiplication in the Hall algebra using the Euler characteristic of the module category.) In proving this result, Green п¬Ѓrst showed that the Hall algebra is naturally a self-dual Hopf algebra. If one starts with a Dynkin quiver Q, then the isomorphism classes of indecomposable modules are in bijection with the set of positive roots О¦+ of the corresponding semisimple complex Lie algebra, the bijection being given by the dimension vector [10]. Thus the set of isomorphism classes of modules is in bijection with the set of functions Оѕ : О¦+ в†’ N0 , and as such is independent of the п¬Ѓeld. In this setting, Ringel proved in [20] that the Hall numbers are given by universal polynomials; that is, given three functions Ој, ОЅ, Оѕ : О¦+ в†’ N0 , there exists a Оѕ в€€ Z[T ] (depending only on Q) such that, for any п¬Ѓnite п¬Ѓeld k with polynomial FОјОЅ |k| = q and any representations M , N and X belonging to the isomorphism classes Ој, ОЅ and Оѕ, respectively, we have X Оѕ FM N = FОјОЅ (q). Received by the editors October 8, 2007. 2010 Mathematics Subject Classiп¬Ѓcation. Primary 16G20. c 2010 American Mathematical Society Reverts to public domain 28 years from publication 355 356 ANDREW HUBERY The proof uses associativity of the multiplication and induction on the AuslanderReiten quiver, reducing to the case when M is isotypic. A similar result holds when Q is an oriented cycle and we consider only nilpotent modules. The indecomposable nilpotent representations are all uniserial, so determined by their simple top and Leowy length. In fact, if Q is the Jordan quiver, consisting of a single vertex and a single loop, then the Ringel-Hall algebra is precisely the classical Hall algebra, so isomorphic to MacdonaldвЂ™s ring of symmetric functions. It is well known that Hall polynomials exist in this context [16]. A natural question to ask, therefore, is whether Hall polynomials exist for more general quivers. Certain results along this line are clear, for example, if all three modules are preprojective or preinjective. Also, some Hall numbers have been calculated for modules over the Kronecker quiver [25] (or equivalently for coherent sheaves over the projective line [3]) and we see that there is again вЂњpolynomial behaviourвЂќ. For example, let us denote the indecomposable preprojectives by Pr and the indecomposable preinjectives by Ir , for r в‰Ґ 0. If R is a regular module of dimension vector (n + 1)Оґ and containing at most one indecomposable summand from each tube, then P , PPRm Inв€’m = (q в€’ 1)aR = PRPm+n+1 m so that P FPRm Inв€’m = aR /(q в€’ 1) and FRPm+n+1 = 1. m Thus, although the regular modules R depend on the п¬Ѓeld, the Hall numbers depend only on the number of automorphisms of R. In general, the isomorphism classes of indecomposable kQ-modules are no longer combinatorially deп¬Ѓned вЂ” they depend intrinsically on the base п¬Ѓeld k. Therefore, some care has to be taken over the deп¬Ѓnition (and meaning) of Hall polynomials. The existence of Hall polynomials has gained importance recently by the relevance of quiver Grassmannians and the numbers Gr X e X FM N := [M ],[N ] dim N =e to cluster algebras. In [6] Caldero and Reineke show for aп¬ѓne quivers that these numbers are given by universal polynomials.1 Also, as shown in [17], the existence of universal polynomials implies certain conditions on the eigenvalues of the Frobenius morphism on l-adic cohomology for the corresponding varieties. The aim of this article is to show how the comultiplication, or GreenвЂ™s Formula, can be used to prove the existence of Hall polynomials. We remark that RingelвЂ™s proof cannot be extended, since the associativity formula alone does not reduce the diп¬ѓculty of the problem вЂ” the middle term remains unchanged. The advantage of GreenвЂ™s Formula is that it reduces the dimension vector of the middle term and hence allows one to apply induction. In this way we can reduce to a situation where the result is clear: for Dynkin quivers we reduce to the case when the middle term is simple; for nilpotent representations of a cyclic quiver we reduce to the case when the middle term is indecomposable; for general aп¬ѓne quivers, we reduce to the case when either all three representations are regular, or else the middle term is regular 1 This is Proposition 5 in [6]. Note, however, that their proof is not quite correct, since the orbit sizes are not constant on each stratum. It is for this reason that we need to reп¬Ѓne the strata; see the remarks at the end of Section 6. Using this reп¬Ѓnement, though, their proof goes through. HALL POLYNOMIALS FOR AFFINE QUIVERS 357 and the end terms are indecomposable. In this latter case, we can use associativity and a result of Schoп¬Ѓeld on exceptional modules [23] to simplify further to the case when M is simple preinjective. After recalling the necessary theory, we apply our reductions in the special cases of Dynkin quivers and nilpotent representations of a cyclic quiver, cases which are of course of particular interest. This gives a new proof of the existence of Hall polynomials in these cases, and the proofs oп¬Ђered here are short and elementary. We then extend this result to all п¬Ѓnite dimensional representations of a cyclic quiver, where we see that Hall polynomials exist with respect to the Segre classes. More precisely, given three Segre symbols О±, ОІ and Оі, there is a universal polynomial Оі such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q, FО±ОІ Оі C FAB = FО±ОІ (q) for all C в€€ S(Оі, k). Aв€€S(О±,k) Bв€€S(ОІ,k) Here, S(О±, k) denotes those k-representations belonging to the Segre class О±. In fact, we may sum over the representations in any two of the three classes and still obtain a polynomial. We illustrate these ideas with an example for the Jordan quiver. Finally, we prove that Hall polynomials exist for all aп¬ѓne quivers with respect to the Bongartz-Dudek decomposition classes [4], following the strategy outlined above. We remark that these decomposition classes are also used by Caldero and Reineke in proving the existence of universal polynomials for quiver Grassmannians [6]. In the п¬Ѓnal section we provide a possible deп¬Ѓnition of what it should mean for Hall polynomials to exist for a wild quiver. This will necessarily be with respect to some combinatorial partition, and we mention some properties this partition should satisfy. 