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Counting Representations of Quivers over Finite Fields
Journal of Algebra 226, 1011–1033 (2000) doi:10.1006/jabr.1999.8220, available online at http://www.idealibrary.com on
Counting Representations of Quivers over Finite Fields1 Jiuzhao Hua School of Mathematics, University of New South Wales, Sydney 2052, Australia Communicated by Wolfgang Soergel Received November 2, 1998
dedicated to professor shaoxue liu on the occasion of his 70th birthday
By counting the numbers of isomorphism classes of representations (indecomposable or absolutely indecomposable) of quivers over finite fields with fixed dimension vectors, we obtain a multi-variable formal identity. If the quiver has no edge-loops, this identity turns out to be a q-analogue of the Kac denominator identity modulus a conjecture of Kac. © 2000 Academic Press
1. INTRODUCTION Let 0 be a finite quiver, i.e., a directed graph with finitely many vertices and finitely many arrows which may include edge-loops. In his remarkable papers [5, 6], Kac introduced a root system 1 associated with the underlying graph of 0, and proved that the dimension vectors of the indecomposable representations of 0 over any algebraically closed field are in one-to-one correspondence with the positive roots of 1. This extends some earlier results on quivers of finite type obtained by Gabriel [2] and those of tame type obtained by Donovan and Freislich [1] and independently by Nazarova [10]. Let q be a finite field of q elements, where q is any prime power. For any α ∈ 1+ , we let A0 α; q be the number of isomorphism classes of absolutely indecomposable representations of 0 over q with fixed dimension
1
This research is supported by an ARC Small Grant. 1011 0021-8693/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.
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vector α. It is proved in Kac [6] that A0 α; q is a polynomial in q with integer coefficients, which are independent of the orientation of 0. Kac also conjectured that those coefficients are always non-negative. If 0 is of finite type (i.e., the underlying graph of 0 is one of the graphs An , Dn , E6 , E7 , and E8 ), then A0 α; q = 1 for all α ∈ 1+ . If 0 is of tame type (i.e., the ˜ n , E˜ 6 , E˜ 7 , and E˜ 8 ), then underlying graph of 0 is one of the graphs A˜n , D A0 α; q = 1 if α is a real root, and A0 α; q = q + n0 if α is an imaginary root, where n0 is equal to the number of vertices of 0 minus one. Thus, Kac’s conjecture is true for quivers of finite type and tame type. For wild quivers, it remains unsolved, and the polynomials A0 α; q are hard to compute in general. Nevertheless, Kac [7] proved that A0 α; q is monic of degree 1 − α; α, where ; is the bilinear form of 1. If 0 has no edge-loops, then the underlying graph of 0 defines a symmetric generalized Cartan matrix, and hence defines a Kac–Moody algebra Ç. It turns out that 1 is isomorphic to the root system of Ç. In Kac [6], Kac also conjectured that the constant term of A0 α; q is equal to the multiplicity of α in Ç, which is defined to be the dimension of the root space corresponding to α. Again, this conjecture can be easily verified for quivers of finite type and tame type. For wild type, it is still open. The primary purpose of this paper is to get a formula for A0 α; q. The starting point is quite simple. First, we introduce two other families of polynomials M0 α; q and I0 α; q (precise definitions are given in Section 2). Together with A0 α; q, these three families of polynomials are nicely linked by three identities. We identify M0 α; q with the number of orbits in a finite set with an action of a finite group, and the Burnside orbit counting formula gives M0 α; q. Instead of calculating M0 α; q individually, we consider the generating function of M0 α; q’s, which turns out to have a “nice” factorization (Theorem 4.3). A formula for A0 α; q is obtained in Theorem 4.6, and several samples of A0 α; q’s are given in Section 5. However, the main result of this paper (Theorem 4.9) is a multi-variable formal identity which interprets an infinite sum as an infinite product spreading over 1+ , which picks up the coefficients of various polynomials A0 α; q as exponents. An important application of this identity is that, by letting q → 0, we reduce the second Kac conjecture mentioned above to a combinatorial identity (Corollary 4.10). 2. REPRESENTATIONS OF QUIVERS OVER FINITE FIELDS A (finite) quiver 0 = 00 ; 01 ; s; e consists of two finite sets 00 , 01 , and two maps s; e: 01 → 00 . The elements of 00 are called vertices, those of 01 called arrows. If γ is an arrow such that sγ = i, eγ = j, then γ is γ commonly presented as ◦ −→ ◦. i
j
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Let be the set 0; 1; 2; 3; : : :, and let be any field. Let 0 = 00 ; 01 ; s; e be a fixed quiver with n vertices. Without loss of generality, we may assume that 00 = 1; 2; : : : ; n. A matrix representation M of the quiver 0 over with dimension vector d1 ; d2 ; : : : ; dn ∈ n is a collection Mγ γ∈01 of -matrices such that Mγ has size di × dj if there γ is an arrow ◦ −→ ◦ in 0. If M and N are matrix representations of 0 over i
j
with dimension vectors d1 ; d2 ; : : : ; dn and d10 ; d20 ; : : : ; dn0 , respectively, then a homomorphism from M to N is an n-tuple of -matrices X1 ; : : : ; Xn such that Xi has size di × di0 for all i and, for each arγ row ◦ −→ ◦ in 0, there holds Xi Nγ = Mγ Xj ; i.e., the following diagram i j commutes: Mγ j i ◦ ◦ Xj
Xi
◦ i
Nγ
◦ j
If all Xi are invertible, then M and N are said to be isomorphic over . If M and N are matrix representations of 0 over , then the direct sum of M and N, denoted by M ⊕ N, is defined by Mγ 0 for all γ ∈ 01 : M ⊕ Nγ = 0 Nγ The notion of representation of quivers is fairly standard in the literature; many authors (e.g., Gabriel [2]) prefer to work with vector spaces and linear transformations. It should be noted that the matrix associated with a linear map f : s → t has size s × t if we deal with row vectors, or size t × s if we deal with column vectors. In this paper, the first convention is used. In what follows, “representation” always means “matrix representation” unless otherwise stated. If M is a representation of 0 over , and Ɛ is an extension field of , then M can be naturally regarded as a representation of 0 over Ɛ, which is denoted by M ⊗ Ɛ. M is said to be indecomposable over if it is non-zero and not isomorphic to a direct sum of two non-zero representations of 0 over . M is said to be absolutely indecomposable over , if for any finite extension field Ɛ of , M ⊗ Ɛ remains indecomposable over Ɛ. A classical theorem asserts that a representation of 0 over is indecomposable over if and only if its endomorphism algebra is a local algebra. For any dimension vector α = α1 ; : : : ; αn ∈ n , let R0 α; be the set of all representations of 0 over with dimension vector α. Thus R0 α; is an affine variety defined over . The algebraic group GLα; = GLα1 ; × · · · × GLαn ; acts on R0 α; as follows: if M ∈ R0 α; and X1 ; : : : ; Xn ∈ GLα; , then X1 ; : : : ; Xn · Mγ = Xi−1 Mγ Xj ;
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for any arrow ◦ −→ ◦ in 0. It is obvious that the isomorphism classes of i
j
representations of 0 over with dimension vector α are in one-to-one correspondence with the orbits in R0 α; under the action of GLα; . From now on, we assume that is the finite field q . In this paper, we are mainly interested in three families of functions which are defined as follows: M0 α; q = the number of isomorphism classes of representations of 0 over q with dimension vector α; I0 α; q = the number of isomorphism classes of indecomposable representations of 0 over q with dimension vector α; A0 α; q = the number of isomorphism classes of absolutely indecomposable representations of 0 over q with dimension vector α: It is proved in [6] that for all α ∈ n , M0 α; q, I0 α; q, and A0 α; q are polynomials in q with rational coefficients, which are independent of the orientation of 0. Moreover, all coefficients of A0 α; q are integers. Let 1 be the root system associated with the underlying graph of 0, W the Weyl group of 1 (for detailed constructions, we refer to [6]). We let 1+ be the set of all positive roots of 1. The following fundamental theorem reveals the deep relation between the representations of quivers and their root systems. It is due to Gabriel [2] in finite type, to Donovan and Freislich [1] and independently to Nazarova [10] in tame type, and to Kac [6] in general. Theorem 2.1. Let 0 be a connected quiver, 1 the root system associated with 0, and W the Weyl group of 1. Then the following statements are valid: (1) The polynomials A0 α; q are independent of the orientation of 0. (2) There exists an absolutely indecomposable representation of 0 over q with dimension vector α if and only if α is a positive root in 1; that is, A0 α; q 6= 0 if and only if α ∈ 1+ . (3) A0 α; q = 1 if and only if α is a positive real root. (4) For any w ∈ W , A0 wα; q = A0 α; q provided α and wα are positive roots. If 0 has no edge-loops then the underlying graph of 0 defines a symmetric generalized Cartan matrix A = cij (where cii = 2 and −cij is the number of edges between i and j in 0 if i 6= j). Thus A defines a Kac–Moody algebra ÇA (for a precise definition, we refer to [8, Chap. 1]). It turns out that 1 is the root system of ÇA. Recall that the multiplicity of α is
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the dimension of the root space corresponding to α, which is denoted by multα. In his paper [6], Kac made the following remarkable conjectures: Kac conjecture I. For any α ∈ 1+ , A0 α; q has no negative coefficients. Kac conjecture II. If 0 has no edge-loops then for any α ∈ 1+ the constant term of A0 α; q is equal to the multiplicity of α, i.e., A0 α; 0 = multα. As mentioned before, these conjectures are valid for quivers of finite type and tame type, but remain unsolved for wild type.
