On the Number of Absolutely Indecomposable Representations of a Quiver

Journal of Algebra 221, 29᎐49 Ž1999. Article ID jabr.1999.7937, available online at http:rrwww.idealibrary.com on

On the Number of Absolutely Indecomposable Representations of a Quiver Bert Sevenhant and Michel Van Den Bergh* Departement WNI, Limburgs Uni¨ ersitair Centrum, Uni¨ ersitaire Campus, Building D, 3590 Diepenbeek, Belgium E-mail: [email protected], [email protected] Communicated by J. T. Stafford Received November 21, 1998

A conjecture of Kac states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac᎐Moody Lie algebra. In this paper we give a combinatorial reformulation of Kac’s conjecture in terms of a property of q-multinomial coefficients. As a side result we give a formula for certain inverse Kostka᎐Foulkes polynomials. 䊚 1999 Academic Press

Key Words: Hall algebra; symmetric functions.

1. INTRODUCTION Throughout Q will be a fixed quiver without loops Žalthough this is not very essential .. We denote the vertices of Q by Q0 . For a given ␣ g ⺞ Q 0 let o␣ Ž q ., i␣ Ž q ., and a␣ Ž q . be, respectively, the number of representations of Q over ⺖q with dimension vector ␣ Žrecall that an indecomposable representation is said to be absolutely indecomposable if it remains indecomposable after extension of the base field.. In w12x Kac proves the following. THEOREM 1.1. o␣ Ž q ., i␣ Ž q ., and a␣ Ž q . are polynomials in q. o␣ Ž q . and a␣ Ž q . ha¨ e integral coefficients. U

Senior researcher at the NFWO. 29 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

30

SEVENHANT AND VAN DEN BERGH

Computing some examples lead Kac to two conjectures regarding a␣ Ž q .. Conjecture 1.2 w12x.

a␣ Ž q . has positive coefficients.

This conjecture is still open. Some positive results are offered in w10, 15, 19x. These papers also yield evidence for the conjecture that o␣ Ž q . has positive coefficients. If true the conjecture would yield some evidence for the hope that absolutely indecomposable representations can be parametrized by a union of affine spaces. However proving the latter seems to be totally out of reach of current techniques. See w1, 2, 8, 9, 16x for results on the generic version. However even without the existence of such a ‘‘cell decomposition’’ one has the feeling that the coefficients of a␣ Ž q . should have an interpretation in terms of representation theory. Such an interpretation is offered for the constant term by Conjecture 1.4 below, which is also due to Kac. To state this conjecture we have to introduce more notation. Recall that associated to Q there is a bilinear form on ⺓ Q 0 given by

½

2

Ž e i , e j . s ya y a ij ji

if i s j otherwise,

where Ž e i . j s ␦ i j and where a i j is the number of edges going from i to j. Let ᒄ be the Kac᎐Moody Lie algebra associated to the bilinear form Ž ᎐, ᎐ .. The following result is proved in w11, 12x Žsee also w14x.. THEOREM 1.3. If ␣ is a real root of ᒄ then a␣ Ž q . s 1. If ␣ is an imaginary root then a␣ Ž q . / 0. If ␣ is not a root then a␣ Ž q . s 0. This last theorem provides some motivation for the following conjecture, which, as was already mentioned above, is also due to Kac. Conjecture 1.4.

a␣ Ž0. is equal to the multiplicity of ␣ in ᒄ.

This conjecture is trivially true in the Dynkin case Žin view of Theorem 1.3., and using the known representation theory of tame quivers w5x, one can show that it is also true in the extended Dynkin case. However it is not known for a single wild quiver. On the other hand there is a lot of positive computer evidence. For example we have checked with little effort Žusing Theorem A below. that for the 3-arrow quiver

o

ª ª ª

o

the conjecture holds up to dimension vector Ž20, 20..