1. Representations of quivers Let kQ be the path algebra of a connected quiver Q over a п¬Ѓeld k (see, for example, [2]). The category mod kQ of п¬Ѓnite dimensional kQ-modules is an hereditary, abelian, Krull-Schmidt category. It is equivalent to the category of k-representations of Q, where a k-representation is given by a п¬Ѓnite dimensional vector space Mi for each vertex i в€€ Q0 together with a linear map Ma : Mt(a) в†’ Mh(a) for each arrow a в€€ Q1 . The dimension vector of M is dim M := (dim Mi )i в€€ ZQ0 and the Euler form on mod kQ satisп¬Ѓes M, N := dimk HomkQ (M, N ) в€’ dimk Ext1kQ (M, N ) = [M, N ] в€’ [M, N ]1 . This depends only on the dimension vectors of M and N ; indeed, di ei в€’ d, e = iв€€Q0 dt(a) eh(a) for all d, e в€€ ZQ0 . aв€€Q1 A split torsion pair on the set ind kQ of indecomposable modules is a decomposition ind kQ = F в€Є T such that Hom(T , F) = 0 = Ext1 (F, T ). The additive subcategories add F and add T are called the torsion-free and torsion classes, respectively. Given a split torsion pair, there exists, for each module X, a 358 ANDREW HUBERY unique submodule Xt в‰¤ X such that Xt в€€ add T and Xf := X/Xt в€€ add F. Hence Xв€ј = Xf вЉ• Xt . Let Q be a Dynkin quiver. The dimension vector map induces a bijection between the set of isomorphism classes of indecomposable representations and the set of positive roots О¦+ of the corresponding semisimple complex Lie algebra [10]. In particular, this description is independent of the п¬Ѓeld k. Moreover, each indecomposable representation X is a brick (End(X) в€ј = k) and rigid (Ext1 (X, X) = 0), thus exceptional. Let Q be the Jordan quiver, having one vertex and one loop. Then kQ = k[t] is a principal ideal domain, so п¬Ѓnite dimensional modules are described by their elementary divisors. In particular, we can associate to a п¬Ѓnite dimensional module M the data {(О»1 , p1 ), . . . , (О»r , pr )} consisting of partitions О»i and distinct monic irreducible polynomials pi в€€ k[t] such that r Mв€ј = M (О»i , pi ), i=1 where, for a partition О» = (1l1 В· В· В· nln ) and monic irreducible polynomial p, we write k[t]/(pr ) M (О», p) = lr . r Let Q be an oriented cycle. Then the category mod0 kQ of nilpotent modules is uniserial, with simple modules parameterised by the vertices of Q [21]. Each indecomposable is determined by its simple top and Loewy length, so the set of all isomorphism classes is in bijection with support-п¬Ѓnite functions (Q0 Г— N) в†’ N0 . Let Q be an extended Dynkin quiver which is not an oriented cycle. The roots of Q are either real or imaginary, О¦ = О¦re в€Є О¦im , with each imaginary root a non-zero integer multiple of a positive imaginary root Оґ [14]. In studying indecomposable representations, Dlab and Ringel [9] showed the importance of the defect map в€‚ : ZQ0 в†’ Z, e в†’ Оґ, e . By deп¬Ѓnition, this map is additive on short exact sequences. We call an indecomposable kQ-module M preprojective if в€‚(M ) < 0, preinjective if в€‚(M ) > 0 and regular if в€‚(M ) = 0. This yields a decomposition of ind kQ into a вЂњsplit torsion tripleвЂќ, ind kQ = P в€Є R в€Є I, where P is the set of indecomposable preprojective modules, I the set of indecomposable preinjective modules and R the set of indecomposable regular modules. In particular, Hom(I, R) = 0, Hom(I, P) = 0, Hom(R, P) = 0, Ext (R, I) = 0, Ext (P, I) = 0, Ext1 (P, R) = 0. 1 1 The indecomposable preprojective and preinjective modules are exceptional. In particular, these indecomposables are determined up to isomorphism by their dimension vectors. Moreover, there is a partial order on the set of indecomposable preinjective modules such that Hom(M, N ) = 0 implies M N. The minimal elements are the simple injective modules. On the other hand, the category of regular modules is an abelian, exact subcategory which decomposes into a direct sum of uniserial categories, or tubes. Each tube HALL POLYNOMIALS FOR AFFINE QUIVERS 359 has a п¬Ѓnite number of quasi-simples, forming a single orbit under the AuslanderReiten translate (say of size p), and thus is equivalent to the category of nilpotent representations of a cyclic quiver (with p vertices) over a п¬Ѓnite п¬Ѓeld extension of k (given by the endomorphism ring of any quasi-simple in the tube). Moreover, there are at most three non-homogeneous tubes (those having period p в‰Ґ 2), and the corresponding quasi-simples are exceptional. Since each tube is a uniserial category, we see that the isomorphism class of a module without homogeneous regular summands can be described combinatorially. A pair (A, B) of modules is called an orthogonal exceptional pair provided that A and B are exceptional modules such that Hom(A, B) = Hom(B, A) = Ext1 (B, A) = 0. We denote by F(A, B) the full subcategory of objects having a п¬Ѓltration with factors A and B. Then F(A, B) is an exact, hereditary, abelian subcategory equivalent to the category of modules over the п¬Ѓnite dimensional, hereditary k-algebra k 0 kd k d = k(В· в€’ в†’ В·), where d = dim Ext1 (A, B) and the quiver above has d arrows from left to right [7]. Theorem 1 (Schoп¬Ѓeld [23]). If M is exceptional but not simple, then M в€€ F(A, B) for some orthogonal exceptional pair (A, B), and M is not a simple object in F(A, B). In fact, there are precisely s(M ) в€’ 1 such pairs, where s(M ) is the size of the support of dim M . Lemma 2. Let (A, B) be an orthogonal exceptional pair of kQ-modules and set d := [A, B]1 . If Q is Dynkin, then d в‰¤ 1, and if Q is extended Dynkin, then d в‰¤ 2. Moreover, if Q is extended Dynkin and d = 2, then dim(A вЉ• B) = Оґ and в€‚(A) = 1. d Proof. If d в‰Ґ 3, then the algebra k0 kk is wild, whereas if d = 2, then this algebra is tame. The п¬Ѓrst result follows. Now let Q be extended Dynkin and d = 2. There are two indecomposable modules R в€ј = R which are both extensions of A by B. Using the description of ind kQ above, we see that dim(A вЉ• B) = dim R = rОґ for some r в‰Ґ 1. Now 1 = A вЉ• B, A = rОґ, dim A = rв€‚(A). Hence r = в€‚(A) = 1. Let Q be extended Dynkin. We п¬Ѓx a preprojective module P and a preinjective module I such that dim P + dim I = Оґ and в€‚(I) = 1. Note that в€‚(P ) = в€’1 and that P and I are necessarily indecomposable. It follows that (I, P ) is an orthogonal exceptional pair such that d = [I, P ]1 = 2. Thus there is an embedding of the module category mod K of the Kronecker algebra K := k 0 k2 k = k(В· в‡’ В·) into mod kQ which sends the simple projective to P and the simple injective to I. Under this embedding we can identify the tubes of mod kQ with those of mod K, and it is well known that the tubes of mod K are parameterised by the closed scheme-theoretic points of the projective line. Moreover, we may assume that the non-homogeneous tubes correspond to some subset of {0, 1, в€ћ}, and if R(x) в€€ 360 ANDREW HUBERY mod kQ is the quasi-simple in the homogeneous tube labelled by x в€€ P1k , then End(R(x)) в€ј = Оє(x) is given by the residue п¬Ѓeld at x and dim R(x) = mОґ where m = deg x = [Оє(x) : k]. In fact, since the indecomposable preprojective and preinjective modules are uniquely determined by their dimension vectors, it is possible to take an open subscheme HZ вЉ‚ P1Z , deп¬Ѓned over the integers, such that for any п¬Ѓeld k, the scheme Hk parameterises the homogeneous tubes in mod kQ. It follows that homogeneous regular modules can be described by pairs consisting of a partition together with a closed point of the scheme Hk . 