3. THE BURNSIDE FORMULA Let 0 be a connected quiver with n vertices 1; 2; : : : ; n. Let bij be the number of arrows from i to j, and aij the number of edges between i and j in the underlying graph of 0. Thus, aii = bii , aij = bij + bji if i 6= j. As already noted in the last section, for each α ∈ n , the finite group GLα; q acts on the finite set R0 α; q and M0 α; q is equal to the number of orbits in R0 α; q . Thus the Burnside orbit counting formula amounts to M0 α; q =
X X Xg 1 Xg = ; GLα; q g∈GLα; Zg g∈Clα; q
q
where Xg = M ∈ R0 α; q g · M = M, the set of representations fixed by g, and Zg = h ∈ GLα; q gh = hg, the centralizer of g in GLα; q , and where Clα; q is a complete set of representatives of the conjugacy classes of GLα; q . Before we can use the Burnside formula to calculate M0 α; q, we have to recall some basic facts about the conjugacy classes of GLm; q . More details can be found in Macdonald’s book [9, p. 138]. Recall that a partition λ = λ1 ; λ2 ; : : : is a finite sequence λ1 ≥ λ2 ≥ · · · of non-negative integers. The integer λ = λ1 + λ2 + · · · is called the weight of λ. In this paper, we use P to denote the set of all partitions including the unique partition of 0. Note that any partition λ can be written in the “exponential form” 1n1 2n2 · · ·, which means that there are exactly ni parts equal to i in λ. Let 8 be the set of all monic irreducible polynomials in the indeterminate t over q with t excluded. For any f ∈ 8, let Jf be the companion matrix of f (for the definition of Jf , we refer to Macdonald [9, p. 140]). For
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each integer r ∈ , we define Jf Id 0 ··· 0 0 Jf Id · · · 0 :: :: :: : : :: ; : Jr f = : : : : 0 0 0 · · · I d 0 0 0 · · · Jf with r diagonal blocks of Jf , where Id means the identity matrix of degree degf . For any f ∈ 8, π = n1 ; n2 ; : : : ∈ P, let Jf; π = Jn1 f ⊕ Jn2 f ⊕ · · · : This is a diagonal block matrix with Jni f (i ≥ 1) in the diagonal. Then any conjugacy class of GLm; q has the Jordan canonical form Jf1 ; π1 ⊕ Jf2 ; π2 ⊕ · · · ⊕ Jfs ; πs ; where f1 ; : : : ; fs are mutually distinct polynomials from P 8, and π1 ; : : : ; πs ∈ P, s is some varying positive integer, which satisfies si=1 degfi πi = m. Now we introduce an “inner product” of the partitions. For λ; µ ∈ P, let λ0 = λ01 ; λ02 ; : : : and µ0 = µ01 ; µ02 ; : :P : be the partitions conjugate to λ and µ, respectively; we define λ; µ = i≥1 λ0i µ0i . Lemma 3.1. For any µ = 1m1 2m2 · · ·; ν = 1n1 2n2 · · · ∈ P, we have XX µ; ν = mini; jmi nj : i≥1 j≥1
Proof. Let µ0 = µ01 ; µ02 ; : : : and ν 0 = νP10 ; ν20 ; : : : be the P partitions conjugate to µ and ν, respectively. Then µ0k = i≥k mi , νk0 = i≥k ni for k ≥ 1. By the definition of µ; ν, we have the following: µ; ν = µ01 ν10 + µ02 ν20 + µ03 ν30 + · · · = m1 + m2 + m3 + · · ·n1 + n2 + n3 + · · · + m2 + m3 + m4 + · · ·n2 + n3 + n4 + · · · + m3 + m4 + m5 + · · ·n3 + n4 + n5 + · · · + · · · ! ! X X X = m1 nj + m2 nj + nj j≥1
+ · · · + mr =
XX i≥1 j≥1
j≥1
X
nj +
j≥1
mini; jmi nj :
X j≥2
j≥2
nj + · · · +
X j≥r
! nj + · · ·
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P Following Macdonald [9], we define nλ = i≥1 i − 1λi for λ = λ1 ; λ2 ; : : : ∈ P. Combining Lemma 1 of Hua [3] and the above lemma, we have the following: Lemma 3.2. For any λ ∈ P, we have λ + 2nλ = λ; λ. Now for any f; g ∈ 8; λ; µ ∈ P, Jf; λ; Jg; µ represents a conjugacy class of the group GLm; q × GLm0 ; q , where m = degf λ and m0 = deggµ. We let XJf;λ;Jg;µ = M ∈ Mm×m0 q Jf; µM = MJg; µ , in which Mm×m0 q means the set of all m × m0 matrices over q . Lemma 3.3. For any f; g ∈ 8; λ; µ ∈ P, we have XJf;λ;Jg;µ = q if f = g; XJf;λ;Jg;µ = 1 otherwise. degf λ;µ
Proof. Suppose λ = 1m1 2m2 · · ·; µ = 1n1 2n2 · · ·, and d = degf ; e = degg. Let A = q x, and A˜ = A ⊗q ¯ q , which is isomorphic to ¯ q x. Given any m × m matrix α over q , we define an A-module structure on qm by setting x · v = αv for v ∈ qm . Let Vα denote this module, and define V˜α = Vα ⊗q ¯ q . By the definition of XJf;λ;Jg;µ , it is easy to see that XJf;λ;Jg;µ = Hom A VJf;λ ; VJg;λ . Note that Hom A VJf;λ ; VJg;λ is a finite dimensional vector space over q , which has the same dimension as Hom A˜ V˜Jf;λ ; V˜Jg;λ over ¯ q . This reduces the calculation to the corresponding calculation over the field ¯ q . Since Jf; µ = J1 f m1 ⊕ J2 f m2 ⊕ m m n · · ·, it follows that V˜Jf;µ ∼ = V˜J1 f1 ⊕ V˜J2 f2 ⊕ · · · : Similarly, V˜Jg;µ ∼ = V˜J1 1g ⊕ n2 V˜J2 g ⊕ · · · : Therefore, nj m Hom A˜ V˜Jf;λ ; V˜Jg;λ ∼ = ⊕di=1 ⊕ej=1 Hom A˜ V˜Ji fi ; V˜Jj g mi nj ∼ : = ⊕di=1 ⊕ej=1 Hom A˜ V˜J f ; V˜J g i
j
The classical theorem that any finite field is separable implies that, for any irreducible monic polynomial f t in q t, there is an invertible matrix X with entries in ¯ q such that XJf X −1 is diagonal with distinct Q diagonal entries ξ1 ; : : : ; ξd . Here f t = di=1 t − ξi . Thus Ji f is similar to Jξ1 ; i ⊕ · · · ⊕ Jξd ; i over ¯ q , where Jξ; i represents the Jordan block matrix of size i with eigenvalue ξ. Similarly, Ji g is similar to Jη1 ; i ⊕ · · · ⊕ Jηe ; i over ¯ q , where η1 ; : : : ; ηe are the roots of g over ¯ q . Therefore, Hom A˜ V˜Ji f ; V˜Jj g ∼ = ⊕ds=1 ⊕et=1 Hom A˜ V˜Jξs ;i ; V˜Jηt ;j :
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˜ Note that every V˜Jδ;i is an indecomposable A-module whose sole ˜ ˜ ˜ composition factor is VJδ;1 . Thus HomA˜ VJξs ;i ; VJηt ;j 6= 0 if and only if ξs = ηt . Also note that f and g have a common root if and only if f = g. Thus if f 6= g, then for all ξs ; ηt (1 ≤ s ≤ d; 1 ≤ t ≤ e), HomA˜ V˜Jξs ;i ; V˜Jηt ;j = 0, and so HomA˜ V˜Jf;λ ; V˜Jg;µ = 0; which implies that XJf;λ;Jg;µ = 1. Now, we suppose that f = g. Thus, HomA˜ V˜Ji f ; V˜Jj g ∼ = ⊕ds=1 HomA˜ V˜Jξs ;i ; V˜Jξs ;j : An elementary calculation with matrices reveals that dim¯ Hom A˜ V˜Jξ ;i ; V˜Jξ ;j = mini; j: s
q
s
Thus, we have dim q XJf;λ;Jg;µ = dim ¯ q Hom A˜ V˜Jf;λ ; V˜Jg;µ XX = mi nj dim ¯ q Hom A˜ V˜Ji f ; V˜Jj g i≥1 j≥1
=
XX
mi nj mini; jd
i≥1 j≥1
= dλ; µ
by Lemma 3:1:
Finally, we have XJf;λ;Jg;µ = qdλ;µ . Note that this formula depends only on the degrees of f and g. For any non-negative integer r, let ϕr q = 1 − q1 − q2 · · · 1 − qr , with convention ϕ0 q = 1. Following Macdonald [9], we let bλ q = Q n1 n2 i≥1 ϕni q for λ = 1 2 · · · ∈ P. Theorem 3.4. Let g = Jf; π1 ; : : : ; Jf; πn , where f ∈ 8 with degf = d, and π1 ; : : : ; πn ∈ P. Then g represents a conjugacy class of the group GLα; q ; here α = dπ1 ; : : : ; dπn , and we have Zg =
n Y i=1
qdπi ;πi bπi q−d
and
Xg =
Y 1≤i≤j≤n
qdaij πi ;πj :
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Proof. Note that similar formulas appeared in [6, p. 90]. For each i, 1 ≤ i ≤ n, ZJf;πi is the number of centralizers of Jf; πi in GLmi ; q , where mi = dπi . Formula (2.6) in [9, p. 139] implies that ZJf;πi = qdπi +2nπi bπi q−d . So Lemma 3.2 implies that Q ZJf;πi = qdπi ;πi bπi q−d . As Zg = ni=1 ZJf;πi , we have Zg =
n Y
qdπi ;πi bπi q−d :
i=1
It is easy to see that Xg = previous lemma that
Qn Qn i=1
Xg =
j=1
XJf;πi ;Jf;πj bij . It follows from the
n n Y Y
qdbij πi ;πj :
i=1 j=1
Since λ; µ = µ; λ for any λ; µ ∈ P, and aii = bii ; aij = bij + bji if i 6= j, we have Y qdaij πi ;πj : Xg = 1≤i≤j≤n
Note that Xg is independent of the orientation of 0. 4. GENERATING FUNCTIONS AND FORMAL IDENTITIES Let 0 be as before, and let X1 ; : : : ; Xn be independent commuting indeterminates. For any α = α1 ; : : : ; αn ∈ n , let X α be the monomial α α X1 1 · · · Xn n . The Krull–Schmidt theorem states that every representation of 0 over a field can be written as a direct sum of indecomposable representations in a unique way up to a permutation of the summands. This implies the following formal identity: Y X M0 α; qX α = 1 − X α −I0 α;q : (4.1) α∈n
α∈n \0
For any α = α1 ; : : : ; αn ∈ n \0, we let α¯ = g:c:d:α1 ; : : : ; αn , the greatest common divisor of α1 ; : : : ; αn . Theorem 4.1. For any α ∈ n \0, the formulas α X 1X d I0 α; q = µ A0 ; qr ; d r d r d d α¯ A0 α; q =
α X 1X µrI0 ; qr d r d d d α¯
are valid, where µ is the classical M¨ obius function.
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Proof. The first formula appeared in [6], and the second one is the M¨ obius inverse of the first. Corollary 4.2. For any α ∈ n \0, we have the following identity: α X 1 α X 1 I0 ;q = A0 ; qd : d d d d d α¯ d α¯ Now we define P0 X1 ; : : : ; Xn ; q =
Q
aij πi ;πj 1≤i≤j≤n q π X1 1 Q πi ;πi −1 q b q πi 1≤i≤n π1 ;:::;πn ∈P
X
π
· · · Xn n :
Thus, P0 X1 ; : : : ; Xn ; q is a formal power series in X1 ; : : : ; Xn with coefficients in q, and the constant term is 1. Note that P0 X1 ; : : : ; Xn ; q is independent of the orientation of 0. Theorem 4.3. With the notation as above, the formal identity X α∈n
M0 α; qX α =
∞ Y
φd q P0 X1d ; · · · ; Xnd ; qd
d=1
holds, where φ1 q = q − 1, and for d ≥ 2, φd q is the number of monic irreducible polynomials of degree d over q . Proof. It follows from the Burnside formula that, for any α ∈ n , we have M0 α; q =
X
Xg
g∈Clα;q
Zg
;
where Xg , Zg , and Clα; q are defined in Section 3. Thus, we have X X X X g X α: M0 α; qX α = Zg α∈n α∈n g∈Clα; q
For f ∈ 8, we let df denote the degree of f . The discussion of Section 3 shows that, for a given α = α1 ; : : : ; αn ∈ n , each conjugacy class of GLα; q is a direct sum of finitely many elements of the form Jfi ; π1i ; Jfi ; π2i ; : : : ; Jfi ; πni , i ≥ 1, where fi ∈ 8; πki ∈ P, which satisfy the following conditions: (1) the fi ’s, i ≥ 1, are mutually distinct; P (2) for a fixed k, 1 ≤ k ≤ n, i≥1 dfi πki = αk .