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

31

Our aim in this note is to reformulate Conjecture 1.4 into a property of Gaussian multinomial coefficients. This is of theoretical interest since such multinomial coefficients are known to have a lot of non-trivial combinatorial properties. Furthermore as was already indicated above, our reformulation is considerably easier to verify by computer than the original statement of Conjecture 1.4. For an entirely different approach to Conjecture 1.4 through the Hall algebra w18x we refer to w20x. To give our reformulation of Conjecture 1.4 we have to introduce some more notation. One easily verifies that the linear transformations sj : ⺓ Q0 ª ⺓ Q0 : ␣ ¬ ␣ y Ž ␣ , e j . e j have order 2 and leave Ž ᎐, ᎐ . invariant. The group generated by the Ž s j . j is called the Weyl group of Q and is denoted by WQ . As usual if w g WQ then ⑀ Ž w . is "1 depending on whether w is a product of an even or odd number of Ž s j .’s. Following w13x we also introduce a formal symbol ␳ and we extend the action of WQ to ⺓ Q 0 [ ⺓ ␳ by putting si ␳ s ␳ y e i . If a, b1 , . . . , bn g ⺞ are such that Ý i bi s a then the corresponding Gaussian multinomial coefficient is defined by

␾a Ž t . a b1 ⭈⭈⭈ bn s ␾ Ž t . ⭈⭈⭈ ␾ Ž t . , b1 bn where ␾ aŽ t . s Ž t a y 1.Ž t ay 1 y 1. ⭈⭈⭈ Ž t y 1.. If ␮ s Ž ␮ 1 , ␮ 2 , . . . . is a partition then we put

␮

w ␮ x s ␮ y ␮ , ␮1 y ␮ ⭈⭈⭈ . 1 2 2 3 By definition a multipartition of ␣ g ⺞ Q 0 is a list of partitions ␭ s Ž ␭ i . i g Q 0 such that < ␭ i < s ␣ i . We view ␭ as a two dimensional list of non-negative integers Ž ␭ i j . i j such that Ý j ␭ i j s ␣ i . We write ␭Ž j . for the element of ⺞ Q 0 given by ␭Ž j . i s ␭ i j . Denote the set of multipartitions of ␣ by P␣ . THEOREM A. The Laurent polynomial p␣ Ž t . s

Ý

␭g P␣

t Ý j Ž ␭Ž j., ␭Ž j..r2

Ł Ž y1. ␭

igQ 0

i1

ty ␭ i1Ž ␭ i1q1.r2 w ␭ i x

32

SEVENHANT AND VAN DEN BERGH

is contained in ⺪w ty1 x. Furthermore, Conjecture 1.4 is equi¨ alent to the statement that p␣ Ž ⬁ . s

½

⑀ Ž w.

if ␣ s ␳ y w␳ , where w g WQ

0

otherwise.

For example in the case of the m-arrow quiver

o

ª .. . ª

o,

pa, b Ž t . would be given by

Ý Ž y1. ␭ q ␮ 1

< ␭
1

t Ý i ␭ i qÝ i ␮ i ymÝ i ␭ i ␮ iy ␭1Ž ␭1q1.r2y ␮ 1Ž ␮ 1q1.r2 w ␭ xw ␮ x . Ž 1.1. 2

2

For the convenience of the reader let us recall one of the standard combinatorial interpretations of Gaussian multinomial coefficients. Let a, b1 , . . . , bn g ⺞. Assume that u1 , . . . , u n are symbols. The cardinality of the set W of all words in the u i with u i appearing exactly bi times is given by the ordinary multinomial coefficient a!rb1! ⭈⭈⭈ bn!. A map ␾ : W ª ⺞ such that

Ý wg W

a t ␾ Ž w . s b ⭈⭈⭈ b 1 n

is called a Mahonian statistic w4, 7x. There are many Mahonian statistics w4, 7x. One example is given by INV Ž u i1 ⭈⭈⭈ u i a . s  i u ) i ¨ N u - ¨ 4 . After choosing a Mahonian statistic, the evaluation of p␣ Ž⬁. amounts to a signed counting of words. Following an established technique in combinatorics it should now be possible to define an involution on these words which inverts almost all signs. It seems very likely that in this way it should be possible to obtain a proof of Conjecture 1.4 but unfortunately we have not succeeded in completing this program. Although Theorem A was meant as a possible step in the proof of Conjecture 1.4 it also has independent interest. For example, it is possible to use it in order to obtain some new identities on Gaussian multinomial coefficients. This is outlined in Section 6.

