Acta Mathematica Sinica, English Series

Mar., 2008, Vol. 24, No. 3, pp. 387–396

Published online: March 5, 2008

DOI: 10.1007/s10114-007-1010-0

Http://www.ActaMath.com

Acta Mathematica Sinica, English Series© The Editorial Office of AMS & Springer-Verlag 2008

Hall Polynomials for Affine Quivers of Type Am(m ≥ 1)

Tao XIEDepartment of Mathematics, Hubei Normal University, Huangshi 435002, P. R. China

E-mail: xietao 1294@sohu.com

Abstract It is well known that Hall polynomials as structural coefficients play an important role in

the structure of Lie algebras and quantum groups. By using the properties of representation categories

of affine quivers, the task of computing Hall polynomials for affine quivers can be reduced to counting

the numbers of solutions of some matrix equations. This method has been applied to obtain Hall

polynomials for indecomposable representations of quivers of type Am(m ≥ 1).

Keywords affine quiver, preprojective module, preinjective module, regular module, Hall polynomial

MR(2000) Subject Classification 16G20, 17B37

1 Introduction

The notion of Hall polynomial was introduced by Steinitz and studied by Hall. In [1], Steinitzshows that the free abelian group with basis indexed by the isomorphism classes of finite p-groups may be endowed with a product by counting the filtrations of finite p-groups. We obtainthe so-called Hall algebra H (Zp) of the ring Zp of p-adic integers. It is a commutative andassociative ring with an identity element and plays an important role in algebra and combina-torics (see [2]). In 1990, Ringel generalized the above definition on a finitary ring (especially thefinite-dimensional algebra over a finite field) and obtained what is now called the Ringel–Hallalgebra. It comes as a surprise that the Ringel–Hall algebra of a finite-dimensional heredi-tary algebra admits a subalgebra, called the composition algebra, which is isomorphic to thepositive part of a quantized enveloping algebra of the corresponding Kac–Moody algebra (see[3–4]). Now the Ringel–Hall algebras serve as a successful model for the realization of quantumgroups. Hence Hall polynomials provide some convenience in calculating the structural coeffi-cients of the corresponding Lie algebras and quantum groups. In [5] Ringel used the theories ofvectorspace categories and of representations of partially ordered sets to reduce the calculationof Hall polynomials for the representation-finite hereditary algebras to 24 cases (including thevalued cases).

The aim of this paper is to calculate the Hall polynomials for affine quivers. We mainlyconsider the Hall polynomials of affine quivers of type Am(m ≥ 1). Given three indecomposablemodules X, Y, Z with X preprojective, by using reflection functors we can assume that themodule X is simple projective. Some phenomena which are different from the Dynkin caseare discovered: one phenomenon is that we can not find three indecomposable preprojective

Received June 29, 2006, Accepted December 29, 2006

388 Xie T.

modules X, Y, Z such that dimY = dim X + dim Z. This means that the Hall polynomialsfor preprojective modules are all zero. Another phenomenon is that the simple projectivemodule can be embedded into infinitely many regular modules such that their cokernels areindecomposable preinjective which means that there may exist infinitely many non-zero Hallpolynomials. Moreover, we find that all these non-zero Hall polynomials can be expressed insome formulas.

2 Preliminaries

Let Q = (Q0, Q1) be an affine quiver of type Am(m ≥ 1), where Q0 is the vertex set which hasm + 1 vertices, Q1 is the arrow set. If ρ ∈ Q1 is an arrow from tail i to head j, we write hρ forj and tρ for i. Thus, we obtain functions h, t : Q1 → Q0. Let Q be the diagram of Q forgettingorientations. Sometimes we also use Q = (Q, Ω) to denote the quiver with an orientation Ωexcluding the acyclic case.

Given a quiver Q, A := KQ denotes the path algebra of Q over a finite field K with q ele-ments and modA denotes the category of finite-dimensional left A-modules. Then, each finite-dimensional representation V of Q, which, by definition, consists of a set of finite-dimensionalvector spaces Vi for each i ∈ Q0 and a set of linear transformations Vρ : Vtρ → Vhρ for eachρ ∈ Q1, is identified with a left A-module. We call dimV :=(dim KVi)i∈Q0 the dimension vec-tor of V . Finally, by RepQ we denote the category of all finite-dimensional representations ofQ. Throughout, we use |M | and [M ] to denote the cardinality of the set M and the isomor-phism class of M, respectively. For unexplained notions concerning the representation theoryof algebras we refer to [6–8].