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It follows from the proof of Lemma 3.3 that, for any f; g ∈ 8 with f 6= g, and for any λi ; µi ∈ P (i = 1; : : : ; n), the following holds: XJf;λ1 ⊕Jg;µ1 ;:::;Jf;λn ⊕Jg;µn = XJf;λ1 ;:::;Jf;λn · XJg;µ1 ;:::;Jg;µn : Therefore, we have the following factorization: X X Xg Xα Z g α∈n g∈Clα; q
Y
X
XJf;π1 ;:::;Jf;πn
f ∈8
π1 ;:::;πn ∈P
ZJf;π1 ;:::;Jf;πn
=
df π1 X1
! df πn : : : Xn
:
Note that, in the above product, each factor is a formal power series with constant term 1. Thus the above product does make sense. It follows from Theorem 3.4 that ! XJf;π1 ;:::;Jf;πn df π1 Y X df πn X · · · Xn ZJf;π1 ;:::;Jf;πn 1 f ∈8 π1 ;:::;πn ∈P ! Q df aij πi ;πj Y X df π1 df πn 1≤i≤j≤n q X1 = · · · Xn Q df πi ;πi bπi q−df 1≤i≤n q f ∈8 π1 ;:::;πn ∈P Y df df P0 X1 ; : : : ; Xn ; qdf = f ∈8
=
∞ Y
φd q P0 X1d ; : : : ; Xnd ; qd :
d=1
The above theorem was obtained independently by Richard Stanley using different notation in an unpublished note. For any α ∈ n \0, we let E0 α; q and H0 α; q be the rational functions in q determined by ! X X α log M0 α; qX = E0 α; qX α /α; ¯ α∈n
log P0 X1 ; : : : ; Xn ; q =
α∈n \0
X
H0 α; qX α /α; ¯
α∈ \0 n
where the “log” means the formal logarithm, i.e., log1 − X = It follows from Theorem 4.3 that α X dφd qH0 ; qd : E0 α; q = d d α¯
P
i≥1 −X
i
/i.
(4.2)
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Lemma 4.4. With notations as above, for any α ∈ n \0, we have α 1 X µdE0 ;q : I0 α; q = α¯ d α¯ d Proof. It follows from identity (4.1) and the definition of E0 α; q that ! X Y E0 α; qX α /α¯ = log 1 − X α −I0 α;q α∈n \0
α∈n \0
=
X
I0 α; q log
α∈n \0
=
X
I0 α; q
α∈ \0 n
=
X
1 1 − Xα
∞ X 1 i=1
i
X iα
∞ X
1 I0 α; qX iα : i n α∈ \0 i=1
By comparing the coefficients of X α on both sides, we have X 1 α 1 E0 α; q = I ;q : α¯ d 0 d d α¯ Thus, E0 α; q =
X α¯ α I ;q : d 0 d d α¯
The M¨ obius inverse of this shows that α 1 X I0 α; q = µdE0 ;q : α¯ d α¯ d
Lemma 4.5. For any α ∈ n \0, we have A0 α; q =
α 1 X µdE0 ; qd : α¯ d α¯ d
Proof. First let us recall two basic techniques on handling sums involving divisors. Suppose that f : → and g: × → are two functions. Then the following hold: X X f n = f n/d; (T1) dn
dn
quivers over finite fields X X
gd; r =
d n r n/d
X X
gd; r:
1023 (T2)
r n d n/r
Now we have the following α X 1X A0 α; q = µrI0 ; qr d r d d d α¯
by Theorem 4:1
α X 1X d X ; qr µr µsE0 by Lemma 4:4 d r d α¯ s α/d ds d α¯ ¯ sα X 1X d X α¯ = µr µ E0 ; qr by T1 d r d α¯ s α/d ds α¯ d α¯ ¯ α¯ sα r 1 XX X = µrµ E0 ;q α¯ d α¯ r d s α/d ds α¯ ¯ sα 1 X X X d = µrµ E0 ; qr by T1 α¯ d α¯ r α/d s α¯ ¯ sd sα r d 1X X X µrµ by T2 E0 ;q = α¯ r α¯ d α/r s α¯ ¯ sd 1X X X α¯ sα r = µrµ E0 ;q by T1 α¯ r α¯ d α/r dsr α¯ ¯ s α/dr ¯ sα 1X X X α¯ = µrµ E0 ; qr by T2 α¯ r α¯ s α/r dsr α¯ ¯ d α/sr ¯ sα X 1X X α¯ r = µrE0 ;q µ : α¯ r α¯ s α/r α¯ dsr ¯ d α/sr ¯ =
Note that the M¨ obius function µ has the following property: X 1 if n = 1; µd = 0 otherwise: dn
Thus we have A0 α; q =
α 1X µrE0 ; qr : α¯ r α¯ r
Theorem 4.6. For any α ∈ n \0, the following identity holds: α q−1 X µdH0 ; qd : A0 α; q = α¯ d α¯ d
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Proof. In fact, we have A0 α; q =
α 1 X ; qd µdE0 α¯ d α¯ d
by Lemma 4:5
=
α X 1 X µd rφr qd H0 ; qdr α¯ d α¯ dr r α/d ¯
=
rα X α¯ 1 X d α/r ¯ φα/dr ; q µd q H 0 α¯ d α¯ dr ¯ α¯ r α/d ¯
=
X X
µd
d α¯ r α/d ¯
by 4:2 by T1
rα 1 ¯ φα/dr qd H0 ; qα/r ¯ dr α¯
rα 1 d α/r ¯ φα/dr ; q q H by T2 0 dr ¯ α¯ r α¯ d α/r ¯ α XX r = µd φr/d qd H0 ; qr by T1 d α¯ r r α¯ d r
=
=
X X
µd
α X r 1X H0 ; qr µd φr/d qd : α¯ r α¯ r d dr
Note that φd q is the number of polynomials in 8 which have degree d. The following is a classical result modified in case n = 1: 1X φn q = µd qn/d − 1 : n dn The M¨ obius inverse of this amounts to the following: Xn µdφn/d qd = µnq − 1: d dn Thus, we have A0 α; q =
α q−1 X µrH0 ; qr : α¯ r α¯ r
As the H0 α; q’s are rational functions in q, so are the A0 α; q’s by the above theorem. As each A0 α; q takes integer value when q is any prime power, A0 α; q must be a polynomial in q with rational coefficients. Thus, Theorem 4.1 and identity (4.1) imply that the I0 α; q’s and M0 α; q’s are all polynomials in q with rational coefficients. Since the formal power series P0 X1 ; : : : ; Xn ; q is independent of the orientation of 0, the functions
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E0 α; q and H0 α; q, and hence the polynomials A0 α; q, I0 α; q, and M0 α; q, are all independent of the orientation of 0. It is proved in [6] that all coefficients of A0 α; q are integers. An alternativePproof of this uα α j tj q with is given in [4]. In what follows, we assume that A0 α; q = j=0 α tj ∈ . Theorem 4.7. The following formal identity holds: X
uα Y Y
M0 α; qX α =
1 − qj X α −tj : α
α∈1+ j=0
α∈n
Proof. The M¨ obius inverse of Lemma 4.5 shows that α X α¯ E0 α; q = A0 ; qd : d d d α¯ By the definition of E0 α; q, we have ! X X α M0 α; qX log =
E0 α; qX α /α¯
α∈ \0
α∈n
n
= =
α X 1 A0 ; qd X α d d α∈n \0 d α¯ X
∞ X 1
X
α∈n \0 i=1
i
A0 α; qi X iα :
By Theorem 2.1, A0 α; q 6= 0 if and only if α ∈ 1+ . Thus, ! ∞ X X X 1 log M0 α; qX α = A0 α; qi X iα i α∈n α∈1+ i=1 = =
uα ∞ X X 1X tjα qij X iα i j=0 α∈1+ i=1 uα X X α∈1+ j=0
=
X X uα
α∈1+ j=0
= log
tjα
∞ X 1 i=1
qij X iα
tjα log 1 − qj X α
uα Y Y α∈1+ j=0
This implies the theorem.
i
1 − qj X α
−1
−tjα
:
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The above theorem implies that M0 α; q ∈ q. Furthermore, if the first Kac conjecture is true, i.e., tjα ≥ 0 for all α ∈ 1+ and j ∈ , then each M0 α; q has no negative coefficients either. Lemma 4.8. The following formal identity is valid: ∞ ∞ X Y 1 Xi = log 1 − qi X: i i q − 1 i=0 i=1
Proof. In fact, we have log
∞ Y
1 − qi X =
i=0
∞ X
log1 − qi X
i=0
=−
∞ X ∞ X 1 i j q X j i=0 j=1
=−
∞ ∞ X 1 jX X qij j i=0 j=1
=
∞ X 1 Xj : j j j=1 q − 1
Now we are able to prove the main result of this paper. Theorem 4.9. The following formal identity holds: Q aij πi ;πj uα ∞ Y X Y Y t α 1≤i≤j≤n q π1 πn X · · · X = 1 − qi+j X α j : Q n 1 πi ;πi −1 bπi q 1≤i≤n q π1 ;:::;πn ∈P α∈1+ i=0 j=0 Proof. The M¨ obius inverse of Theorem 4.6 implies that α X α¯ 1 H0 α; q = A0 ; qd : d dq −1 d d α¯ Thus, according to the definition of H0 α; q, we have X H0 α; qX α /α¯ log P0 X1 ; : : : ; Xn ; q = α∈n \0
= =
α X 1 1 d A ; q Xα 0 d d d q − 1 n α∈ \0 d α¯ X
X
∞ X
1 A0 α; qi X iα : iq − 1 n α∈ \0 i=1 i
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By Theorem 2.1, A0 α; q 6= 0 if and only if α ∈ 1+ . Thus, log P0 X1 ; : : : ; Xn ; q = = =
∞ X X
1 A0 α; qi X iα i iq − 1 α∈1+ i=1 ∞ X X
uα X 1 tjα qij X iα i + i=1 iq − 1 j=0 α∈1 uα X X α∈1+ j=0
=
uα X X +
α∈1 j=0
= log
tjα tjα
∞ X qj X α i i i=1 iq − 1 ∞ X
log1 − qi qj X α
i=0
uα ∞ Y Y Y
1−q
i+j
X
α α tj
by Lemma 4:8 ! :
α∈1+ i=0 j=0
This implies that P0 X1 ; : : : ; Xn ; q =
uα ∞ Y Y Y
1 − qi+j X α
tjα
:
α∈1+ i=0 j=0
The theorem now follows from the definition of P0 X1 ; : : : ; Xn ; q. As noted before, if 0 has no edge-loops, then 0 defines a Kac-Moody algebra Ç with root system 1 and Weyl group W . Recall that the Kac denominator identity of Ç can be written in the form (see [5, p. 65]) X Y −1lw X sw = 1 − X α multα ; w∈W
α∈1+
where lw is the length of w and sw is the sum of positive roots mapped into negative roots by w−1 . Now let q approach 0 in Theorem 4.9; we have ! Q aij πi ;πj X Y 1≤i≤j≤n q π1 πn lim X1 · · · Xn = 1 − X α A0 α;0 : Q π ;π −1 i i q→0 bπi q + 1≤i≤n q π1 ;:::;πn ∈P α∈1 If Kac conjecture II is true, i.e., A0 α; 0 = multα for all α ∈ 1+ , then we have Y Y 1 − X α A0 α;0 = 1 − X α multα : α∈1+
α∈1+
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Thus, Theorem 4.9 immediately implies the following Corollary 4.10. The second Kac conjecture is true if and only if the following identity holds:
lim
q→0
Q
aij πi ;πj 1≤i≤j≤n q π X1 1 Q πi ;πi −1 bπi q 1≤i≤n q π1 ;:::;πn ∈P
X
! π · · · Xn n
X
=
−1lw X sw :
w∈W
We remark that another equivalent form of the second Kac conjecture, also stated in combinatorial identities, is obtained by Sevenhant and Van den Bergh [11]. Their form is essentially equivalent to the form given above.