33

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

Ž . In our proof of Theorem A we needed the entries Ky1 ␭, 1 n t of the inverse Kostka᎐Foulkes matrix in one variable. A combinatorial interpretation for Ž . w x Ky1 ␭, ␮ 1 was given in 6 but the methods used in that paper do not seem to extend to the one-parameter case. Using another method instead, based on a recurrence relation ŽA.3. derived from Pieri’s formula, we obtain the following result. THEOREM B.

Assume that ␮ s Ž c, 1d . is a hook. Then X

Ky1 ␭, ␮

Ž t. s Ž ᎐.

l Ž ␭ .ql Ž ␮ . Ý i G 2 ␭Xi Ž ␭Xiq1.r2yÝ cjs 2 ␭Xj

t

w␭ x X

1 y t ␭c X

1 y t ␭1

.

Ž 1.2.

Our proof of this result is reproduced in Appendix A. We were informed by Richard Stanley that in the case relevant to us Ž ␮ s 1n . Ž1.2. was conjectured by Carbonara, and subsequently proved by Macdonald, using a different method w3x. It is not clear to us that Macdonald’s method extends to the more general case of hooks. On the other hand one of the referees of this paper indicated a different Žand ingenious. proof of Ž1.2. which is also valid for hooks, but which uses somewhat more sophisticated machinery than our naive proof. We wish to thank V. Kac for commenting on a first version of this paper. In particular he informed us that the Ph.D. thesis of Jiuzhao Hua w10x contains related results. We also wish to thank Stanley for showing us his very interesting notes on the generating function of o␣ Ž q . w21x. Both Stanley’s notes and Hua’s thesis contain Žin a different notation. the rational functions r␣ Ž q ., defined herein. 2. A RECURRENCE RELATION FOR o␣ Ž0. Let k be an arbitrary field. A quiver Q is a quadruple Ž Q0 , Q1 , t, h., where Q0 and Q1 denote, respectively, the vertices and the arrows of Q, and h, t: Q1 ª Q0 are maps associating an arrow with its head and tail. We assume throughout that Q0 , Q1 are finite sets and that our quivers have no loops. A representation V of Q is a pair ŽŽ Vi . i g Q 0 , Ž ␾ e .e g Q 1 ., where the Vi are finite dimensional vector spaces and the ␾ e are maps VtŽ e. ª VhŽ e. . The element of ⺞ Q 0 given by Ždim Vi . i is the dimension ¨ ector of the representation. Homomorphisms between representations are defined in the usual way. Define R Ž Q, ␣ , k . s Ł e g Q i HomŽ k ␣ t Ž e . , k ␣ hŽ e . . and GLŽ ␣ , k . s Ł i g Q 0 GLŽ ␣ i , k .. Every element x g RŽ Q, ␣ , q . defines a representation Vx of Q. It is clear that GLŽ ␣ , k . acts on RŽ Q, ␣ , k . by conjugation

34

SEVENHANT AND VAN DEN BERGH

of matrices and furthermore Vx ( Vy if and only if x, y are in the same GLŽ ␣ , k .-orbit. Thus the isomorphism classes of representations of dimension vector ␣ are in one᎐one correspondence with the orbits RŽ Q, ␣ , k .rGLŽ ␣ , k .. From now on k will be a finite field with q elements. We write RŽ Q, ␣ , q . for RŽ Q, ␣ , k . and GLŽ ␣ , q . for GLŽ ␣ , k .. For ␣ g ⺞ Q 0 let UŽ ␣ , q . be the set of unipotent elements in GLŽ ␣ , q .. We define r␣ Ž q . s

1

 Ž u, x . g U Ž ␣ , q . = R Ž Q, ␣ , q . N ux s x 4 .

GL Ž ␣ , q .