For M, N1, . . . , Nt ∈ modA, we denote by FMN1,...,Nt

the number of filtrations

0 = U0 ⊆ U1 ⊆ · · · ⊆ Ut = M

of M such that Ui/Ui−1 � Ni for all 1 ≤ i ≤ t. In particular, FML,N is the number of the

submodules U of M such that U � N and M/U � L. Note that if dim M �= dim L + dim N,

then FML,N = 0.

Let K be the algebraic closure of K, and K ′ be a field extension of K. For any K-spaceV , we denote by V K′

the K ′-vector space V ⊗K K ′. In particular, AK′naturally becomes a

K ′-algebra and for each A-module M , MK′becomes an AK′

-module. The field extension K ′

is said to be conservative for an A-module M if, for any indecomposable summand N of M ,NK′

is an indecomposable AK′-module. Given three A-modules X, Y, Z ∈ modA, let

ΩX,Y,Z = {K ′|K ⊆ K ′ ⊆ K and K ′ is conservative for all X, Y, Z}.Note that ΩX,Y,Z is always infinite. If there exits a polynomial ϕY

Z,X ∈ Z[T ] such that, forany K ′ ∈ ΩX,Y,Z ,

ϕYZ,X(|K ′|) = FY K′

ZK′ , XK′ ,

we say that the Hall polynomial exists for X, Y, Z. Such a ϕYZ,X is called a Hall polynomial.

Recall from [3] that the Ringel–Hall algebra H (A) of A is the free abelian group with basisthe set of isoclasses [M ] of all finite A-modules, with multiplication given by

[M ] · [N ] :=∑[L]

FLM,N [L].

Hall Polynomials for Affine Quivers of Type Am(m ≥ 1) 389

For A-modules X, Y , define〈dimX, dimY 〉 := dim kHomA(X, Y ) − dim kExt1A(X, Y ).

This gives a bilinear form 〈−,−〉 : Zm+1 × Z

m+1 → Z, called the Euler form of A. Denote by(−,−) the symmetrization of 〈−,−〉, i.e.

(dimX, dimY ) := 〈dimX, dimY 〉 + 〈dimX, dimY 〉.Furthermore, let qA be the quadratic form on Z

m+1 given by qA(dimX) = 〈dimX, dimX〉. Itis well known that qA is positive semi-definite but not positive definite. The vector x satisfyingqA(x) = 1 is called a real root of Q. Define the null space N = {z ∈ Z

m+1|qA(z) = 0} and weknow that N = Zn where n is the minimal positive imaginary root of Q (see [9]).

Choose the standard basis e1 = (1, 0, . . . , 0, 0), e2 = (0, 1, 0, . . . , 0), . . . , em+1 = (0, 0, . . . ,

0, 1) of Zm+1. For each 1 ≤ i ≤ m + 1, there is a transformation si on Z

m+1, called a refection,defined by:

si(x) = x − 2(x, ei)(ei, ei)

ei = x − (x, ei)ei, ∀x ∈ Zm+1.

It is easy to see that s2i = id and (six, siy) = (x,y) for all x,y ∈ Z

m+1. Furthermore, si(x) = x

if and only if (x, ei) = 0.The subgroup of GL(Zm+1) generated by s1, s2, . . . , sm+1 is called the Weyl group of Q and

denoted by W . And the group W of all linear transformations of Zm+1/N induced by the

transformations of W is finite and N is stable under the action of W .The element in the Weyl group W of the form c = si1si2 · · · sim+1 , where (i1, i2, . . . , im+1) is

a permutation of (1, 2, . . . , m+1), is called a Coxeter transformation. There is a very importantconcept of the defect ∂c(x) of a vector x ∈ Z

m+1 with respect to a Coxeter transformation c ∈ W

(see [9]): If c ∈ W is of the order h, then, given x ∈ Zm+1,

ch(x) = x + ∂c(x)n.