2. Hall algebras of quiver representations Now let k be a п¬Ѓnite п¬Ѓeld. In [19] Ringel introduced the Ringel-Hall algebra H = H(mod kQ), a free abelian group with basis the set of isomorphism classes of п¬Ѓnite dimensional kQ-modules and multiplication X FM N [X]. [M ][N ] := [X] X The structure constants FM N are called Hall numbers and are given by X PM X N в€ј в€ј FM , N := |{U в‰¤ X : U = N, X/U = M }| = aM aN f g X where aM := | Aut(M )| and PM в†’Xв€’ в†’ M в†’ 0 exact}|. We N := |{(f, g) : 0 в†’ N в€’ also have RiedtmannвЂ™s Formula [18], X FM N = aX | Ext1 (M, N )X | В· , | Hom(M, N )| aM aN where Ext1 (M, N )X is the set of classes of extensions of M by N with middle term isomorphic to X. Dually, there is a natural comultiplication О” : H в†’ H вЉ— H, О”([X]) = [M ],[N ] X PM N [M ] вЉ— [N ], aX and H is both an associative algebra with unit [0], and a coassociative coalgebra with counit ([M ]) = ОґM 0 . Both these statements follow from the identity X M FAB FXC = [X] M X FAX FBC , [X] a consequence of the pull-back/push-out constructions. A milestone in the theory of Ringel-Hall algebras was the proof by Green that the Ringel-Hall algebra is a twisted bialgebra ([12, 22]). That is, we deп¬Ѓne a new multiplication on the tensor product H вЉ— H via ([A] вЉ— [B]) В· ([C] вЉ— [D]) := q A,D [A][C] вЉ— [B][D]. Then the multiplication and comultiplication are compatible with respect to this new multiplication on the tensor product; that is, О”([M ][N ]) = О”([M ]) В· О”([N ]). HALL POLYNOMIALS FOR AFFINE QUIVERS 361 The proof reduces to GreenвЂ™s Formula qв€’ E E FM N FXY /aE = [E] A,D M N X Y FAB FCD FAC FBD [A],[B],[C],[D] aA aB aC aD . aM aN aX aY There is also a positive-deп¬Ѓnite pairing on the Ringel-Hall algebra, and (after twisting the multiplication by the Euler form) the Ringel-Hall algebra is naturally в€љ isomorphic to the specialisation at v = q of the quantised enveloping algebra of the positive part of a (generalised) Kac-Moody Lie algebra [24]. From this one can deduce part of KacвЂ™s Theorem that the set of dimension vectors of indecomposable modules coincides with the set of positive roots О¦+ of the quiver [8, 13]. Suppose that M is exceptional. Then Aut(M m ) в€ј = GLm (k) and RiedtmannвЂ™s Formula implies that M в€’m FM Mm = q m+1 M,M aM m+1 q m+1 в€’ 1 . = aM aM m qв€’1 m+1 в€ј Alternatively we can use the quiver Grassmannian Gr MM = Pm k , which paramm+1 eterises subrepresentations M в‰¤ M . If (F, T ) is a split torsion pair and X в€ј = Xf вЉ• Xt is a module with Xf в€€ add F and Xt в€€ add T , then E FX = ОґEX f Xt and a X = a Xf a Xt q Xf ,Xt . These observations will be used repeatedly. 3. Representation-finite hereditary algebras Let Q be a Dynkin quiver and denote by О¦+ the set of positive roots of the corresponding п¬Ѓnite dimensional, semisimple complex Lie algebra. Recall that for any п¬Ѓeld k the isomorphism classes of kQ-modules are in bijection with functions О± : О¦+ в†’ N0 . In this section, we provide a new proof of the existence of Hall polynomials for Dynkin quivers. The proof relies on combining GreenвЂ™s Formula with split torsion pairs. We begin with an easy lemma. Lemma 3. For each function О± : О¦+ в†’ N0 there exists a monic polynomial aО± в€€ Z[T ] such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q and any k-representation A with [A] = О±, we have aО± (q) = aA = | Aut(A)|. Proof. Each indecomposable is exceptional and the dimension of homomorphisms between indecomposables is given via the Auslander-Reiten quiver and the mesh relations, hence is independent of the п¬Ѓeld k. Given a split torsion pair (F, T ) and a module X в€ј = Xf вЉ• Xt , we can simplify the left-hand side of GreenвЂ™s Formula as follows: E E X X FM N FXf Xt /aE = FM N /aX = FM N /aXf aXt q Xf ,Xt . E (Note that the sum is actually over isomorphism classes of representations, though we shall often use this more convenient notation.) If we now consider the right-hand side of GreenвЂ™s Formula, we see that all Hall numbers involve middle terms with 362 ANDREW HUBERY dimension vector strictly smaller than dim X (provided that M , N , Xf and Xt are all non-zero, of course): qв€’ A,D X Xt M N FAB FCD FACf FBD A,B,C,D aA aB aC aD . a M a N a Xf a Xt Thus X FM N = q Xf ,Xt в€’ A,D X Xt M N FAB FCD FACf FBD A,B,C,D aA aB aC aD . aM aN Theorem 4 (Ringel [20]). Hall polynomials exist for Q; that is, given Ој, ОЅ and Оѕ Оѕ, there exists an integer polynomial FОјОЅ в€€ Z[T ] such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q and any k-representations M , N and X with [M ] = Ој, [N ] = ОЅ and [X] = Оѕ, we have Оѕ X (q) = FM FОјОЅ N. X Proof. We wish to show that FM N is given by some universal integer polynomial. m1 mr Let X = I1 вЉ• В· В· В· вЉ• Ir be a decomposition of X into pairwise non-isomorphic indecomposable modules Ii . Then (up to reordering) there exists a split torsion pair such that Xt = Irmr . We now use the formula above together with inducX tion on dimension vector to deduce that FM N is of the form (polynomial)/(monic polynomial). Since this must take integer values at all prime powers, we have that X X FM N is given by some universal polynomial FM N (T ) в€€ Z[T ]. It is thus enough to m consider the case when X = I is isotypic. Suppose X = I m+1 with I indecomposable. Since I is exceptional, we can simplify the left-hand side of GreenвЂ™s Formula as m+1 E E I m FM N FII m /aE = FM N /q aI aI m , E whence m+1 I FM N = q mв€’ A,D m M N I I FAB FCD FAC FBD A,B,C,D aA aB aC aD . aM aN m+1 I By induction on dimension vector, FM N is given by a universal integer polynomial. We thus reduce to the case when X = I is indecomposable. GabrielвЂ™s Theorem tells us that the indecomposable modules are determined up to isomorphism by their dimension vectors, so if dim E = dim X and E в€ј = X, then E must be decomposable. Set R := rad X and T := X/R, so that FTXR = 1. Then E E FM N FT R /aE в€’ X FM N /aX = E E E FM N FT R /aE . E decomp The second sum on the right-hand side is over decomposable modules, hence is of the form (polynomial)/(monic polynomial), as is the п¬Ѓrst sum by GreenвЂ™s Formula X and induction on dimension vector. Thus FM N is given by a universal integer polynomial. Alternatively, Theorem 1 gives an exact sequence 0 в†’ B в†’ X в†’ A в†’ 0 for some orthogonal exceptional pair (A, B) with [A, B]1 = 1. Either way, we reduce to the case when X is simple, where the result is trivial. HALL POLYNOMIALS FOR AFFINE QUIVERS 