By partitioning UŽ q, ␣ . into conjugacy classes one easily shows that r␣ is a rational function of q Žbut in general not a polynomial.. In this section we prove the following result. PROPOSITION 2.1.

r␣ Ž q . has no pole in 0. Furthermore one has r␤ Ž 0 . o␥ Ž 0 . s ␦ 0 ␣ .

Ý

Ž 2.1.

␤ , ␥ g⺞ Q0 ␤q ␥s ␣

Proof. The case ␣ s 0 is trivial, so we consider the case ␣ / 0. By the Burnside formula we have the following expression for o␣ Ž q .: o␣ Ž q . s

1 GL Ž ␣ , q .

Aut Ž Vx . .

Ž 2.2.

End Ž Vx . .

Ž 2.3.

Ý

xgR Ž Q , ␣ , q .

We will study a slightly modified version of Ž2.2., t␣ Ž q . s

1 GL Ž ␣ , q .

Ý

xgR Ž Q , ␣ , q .

Giving an element g of EndŽ Vx . amounts to giving a decomposition of Vx s V1, x [ V2, x , g s Ž g 1 , g 2 . such that g 1 acts nilpotently on V1, x and g 2 g AutŽ V2, x .. Denote by NilŽ V . the nilpotent endomorphisms of a representation V. We find t␣ Ž q . s

1 GL Ž ␣ , q .

Ý

␤q ␥ s ␣

␺␤ , ␥

Ý

Nil Ž Vy . ⭈ Aut Ž Vz . .

ygR Ž Q , ␤ , q . zgR Ž Q , ␥ , q .

Here ␺␤ , ␥ s Ł i g Q 0 ␺␤ i , ␥ i and ␺␤ i , ␥ i is equal to the number of decompositions k ␣ i s B [ C where B and C are, respectively, vector spaces of

35

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

dimension ␤i and ␥ i . It is elementary to see that GL Ž ␣ i , q .

␺␤ i , ␥ i s

.

GL Ž ␤i , q . ⭈ GL Ž ␥ i , q .

Substituting yields t␣ Ž q . s

Ý

␤q ␥ s ␣

⭈

ž

ž

1 ygR Ž Q , ␤ , q .

1 GL Ž ␥ , q .

Nil Ž Vy .

Ý

GL Ž ␤ , q .

Ý

/

Aut Ž Vz .

zgR Ž Q , ␥ , q .

/

.

Thus we find t␣ Ž q . s

Ý

␤q ␥ s ␣

r␤ Ž q . o␥ Ž q .

Ž 2.4.

Žwhere we have used that one can go from a nilpotent endomorphism to a unipotent endomorphism and back by adding and substracting the identity endomorphism.. So in particular t␣ is a rational function Žwhich could have been shown directly.. Now we consider a different way of evaluating t␣ . Let V1 , . . . , Vn represent the isomorphism classes of representations of Q over ⺖q , with dimension vector ␣ . Grouping RŽ Q, ␣ , q . into GLŽ ␣ , q . orbits yields n

t␣ Ž q . s

Ý is1

End Ž Vi . Aut Ž Vi .

.

Ž 2.5.

Now let W be an arbitrary representation of Q over ⺖q and let W s W1[a1 [ ⭈⭈⭈ [ Wp[a p be its decomposition into indecomposables. Then End Ž W . s

Hom Ž Wi , Wj .

Ł

GL a lŽ End Ž Wl . . ⭈

i, j

Aut Ž W . s

ai a j

Ł l

Ł i/j

Hom Ž Wi , Wj .

whence End Ž W . Aut Ž W .

s

Ł l

Ma lŽ End Ž Wl . . GL a lŽ End Ž Wl . .

.

ai a j

,

36

SEVENHANT AND VAN DEN BERGH

EndŽWl . is a local and its residue field ⺖ l is a finite extension of ⺖q . Since GL a lŽEndŽWl .. is the unit group of Ma lŽEndŽWl .., and since this unit group is the inverse image of GL a lŽ⺖ l . we find that Ma lŽ End Ž Wl . .

M a lŽ ⺖ l .

s

GL a lŽ End Ž Wl . .

GL a lŽ ⺖ l .

.