5. EXAMPLES Theorem 4.6 provides a practical way to compute the polynomials A0 α; q. The following examples were carried out by Maple based on Theorem 4.6; more examples can be found in [4, Appendix A]. As the A0 α; q’s, M0 α; q’s, and I0 α; q’s are all independent of the orientation of 0, we can just refer to 0 as a graph. If 0 is the hyperbolic graph ◦ ≡≡≡≡ ◦, then we have the following: 2
1
A0 1; 1; q = q2 + q + 1 A0 2; 3; q = q6 + q5 + 3q4 + 4q3 + 5q2 + 3q + 2 A0 3; 6; q = q10 + q9 + 3q8 + 5q7 + 8q6 + 12q5 + 17q4 + 16q3 + 14q2 + 7q + 3 I0 1; 1; q = q2 + q + 1 I0 2; 3; q = q6 + q5 + 3q4 + 4q3 + 5q2 + 3q + 2 I0 3; 6; q = q10 + q9 + 3q8 + 5q7 + +
49 3 q 3
+
41 2 q 3
+
20 q 3
25 6 q 3
+ 12q5 + 17q4
+3
M0 1; 1; q = q2 + q + 2 M0 2; 3; q = q6 + 2q5 + 6q4 + 10q3 + 16q2 + 12q + 10 M0 3; 6; q = 2q12 + 2q11 + 8q10 + 12q9 + 26q8 + 44q7 + 83q6 + 124q5 + 180q4 + 194q3 + 182q2 + 104q + 54:
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If 0 is the hyperbolic graph ◦ ==== ◦−−−−◦, then we have the following: 1
2
3
A0 2; 2; 0; q = q + 1 A0 3; 3; 3; q = q + 1 A0 4; 4; 2; q = q5 + 2q4 + 6q3 + 10q2 + 13q + 7 I0 2; 2; 0; q = 21 q2 + 21 q + 1 I0 3; 3; 3; q = 13 q3 + 23 q + 1 I0 4; 4; 2; q = q5 + 25 q4 + 6q3 +
21 2 q 2
+ 12q + 7
M0 2; 2; 0; q = q + 3q + 6 2
M0 3; 3; 3; q = 8q3 + 32q2 + 80q + 88 M0 4; 4; 2; q = q5 + 13q4 + 56q3 + 170q2 + 287q + 240: If 0 is the wild quiver •, then the following hold: A0 1; q = q2 A0 2; q = q5 + q3 A0 3; q = q10 + q8 + q7 + q6 + q5 + q4 A0 4; q = q17 + q15 + q14 + 2q13 + q12 + 3q11 + 2q10 + 4q9 + 2q8 + 3q7 + q6 + q5 I0 1; q = q2 I0 2; q = q5 + 21 q4 + q3 − 21 q2 I0 3; q = q10 + q8 + q7 + 43 q6 + q5 + q4 − 13 q2 I0 4; q = q17 + q15 + q14 + 2q13 + q12 + 3q11 + 25 q10 + 4q9 + 94 q8 + 3q7 + 23 q6 + 21 q5 − 14 q4 − 21 q3 M0 1; q = q2 M0 2; q = q5 + q4 + q3 M0 3; q = q10 + q8 + 2q7 + 2q6 + 2q5 + q4 M0 4; q = q17 + q15 + q14 + 2q13 + 2q12 + 3q11 + 4q10 + 6q9 + 5q8 + 5q7 + 3q6 + q5 : All of the above polynomials are consistent with Kac’s conjectures. These examples show that I0 α; q may have fractional coefficients, and also suggest the following:
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Conjecture. If 0 has no edge-loops, then the coefficients of the polynomial I0 α; q (α ∈ 1+ ) are all non-negative. It is worth noticing that the above conjecture can be easily verified for quivers of finite type and tame type, see examples below. In the rest of this section, we specialize Theorems 4.7 and 4.9 to several graphs of finite type or tame type, as their representations have been completely classified; this would generate some explicit identities which seem to be new. Example I (the trivial graph A1 ). A1 has a single vertex but no edges. A representation of A1 is just a vector space. Thus every representation of dimension bigger than one is decomposable. Consequently, AA1 1; q = 1 and AA1 n; q = 0 for n > 1. Thus Theorem 4.9 implies the following identity: X λ∈P
∞ Y X λ = 1 − qi X: qλ;λ bλ q−1 i=0
(5.1)
Note that the Euler identity shows that ∞ X
∞ Y qii−1/2 X i = 1 + qi X: 2 i i=0 1 − q1 − q · · · 1 − q i=0
Upon the Euler identity, (5.1) implies the following: X qλ;λ qn : = 1 − q1 − q2 · · · 1 − qn λ`n bλ q With the above formula, one can easily show that the number of nilpotent n × n matrices over q of is equal to qnn−1 . Example II (the Jordan graph A˜0 ). A˜0 consists of a single node plus a single loop. It follows from the Jordan normal form theorem that AA˜0 n; q = q for n ≥ 1. Thus Theorem 4.9 shows that X λ∈P
∞ ∞ Y Y X λ = 1 − qi X j ; bλ q−1 i=1 j=1
(5.2)
while Theorem 4.7 shows that 1+
∞ X
MA˜0 i; qX i =
i=1
∞ Y
1 − qX i −1 :
i=1
It follows that MA˜0 n; q =
X λ`n
qlλ ;
(5.3)
quivers over finite fields
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P where lλ is the length of λ, i.e., lλ = i≥0 ni if λ = 1n1 2n2 · · · ∈ P. This has already been proved in [3]. Now, for any positive integer n, we let Gn; q be the polynomial 1 X n d Gn; q = φ q ; n dn d where φn is the classical Euler function; i.e., φn is the number of numbers up to n which are relatively prime to n. An easy exercise shows the following: ∞ ∞ Y Y 1 − qX i −1 = 1 − X i −Gi;q : (5.4) i=1
i=1
Now identities (4.1), (5.3), and (5.4) imply that 1 X n d IA˜0 n; q = φ q : n dn d Thus, the IA˜0 n; q’s are polynomials in q with non-negative rational coefficients. Example III (the graph A2 ). A2 is the graph ◦−−−−◦. The root system associated with A2 has three positive roots 1; 0, 0; 1, and 1; 1. It follows from Theorem 4.7 that ∞ X ∞ X MA2 m; n; qX m Y n = 1 − X−1 1 − Y −1 1 − XY −1 : m=0 n=0
This implies that MA2 m; n; q = minm; n. Theorem 4.9 implies that ∞ Y qλ;µ X λ Y µ = 1 − qi X1 − qi Y 1 − qi XY : qλ;λ+µ;µ bλ q−1 bµ q−1 i=0
X λ;µ∈P
Since, for any quiver of finite type, every indecomposable representation over q is absolutely indecomposable, we have I0 α; q = A0 α; q = 1 for all α ∈ 1+ . Example IV (the Kronecker graph A˜1 ). A˜1 is the graph ◦ ==== ◦, which is of tame type. The real roots of the root system associated with A˜1 are of the forms i; i + 1 and i; i − 1, i ∈ , while the imaginary roots are of the form i; i, i ∈ and i 6= 0. It follows from the classification theorem of representations of tame quivers that AA˜1 i; i = q + 1. Thus Theorem 4.7 amounts to ∞ X ∞ X MA˜1 i; j; qX i Y j i=0 j=0
=
∞ Y
(5.5)
1 − X Y
j=1
j
j−1 −1
1 − X
j−1
j −1
j −1
j −1
Y 1 − X Y 1 − qX Y ; j
j
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while Theorem 4.9 shows that X
q2λ;µ X λ Y µ qλ;λ+µ;µ bλ q−1 bµ q−1
λ;µ∈P
=
∞ ∞ Y Y
1 − qi−1 X j Y j−1 1 − qi−1 X j−1 Y j 1 − qi−1 X j Y j 1 − qi X j Y j :
i=1 j=1
(5.6) Let q → 0 in the last identity; then the right hand side appears in the following Jacobi triple product identity: ∞ X Y −1i X ii−1/2 Y ii+1/2 = 1 − X i Y i−1 1 − X i−1 Y i 1 − X i Y i : i∈
i=1
Thus the following holds: X
lim
q→0
λ;µ∈P
X q2λ;µ X λ Y µ = −1i X ii−1/2 Y ii+1/2 : λ;λ+µ;µ −1 −1 q bλ q bµ q i∈
And so identity (5.6) can be considered as a q-analogue of the Jacobi triple product identity. Now identities (4.1), (5.4), and (5.5) imply that Y 1 − X α −IA˜1 α;q α∈1+
=
∞ Y
1 − X j Y j−1 −1 1 − X j−1 Y j −1 1 − X j Y j −1 1 − qX j Y j −1
j=1
=
∞ Y
1 − X j Y j−1 −1 1 − X j−1 Y j −1 1 − X j Y j −Gj;q−1 :
j=1
It follows that
1; Gm; q + 1; IA˜1 m; n; q = 0;
if m − n = 1; if m − n = 0; if m − n > 1:
Thus, the IA˜1 m; n; q’s are polynomials in q with non-negative rational coefficients. Similar identities could be written down for any graph of finite type or tame type, this is left to the readers. ACKNOWLEDGMENTS This paper is based on the author’s Ph.D. thesis done at the University of New South Wales. The author thanks his supervisor, Dr. Peter Donovan, for valuable guidance. He also thanks the referee for many helpful comments and suggestions.
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REFERENCES 1. P. W. Donovan and M. R. Freislich, The representation theory of finite graphs and associated algebras, Carleton Math. Lecture Notes 5 (1973). 2. P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71–103. 3. J. Hua, Generalizing the recursion relationship for the partition function, J. Combin. Theory Ser. A 79 (1997), 105–117. 4. J. Hua, “Representations of Quivers over Finite Fields,” Ph.D. thesis, University of New South Wales, 1998. 5. V. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57–92. 6. V. Kac, “Root Systems, Representations of Quivers and Invariant Theory,” Lecture Notes in Mathematics 996, pp. 74–108, Springer-Verlag, Berlin/New York. 7. V. Kac, Infinite root systems, representations of graphs and invariant theory, II, J. Algebra 78 (1982), 141–162. 8. V. Kac, “Infinite Dimensional Lie Algebras,” Progress in Math., Vol. 44, Birkh¨auser, Boston, 1983. 9. I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” Clarendon, Oxford, 1979. 10. L. A. Nazarova, Representations of quivers of infinite type, Math. USSR Izv. Ser. Mat. 37 (1973), 752–791. 11. B. Sevenhant and M. Van den Bergh, “On the Numbers of Absolutely Indecomposable Representations of Quivers,” preprint.