In this fashion, ∂c : Zn+1 −→ Z

n+1 defines a linear form and thus we have

∂c(x) =∑

i

∂c(ei)xi.

Moreover, one can easily see that c−1 is also a transformation and

∂c(cx) = ∂c(x).For each 1 ≤ i ≤ m + 1, define pi = sm+1sn · · · si+1ei, qi = s1s2 · · · si−1ei, then qi = −cpi,where c = s1s2 · · · sm+1.

Lemma 2.1 [9] Let Q be an affine quiver. Then :(1) The vectors x = c−rpkt

0 ≤ r, 1 ≤ t ≤ m+1 are just all positive roots of Q of negativedefect with respect to c;

(2) The vectors x = c−rqkt0 ≤ r, 1 ≤ t ≤ m + 1 are just all positive roots of Q of positive

defect with respect to c;(3) The vectors x = x0 + rn 0 ≤ r with ∂c(x0) = 0 are just all positive roots of Q of zero

defect with respect to c.

Corollary 2.2 If x = c−rpkt0 ≤ r, 1 ≤ t ≤ m+1, then ∂c(x) =∂c(pkt

) for all 1 ≤ t ≤ m+1.

Lemma 2.3 [9] The defect vectors of Am(m ≥ 1), with respect to the given orientation of Q,

are indicated in the following graphs :

390 Xie T.

n + 10

0 0

0

0

0−(n + 1).

Lemma 2.4 [7] Let M be a directing A-module. Then EndA(M) = K and Ext1A(M, M) = 0for all i ≥ 1. Also, if M ′ is an indecomposable A-module with dim M = dim M ′, then M � M ′.

Lemma 2.5 [9–10] If Q = (Q, Ω) is the quiver of type Am(m ≥ 1) then, for any other orienta-tion Ω′, there exists a (−)-admissible sequence i1, i2, . . . , ir such that Ω′ = σir

σir−1 · · ·σi1Ω, andΩ′ has only one sink and one source. Moreover, if cl is a Coxeter transformation determinedby the admissible orientation of σil−1σir−2 · · ·σi1Ω, then cl = sil

cl−1sil.

Corollary 2.6 The defect of projective modules do not depend on the orientation of Q.

Lemma 2.7 [5] Given three indecomposable modules X, Y , Z of modA with dimensionvectors satisfying dim Y = dim X + dim Z, let i be a sink (resp. source) of Q = (Q, Ω), anddim X be not the simple root ei corresponding to i. Let R = R+

i (resp. R−i ) be the reflection

corresponding to i. ThenFY

Z,X = FRYRZ,RX .

Remark We are going to calculate the Hall polynomials for indecomposable modules ofA = KQ, where Q are affine quivers of type Am(m ≥ 1). From the above lemmas, we haveto consider only affine quivers with a specific orientation and calculate ϕY

Z,X , where Y , Z areindecomposable A-modules, and X is a simple projective or a simple injective A-module.

3 Hall Polynomials for Affine Quivers of Type Am (m ≥ 1)

We know that indecomposable A-modules (i.e., the indecomposable representations of Q) aredivided into three classes (see [11]): preprojective modules, regular modules, and preinjectivemodules. We use P, R, I to denote the sets of isomorphism class of indecomposable pre-projective modules, regular modules, injective modules, respectively. And the set of all theisomorphism class of regular modules is an infinite union of tubular components:

R = T (0) ∪ T (∞) ∪⋃f∈I

T (f),

where I is the set of all monic irreducible polynomials of K[T ] excluding the polynomial T . Inparticular, we use T (λ) to denote T (f) when f = T − λ ∈ I.

In this section, we will calculate Hall polynomials for indecomposable modules of A = KQ,

where Q is the affine quiver of type Am(m ≥ 1). First, we study the simple case A1, someHall polynomials in this case are obtained, and we find that the method in this case can beapplied to the general case. Next, we consider the general case Am. By describing preciselythe indecomposable regular modules in non-homogeneous tubes and homogeneous tubes T (λ),some Hall polynomials in these cases are obtained.

Let A = KQ be the path algebra where Q is the affine quiver of type Am, and assume X,Y , Z are three indecomposable A-modules satisfying dim Y = dim X + dim Z.