It is an easy exercise to show that the right-hand side of the above equation is always divisible by q Žas a rational number.. We conclude that
žÝ ␤

r␤ Ž 0 . e Ž ␤ . ⭈

/ žÝ ␥

o␥ Ž 0 . e Ž ␥ . s 1.

/

Ž 2.6.

Let us now consider the relation between o␣ Ž q . and i␣ Ž q .. Since every representation has a unique decomposition into idecomposables, we find o␣ Ž q . s

Ý Ł

␣ sÝ r i ␤ i ␤i/ ␤ j

i

ž

i␤ Ž q . q r i y 1 ri

/

This formula becomes more elegant in terms of generating functions

Ý o␣ Ž q . e Ž ␣ . s ␣

1 Ł␤ Ž 1 y e Ž ␤ . .

i␤ Ž q .

.

Ž 2.7.

Now combining Ž2.6. and Ž2.7., together with the fact that i␣ Ž0. s a␣ Ž0. w12x we find

Ł Ž 1 y eŽ ␣ . . a ␣

␣ Ž0 .

s

Ý r␣ Ž 0. e Ž ␤ . .

Ž 2.8.

␣

Let ⌬q; ⺓ Q 0 be the positive roots associated to the bilinear form Ž ᎐, ᎐ . w13x.

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

37

For ␣ g ⌬q let m␣ be the corresponding multiplicity, m␣ is determined by the formal identity m Ł Ž 1 y eŽ ␣ . .

␣g⌬ q

␣

Ý ⑀ Ž w. eŽ ␳ y wŽ ␳ . . .

s

Ž 2.9.

wgWQ

Comparing Ž2.8. and Ž2.9. yields a first reformulation of Conjecture 1.4. PROPOSITION 2.2. Conjecture 1.4 is equi¨ alent to r␣ Ž 0 . s

½

⑀ Ž w. 0

if ␣ s ␳ y w␳ otherwise.

Ž 2.10.

The above proposition can also be obtained from the results in Hua’s thesis w10x. Below we fill further simplify the computation of r␣ Ž0.. This will yield a proof of Theorem A.

3. SOME REMARKS ON SYMMETRIC FUNCTIONS Let Pn be the set of partitions of a natural number n. If ␭ g Pn then s␭Ž x . and P␭Ž x; t . denote, respectively, the Schur function and the Hall᎐Littlewood function associated to ␭ Žin an infinite number of variables. w17x. As usual we denote by l Ž ␭. the largest i such that ␭ i / 0, < ␭ < s Ý ␭ i , and nŽ ␭. s ÝŽ i y 1. ␭ i . ␭X is the conjugate partition to ␭. We also associate with ␭ its diagram in ⺞ 2 . The notation y g ␭ means that y is one of the boxes in this diagram. In that case hŽ y . is the hook length of y. The relation between s␭ and P␭ is given by s␭ Ž x . s

Ý K␭␮ Ž t . P␮ Ž x ; t . ,

Ž 3.1.

␮

where K␭␮Ž t . are the Kostka᎐Foulkes polynomials in one variable. For use below it is convenient to introduce the modified version K˜␭␮ Ž q . s q nŽ ␮ . K␭␮ Ž qy1 . . If ␮ is a partition of n then below u␮ will denote an arbitrary unipotent element of GLŽ n, q . corresponding to ␮. We let C Ž u␮ . be its centralizer.

38

SEVENHANT AND VAN DEN BERGH

According to w17, Example III.3.2x, q nŽ ␮ .

P␮ Ž qy1 , qy2 , . . . ; qy1 . s

C Ž u␮ .

as a formal power series in qy1 . On the other hand by w17, Example I.3.2x Žreplacing q by qy1 and multiplying by qy< ␭ < . we find y1

s␭ Ž q

y2

,q

X

qy< ␭
,.... s

Ł y g ␭ Ž 1 y qyhŽ y. .

s

q nŽ ␭ . Ł y g ␭ Ž q hŽ y . y 1 .

.

Substituting this in Ž3.1. yields

Ý ␮

K˜␭␮ Ž q . C Ž u␮ .