Counting Representations of Quivers over Finite Fields1 Jiuzhao Hua School of Mathematics, University of New South Wales, Sydney 2052, Australia Communicated by Wolfgang Soergel Received November 2, 1998
dedicated to professor shaoxue liu on the occasion of his 70th birthday
By counting the numbers of isomorphism classes of representations (indecomposable or absolutely indecomposable) of quivers over finite fields with fixed dimension vectors, we obtain a multi-variable formal identity. If the quiver has no edge-loops, this identity turns out to be a q-analogue of the Kac denominator identity modulus a conjecture of Kac. © 2000 Academic Press
1. INTRODUCTION Let 0 be a finite quiver, i.e., a directed graph with finitely many vertices and finitely many arrows which may include edge-loops. In his remarkable papers [5, 6], Kac introduced a root system 1 associated with the underlying graph of 0, and proved that the dimension vectors of the indecomposable representations of 0 over any algebraically closed field are in one-to-one correspondence with the positive roots of 1. This extends some earlier results on quivers of finite type obtained by Gabriel [2] and those of tame type obtained by Donovan and Freislich [1] and independently by Nazarova [10]. Let q be a finite field of q elements, where q is any prime power. For any α ∈ 1+ , we let A0 α; q be the number of isomorphism classes of absolutely indecomposable representations of 0 over q with fixed dimension
1
This research is supported by an ARC Small Grant. 1011 0021-8693/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.
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vector α. It is proved in Kac [6] that A0 α; q is a polynomial in q with integer coefficients, which are independent of the orientation of 0. Kac also conjectured that those coefficients are always non-negative. If 0 is of finite type (i.e., the underlying graph of 0 is one of the graphs An , Dn , E6 , E7 , and E8 ), then A0 α; q = 1 for all α ∈ 1+ . If 0 is of tame type (i.e., the ˜ n , E˜ 6 , E˜ 7 , and E˜ 8 ), then underlying graph of 0 is one of the graphs A˜n , D A0 α; q = 1 if α is a real root, and A0 α; q = q + n0 if α is an imaginary root, where n0 is equal to the number of vertices of 0 minus one. Thus, Kac’s conjecture is true for quivers of finite type and tame type. For wild quivers, it remains unsolved, and the polynomials A0 α; q are hard to compute in general. Nevertheless, Kac [7] proved that A0 α; q is monic of degree 1 − α; α, where ; is the bilinear form of 1. If 0 has no edge-loops, then the underlying graph of 0 defines a symmetric generalized Cartan matrix, and hence defines a Kac–Moody algebra Ç. It turns out that 1 is isomorphic to the root system of Ç. In Kac [6], Kac also conjectured that the constant term of A0 α; q is equal to the multiplicity of α in Ç, which is defined to be the dimension of the root space corresponding to α. Again, this conjecture can be easily verified for quivers of finite type and tame type. For wild type, it is still open. The primary purpose of this paper is to get a formula for A0 α; q. The starting point is quite simple. First, we introduce two other families of polynomials M0 α; q and I0 α; q (precise definitions are given in Section 2). Together with A0 α; q, these three families of polynomials are nicely linked by three identities. We identify M0 α; q with the number of orbits in a finite set with an action of a finite group, and the Burnside orbit counting formula gives M0 α; q. Instead of calculating M0 α; q individually, we consider the generating function of M0 α; q’s, which turns out to have a “nice” factorization (Theorem 4.3). A formula for A0 α; q is obtained in Theorem 4.6, and several samples of A0 α; q’s are given in Section 5. However, the main result of this paper (Theorem 4.9) is a multi-variable formal identity which interprets an infinite sum as an infinite product spreading over 1+ , which picks up the coefficients of various polynomials A0 α; q as exponents. An important application of this identity is that, by letting q → 0, we reduce the second Kac conjecture mentioned above to a combinatorial identity (Corollary 4.10). 2. REPRESENTATIONS OF QUIVERS OVER FINITE FIELDS A (finite) quiver 0 = 00 ; 01 ; s; e consists of two finite sets 00 , 01 , and two maps s; e: 01 → 00 . The elements of 00 are called vertices, those of 01 called arrows. If γ is an arrow such that sγ = i, eγ = j, then γ is γ commonly presented as ◦ −→ ◦. i
j
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1013
Let be the set 0; 1; 2; 3; : : :, and let be any field. Let 0 = 00 ; 01 ; s; e be a fixed quiver with n vertices. Without loss of generality, we may assume that 00 = 1; 2; : : : ; n. A matrix representation M of the quiver 0 over with dimension vector d1 ; d2 ; : : : ; dn ∈ n is a collection Mγ γ∈01 of -matrices such that Mγ has size di × dj if there γ is an arrow ◦ −→ ◦ in 0. If M and N are matrix representations of 0 over i
j
with dimension vectors d1 ; d2 ; : : : ; dn and d10 ; d20 ; : : : ; dn0 , respectively, then a homomorphism from M to N is an n-tuple of -matrices X1 ; : : : ; Xn such that Xi has size di × di0 for all i and, for each arγ row ◦ −→ ◦ in 0, there holds Xi Nγ = Mγ Xj ; i.e., the following diagram i j commutes: Mγ j i ◦ ◦ Xj
Xi
◦ i
Nγ
◦ j
If all Xi are invertible, then M and N are said to be isomorphic over . If M and N are matrix representations of 0 over , then the direct sum of M and N, denoted by M ⊕ N, is defined by Mγ 0 for all γ ∈ 01 : M ⊕ Nγ = 0 Nγ The notion of representation of quivers is fairly standard in the literature; many authors (e.g., Gabriel [2]) prefer to work with vector spaces and linear transformations. It should be noted that the matrix associated with a linear map f : s → t has size s × t if we deal with row vectors, or size t × s if we deal with column vectors. In this paper, the first convention is used. In what follows, “representation” always means “matrix representation” unless otherwise stated. If M is a representation of 0 over , and Ɛ is an extension field of , then M can be naturally regarded as a representation of 0 over Ɛ, which is denoted by M ⊗ Ɛ. M is said to be indecomposable over if it is non-zero and not isomorphic to a direct sum of two non-zero representations of 0 over . M is said to be absolutely indecomposable over , if for any finite extension field Ɛ of , M ⊗ Ɛ remains indecomposable over Ɛ. A classical theorem asserts that a representation of 0 over is indecomposable over if and only if its endomorphism algebra is a local algebra. For any dimension vector α = α1 ; : : : ; αn ∈ n , let R0 α; be the set of all representations of 0 over with dimension vector α. Thus R0 α; is an affine variety defined over . The algebraic group GLα; = GLα1 ; × · · · × GLαn ; acts on R0 α; as follows: if M ∈ R0 α; and X1 ; : : : ; Xn ∈ GLα; , then X1 ; : : : ; Xn · Mγ = Xi−1 Mγ Xj ;
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for any arrow ◦ −→ ◦ in 0. It is obvious that the isomorphism classes of i
j
representations of 0 over with dimension vector α are in one-to-one correspondence with the orbits in R0 α; under the action of GLα; . From now on, we assume that is the finite field q . In this paper, we are mainly interested in three families of functions which are defined as follows: M0 α; q = the number of isomorphism classes of representations of 0 over q with dimension vector α; I0 α; q = the number of isomorphism classes of indecomposable representations of 0 over q with dimension vector α; A0 α; q = the number of isomorphism classes of absolutely indecomposable representations of 0 over q with dimension vector α: It is proved in [6] that for all α ∈ n , M0 α; q, I0 α; q, and A0 α; q are polynomials in q with rational coefficients, which are independent of the orientation of 0. Moreover, all coefficients of A0 α; q are integers. Let 1 be the root system associated with the underlying graph of 0, W the Weyl group of 1 (for detailed constructions, we refer to [6]). We let 1+ be the set of all positive roots of 1. The following fundamental theorem reveals the deep relation between the representations of quivers and their root systems. It is due to Gabriel [2] in finite type, to Donovan and Freislich [1] and independently to Nazarova [10] in tame type, and to Kac [6] in general. Theorem 2.1. Let 0 be a connected quiver, 1 the root system associated with 0, and W the Weyl group of 1. Then the following statements are valid: (1) The polynomials A0 α; q are independent of the orientation of 0. (2) There exists an absolutely indecomposable representation of 0 over q with dimension vector α if and only if α is a positive root in 1; that is, A0 α; q 6= 0 if and only if α ∈ 1+ . (3) A0 α; q = 1 if and only if α is a positive real root. (4) For any w ∈ W , A0 wα; q = A0 α; q provided α and wα are positive roots. If 0 has no edge-loops then the underlying graph of 0 defines a symmetric generalized Cartan matrix A = cij (where cii = 2 and −cij is the number of edges between i and j in 0 if i 6= j). Thus A defines a Kac–Moody algebra ÇA (for a precise definition, we refer to [8, Chap. 1]). It turns out that 1 is the root system of ÇA. Recall that the multiplicity of α is
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the dimension of the root space corresponding to α, which is denoted by multα. In his paper [6], Kac made the following remarkable conjectures: Kac conjecture I. For any α ∈ 1+ , A0 α; q has no negative coefficients. Kac conjecture II. If 0 has no edge-loops then for any α ∈ 1+ the constant term of A0 α; q is equal to the multiplicity of α, i.e., A0 α; 0 = multα. As mentioned before, these conjectures are valid for quivers of finite type and tame type, but remain unsolved for wild type.