Firstly, we want to solve the case A1, some indecomposable modules (or representations)are described as in the following (see for example [7]):

Hall Polynomials for Affine Quivers of Type Am(m ≥ 1) 391

1 2,β

α

A1:

Kn Kn+1,

(In 0)

(0 In)

(P)

Kn+1 Kn,

(In

0

)

(0In

)

(I )

Kn Kn,In

Jn(0)

T (0)

Kn Kn,

Jn(0)

In

T (∞)

Kn Kn.Jn(λ)

In

T (λ)

From the final remark of the above section, we can suppose that X is the simple projective:

0 K.0

0

Now we consider the case when X is embedded into indecomposable preprojective modules.Since the dimension vectors of indecomposable preprojective modules are given by (n, n + 1)(n ≥ 0), we can not find another two indecomposable preprojective modules Y , Z satisfyingdim Y = dim X + dim Z. Hence, FY

Z,X = 0, where X, Y , Z are indecomposable preprojectivemodules. Dually, we will get that FY

Z,X = 0 for any X, Y , Z indecomposable preinjectivemodules. From the above analysis, we get the following result:

Proposition 3.1 If Q = A1, then, for any X, Y, Z ∈ P or I , ϕYZ,X = 0.

Next, we consider the case when X is embedded into tubes T (λ) and T (0). For the conve-nience of discussion we denote the two components of regular modules by T (λ)(λ ∈ K); andthe homomorphism is given by φ = (0, φ2) with the following commutative diagram denotingthe process of calculating the Cokernel of φ :

392 Xie T.

0 Kn Kn

K Kn Kn−1,

0 In

0 0 Jn(λ) In A B

φ2ω

where φ2 = (λ1, λ2, . . . , λn)T (λ1, . . . , λn ∈ K).From the above commutative diagram we get that A = ωJn(λ) and B = ω, i.e., we can

describe the Cokernel of the morphism φ.Since the indecomposable preinjective modules are determined by their dimension vectors

up to an isomorphism, the problem of calculating ϕYZ,X is reduced to counting the number of

sequences (λ1, λ2, λ3, . . . , λn) (λi ∈ K, i = 1, 2, . . . , n) such that Cokerφ is indecomposable.Note that a module M is indecomposable if and only if its endomorphism algebra EndA(M)

is local. So we need to calculate the endomorphism algebra of Cokerφ, which is an algebrainvolving λ1, λ2, . . . , λn as parameters. So the problem of calculating ϕY

Z,X is reduced to findinghow many (λ1, λ2, . . . , λn) (λi ∈ K, i = 1, 2, . . . , n) such that EndA(Cokerφ) is local.

By choosing suitable basis for Kn and Kn−1, EndA(Cokerφ) can be described as a set ofpairs (C, D), where C is an n × n matrix (ci,j) and D is an (n − 1) × (n − 1) matrix (di,j)satisfying the following matric equations:

DW = WC,

DWJn(λ) = WJn(λ)C.

When φ2 = (1, λ2, . . . , λn)T (λ2, . . . , λn ∈ K), then

ω =

⎛⎜⎜⎝

−λ2 1 0 · · · 0 0

−λ3 0 1 · · · 0 0

.

.

....

.

.

.. . .

.

.

....

−λn 0 0 · · · 0 1

⎞⎟⎟⎠ .

From the above matrix equalities we obtain the following:

cij = 0 (i �= j),

c11 = c22 = · · · = cnn,

dij = cij (1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1).

We get that C and D are quantitative matrices. That is to say End(Cokerφ) is localand from the above calculating process, End(Cokerφ) is local which does not depend on thechoice of λ2, λ3, . . . , λn. Moreover, applying the above method, we can check that, when φ2 =(0, λ2, λ3, . . . , λn) (λ2, . . . , λn ∈ K), End(Cokerφ) is not local, i.e., Cokerφ is decomposable. So,we obtain the following result:

ϕYZ,X = Tn−1(n ≥ 1),

where X ∈ P, Y ∈ T (λ)(λ �= 0), Z ∈ I .