363 4. Cyclic quivers Let Q be an oriented cycle and k an arbitrary п¬Ѓeld. Recall that the set of isomorphism classes of nilpotent modules is in bijection with support-п¬Ѓnite functions (Q0 Г— N) в†’ N0 . It is known from [21] that Hall polynomials exist in this context. We provide a new proof of this fact, using GreenвЂ™s Formula and induction on partitions. Given a module X denote by О»(X) the partition formed by taking the Loewy lengths of its indecomposable summands. We order partitions via the reverse lexicogarphic ordering; that is, if О» = (1l1 2l2 В· В· В· nln ) and Ој = (1m1 2m2 В· В· В· nmn ) are written in exponential form, then О»<Ој if there exists i such that li > mi and lj = mj for all j > i. The cup product О» в€Є Ој is the partition (1l1 +m1 В· В· В· nln +mn ). As in the representation-п¬Ѓnite case, we have the following lemma. Lemma 5. The dimension dim Hom(A, B) depends only on О± = [A] and ОІ = [B] and not on the choice of п¬Ѓeld k. Moreover, there exists a monic polynomial aО± в€€ Z[T ] such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q and any k-representation A with [A] = О±, we have aО± (q) = aA . Proof. Since indecomposables are uniserial, it is clear that the dimension of the space of homomorphisms between indecomposables is independent of the п¬Ѓeld. The lemma follows easily. Lemma 6 ([15]). If X is an extension of M by N , then О»(X) в‰¤ О»(M ) в€Є О»(N ) with equality if and only if X в€ј = M вЉ• N. Theorem 7. Hall polynomials exist for nilpotent representations of cyclic quivers. Proof. Let X = X1 вЉ• X2 be decomposable. Then E E FM N FX1 X2 /aE в€’ X X FM N FX1 X2 /aX = E E E FM N FX1 X2 /aE . О»(E)<О»(X) By induction on О»(X) we know that the second sum on the right-hand side is of the form (polynomial)/(monic polynomial), as is the п¬Ѓrst sum by GreenвЂ™s Formula and induction on dimension vector. Moreover, by RiedtmannвЂ™s Formula, we know that X FX /aX = 1/q [X1 ,X2 ] aX1 aX2 , 1 X2 X the reciprocal of a monic polynomial. Hence FM N is given by a universal integer polynomial. We are reduced to proving the formula when X is indecomposable. Since the category is uniserial, this implies that both M and N are indecomposable, in which case the Hall number is either 1 or 0 and is independent of the п¬Ѓeld. Remark. The same induction can be used to show that X 2 deg FM N в‰¤ deg aX в€’ deg(aM aN ). For, if X в€ј = X1 вЉ• X2 , then induction gives X 2 deg FM N в‰¤ deg(aX1 aX2 ) + 2[X1 , X2 ] в€’ deg(aM aN ), and obviously deg aX = deg(aX1 aX2 ) + [X1 , X2 ] + [X2 , X1 ]. 364 ANDREW HUBERY Hence taking such a decomposition with [X1 , X2 ] в‰¤ [X2 , X1 ] gives the result. In fact, for the Jordan quiver, we always have equality and the leading coeп¬ѓcient is given by the corresponding Littlewood-Richardson coeп¬ѓcient [16]. Now consider the Jordan quiver Q, so that kQ = k[t]. Recall that a п¬Ѓnite dimensional module M is determined by the data {(О»1 , p1 ), . . . , (О»r , pr )} consisting of partitions О»i and distinct monic irreducible polynomials pi в€€ k[t] such that Mв€ј = r M (О»i , pi ). i=1 Clearly the primes pi depend on the п¬Ѓeld, but we can partition the set of isomorphism classes by considering just their degrees. This is called the Segre decomposition. More precisely, a Segre symbol is a multiset Пѓ = {(О»1 , d1 ), . . . , (О»r , dr )} of pairs (О», d) consisting of a partition О» and a positive integer d. The corresponding Segre class S(Пѓ, k) consists of those isomorphism classes of modules of type {(О»1 , p1 ), . . . , (О»r , pr )}, where the pi в€€ k[t] are distinct monic irreducible polynomials with deg pi = di . Theorem 8 ([1, 4, 11]). Let k be an algebraically closed п¬Ѓeld. Then the Segre classes stratify the variety End(km ) into smooth, irreducible, GLm (k)-stable subvarieties, each admitting a smooth, rational, geometric quotient. Moreover, the stabilisers of any two matrices in the same Segre class are conjugate inside GLm (k). Lemma 9. Given a Segre symbol Пѓ, there exists a monic polynomial aПѓ в€€ Z[T ] and a polynomial nПѓ в€€ Q[T ] such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q, we have aПѓ (q) = aM for any M в€€ S(Пѓ, k), and Proof. If Пѓ = {(О»1 , d1 ), . . . , (О»r , dr )}, then aПѓ (T ) = tion О» = (1l1 В· В· В· nln ), we have aО» (T ) := T i,j nПѓ (q) = |S(Пѓ, k)|. i aО»i (T di ) where, for a parti- (1 в€’ T в€’1 ) В· В· В· (1 в€’ T в€’li ). min{i,j}li lj i This polynomial occurs in [16] as the size of the automorphism group of the module M (О», t). We write Пѓ(d) to be the Segre symbol formed by those pairs (О»i , di ) in Пѓ with di = d. To obtain the formula for nПѓ , let us п¬Ѓrst suppose that Пѓ = Пѓ(d) for some d. Then nПѓ = П†d (П†d в€’ 1) В· В· В· (П†d в€’ r + 1)/zПѓ , where П†d is the number of monic irreducible polynomials of degree d, r = |Пѓ|, and zПѓ is the size of the stabiliser for the natural action of the symmetric group Sr on (О»1 , . . . , О»r ) given by place permutation. In general we can write Пѓ = d Пѓ(d), and nПѓ = d nПѓ(d) . Theorem 10. Given three Segre symbols ПЃ, Пѓ and П„ , there exists an integer polyП„ в€€ Z[T ] such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q, we have nomial FПЃПѓ П„ FПЃПѓ (q) = T FRS Rв€€S(ПЃ,k) Sв€€S(Пѓ,k) for all T в€€ S(П„, k). HALL POLYNOMIALS FOR AFFINE QUIVERS 365 Moreover, we have the identities П„ (q) = nПЃ (q) nП„ (q)FПЃПѓ T FRS for all R в€€ S(ПЃ, k) T FRS for all S в€€ S(Пѓ, k). Sв€€S(Пѓ,k) T в€€S(П„,k) = nПѓ (q) Rв€€S(ПЃ,k) T в€€S(П„,k) It follows that both cients). nП„ П„ nПЃ FПЃПѓ and nП„ П„ nПѓ FПЃПѓ are polynomials (but with rational coeп¬ѓ- Proof. The proof is similar to that where we just considered nilpotent modules (i.e. the single irreducible polynomial p(t) = t). We begin by noting that there are no homomorphisms between modules corresponding to distinct irreducible polynomials. In particular, we can decompose ПЃ = d ПЃ(d), and given R в€€ S(ПЃ, k), there is a unique decomposition R = d R(d) such that R(d) в€€ S(ПЃ(d), k). We deduce that T (d) T = FRS FR(d)S(d) . d In particular, we can reduce to the case when all degrees which occur in ПЃ, Пѓ and П„ equal some п¬Ѓxed integer d. Let us п¬Ѓx T := M (ОЅ1 , p1 ) вЉ• В· В· В· вЉ• M (ОЅm , pm ) в€€ S(П„, k). By adding in copies of the zero partition, we may assume that ПЃ = (О»1 , . . . , О»m ) and Пѓ = (Ој1 , . . . , Ојm ) for the same m (we have simpliп¬Ѓed the notation by omitting the number d). Since there are no homomorphisms between modules corresponding T is non-zero for some R в€€ S(ПЃ, k) and to distinct irreducible polynomials, if FRS S в€€ S(Пѓ, k), then there exist permutations r, s в€€ Sm such that M (О»r(i) , pi ), R= i S= M (Ојs(i) , pi ) and d i FО»ОЅr(i) Ојs(i) (q ), T FRS = i i ОЅ is the classical Hall polynomial. where FО»Ој It follows