3. THE BURNSIDE FORMULA Let 0 be a connected quiver with n vertices 1; 2; : : : ; n. Let bij be the number of arrows from i to j, and aij the number of edges between i and j in the underlying graph of 0. Thus, aii = bii , aij = bij + bji if i 6= j. As already noted in the last section, for each α ∈ n , the finite group GLα; q acts on the finite set R0 α; q and M0 α; q is equal to the number of orbits in R0 α; q . Thus the Burnside orbit counting formula amounts to M0 α; q =
X X Xg 1 Xg = ; GLα; q g∈GLα; Zg g∈Clα; q
q
where Xg = M ∈ R0 α; q g · M = M, the set of representations fixed by g, and Zg = h ∈ GLα; q gh = hg, the centralizer of g in GLα; q , and where Clα; q is a complete set of representatives of the conjugacy classes of GLα; q . Before we can use the Burnside formula to calculate M0 α; q, we have to recall some basic facts about the conjugacy classes of GLm; q . More details can be found in Macdonald’s book [9, p. 138]. Recall that a partition λ = λ1 ; λ2 ; : : : is a finite sequence λ1 ≥ λ2 ≥ · · · of non-negative integers. The integer λ = λ1 + λ2 + · · · is called the weight of λ. In this paper, we use P to denote the set of all partitions including the unique partition of 0. Note that any partition λ can be written in the “exponential form” 1n1 2n2 · · ·, which means that there are exactly ni parts equal to i in λ. Let 8 be the set of all monic irreducible polynomials in the indeterminate t over q with t excluded. For any f ∈ 8, let Jf be the companion matrix of f (for the definition of Jf , we refer to Macdonald [9, p. 140]). For
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each integer r ∈ , we define Jf Id 0 ··· 0 0 Jf Id · · · 0 :: :: :: : : :: ; : Jr f = : : : : 0 0 0 · · · I d 0 0 0 · · · Jf with r diagonal blocks of Jf , where Id means the identity matrix of degree degf . For any f ∈ 8, π = n1 ; n2 ; : : : ∈ P, let Jf; π = Jn1 f ⊕ Jn2 f ⊕ · · · : This is a diagonal block matrix with Jni f (i ≥ 1) in the diagonal. Then any conjugacy class of GLm; q has the Jordan canonical form Jf1 ; π1 ⊕ Jf2 ; π2 ⊕ · · · ⊕ Jfs ; πs ; where f1 ; : : : ; fs are mutually distinct polynomials from P 8, and π1 ; : : : ; πs ∈ P, s is some varying positive integer, which satisfies si=1 degfi πi = m. Now we introduce an “inner product” of the partitions. For λ; µ ∈ P, let λ0 = λ01 ; λ02 ; : : : and µ0 = µ01 ; µ02 ; : :P : be the partitions conjugate to λ and µ, respectively; we define λ; µ = i≥1 λ0i µ0i . Lemma 3.1. For any µ = 1m1 2m2 · · ·; ν = 1n1 2n2 · · · ∈ P, we have XX µ; ν = mini; jmi nj : i≥1 j≥1
Proof. Let µ0 = µ01 ; µ02 ; : : : and ν 0 = νP10 ; ν20 ; : : : be the P partitions conjugate to µ and ν, respectively. Then µ0k = i≥k mi , νk0 = i≥k ni for k ≥ 1. By the definition of µ; ν, we have the following: µ; ν = µ01 ν10 + µ02 ν20 + µ03 ν30 + · · · = m1 + m2 + m3 + · · ·n1 + n2 + n3 + · · · + m2 + m3 + m4 + · · ·n2 + n3 + n4 + · · · + m3 + m4 + m5 + · · ·n3 + n4 + n5 + · · · + · · · ! ! X X X = m1 nj + m2 nj + nj j≥1
+ · · · + mr =
XX i≥1 j≥1
j≥1
X
nj +
j≥1
mini; jmi nj :
X j≥2
j≥2
nj + · · · +
X j≥r
! nj + · · ·
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P Following Macdonald [9], we define nλ = i≥1 i − 1λi for λ = λ1 ; λ2 ; : : : ∈ P. Combining Lemma 1 of Hua [3] and the above lemma, we have the following: Lemma 3.2. For any λ ∈ P, we have λ + 2nλ = λ; λ. Now for any f; g ∈ 8; λ; µ ∈ P, Jf; λ; Jg; µ represents a conjugacy class of the group GLm; q × GLm0 ; q , where m = degf λ and m0 = deggµ. We let XJf;λ;Jg;µ = M ∈ Mm×m0 q Jf; µM = MJg; µ , in which Mm×m0 q means the set of all m × m0 matrices over q . Lemma 3.3. For any f; g ∈ 8; λ; µ ∈ P, we have XJf;λ;Jg;µ = q if f = g; XJf;λ;Jg;µ = 1 otherwise. degf λ;µ
Proof. Suppose λ = 1m1 2m2 · · ·; µ = 1n1 2n2 · · ·, and d = degf ; e = degg. Let A = q x, and A˜ = A ⊗q ¯ q , which is isomorphic to ¯ q x. Given any m × m matrix α over q , we define an A-module structure on qm by setting x · v = αv for v ∈ qm . Let Vα denote this module, and define V˜α = Vα ⊗q ¯ q . By the definition of XJf;λ;Jg;µ , it is easy to see that XJf;λ;Jg;µ = Hom A VJf;λ ; VJg;λ . Note that Hom A VJf;λ ; VJg;λ is a finite dimensional vector space over q , which has the same dimension as Hom A˜ V˜Jf;λ ; V˜Jg;λ over ¯ q . This reduces the calculation to the corresponding calculation over the field ¯ q . Since Jf; µ = J1 f m1 ⊕ J2 f m2 ⊕ m m n · · ·, it follows that V˜Jf;µ ∼ = V˜J1 f1 ⊕ V˜J2 f2 ⊕ · · · : Similarly, V˜Jg;µ ∼ = V˜J1 1g ⊕ n2 V˜J2 g ⊕ · · · : Therefore, nj m Hom A˜ V˜Jf;λ ; V˜Jg;λ ∼ = ⊕di=1 ⊕ej=1 Hom A˜ V˜Ji fi ; V˜Jj g mi nj ∼ : = ⊕di=1 ⊕ej=1 Hom A˜ V˜J f ; V˜J g i
j
The classical theorem that any finite field is separable implies that, for any irreducible monic polynomial f t in q t, there is an invertible matrix X with entries in ¯ q such that XJf X −1 is diagonal with distinct Q diagonal entries ξ1 ; : : : ; ξd . Here f t = di=1 t − ξi . Thus Ji f is similar to Jξ1 ; i ⊕ · · · ⊕ Jξd ; i over ¯ q , where Jξ; i represents the Jordan block matrix of size i with eigenvalue ξ. Similarly, Ji g is similar to Jη1 ; i ⊕ · · · ⊕ Jηe ; i over ¯ q , where η1 ; : : : ; ηe are the roots of g over ¯ q . Therefore, Hom A˜ V˜Ji f ; V˜Jj g ∼ = ⊕ds=1 ⊕et=1 Hom A˜ V˜Jξs ;i ; V˜Jηt ;j :
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˜ Note that every V˜Jδ;i is an indecomposable A-module whose sole ˜ ˜ ˜ composition factor is VJδ;1 . Thus HomA˜ VJξs ;i ; VJηt ;j 6= 0 if and only if ξs = ηt . Also note that f and g have a common root if and only if f = g. Thus if f 6= g, then for all ξs ; ηt (1 ≤ s ≤ d; 1 ≤ t ≤ e), HomA˜ V˜Jξs ;i ; V˜Jηt ;j = 0, and so HomA˜ V˜Jf;λ ; V˜Jg;µ = 0; which implies that XJf;λ;Jg;µ = 1. Now, we suppose that f = g. Thus, HomA˜ V˜Ji f ; V˜Jj g ∼ = ⊕ds=1 HomA˜ V˜Jξs ;i ; V˜Jξs ;j : An elementary calculation with matrices reveals that dim¯ Hom A˜ V˜Jξ ;i ; V˜Jξ ;j = mini; j: s
q
s
Thus, we have dim q XJf;λ;Jg;µ = dim ¯ q Hom A˜ V˜Jf;λ ; V˜Jg;µ XX = mi nj dim ¯ q Hom A˜ V˜Ji f ; V˜Jj g i≥1 j≥1
=
XX
mi nj mini; jd
i≥1 j≥1
= dλ; µ
by Lemma 3:1:
Finally, we have XJf;λ;Jg;µ = qdλ;µ . Note that this formula depends only on the degrees of f and g. For any non-negative integer r, let ϕr q = 1 − q1 − q2 · · · 1 − qr , with convention ϕ0 q = 1. Following Macdonald [9], we let bλ q = Q n1 n2 i≥1 ϕni q for λ = 1 2 · · · ∈ P. Theorem 3.4. Let g = Jf; π1 ; : : : ; Jf; πn , where f ∈ 8 with degf = d, and π1 ; : : : ; πn ∈ P. Then g represents a conjugacy class of the group GLα; q ; here α = dπ1 ; : : : ; dπn , and we have Zg =
n Y i=1
qdπi ;πi bπi q−d
and
Xg =
Y 1≤i≤j≤n
qdaij πi ;πj :
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Proof. Note that similar formulas appeared in [6, p. 90]. For each i, 1 ≤ i ≤ n, ZJf;πi is the number of centralizers of Jf; πi in GLmi ; q , where mi = dπi . Formula (2.6) in [9, p. 139] implies that ZJf;πi = qdπi +2nπi bπi q−d . So Lemma 3.2 implies that Q ZJf;πi = qdπi ;πi bπi q−d . As Zg = ni=1 ZJf;πi , we have Zg =
n Y
qdπi ;πi bπi q−d :
i=1
It is easy to see that Xg = previous lemma that
Qn Qn i=1
Xg =
j=1
XJf;πi ;Jf;πj bij . It follows from the
n n Y Y
qdbij πi ;πj :
i=1 j=1
Since λ; µ = µ; λ for any λ; µ ∈ P, and aii = bii ; aij = bij + bji if i 6= j, we have Y qdaij πi ;πj : Xg = 1≤i≤j≤n
Note that Xg is independent of the orientation of 0. 4. GENERATING FUNCTIONS AND FORMAL IDENTITIES Let 0 be as before, and let X1 ; : : : ; Xn be independent commuting indeterminates. For any α = α1 ; : : : ; αn ∈ n , let X α be the monomial α α X1 1 · · · Xn n . The Krull–Schmidt theorem states that every representation of 0 over a field can be written as a direct sum of indecomposable representations in a unique way up to a permutation of the summands. This implies the following formal identity: Y X M0 α; qX α = 1 − X α −I0 α;q : (4.1) α∈n
α∈n \0
For any α = α1 ; : : : ; αn ∈ n \0, we let α¯ = g:c:d:α1 ; : : : ; αn , the greatest common divisor of α1 ; : : : ; αn . Theorem 4.1. For any α ∈ n \0, the formulas α X 1X d I0 α; q = µ A0 ; qr ; d r d r d d α¯ A0 α; q =
α X 1X µrI0 ; qr d r d d d α¯
are valid, where µ is the classical M¨ obius function.
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Proof. The first formula appeared in [6], and the second one is the M¨ obius inverse of the first. Corollary 4.2. For any α ∈ n \0, we have the following identity: α X 1 α X 1 I0 ;q = A0 ; qd : d d d d d α¯ d α¯ Now we define P0 X1 ; : : : ; Xn ; q =
Q
aij πi ;πj 1≤i≤j≤n q π X1 1 Q πi ;πi −1 q b q πi 1≤i≤n π1 ;:::;πn ∈P
X
π
· · · Xn n :
Thus, P0 X1 ; : : : ; Xn ; q is a formal power series in X1 ; : : : ; Xn with coefficients in q, and the constant term is 1. Note that P0 X1 ; : : : ; Xn ; q is independent of the orientation of 0. Theorem 4.3. With the notation as above, the formal identity X α∈n
M0 α; qX α =
∞ Y
φd q P0 X1d ; · · · ; Xnd ; qd
d=1
holds, where φ1 q = q − 1, and for d ≥ 2, φd q is the number of monic irreducible polynomials of degree d over q . Proof. It follows from the Burnside formula that, for any α ∈ n , we have M0 α; q =
X
Xg
g∈Clα;q
Zg
;
where Xg , Zg , and Clα; q are defined in Section 3. Thus, we have X X X X g X α: M0 α; qX α = Zg α∈n α∈n g∈Clα; q
For f ∈ 8, we let df denote the degree of f . The discussion of Section 3 shows that, for a given α = α1 ; : : : ; αn ∈ n , each conjugacy class of GLα; q is a direct sum of finitely many elements of the form Jfi ; π1i ; Jfi ; π2i ; : : : ; Jfi ; πni , i ≥ 1, where fi ∈ 8; πki ∈ P, which satisfy the following conditions: (1) the fi ’s, i ≥ 1, are mutually distinct; P (2) for a fixed k, 1 ≤ k ≤ n, i≥1 dfi πki = αk .