X

s

q nŽ ␭ . Ł y g ␭ Ž q hŽ y . y 1 .

.

Thus we find that the left-hand side of the above equation does not have a pole in q s 0. Furthermore

Ý ␮

K˜␭␮ Ž q . C Ž u␮ .

s qs 0

½

Ž y1.

n

0

if ␭ s Ž 1n . otherwise.

Ž 3.2.

4. A LEMMA FROM THE REPRESENTATION THEORY OF GLŽ n, q . By definition a multipartition of ␣ g ⺞ Q 0 is a list of partitions ␭ s Ž ␭ i . i g Q such that < ␭ i < s ␣ i . We view ␭ as a two dimensional list of 0 non-negative integers Ž ␭ i j . i j such that Ý j ␭ i j s ␣ i . We write ␭Ž j . for the element of ⺞ Q 0 given by ␭Ž j . i s ␭ i j . Denote the set of multipartitions of ␣ by P␣ . For ␭ g P␣ , u␭ will be an element of GLŽ ␣ , q . s Ł i GLŽ ␣ i , q . of the form Ž u␭i .. We also put f␭ Ž ␮ . s

Ł K˜␭ ␮ Ž q . . i

i

i

We view the f␭ as functions from P␣ to ⺪w q x. The following result is easy to see. LEMMA 4.1. Let W be a representation of GLŽ ␣ , ⺖p . o¨ er ⺖p . Then the u␮ function ␮ ¬ q dim ⺖ pW is a linear combination of the functions f␭ with coefficients in ⺪w q x.

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

39

Proof. Since the f␭Ž ␮ . form an upper triangular matrix with q-powers u␮ on the diagonal it is clear that we can express q dim ⺖ pW as a linear y1 combination of the f␭ with coefficients in ⺪w q , q x. Now let q be a fixed power of p and let V be the permutation u␮ representation of W m⺖ p ⺖q Žover ⺓.. Then q dim W s TrŽ u␮ , V .. Now by Green’s formula for the irreducible characters of GLŽ ␣ , q . we can express the values of the character of V at unipotent elements as a ⺡-linear combination of Žproducts of. Green polynomials w22, p. 135x. Furthermore the denominators of the coefficients are bounded in terms of ␣ . Expressing these Green polynomials further in terms of the f␭ w17, III.Ž7.11.x we u␮ find that for a fixed q, q dim W is a linear combination of the f␭Ž q . with coefficients in ⺡, whose denominator is still bounded in terms of ␣ . If we now let q go to ⬁ we see that if we consider q as a variable again, the remark of the first paragraph yields that the coefficients must be in ⺪w q x.

5. PROOF OF THEOREM A In this section we use all the notations from the previous sections. In particular Q is a fixed quiver without oriented cycles. It is easy to see that we have r␣ Ž q . s

R Ž Q, ␣ , q .

Ý

u␭

.

C Ž u␭ .

␭g P␣

Now by Lemma 4.1 we have R Ž Q, ␣ , q .

u␭

s

Ý c␮ f␮ Ž ␭. , ␮

where c␮ g ⺪w q x. This yields r␣ Ž q . s

Ý

␭ , ␮ g P␣

c␮

f␮ Ž ␭ . C Ž u␭ .

.

Let us rewrite this using the definition of f␮ , r␣ Ž q . s

Ý

␮g P␣

c␮

Ł Ý

igQ 0 ␭ g P i ␣i

K˜␮ i , ␭ iŽ q . C Ž u␭i .

.

40

SEVENHANT AND VAN DEN BERGH

Using Ž3.2. we now find r␣ Ž 0 . s Ž y1 .

Ýi ␣ i

c1 ␣ Ž 0 . ,

Ž 5.1.

where Ž1␣ . i s 1␣ i . Hence the key point is to find c1␣ . Going back to the definition of f␮ we have that R Ž Q, ␣ , q .

u␭

s

Ý c␮ Ł ␮

igQ 0

q nŽ ␭ i . K␮ i␭ iŽ qy1 . .