that d i FО»ОЅr(i) Ојs(i) (q ), T FRS = r,s Rв€€S(ПЃ),Sв€€S(Пѓ) i where the sum is taken over all permutations r and s yielding non-isomorphic modules; that is, r runs through the cosets in Sm with respect to the stabiliser of (О»1 , . . . , О»m ), and similarly for s. It is now clear that this number is described by П„ . a universal polynomial over the integers, which we denote by FПЃПѓ Suppose instead that we п¬Ѓx R = M (О»1 , p1 ) вЉ• В· В· В· вЉ• M (О»m , pm ) в€€ S(ПЃ), where О»1 , . . . , О»m are all non-zero. Assume п¬Ѓrst that П„ = (ОЅ1 , . . . , ОЅm ) consists of precisely m partitions. By adding in copies of the zero partition, we may further assume that Пѓ = (Ој1 , . . . , Ојm ) also consists of m partitions. It follows as before that ОЅ T FRS = Sв€€S(Пѓ),T в€€S(П„ ) d FО»it(i) Ојs(i) (q ). s,t i 366 ANDREW HUBERY T In general, П„ will consist of m + n partitions, and if FRS = 0, then we can write T = T вЉ•X and S = S вЉ•X such that T contains all summands of T corresponding T T to the polynomials pi occurring in R. We observe that FRS = FRS . It follows that T FRS = Пѓ=Пѓ в€ЄОѕ П„ =П„ в€ЄОѕ Sв€€S(Пѓ) T в€€S(П„ ) T FRS . NОѕ S в€€S(Пѓ ) T в€€S(П„ ) The number NОѕ equals the number of isomorphism classes of X = i M (Оѕi , xi ) в€€ S(Оѕ) such that the polynomials p1 , . . . , pm , x1 , . . . , xn are pairwise distinct. Thus NОѕ = (П†d в€’ m) В· В· В· (П†d в€’ m в€’ n + 1)/zОѕ and hence the number immediately that S,T T FRS is again given by a universal polynomial. It follows T FRS = nП„ R,S T FRS = nПЃ R,S,T An analogous argument works for the sum T FRS . S,T T R,T FRS . 5. An example Theorem 10 gives a good generalisation of Hall polynomials in the case of arbitrary k[t]-modules. In this section we illustrate the proof of Theorem 10 and in so doing, show that it is not possible to п¬Ѓx two modules and still obtain a universal polynomial. We п¬Ѓx the degree d = 1 and the Segre symbols ПЃ := {(1, 1), (1, 1, 1), (2, 1)}, Пѓ := {(1), (1)} and П„ := {(1, 1, 1), (2, 1, 1), (2, 1)}. Note that 1 q(q в€’ 1) and nПЃ = nП„ = q(q в€’ 1)(q в€’ 2). 2 T We are interested in the numbers FRS for the modules nПѓ = R := M ((1, 1), x) вЉ• M ((1, 1, 1), y) вЉ• M ((2, 1), z) S := M ((1), x ) вЉ• M ((1), y ) T := M ((1, 1, 1), x ) вЉ• M ((2, 1, 1), y ) вЉ• M ((2, 1), z ), where x, y and z are distinct elements of k, as are x and y , and x , y and z . We begin by computing the possible Hall polynomials that can appear. We have (1,1,1) F(1,1)(1) = q 2 + q + 1 and (2,1,1) F(2,1)(1) = q(q + 1). Both of these sequences are split, so we can apply RiedtmannвЂ™s Formula. Also, (2,1) F(1,1)(1) = 1 and (2,1,1) F(1,1,1)(1) = 1. These are clear, since in both cases we just have the top and radical of the middle term. We deduce that вЋ§ 2 вЋЄ вЋЁq + q + 1 if (x, y, z) = (x , y , z ) and {x , y } = {x, y}; T FRS = q 2 + q if (x, y, z) = (z , x , y ) and {x , y } = {x, z}; вЋЄ вЋ© 0 otherwise. HALL POLYNOMIALS FOR AFFINE QUIVERS Hence T FRS R T FRS S T FRS T вЋ§ 2 вЋЄ вЋЁq + q + 1 = q2 + q вЋЄ вЋ© 0 вЋ§ 2 вЋЄ вЋЁq + q + 1 = q2 + q вЋЄ вЋ© 0 вЋ§ 2 вЋЄ вЋЁq + q + 1 = q2 + q вЋЄ вЋ© 0 367 if {x , y } = {x , y }; if {x , y } = {y , z }; otherwise, if (x, y, z) = (x , y , z ); if (x, y, z) = (z , x , y ); otherwise, if {x , y } = {x, y}; if {x , y } = {x, z}; otherwise. Finally, T FRS = 2q 2 + 2q + 1, R,S T FRS = 2q 2 + 2q + 1, S,T T FRS = 2(2q 2 + 2q + 1)(q в€’ 2). R,T Thus we only get universal polynomials if we sum over two of the Segre classes. 6. Tame hereditary algebras Let Q be an extended Dynkin quiver which is not an oriented cycle and k a п¬Ѓnite п¬Ѓeld. Recall that we have the вЂњsplit torsion tripleвЂќ ind kQ = P в€Є R в€Є I given by the indecomposable preprojective, regular and preinjective modules. Moreover, the category of regular modules decomposes into a direct sum of tubes indexed by the projective line in such a way that each regular simple module R in the tube labelled by x satisп¬Ѓes End(R) в€ј = Оє(x) and dim R = (deg x)Оґ. Finally, we may also assume that the non-homogeneous tubes are labelled by some subset of {0, 1, в€ћ}, whereas the homogeneous tubes are labelled by the closed points of the scheme HZ вЉ— k for some open integral subscheme HZ вЉ‚ P1Z . The indecomposable preprojective and preinjective modules are all exceptional, as are the regular simple modules in the non-homogeneous tubes. Hence the isomorphism class of a module without homogeneous regular summands can be described combinatorially, whereas homogeneous regular modules are determined by pairs consisting of a partition together with a closed point of the scheme Hk . We are now in a position to deп¬Ѓne the partition of Bongartz and Dudek [4]. A decomposition symbol is a pair О± = (Ој, Пѓ) such that Ој speciп¬Ѓes a module without homogeneous regular summands and Пѓ = {(О»1 , d1 ), . . . , (О»r , dr )} is a Segre symbol. Given a decomposition symbol О± = (Ој, Пѓ) and a п¬Ѓeld k, we deп¬Ѓne S(О±, k) to be the set of isomorphism classes of modules of the form M (Ој, k) вЉ• R, where M (Ој, k) is the kQ-module determined by Ој and R = R(О»1 , x1 ) вЉ• В· В· В· вЉ• R(О»r , xr ) for some distinct points x1 , . . . , xr в€€ Hk such that deg xi = di . 368 ANDREW HUBERY Theorem 11 (Bongartz and Dudek [4]). If k is algebraically closed, each decomposition class S(О±, k) determines a smooth, irreducible, GL-invariant subvariety of the corresponding representation variety, which furthermore admits a smooth, rational geometric quotient. Note that it is still open as to whether the closure of a decomposition class is again the union of decomposition classes. Also, unlike in the classical case, the endomorphism algebras of modules in the same decomposition class are not necessarily isomorphic as algebras. We do, however, have the following result. Lemma 12. Given a decomposition symbol О± = (Ој, Пѓ), there exist universal polynomials aО± and nО± such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q, aО± (q) = | Aut(A)| for all A в€€ S(О±, k), and nО± (q) = |S(О±, k)|. Moreover, aО± is a monic integer polynomial. We can now state the main result of this paper. Main Theorem. Hall polynomials exist with respect to the decomposition classes described above; that is, given decomposition classes О±, ОІ and Оі, there exists a Оі such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q, rational polynomial FО±ОІ Оі FО±ОІ (q) = C FAB for all C в€€ S(Оі, k) Aв€€S(О±,k) Bв€€S(ОІ,k) and, moreover, Оі nОі (q)FО±ОІ (q) = nО± (q) C FAB for any A в€€ S(О±, k) C FAB for any B в€€ S(ОІ, k). Bв€€S(ОІ,k) Cв€€S(Оі,k) = nОІ (q) Aв€€S(О±,k) Cв€€S(Оі,k) Remarks. (1) When О± = (в€…, ПЃ), ОІ = (в€…, Пѓ) and Оі = (в€…, П„ ), so that S(О±, k), S(ОІ, k) and S(Оі, k) contain only homogeneous regular modules, the result follows from Theorem 