Hall Polynomials for Affine Quivers of Type Am(m ≥ 1) 393

We will get the same result with the above case after applying a similar method to the caseof λ = 0. As conclusion, we get the following proposition:

Proposition 3.2 If Q = A1, then, for any X ∈ P, Y ∈ T (λ)(λ ∈ K), Z ∈ I , the nonzeroHall polynomials are ϕY

Z,X = Tn−1(n ≥ 1).

Next we consider the general case Am. From the final remark of Section 2 we have toconsider only the following quiver:

a

d1 d2

c1 c2cq

dq

b,

where p + q = m − 1

Theorem 3.3 If Q is the affine quiver of type of Am(m ≥ 1), then, for any X, Y, Z ∈ P

or I , ϕYZ,X = 0.

Proof Since the categories of modA and modAop are dual with each other, the calculationof Hall polynomials for the preinjective component of modA can be reduced to that for thepreprojective component.

On one hand, we suppose that we can find three indecomposable preprojective A-modulesX,Y,Z such that dim Y = dim X + dim Z, for any orientation Ω of Q with the Coxetertransformation c, we have ∂c(dim Y ) = ∂c(dim X) + ∂c(dim Z). On the other hand, fromCorollary 2.1, we can suppose that ∂c(dim Y ) = ∂c(dim p(i)), ∂c(dim X) = ∂c(dim p(j)),∂c(dim Z) = ∂c(dim p(k)), i, j, k ∈ {1, 2, . . . , m + 1}, p(i), p(j), p(k) are three indecompos-able projective modules. Then ∂c(dim p(i)) = ∂c(dim p(j)) + ∂c(dim p(k)), which contradictsLemma 2.3. Thus, there do not exist three indecomposable preprojective (resp. preinjective)A-modules X, Y, Z such that dim Y = dim X+dim Z, i.e., for any X, Y, Z ∈ P or I , ϕY

Z,X = 0.It is well known that the family of pairwise non-isomorphic indecomposable regular modules

in T (λ) is described as follows:

Kn

KnIn

KnIn

Kn

Kn

In

In

In

In

Kn

Kn.Kn.

In

Jn(λ)

Next we will consider the case when a simple projective module is embedded into the indecom-posable modules given above, the simple module id given below:

00

0

00

0

0

0

0

0

0

0

0K

0

0

394 Xie T.

By choosing a suitable basis for each vector space, then the element of EndA(Cokerφ) canbe described as (E, C1, C2, . . . , Cp, D1, D2, . . . , Dq, B), where all of these denote matrices ofcorresponding sizes, and satisfy the following matric equations:

InA = C1In, InC1 = C2In, · · · , InCp−1 = CpIn,

InA = D1In, InD1 = D2In, · · · , InDq−1 = DqIn,

Bω = ωDq, BωJn(λ) = ωJn(λ)Cp.

Then we have

E = C1 = C2 · · · = Cp = D1 = D2 · · · = Dq.

Finally we have

BωJn(λ) = ωJn(λ)E,

Bω = ωE.

Let us recall that these matric equations are the same as in the case A1. On the basis ofthe results of case A1 we know that the Hall polynomials in this case are the same as in thecase A1. Then we obtain the following result:

Theorem 3.4 If Q is the affine quiver of type Am(m ≥ 1), then, for any X ∈ P, Y ∈ T (λ),Z ∈ I , the nonzero polynomials are ϕY

Z,X = Tn−1(n ≥ 1).

Remark From the above process of calculation, the Hall polynomials for an algebra arecompletely determined by some matric equations, which leads to the following definition:

Definition Given a homomorphism φ : X → Y, where X is a simple projective A-module,Y is an indecomposable A-module, via choosing a basis for each vector space, each element ofEndA(Cokerφ) can be regarded as a sequence of matrices, which satisfies some matrix equations.Those matric equations are called the characteristic matric equations.

Remark In many cases, we may find that the matric equations are exactly the same with eachother, so their Hall polynomials must be identical. In this manner, we have to find only differentcharacteristic matric equations and determine in which cases we have the same characteristicmatric equations.