Ž y1 . and summing over ␭ yields Multiplying with Ł i g Q 0 qyn Ž ␭ i . Ky1 ␭ i1␣ i q c1 ␣ s

u Ý R Ž Q, ␣ , q . Ł

y1 qyn Ž ␭ i . Ky1 .. ␭ i 1␣ i Ž q

␭

␭

igQ 0

Substituting into Ž5.1. and combining with Ž1.2. Žwhich is proved in Appendix A. we find

Ž y1.

Ýi ␣ i

c1 ␣ s

X i1

Ý Ž y1. Ý ␭ i

␭ g P␣

qye␭ Ł w ␭Xi x Ž qy1 . , i

where e␭ s ydim R Ž Q, ␣ , q .

u␭

q

nŽ ␭i . q

Ý igQ 0

s ydim R Ž Q, ␣ , q .

u␭

q

Ý

2

igQ 0 jG2

␭Xi2j

Ý

␭Xi j Ž ␭Xi j q 1 .

y

igQ 0 jG0

Ý

␭Xi1 Ž ␭Xi1 q 1 . 2

igQ 0

.

On the other hand it is easy to see that dim R Ž Q, ␣ , q .

u␭

s

Ý

a i k ␭Xi j ␭Xk j ,

i , kgQ 0 jG0

where a i k is the number of arrows going from i to k. Define ␭Ž j . as the element of ⺞ ⺡ 0 given by ␭Ž j . i s ␭ i j . Then the expression for e␭ can be further rewritten as e␭ s

1 2

X

X

Ý Ž ␭ Ž j. , ␭ Ž j. . y Ý jG0

igQ 0

␭Xi1 Ž ␭Xi1 q 1 . 2

.

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

41

So now our final expression for Žy1. Ý i ␣ i c1␣ becomes X

Ý

X

qyÝ j Ž ␭ Ž j., ␭ Ž j..r2

X i1

Ł Ž y1. ␭

igQ 0

␭g P␣

X

X

q Ž ␭ i1Ž ␭ i1q1..r2 w ␭Xi x Ž qy1 . .

Ž 5.2.

We now put p␣ Ž t . s Žy. Ý i ␣ i c1␣ Ž ty1 .. Invoking Ž5.1. together with Proposition 2.2 and furthermore replacing ␭X by ␭ and qy1 by t finishes the proof of Theorem A.

6. SOME COMBINATORIAL CONSIDERATIONS As a warm-up we will consider Theorem A in the case that Q is a one-vertex quiver with no loops. Then Theorem A asserts that pa Ž t . s

Ý Ž y1. ␭ t Ý ␭ y ␭ Ž ␭ q1.r2 w ␭ x 1

i

2 i

1

1

Ž 6.1.

< ␭
is a polynomial in ty1 . On the other hand inspection reveals that if a ) 1 then paŽ t . g t⺪w t x. Combining this we obtain: PROPOSITION 6.1. The Laurent polynomial paŽ t . gi¨ en by Ž6.1. is identically zero. This result is probably known in some form. In any case the authors found it a pleasant exercise to prove it directly. Now let us consider the m-arrow quiver. In that case we obtain using the same method as above: PROPOSITION 6.2. The Laurent polynomial pa, b Ž t . defined by Ž1.1. is identically zero if a 4 b. Computer computations show that probably the following conjecture is true. Conjecture 6.3. The Laurent polynomial pa, b Ž t . is identically zero if and only if a G mb q 2 or b G ma q 2. It is clear that a similar reasoning can be applied to more complicated quivers.

APPENDIX A: ON THE INVERSE OF THE KOSTKA᎐FOULKES MATRIX In this Appendix we prove formula Ž1.2.. As was already pointed out in the Introduction, this result was also proved by Macdonald w3x in the special case that ␮ s 1n.