10. (2) It is not true that each Hall polynomial has integer coeп¬ѓcients. In fact, we can easily construct a counterexample for the Kronecker quiver. Let ПЂr be the decomposition class corresponding to the indecomposable preprojective Pr and consider the three Segre symbols Пѓ1 := (1), 1 , (1), 1 , Пѓ2 := (2), 1 and Пѓ3 := (1), 2 . Consider decomposition classes О±i := (в€…, Пѓi ). As mentioned in the introP2 = 1 for all regular modules R containing at most one indeduction, FRP 0 composable from each tube, and is zero otherwise. Therefore FО±ПЂ12ПЂ0 = nО±1 = q(q + 1)/2, FО±ПЂ22ПЂ0 = nО±2 = q + 1, FО±ПЂ32ПЂ0 = nО±3 = q(q в€’ 1)/2. This does not contradict Proposition 6.1 of [17] since we are not counting the rational points of any scheme. In fact, let Dr be the locally closed subscheme consisting of those representations X such that dim X = 2Оґ 2 в€ј 2 and dim rad End(X) = r. The quiver Grassmannian Gr P P0 = P parameterises all submodules of P2 isomorphic to P0 , and decomposes into an open HALL POLYNOMIALS FOR AFFINE QUIVERS 369 subscheme consisting of those submodules whose cokernel lies in D0 (with FО±ПЂ12ПЂ0 + FО±ПЂ32ПЂ0 = q 2 rational points) and a closed subscheme consisting of those points whose cokernel lies in D1 (with FО±ПЂ22ПЂ0 = q + 1 rational points). Let us call a decomposition class О± discrete if nО± = 1 (hence О± = (Ој, в€…)). For Оі will satisfy example, if О± is discrete, then the Hall polynomial FО±ОІ Оі FО±ОІ (q) = C FAB for all C в€€ S(Оі, k). Bв€€S(ОІ,k) We call a module discrete if it is the unique module (up to isomorphism) in a discrete decomposition class, which is if and only if it contains no homogeneous regular summand. 7. Reductions using GreenвЂ™s Formula In this section we show how GreenвЂ™s Formula together with split torsion pairs can be used to set up an induction on dimension vector. This reduces the problem X of existence of Hall polynomials to the special case of FM N with X regular and either homogeneous or non-homogeneous lying in a single tube. Let (T , F) be a split torsion pair and suppose that each indecomposable homogeneous regular module is contained in T . We can decompose any module A в€ј = Af вЉ•At with Af в€€ add F and At в€€ add T , and Af is discrete. Thus every decomposition class О± can be written as О±f вЉ• О±t with О±f discrete. A dual result clearly holds if each indecomposable homogeneous regular module is contained in F. Proposition 13. Suppose that Hall polynomials exist for all modules of dimension vector smaller than d and let Оѕ be a decomposition class of dimension vector d. Let (T , F) be a split torsion pair such that Оѕf and Оѕt are both non-zero and such that all indecomposable homogeneous regular modules are contained in either F or T . Оѕ exists and is given by Then the Hall polynomial FОјОЅ nОј nОЅ Оѕ Ој Оѕt aО± aОІ aОі aОґ Оѕ ОЅ FОјОЅ = q Оѕf ,Оѕt в€’ О±,Оґ FО±ОІ FОіОґ FО±Оіf FОІОґ . aОј aОЅ nО± nОІ nОі nОґ О±,ОІ,Оі,Оґ Proof. The proof involves analysing the inп¬‚uence of the split torsion pair on GreenвЂ™s Formula. We then sum in a suitable way to deduce the existence of Hall polynomials. Suppose that all indecomposable homogeneous regular modules are contained in T and consider GreenвЂ™s Formula. On the left we have E E X X FM N FXf Xt /aE = FM N /aX = FM N /aXf aXt q Xf ,Xt E whereas the right-hand side reads qв€’ A,D X Xt M N FAB FCD FACf FBD A,B,C,D aA aB aC aD . a M a N a Xf a Xt We thus have the equality X FM N = q Xf ,Xt в€’ A,D A,B,C,D Observe that C = Cf and B = Bt . X Xt M N FAB FCD FACf FBD aA aB aC aD . aM aN 370 ANDREW HUBERY Writing A = Af вЉ• At and M = Mf вЉ• Mt , associativity of Hall numbers implies M FAB = t M FALf At FLB = t L Similarly, FAMf Lt FALttBt = ОґAf Mf FAMttBt . FAMf L FALt Bt = L FCNf D = N ОґDt Nt FCffDf , X FM N = q Lt so that Xf ,Xt в€’ A,D N X FAMttBt FCffDf FACf f FBXttD At ,Bt ,Cf ,Df aA aBt aCf aD , aM aN where A = Mf вЉ• At and D = Df вЉ• Nt . Therefore X FM N = M в€€S(Ој) N в€€S(ОЅ) Оѕf ,Оѕt в€’ О±,Оґ q N X FAMttBt FCffDf FACf f At в€€S(О±t ) Mt в€€S(Ојt ) О±t ,ОІt Оіf ,Оґf FBXttD Bt в€€S(ОІt ) Nt в€€S(ОЅt ) aО± aОІt aОіf aОґ , aОј aОЅ where we have written О± = Ојf вЉ• О±t and, for At в€€ S(О±t ), A = Mf вЉ• At . Similarly, Оґ = Оґf вЉ• ОЅt and, for Nt в€€ S(ОЅt ), D = Df вЉ• Nt . Оѕ ОЅ By induction, we have the Hall polynomials FО±ОјttОІt , FОІОѕttОґ , FО±Оіf f and FОіffОґf so that, for any X в€€ S(Оѕ), aО± aОІt aОіf aОґ ОЅ Оѕ X FM q Оѕf ,Оѕt в€’ О±,Оґ FО±ОјttОІt nОјt /nОІt FОіffОґf FО±Оіf f /nО± FОІОѕttОґ N = aОј aОЅ M в€€S(Ој) N в€€S(ОЅ) О±t ,ОІt Оіf ,Оґf = q Оѕf ,Оѕt в€’ О±,Оґ ОЅ Оѕ FО±ОјttОІt FОіffОґf FО±Оіf f FОІОѕttОґ О±t ,ОІt Оіf ,Оґf aО± aОІt aОіf aОґ nОјt . aОј aОЅ nО± nОІt Оѕt We now note that the polynomial FОІОґ is non-zero only if ОІ = ОІt . In this case, we Ој Ојt also have the identity FО±ОІ = ОґО±f Ојf FО±t ОІt exactly as for modules. Similarly, Оі = Оіf ОЅ ОЅ and FОіОґ = ОґОЅt Оґt FОіffОґf . Hence we can simplify the above expression to get Оѕ = FОјОЅ q Оѕf ,Оѕt в€’ О±,Оґ Оѕ Ој Оѕt ОЅ FО±ОІ FОіОґ FО±Оіf FОІОґ О±,ОІ,Оі,Оґ aО± aОІ aОі aОґ nОј . aОј aОЅ nО± nОІ Similarly, X FM N = M в€€S(Ој) Xв€€S(Оѕ) q Оѕf ,Оѕt в€’ О±,Оґ X At в€€S(О±t ) Mt в€€S(Ојt ) О±t ,ОІt Оіf ,Оґf = N FAMttBt FCffDf FACf f q Оѕf ,Оѕt в€’ О±,Оґ Bt в€€S(ОІt ) Xt в€€S(Оѕt ) ОЅ Оѕ FО±ОјttОІt FОіffОґf FО±Оіf f FОІОѕttОґ О±t ,ОІt Оіf ,Оґf = q Оѕf ,Оѕt в€’ О±,Оґ Оѕ Ој Оѕt ОЅ FО±ОІ FОіОґ FО±Оіf FОІОґ О±,ОІ,Оі,Оґ FBXttD aО± aОІt aОіf aОґ aОј aОЅ aО± aОІt aОіf aОґ nОјt nОѕt aОј aОЅ nО± nОІt nОЅt aО± aОІ aОі aОґ nОј nОѕ aОј aОЅ nО± nОІ nОЅ Оѕ = FОјОЅ nОѕ /nОЅ . Dually, X Оѕ FM N = FОјОЅ nОѕ /nОј . N в€€S(ОЅ) Xв€€S(Оѕ) Оѕ . Note also that nОЅ = nОґ and This proves the existence of the Hall polynomial FОјОЅ Оѕ nОі = 1, so that we can write FОјОЅ as in the statement of the proposition. HALL POLYNOMIALS FOR AFFINE QUIVERS 371 In the case that all indecomposable homogeneous regular modules are contained in F, the proof goes through mutatis mutandis. Оѕ when Оѕ is inLemma 14. It is enough to prove the existence of polynomials FОјОЅ decomposable preprojective, indecomposable preinjective, non-homogeneous regular in a single tube, or homogeneous regular. Proof. If Оѕ can be expressed as Оѕf вЉ• Оѕt for some split torsion pair (T , F) with both Оѕf and Оѕt non-zero and with all indecomposable homogeneous regular modules contained in either F or T , then the previous proposition together with induction on dimension vector implies the result. In particular, we immediately reduce to the cases when Оѕ is either preprojective, non-homogeneous regular in a single tube, homogeneous regular or preinjective. If Оѕ is decomposable preprojective, then we can п¬Ѓnd a section of the AuslanderReiten quiver containing just one isotypic summand of X в€€ S(Оѕ), and with all other summands