The next aim is to consider the case when the simple projective module is embedded into theindecomposable modules in non-homogeneous tubes T (0) and T (∞). From the Auslander–Reiten quiver of the regular modules in non-homogeneous tubes, we may find that each quasi-simple module (which is lying on the mouth of a tube) determines a chain of indecomposablemodules, and every indecomposable module in this chain is determined by its top and length.From the method introduced in [11, §10], we can describe all the indecomposable modules inthe tubes. Applying the same method of calculating Hall polynomials as above, we find thatthere are only six different characteristic matrix equations. Moreover, after calculating the sixcases, we find that only two cases yield non-zero Hall polynomials. So the next main work isto deal with these two cases.

The characteristic matrix equations of the first case are given in the following:

Hall Polynomials for Affine Quivers of Type Am(m ≥ 1) 395

(1) Cω = ωD,

(2) (0, Is)D = E(0, Is),

(3) BE = CB

.

When φ2 = (1, λ2, . . . , λn)T (λ2, . . . , λn ∈ K), then

ω =

⎛⎜⎜⎜⎜⎜⎝

−λ2 1 0 · · · 0 0−λ3 0 1 · · · 0 0

......

.... . .

......

−λn 0 0 · · · 0 1

⎞⎟⎟⎟⎟⎟⎠

, B = ω(Is, 0)T ,

C = (cij)s×s, D = (dij)(s+1)×(s+1), E = (eij)s×s.

From (2) we have

di,1 = 0 (i = 2, . . . , s + 1),

eij = di+1,j+1 (i, j = 1, 2, . . . , s).

From (3) and (1) we have

−s∑

i=1

λi+1cji = −λj+1e11 + ej1 = −λj+1d22 + dj+1,2 = −λj+1d11,

(j = 1, 2, . . . , s − 1),

−s∑

i=1

λi+1csi = −λj+1e11 = −λj+1d22 = −λj+1d11,

cij = −λi+1e1,j+1 + ei+1,j+1 = −λi+1d2,j+2 + di+2,j+2

= −λi+1d1,j+1 + di+1,j+1, (i, j = 1, 2, . . . , s − 1),

cs,j = −λs+1e1,j+1 = −λs+1d2,j+2 = −λs+1d1,j+1 + ds+1,j+1,

(j = 1, 2, . . . , s − 1),

cis = −λi+1d1,s+1 + di+1,s+1, (i = 1, 2, . . . , s).

If λs+1 �= 0, then

dij = 0 (i > j), dij = di+1,j+1 (i ≤ j and i, j = 1, 2, . . . , s + 1).

If λ2 = λ3 = · · · = λs = 0, then

dij = 0 (i �= j) and dii = di+1,i+1 (i = 1, 2, . . . , s).

If one of λ2, λ3, . . . , λs is non-zero, then Cokerφ is decomposable. This can be easily checked.When φ = (0, λ2, λ3, . . . , λs+1)T , then it is also easy to check that Cokerφ is decomposable.So, the nonzero Hall polynomial for this case is T − 1.Our next work is to calculate the second case. The characteristic matrix equations are

Cω = ωD,

CωJs+1(0) = ωJs+1(0)D.

Recall that these characteristic matrix equations are the same as in the case of A1, so thenonzero Hall polynomial in this case is Tn−1 (n ≥ 1). Consequently, we get the followingtheorem:

396 Xie T.

Theorem 3.5 If Q is the affine quiver of type Am (m ≥ 1), then for any X ∈ P, Y ∈ T (0) orY ∈ T (∞), Z ∈ I , the nonzero Hall polynomials are ϕY

Z,X = T −1 and ϕYZ,X = Tn−1 (n ≥ 1).

Theorem 3.6 If Q = Am (m ≥ 1), then, for any X, Y, Z ∈ T belonging to the same tube,the nonzero Hall polynomial is ϕY

Z,X = 1.

Proof From the fact that any indecomposable regular module M is regular uniserial, M isuniquely determined by its quasi-top and quasi-length. Then the theorem follows from thedefinition of Hall polynomials.

Theorem 3.7 Let Q be the affine quiver of type of Am (m ≥ 1). Then for any X, Y, Z ∈ R

lying in distinct tubes, ϕYZ,X = 0.

Proof This theorem can be easily obtained from the fact that there are no non-zero morphismsbetween two modules in distinct tubes.

Acknowledgements The author is grateful to Bangming Deng for his help.

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