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It follows from the proof of Lemma 3.3 that, for any f; g ∈ 8 with f 6= g, and for any λi ; µi ∈ P (i = 1; : : : ; n), the following holds: XJf;λ1 ⊕Jg;µ1 ;:::;Jf;λn ⊕Jg;µn = XJf;λ1 ;:::;Jf;λn · XJg;µ1 ;:::;Jg;µn : Therefore, we have the following factorization: X X Xg Xα Z g α∈n g∈Clα; q
Y
X
XJf;π1 ;:::;Jf;πn
f ∈8
π1 ;:::;πn ∈P
ZJf;π1 ;:::;Jf;πn
=
df π1 X1
! df πn : : : Xn
:
Note that, in the above product, each factor is a formal power series with constant term 1. Thus the above product does make sense. It follows from Theorem 3.4 that ! XJf;π1 ;:::;Jf;πn df π1 Y X df πn X · · · Xn ZJf;π1 ;:::;Jf;πn 1 f ∈8 π1 ;:::;πn ∈P ! Q df aij πi ;πj Y X df π1 df πn 1≤i≤j≤n q X1 = · · · Xn Q df πi ;πi bπi q−df 1≤i≤n q f ∈8 π1 ;:::;πn ∈P Y df df P0 X1 ; : : : ; Xn ; qdf = f ∈8
=
∞ Y
φd q P0 X1d ; : : : ; Xnd ; qd :
d=1
The above theorem was obtained independently by Richard Stanley using different notation in an unpublished note. For any α ∈ n \0, we let E0 α; q and H0 α; q be the rational functions in q determined by ! X X α log M0 α; qX = E0 α; qX α /α; ¯ α∈n
log P0 X1 ; : : : ; Xn ; q =
α∈n \0
X
H0 α; qX α /α; ¯
α∈ \0 n
where the “log” means the formal logarithm, i.e., log1 − X = It follows from Theorem 4.3 that α X dφd qH0 ; qd : E0 α; q = d d α¯
P
i≥1 −X
i
/i.
(4.2)
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Lemma 4.4. With notations as above, for any α ∈ n \0, we have α 1 X µdE0 ;q : I0 α; q = α¯ d α¯ d Proof. It follows from identity (4.1) and the definition of E0 α; q that ! X Y E0 α; qX α /α¯ = log 1 − X α −I0 α;q α∈n \0
α∈n \0
=
X
I0 α; q log
α∈n \0
=
X
I0 α; q
α∈ \0 n
=
X
1 1 − Xα
∞ X 1 i=1
i
X iα
∞ X
1 I0 α; qX iα : i n α∈ \0 i=1
By comparing the coefficients of X α on both sides, we have X 1 α 1 E0 α; q = I ;q : α¯ d 0 d d α¯ Thus, E0 α; q =
X α¯ α I ;q : d 0 d d α¯
The M¨ obius inverse of this shows that α 1 X I0 α; q = µdE0 ;q : α¯ d α¯ d
Lemma 4.5. For any α ∈ n \0, we have A0 α; q =
α 1 X µdE0 ; qd : α¯ d α¯ d
Proof. First let us recall two basic techniques on handling sums involving divisors. Suppose that f : → and g: × → are two functions. Then the following hold: X X f n = f n/d; (T1) dn
dn
quivers over finite fields X X
gd; r =
d n r n/d
X X
gd; r:
1023 (T2)
r n d n/r
Now we have the following α X 1X A0 α; q = µrI0 ; qr d r d d d α¯
by Theorem 4:1
α X 1X d X ; qr µr µsE0 by Lemma 4:4 d r d α¯ s α/d ds d α¯ ¯ sα X 1X d X α¯ = µr µ E0 ; qr by T1 d r d α¯ s α/d ds α¯ d α¯ ¯ α¯ sα r 1 XX X = µrµ E0 ;q α¯ d α¯ r d s α/d ds α¯ ¯ sα 1 X X X d = µrµ E0 ; qr by T1 α¯ d α¯ r α/d s α¯ ¯ sd sα r d 1X X X µrµ by T2 E0 ;q = α¯ r α¯ d α/r s α¯ ¯ sd 1X X X α¯ sα r = µrµ E0 ;q by T1 α¯ r α¯ d α/r dsr α¯ ¯ s α/dr ¯ sα 1X X X α¯ = µrµ E0 ; qr by T2 α¯ r α¯ s α/r dsr α¯ ¯ d α/sr ¯ sα X 1X X α¯ r = µrE0 ;q µ : α¯ r α¯ s α/r α¯ dsr ¯ d α/sr ¯ =
Note that the M¨ obius function µ has the following property: X 1 if n = 1; µd = 0 otherwise: dn
Thus we have A0 α; q =
α 1X µrE0 ; qr : α¯ r α¯ r
Theorem 4.6. For any α ∈ n \0, the following identity holds: α q−1 X µdH0 ; qd : A0 α; q = α¯ d α¯ d
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Proof. In fact, we have A0 α; q =
α 1 X ; qd µdE0 α¯ d α¯ d
by Lemma 4:5
=
α X 1 X µd rφr qd H0 ; qdr α¯ d α¯ dr r α/d ¯
=
rα X α¯ 1 X d α/r ¯ φα/dr ; q µd q H 0 α¯ d α¯ dr ¯ α¯ r α/d ¯
=
X X
µd
d α¯ r α/d ¯
by 4:2 by T1
rα 1 ¯ φα/dr qd H0 ; qα/r ¯ dr α¯
rα 1 d α/r ¯ φα/dr ; q q H by T2 0 dr ¯ α¯ r α¯ d α/r ¯ α XX r = µd φr/d qd H0 ; qr by T1 d α¯ r r α¯ d r
=
=
X X
µd
α X r 1X H0 ; qr µd φr/d qd : α¯ r α¯ r d dr
Note that φd q is the number of polynomials in 8 which have degree d. The following is a classical result modified in case n = 1: 1X φn q = µd qn/d − 1 : n dn The M¨ obius inverse of this amounts to the following: Xn µdφn/d qd = µnq − 1: d dn Thus, we have A0 α; q =
α q−1 X µrH0 ; qr : α¯ r α¯ r
As the H0 α; q’s are rational functions in q, so are the A0 α; q’s by the above theorem. As each A0 α; q takes integer value when q is any prime power, A0 α; q must be a polynomial in q with rational coefficients. Thus, Theorem 4.1 and identity (4.1) imply that the I0 α; q’s and M0 α; q’s are all polynomials in q with rational coefficients. Since the formal power series P0 X1 ; : : : ; Xn ; q is independent of the orientation of 0, the functions
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E0 α; q and H0 α; q, and hence the polynomials A0 α; q, I0 α; q, and M0 α; q, are all independent of the orientation of 0. It is proved in [6] that all coefficients of A0 α; q are integers. An alternativePproof of this uα α j tj q with is given in [4]. In what follows, we assume that A0 α; q = j=0 α tj ∈ . Theorem 4.7. The following formal identity holds: X
uα Y Y
M0 α; qX α =
1 − qj X α −tj : α
α∈1+ j=0
α∈n
Proof. The M¨ obius inverse of Lemma 4.5 shows that α X α¯ E0 α; q = A0 ; qd : d d d α¯ By the definition of E0 α; q, we have ! X X α M0 α; qX log =
E0 α; qX α /α¯
α∈ \0
α∈n
n
= =
α X 1 A0 ; qd X α d d α∈n \0 d α¯ X
∞ X 1
X
α∈n \0 i=1
i
A0 α; qi X iα :
By Theorem 2.1, A0 α; q 6= 0 if and only if α ∈ 1+ . Thus, ! ∞ X X X 1 log M0 α; qX α = A0 α; qi X iα i α∈n α∈1+ i=1 = =
uα ∞ X X 1X tjα qij X iα i j=0 α∈1+ i=1 uα X X α∈1+ j=0
=
X X uα
α∈1+ j=0
= log
tjα
∞ X 1 i=1
qij X iα
tjα log 1 − qj X α
uα Y Y α∈1+ j=0
This implies the theorem.
i
1 − qj X α
−1
−tjα
:
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The above theorem implies that M0 α; q ∈ q. Furthermore, if the first Kac conjecture is true, i.e., tjα ≥ 0 for all α ∈ 1+ and j ∈ , then each M0 α; q has no negative coefficients either. Lemma 4.8. The following formal identity is valid: ∞ ∞ X Y 1 Xi = log 1 − qi X: i i q − 1 i=0 i=1
Proof. In fact, we have log
∞ Y
1 − qi X =
i=0
∞ X
log1 − qi X
i=0
=−
∞ X ∞ X 1 i j q X j i=0 j=1
=−
∞ ∞ X 1 jX X qij j i=0 j=1
=
∞ X 1 Xj : j j j=1 q − 1
Now we are able to prove the main result of this paper. Theorem 4.9. The following formal identity holds: Q aij πi ;πj uα ∞ Y X Y Y t α 1≤i≤j≤n q π1 πn X · · · X = 1 − qi+j X α j : Q n 1 πi ;πi −1 bπi q 1≤i≤n q π1 ;:::;πn ∈P α∈1+ i=0 j=0 Proof. The M¨ obius inverse of Theorem 4.6 implies that α X α¯ 1 H0 α; q = A0 ; qd : d dq −1 d d α¯ Thus, according to the definition of H0 α; q, we have X H0 α; qX α /α¯ log P0 X1 ; : : : ; Xn ; q = α∈n \0
= =
α X 1 1 d A ; q Xα 0 d d d q − 1 n α∈ \0 d α¯ X
X
∞ X
1 A0 α; qi X iα : iq − 1 n α∈ \0 i=1 i
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By Theorem 2.1, A0 α; q 6= 0 if and only if α ∈ 1+ . Thus, log P0 X1 ; : : : ; Xn ; q = = =
∞ X X
1 A0 α; qi X iα i iq − 1 α∈1+ i=1 ∞ X X
uα X 1 tjα qij X iα i + i=1 iq − 1 j=0 α∈1 uα X X α∈1+ j=0
=
uα X X +
α∈1 j=0
= log
tjα tjα
∞ X qj X α i i i=1 iq − 1 ∞ X
log1 − qi qj X α
i=0
uα ∞ Y Y Y
1−q
i+j
X
α α tj
by Lemma 4:8 ! :
α∈1+ i=0 j=0
This implies that P0 X1 ; : : : ; Xn ; q =
uα ∞ Y Y Y
1 − qi+j X α
tjα
:
α∈1+ i=0 j=0
The theorem now follows from the definition of P0 X1 ; : : : ; Xn ; q. As noted before, if 0 has no edge-loops, then 0 defines a Kac-Moody algebra Ç with root system 1 and Weyl group W . Recall that the Kac denominator identity of Ç can be written in the form (see [5, p. 65]) X Y −1lw X sw = 1 − X α multα ; w∈W
α∈1+
where lw is the length of w and sw is the sum of positive roots mapped into negative roots by w−1 . Now let q approach 0 in Theorem 4.9; we have ! Q aij πi ;πj X Y 1≤i≤j≤n q π1 πn lim X1 · · · Xn = 1 − X α A0 α;0 : Q π ;π −1 i i q→0 bπi q + 1≤i≤n q π1 ;:::;πn ∈P α∈1 If Kac conjecture II is true, i.e., A0 α; 0 = multα for all α ∈ 1+ , then we have Y Y 1 − X α A0 α;0 = 1 − X α multα : α∈1+
α∈1+
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Thus, Theorem 4.9 immediately implies the following Corollary 4.10. The second Kac conjecture is true if and only if the following identity holds:
lim
q→0
Q
aij πi ;πj 1≤i≤j≤n q π X1 1 Q πi ;πi −1 bπi q 1≤i≤n q π1 ;:::;πn ∈P
X
! π · · · Xn n
X
=
−1lw X sw :
w∈W
We remark that another equivalent form of the second Kac conjecture, also stated in combinatorial identities, is obtained by Sevenhant and Van den Bergh [11]. Their form is essentially equivalent to the form given above.