lying to the left. We can use this section to deп¬Ѓne a torsion pair, and so we can again apply our reduction. Suppose that Оѕ is isotypic preprojective, say X = P r+1 в€€ S(Оѕ) for some indecomposable preprojective P . Since P is exceptional, we have (as in the proof of Theorem 4) that r+1 P FM N = q rв€’ A,D r M N P P FAB FCD FAC FBD aA aB aC aD /aM aN . A,B,C,D We note that N , C and D are all preprojective, so discrete. Thus we can again use induction to deduce that r+1 P FM N = q rв€’ О±,Оґ r Ој ОЅ ПЂ ПЂ FО±ОІ FОіОґ FО±Оі FОІОґ О±,ОІ,Оі,Оґ aО± aОІ aОі aОґ 1 aОј aОЅ nО± nОІ for all M в€€ S(Ој). Thus it is enough to assume that Оѕ is indecomposable preprojective. Similarly, it is enough to consider indecomposable preinjective Оѕ. Оѕ when Оѕ is regular and either Lemma 15. It is enough to prove the existence of FОјОЅ homogeneous or else lying in a single non-homogeneous tube. Proof. Suppose that X в€€ S(Оѕ) is indecomposable preprojective. Let R = rad(X) and T = top(X), so R is preprojective and T is semisimple, and both are discrete. If R = 0, then X is simple projective and the result is clear, so assume that both R and T are non-zero. Consider the left-hand side of GreenвЂ™s Formula. We have that if FTER is non-zero, then either E в€ј = X and FTXR = 1, or else E is decomposable and contains a preprojective summand (since it has the same defect as X). Thus E E FM N FT R /aE в€’ X FM N /aX = E L L FM N FT R /aL . L decomp Consider the second sum. Since L is decomposable and contains a preprojective summand, and since R and T are discrete, we can follow the proof of the previous lemma and apply induction to deduce the existence of a universal polynomial FП„О»ПЃ such that FП„О»ПЃ = FTLR for all L в€€ S(О»). 372 ANDREW HUBERY Similarly, since N is preprojective, so discrete, there exists a universal polynomial О» such that FОјОЅ О» nО» /nОј = FОјОЅ L FM N for all M в€€ S(Ој). Lв€€S(О») Hence L L FM N FT R /aL = L decomp О» FОјОЅ FП„О»ПЃ nО» /nОј aО» . О»=Оѕ Now consider the п¬Ѓrst sum. Using GreenвЂ™s Formula, we can rewrite it as M N T R aA aB aC aD q в€’ A,D FAB FCD FAC FBD . aM aN aT aR A,B,C,D We note that A and C must both be semisimple and D must be preprojective, so all three are discrete. Hence by induction we have universal polynomials such that ПЃ R FBD = FОІОґ /nОІ , T П„ FAC = FО±Оі , Ој M FAB = FО±ОІ , N ОЅ FCD = FОіОґ , Bв€€S(ОІ) where the п¬Ѓrst holds for all B в€€ S(ОІ) and the last holds for all M в€€ S(Ој). X Оѕ Putting this together we obtain a universal polynomial such that FM N = FОјОЅ for all M в€€ S(Ој). If Оѕ is indecomposable preinjective, then we can apply a similar argument using soc(X) and X/soc(X). Remark. In fact, if X lies in a single non-homogeneous tube, we may further assume that X is indecomposable. For X is discrete and if X = X1 вЉ• X2 is decomposable, then every other extension E of X2 by X1 lies in the same tube and satisп¬Ѓes О»(E) < О»(X) (in the notation of Lemma 6). We can thus use induction on О»(X) and GreenвЂ™s Formula as in the proof of Theorem 7. 8. Reductions using associativity In the previous section we reduced the problem of п¬Ѓnding Hall polynomials to X the special case of FM N for X either homogeneous regular or non-homogeneous regular lying in a single tube. We now wish to improve this to the case when M is simple preinjective and N is indecomposable preprojective. Unfortunately, it seems diп¬ѓcult to do this by applying reп¬‚ection functors, since we have often used induction on dimension vector. Instead we will use associativity together with Theorem 1. Оѕ when ОЅ is indecomposable preprojective, Lemma 16. It is enough to consider FОјОЅ Ој is indecomposable preinjective and Оѕ is either homogeneous regular or else nonhomogeneous regular in a single tube. Proof. We showed in the last section that it is enough to consider the case when Оѕ is homogeneous regular or non-homogeneous regular in a single tube. We use induction on the dimension vector of Оѕ and the defect of Ој. We note that if в€‚(Ој) = 0, then both M and N are regular. Thus either M , N and X are all homogeneous regular, so there exists a universal polynomial by Theorem 10, or else M , N and X all lie in a single non-homogeneous tube, so there exists a universal polynomial by Theorem 7. HALL POLYNOMIALS FOR AFFINE QUIVERS 373 X Now consider FM N for X в€€ S(Оѕ) and assume that в€‚(Ој) > 0. Suppose further that we can п¬Ѓnd a split torsion pair such that Ојf and Ојt are both non-zero, and such that all homogeneous regulars are torsion-free. Note that X FM N = X FM FL . f L Mt N L Since dim L < dim X and Mt is discrete, there exists a Hall polynomial FОјО»t ОЅ such that L FM = FОјО»t ОЅ for all L в€€ S(О»). tN N в€€S(ОЅ) Since 0 в‰¤ в€‚(Mf ) < в€‚(M ), there exists a Hall polynomial FОјОѕf О» such that X FM = FОјОѕf О» fL for all X в€€ S(Оѕ). Mf в€€S(Ојf ) Lв€€S(О») Thus Оѕ FОјОЅ := FОјОѕf О» FОјО»t ОЅ . X FM N = M в€€S(Ој) N в€€S(ОЅ) О» Similarly, X Оѕ FM N = FОјОЅ nОѕ /nОЅ X Оѕ FM N = FОјОЅ nОѕ /nОј . and M в€€S(Ој) Xв€€S(Оѕ) N в€€S(ОЅ) Xв€€S(Оѕ) Оѕ . This proves the existence of the Hall polynomial FОјОЅ We have reduced to the case when M is isotypic preinjective. Suppose that M = I r+1 with I indecomposable preinjective. Since I is exceptional, L FII r = ОґLM q r+1 в€’ 1 , qв€’1 so associativity gives X FM N = qв€’1 q r+1 в€’ 1 X L FIL FI r N . L Again, dim L < dim X, so we have the Hall polynomial FО№О»r ОЅ . Also, 0 в‰¤ в€‚(I) < Оѕ Оѕ в€‚(M ), so we have the Hall polynomial FО№О» , and hence the Hall polynomial FОјОЅ . The dual arguments clearly work for N , and since 0 = в€‚(X) = в€‚(M ) + в€‚(N ), we see that we can always reduce to the case when both M is indecomposable preinjective and N is indecomposable preprojective. The next lemma is a nice generalisation of what happens in the Kronecker case [25]. Lemma 17. Let X be either homogeneous regular or non-homogeneous regular lying in a single tube. If M and N are indecomposable with в€‚(M ) = 1 and dim M > Оґ, then X X FM N = FM N , where M and N are indecomposable and dim M в€’ dim M = Оґ = dim N в€’ dim N . 374 ANDREW HUBERY Proof. By assumption, there exists an embedding of the module category of the Kronecker quiver whose image contains M . (For, take an indecomposable preinjective module I such that dim I < Оґ and dim M в€’ dim I в€€ ZОґ. Then there exists an orthogonal exceptional pair (I, P ) with [I, P ]1 = 2.) In this way, we see that there exists a short exact sequence of the form 0в†’Rв†’M в†’M в†’0 for any indecomposable regular R such that dim R = Оґ and [M, T ]1 = 0, where T is the regular top of R; equivalently [S, M ] = 0, where S is the regular socle of R. This occurs for precisely one such indecomposable in each non-homogeneous tube. It is easily seen from RiedtmannвЂ™s Formula that RвЉ•M FM =q R M FM R = 1, RвЉ•M and FRM = 1. Dually, we have a short exact sequence 0в†’N в†’N в†’R в†’0 for any indecomposable regular R such that [S , N ]1 = 0, where S is the regular socle of R . In particular, we can take R = R for any homogeneous regular indecomposable R of dimension vector Оґ. It may be, however, that there are no such homogeneous regular modules of dimension vector Оґ. For example, take Q a quiver of type D4 and k a п¬Ѓeld with two elements. Suppose, therefore, that there are no homogeneous regular modules of dimension vector Оґ. By assumption we have a short exact sequence 0в†’N в†’Xв†’M в†’0 with X either homogeneous regular or non-homogeneous regular lying in a single tube. Take a regular simple S in a non-homogeneous tube diп¬Ђerent from that containing X, and apply Hom(S, в€’). Since there are no homomorphisms or extensions between regular modules in distinct tubes, we see that [S, M ] = [S, N ]1 . Take S such that [S, M ] = 1 and set R to be the indecomposable regular module of dimension vector Оґ and with regular socle S. In all cases, we have found a module R for which there exist exact sequences 0 в†’ R в†’ M в†’ M в†’ 0 and 0 в†’ N в†’ N в†’ R в†’ 0. Associativity of Hall numbers now gives X X FM N + qFRвЉ•M N L FM = X R FLN X L X FM L FRN = FM = L N X + qFM N вЉ•R . L Similarly, X FRвЉ•M N L X FRM FLN = = L X L FRL FM N L and X FM N вЉ•R X L FM L FN R = = L L FM X N FLR . L Thus X X FM N в€’ FM N L FM =q N X X FLR . в€’ FRL L If R is homogeneous regular, then it is well known (using the natural duality on X X representations) that the Hall numbers are вЂњsymmetricвЂќ; that is, FLR = FRL . On HALL POLYNOMIALS FOR AFFINE QUIVERS 375 the other hand, if R is non-homogeneous and lying in a diп¬Ђerent tube from X, then X X = FRL = 0. In all cases we get that FLR X X FM N = FM N as required. Lemma 18. It is enough to prove the existence of Hall polynomials in the case when Ој is simple preinjective, ОЅ is indecomposable preprojective and Оѕ is regular and either homogeneous or contained in a single non-homogeneous tube. Proof. By Lemma 17, we may assume that M is indecomposable preinjective and either в€‚(M ) в‰Ґ 2 or else в€‚(M ) = 1 and dim M < Оґ. If M is not simple, then Theorem 1 yields an orthogonal exceptional pair (A, B) and a short exact sequence of the form 0 в†’ B b в†’ M в†’ Aa в†’ 0. If d = dim Ext1 (A, B) = 2, then by Lemma 2 and using that M is indecomposable preinjective, we have dim M = rОґ + dim A > Оґ where r = b = a в€’ 1, and в€‚(M ) = в€‚(A) = 1, a contradiction. Thus d = 1 and, since there are only three L = 1 for indecomposable modules in F(A, B), a = b = 1. It is easily seen that FAB both L = M and L = A вЉ• B. Hence X X FM N + FAвЉ•BN = L X FAB FLN = L X L FAL FBN . L Note also that X FAвЉ•BN = L X FBA FLN = X L FBL FAN . L L Putting this together we obtain X L FAL FBN в€’ X FM N = L X L FBL FAN . L О» О» and FО±ОЅ . Now, since in both sums dim L < dim X, we have Hall polynomials FОІОЅ Since M A, we can use induction on the partial order for indecomposable preinОѕ jective modules to deduce the existence of the Hall polynomial FО±О» . Finally, there X are three possibilities for B: if B is preprojective, then FBL = 0; if B is (nonОѕ homogeneous) regular, then there is a Hall polynomial FОІО» and ОІ is discrete since all three modules are regular; if B is preinjective, then в€‚(M ) > в€‚(A), в€‚(B) > 0 and Оѕ we can use induction on the defect to obtain a Hall polynomial FОІО» . We have shown that, for all X в€€ S(Оѕ), Оѕ О» FО±О» FОІОЅ в€’ X FM N = О» and hence the Hall polynomial Оѕ FОјОЅ Оѕ О» FОІО» FО±ОЅ , О» exists. We now show that Hall polynomials exist when M is simple preinjective, N is indecomposable preprojective, and X is regular and either homogeneous or contained in a single non-homogeneous tube, thus completing the proof of the Main Theorem. Let d = [N, M ]. Then there is an epimorphism N M d , unique up to an d automorphism of M . Let P be the kernel of this map. We thus have a short exact sequence 0 в†’ P в†’ N в†’ Md в†’ 0 376 ANDREW HUBERY N and FM d P = 1. Applying Hom(в€’, M ) yields [P, M ] = 0, whereas Hom(в€’, N ) yields 1 [P, N ] = 0. Finally, using Hom(P, в€’), we obtain that [P, P ]1 = 0 and hence that P is rigid. Now consider the push-out diagram P вЏђ вЏђ P вЏђ вЏђ 0 в€’в€’в€’в€’в†’ N в€’в€’в€’в€’в†’ вЏђ вЏђ X вЏђ вЏђ в€’в€’в€’в€’в†’ M в€’в€’в€’в€’в†’ 0 0 в€’в€’в€’в€’в†’ M d в€’в€’в€’в€’в†’ M d+1 в€’в€’в€’в€’в†’ M в€’в€’в€’в€’в†’ 0 where the bottom row is split since M is exceptional. We note that [X, M ] = d + 1 X and FM d+1 P = 1. By associativity, we get X FM N + X L FM L FM d P = X L FM L FM d P = L L decomp so that X FM N = q d+1 в€’ 1 в€’ qв€’1 L X FM M d FLP = L q d+1 в€’ 1 , qв€’1 X L FM L FM d P . L decomp Since dim L < dim X and M and P are discrete, there exists a Hall polynomial FОјО»d ПЂ such that L О» FM d P = FОј d ПЂ for all L в€€ S(О»). Since L is decomposable and has a preprojective summand (в€‚(L) = в€‚(N ) < 0), Оѕ Lemma 16 gives us the Hall polynomial FОјО» . Thus there exists a Hall polynomial Оѕ FОјОЅ as required. This completes the proof of the Main Theorem. 9. Wild quivers We п¬Ѓnish by oп¬Ђering a deп¬Ѓnition of Hall polynomials for wild quivers. Let Q be an arbitrary quiver. We п¬Ѓrst need a combinatorial partition of the set of isomorphism classes of kQ-modules, by which we mean some combinatorial вЂњpropertiesвЂќ of modules such that the sets S(О±, k) yield a partition of the set of isomorphism classes of modules, where S(О±, k) consists of those kQ-modules having property О±. Moreover, we need universal polynomials aО± and nО± for each О± such that, for any п¬Ѓnite п¬Ѓeld k with |k| = q, aО± (q) = | Aut(A)| for each A в€€ S(О±, k) and nО± (q) = |S(О±, k)|. Finally, we would like that there are only п¬Ѓnitely many such classes of a given dimension vector. Some further properties which would clearly be of interest are that, over an algebraically closed п¬Ѓeld k, each S(О±, k) is a locally closed, smooth, irreducible, GL-invariant subvariety of the representation variety, that each S(О±, k) admits a smooth, rational, geometric quotient (in which case nО± counts the number of rational points of this quotient), and that the S(О±, k) yield a stratiп¬Ѓcation of the representation varieties. HALL POLYNOMIALS FOR AFFINE QUIVERS 377 We say that Hall polynomials exist with respect to such a family if the following Оі conditions are satisп¬Ѓed. Given О±, ОІ and Оі, there exists a universal polynomial FО±ОІ such that Оі (q) = FО±ОІ C FAB for all C в€€ S(Оі, k) Aв€€S(О±,k) Bв€€S(ОІ,k) and further that Оі nОі (q)FО±ОІ (q) = nО± (q) C FAB for all A в€€ S(О±, k) C FAB for all B в€€ S(ОІ, k). 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