5. EXAMPLES Theorem 4.6 provides a practical way to compute the polynomials A0 α; q. The following examples were carried out by Maple based on Theorem 4.6; more examples can be found in [4, Appendix A]. As the A0 α; q’s, M0 α; q’s, and I0 α; q’s are all independent of the orientation of 0, we can just refer to 0 as a graph. If 0 is the hyperbolic graph ◦ ≡≡≡≡ ◦, then we have the following: 2
1
A0 1; 1; q = q2 + q + 1 A0 2; 3; q = q6 + q5 + 3q4 + 4q3 + 5q2 + 3q + 2 A0 3; 6; q = q10 + q9 + 3q8 + 5q7 + 8q6 + 12q5 + 17q4 + 16q3 + 14q2 + 7q + 3 I0 1; 1; q = q2 + q + 1 I0 2; 3; q = q6 + q5 + 3q4 + 4q3 + 5q2 + 3q + 2 I0 3; 6; q = q10 + q9 + 3q8 + 5q7 + +
49 3 q 3
+
41 2 q 3
+
20 q 3
25 6 q 3
+ 12q5 + 17q4
+3
M0 1; 1; q = q2 + q + 2 M0 2; 3; q = q6 + 2q5 + 6q4 + 10q3 + 16q2 + 12q + 10 M0 3; 6; q = 2q12 + 2q11 + 8q10 + 12q9 + 26q8 + 44q7 + 83q6 + 124q5 + 180q4 + 194q3 + 182q2 + 104q + 54:
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If 0 is the hyperbolic graph ◦ ==== ◦−−−−◦, then we have the following: 1
2
3
A0 2; 2; 0; q = q + 1 A0 3; 3; 3; q = q + 1 A0 4; 4; 2; q = q5 + 2q4 + 6q3 + 10q2 + 13q + 7 I0 2; 2; 0; q = 21 q2 + 21 q + 1 I0 3; 3; 3; q = 13 q3 + 23 q + 1 I0 4; 4; 2; q = q5 + 25 q4 + 6q3 +
21 2 q 2
+ 12q + 7
M0 2; 2; 0; q = q + 3q + 6 2
M0 3; 3; 3; q = 8q3 + 32q2 + 80q + 88 M0 4; 4; 2; q = q5 + 13q4 + 56q3 + 170q2 + 287q + 240: If 0 is the wild quiver •, then the following hold: A0 1; q = q2 A0 2; q = q5 + q3 A0 3; q = q10 + q8 + q7 + q6 + q5 + q4 A0 4; q = q17 + q15 + q14 + 2q13 + q12 + 3q11 + 2q10 + 4q9 + 2q8 + 3q7 + q6 + q5 I0 1; q = q2 I0 2; q = q5 + 21 q4 + q3 − 21 q2 I0 3; q = q10 + q8 + q7 + 43 q6 + q5 + q4 − 13 q2 I0 4; q = q17 + q15 + q14 + 2q13 + q12 + 3q11 + 25 q10 + 4q9 + 94 q8 + 3q7 + 23 q6 + 21 q5 − 14 q4 − 21 q3 M0 1; q = q2 M0 2; q = q5 + q4 + q3 M0 3; q = q10 + q8 + 2q7 + 2q6 + 2q5 + q4 M0 4; q = q17 + q15 + q14 + 2q13 + 2q12 + 3q11 + 4q10 + 6q9 + 5q8 + 5q7 + 3q6 + q5 : All of the above polynomials are consistent with Kac’s conjectures. These examples show that I0 α; q may have fractional coefficients, and also suggest the following:
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Conjecture. If 0 has no edge-loops, then the coefficients of the polynomial I0 α; q (α ∈ 1+ ) are all non-negative. It is worth noticing that the above conjecture can be easily verified for quivers of finite type and tame type, see examples below. In the rest of this section, we specialize Theorems 4.7 and 4.9 to several graphs of finite type or tame type, as their representations have been completely classified; this would generate some explicit identities which seem to be new. Example I (the trivial graph A1 ). A1 has a single vertex but no edges. A representation of A1 is just a vector space. Thus every representation of dimension bigger than one is decomposable. Consequently, AA1 1; q = 1 and AA1 n; q = 0 for n > 1. Thus Theorem 4.9 implies the following identity: X λ∈P
∞ Y X λ = 1 − qi X: qλ;λ bλ q−1 i=0
(5.1)
Note that the Euler identity shows that ∞ X
∞ Y qii−1/2 X i = 1 + qi X: 2 i i=0 1 − q1 − q · · · 1 − q i=0
Upon the Euler identity, (5.1) implies the following: X qλ;λ qn : = 1 − q1 − q2 · · · 1 − qn λ`n bλ q With the above formula, one can easily show that the number of nilpotent n × n matrices over q of is equal to qnn−1 . Example II (the Jordan graph A˜0 ). A˜0 consists of a single node plus a single loop. It follows from the Jordan normal form theorem that AA˜0 n; q = q for n ≥ 1. Thus Theorem 4.9 shows that X λ∈P
∞ ∞ Y Y X λ = 1 − qi X j ; bλ q−1 i=1 j=1
(5.2)
while Theorem 4.7 shows that 1+
∞ X
MA˜0 i; qX i =
i=1
∞ Y
1 − qX i −1 :
i=1
It follows that MA˜0 n; q =
X λ`n
qlλ ;
(5.3)
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P where lλ is the length of λ, i.e., lλ = i≥0 ni if λ = 1n1 2n2 · · · ∈ P. This has already been proved in [3]. Now, for any positive integer n, we let Gn; q be the polynomial 1 X n d Gn; q = φ q ; n dn d where φn is the classical Euler function; i.e., φn is the number of numbers up to n which are relatively prime to n. An easy exercise shows the following: ∞ ∞ Y Y 1 − qX i −1 = 1 − X i −Gi;q : (5.4) i=1
i=1
Now identities (4.1), (5.3), and (5.4) imply that 1 X n d IA˜0 n; q = φ q : n dn d Thus, the IA˜0 n; q’s are polynomials in q with non-negative rational coefficients. Example III (the graph A2 ). A2 is the graph ◦−−−−◦. The root system associated with A2 has three positive roots 1; 0, 0; 1, and 1; 1. It follows from Theorem 4.7 that ∞ X ∞ X MA2 m; n; qX m Y n = 1 − X−1 1 − Y −1 1 − XY −1 : m=0 n=0
This implies that MA2 m; n; q = minm; n. Theorem 4.9 implies that ∞ Y qλ;µ X λ Y µ = 1 − qi X1 − qi Y 1 − qi XY : qλ;λ+µ;µ bλ q−1 bµ q−1 i=0
X λ;µ∈P
Since, for any quiver of finite type, every indecomposable representation over q is absolutely indecomposable, we have I0 α; q = A0 α; q = 1 for all α ∈ 1+ . Example IV (the Kronecker graph A˜1 ). A˜1 is the graph ◦ ==== ◦, which is of tame type. The real roots of the root system associated with A˜1 are of the forms i; i + 1 and i; i − 1, i ∈ , while the imaginary roots are of the form i; i, i ∈ and i 6= 0. It follows from the classification theorem of representations of tame quivers that AA˜1 i; i = q + 1. Thus Theorem 4.7 amounts to ∞ X ∞ X MA˜1 i; j; qX i Y j i=0 j=0
=
∞ Y
(5.5)
1 − X Y
j=1
j
j−1 −1
1 − X
j−1
j −1
j −1
j −1
Y 1 − X Y 1 − qX Y ; j
j
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while Theorem 4.9 shows that X
q2λ;µ X λ Y µ qλ;λ+µ;µ bλ q−1 bµ q−1
λ;µ∈P
=
∞ ∞ Y Y
1 − qi−1 X j Y j−1 1 − qi−1 X j−1 Y j 1 − qi−1 X j Y j 1 − qi X j Y j :
i=1 j=1
(5.6) Let q → 0 in the last identity; then the right hand side appears in the following Jacobi triple product identity: ∞ X Y −1i X ii−1/2 Y ii+1/2 = 1 − X i Y i−1 1 − X i−1 Y i 1 − X i Y i : i∈
i=1
Thus the following holds: X
lim
q→0
λ;µ∈P
X q2λ;µ X λ Y µ = −1i X ii−1/2 Y ii+1/2 : λ;λ+µ;µ −1 −1 q bλ q bµ q i∈
And so identity (5.6) can be considered as a q-analogue of the Jacobi triple product identity. Now identities (4.1), (5.4), and (5.5) imply that Y 1 − X α −IA˜1 α;q α∈1+
=
∞ Y
1 − X j Y j−1 −1 1 − X j−1 Y j −1 1 − X j Y j −1 1 − qX j Y j −1
j=1
=
∞ Y
1 − X j Y j−1 −1 1 − X j−1 Y j −1 1 − X j Y j −Gj;q−1 :
j=1
It follows that
1; Gm; q + 1; IA˜1 m; n; q = 0;
if m − n = 1; if m − n = 0; if m − n > 1:
Thus, the IA˜1 m; n; q’s are polynomials in q with non-negative rational coefficients. Similar identities could be written down for any graph of finite type or tame type, this is left to the readers. ACKNOWLEDGMENTS This paper is based on the author’s Ph.D. thesis done at the University of New South Wales. The author thanks his supervisor, Dr. Peter Donovan, for valuable guidance. He also thanks the referee for many helpful comments and suggestions.
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REFERENCES 1. P. W. Donovan and M. R. Freislich, The representation theory of finite graphs and associated algebras, Carleton Math. Lecture Notes 5 (1973). 2. P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71–103. 3. J. Hua, Generalizing the recursion relationship for the partition function, J. Combin. Theory Ser. A 79 (1997), 105–117. 4. J. Hua, “Representations of Quivers over Finite Fields,” Ph.D. thesis, University of New South Wales, 1998. 5. V. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57–92. 6. V. Kac, “Root Systems, Representations of Quivers and Invariant Theory,” Lecture Notes in Mathematics 996, pp. 74–108, Springer-Verlag, Berlin/New York. 7. V. Kac, Infinite root systems, representations of graphs and invariant theory, II, J. Algebra 78 (1982), 141–162. 8. V. Kac, “Infinite Dimensional Lie Algebras,” Progress in Math., Vol. 44, Birkh¨auser, Boston, 1983. 9. I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” Clarendon, Oxford, 1979. 10. L. A. Nazarova, Representations of quivers of infinite type, Math. USSR Izv. Ser. Mat. 37 (1973), 752–791. 11. B. Sevenhant and M. Van den Bergh, “On the Numbers of Absolutely Indecomposable Representations of Quivers,” preprint.