42

SEVENHANT AND VAN DEN BERGH

Ž . Ž . Denote by Ky1 ␭␮ t the inverse of the Kostka᎐Foulkes matrix K␭␮ t . Thus by definition P␭ Ž x ; t . s

Ý Ky1 ␭␮ Ž t . s␮ Ž x . . ␮

Our basic tool will be the fact that both P␮ and s␮ satisfy a version of Pieri’s formula for multiplication by s1m s P1m s e m Žthe mth elementary symmetric function w17x.. This will eventually lead to a recursion formula for the entries of Ky1 Ž t .. Let us use the notation ␮ -m ␯ to signify that ␯ y ␮ is vertical strip of length m Ža vertical strip is a skew diagram such that every horizontal line cuts the diagram at most once.. Pieri’s formula for s␮ is classical w17, I.Ž5.17.x: s␮ e m s

s␯ .

Ž A.1.

f␮␯, 1m Ž t . P␯

Ž A.2.

Ý

␮-m ␯

Pieri’s formula for P␮ is similar w17, III.3x, P␮ e m s

Ý

␮-m ␯

where f␮␯ , 1m Ž t . s

Ł i

X ␯ iX y ␯ iq1 X ␯ i y ␮Xi .

Here and below w ab x is an abbreviation w b a ya b x. For simplicity we will routinely use the convention that a Gaussian binomial coefficient w ab x is zero if b - 0. This allows us to be somewhat informal with regard to summation bounds. y1 Ž . Ž . Ž . Substituting P␯ s Ý ␣ F ␯ Ky1 ␯ ␣ t s␣ , P␮ s Ý ␤ F ␮ K␮ ␤ t s␤ in A.2 yields

Ý

␤F ␮

Ky1 ␮␤ Ž t . s␤ e m s

Ý Ý

␮-m ␯ ␣ F ␯

f␮␯, 1m Ž t . Ky1 ␯ ␣ Ž t . s␣ .

Using Pieri’s formula for s␤ now yields

Ý Ý

␤ F ␮ ␤ -m ␦

Ky1 ␮␤ Ž t . s␦ s

Ý Ý

␮-m ␯ ␣ F ␯

f␮␯, 1m Ž t . Ky1 ␯ ␣ Ž t . s␣ .

Equating the coefficients of s␦ in this equation we obtain for all pairs of partitions ␮ , ␦ such that < ␦ < s m q < ␮ < a relation

Ý Ý

␤ F ␮ ␤ -m ␦

Ky1 ␮␤ Ž t . s

Ý Ý

␮-m ␯ ␦ F ␯

f␮␯, 1m Ž t . Ky1 ␯␦ Ž t . .

Ž A.3.

43

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

Ž . It is easy to see that this relation determines Ky1 ␺⑀ t uniquely. To see this apply ŽA.3. with ␦ s ⑀ and ␮ equal to ␺ minus the last column. Then y1 Ž . y1 Ž . < < < < ŽA.3. expresses Ky1 ␺⑀ t in terms of K␮ ␤ t with ␮ - ␺ and K␯⑀ with ␯ - ␺ . All of these may be assumed to be known by induction. Equation ŽA.3. simplifies considerably if we apply it in the case ␦ s Ž c, 1d .. We find y1 Ky1 ␮ , Ž cy1, 1 dy mq 1 . Ž t . q K␮ , Ž c , 1 dy m . Ž t . s

Ý

␮-m ␯

f␮␯, 1m Ž t . Ky1 ␯ , Ž c , 1 d . Ž t . . Ž A.4 .

It follows that in order to prove our theorem, it is sufficient to show that Ž1.2. gives the correct value if ␭ s 1n and satisfies the recursion relation ŽA.4.. The first statement is obvious, so we will prove the second one. We will do this in the case c ) 1. The case c s 1 is similar but requires fewer steps. So assume c ) 1. Making the appropriate substitutions and replacing the partitions ␮ , ␯ by their conjugate ones we find that we have to prove the identity

ž

Ž y1.

1 ␮ 1 qdymq2 Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ cy js 2 ␮ j

t

=t Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ js 2 ␮ j c

Ý Ž y1. ␯ qdq1 t Ý

s

1

1 y t ␮ cy 1 1 y t ␮1

1 y t ␮c 1 y t ␮1

/

=

c i G 2 Ž ␯ i Ž ␯ iq1..r2yÝ js 2 ␯ j

␮-m ␯

=

1 y t ␯c ␯ 1 y ␯ 2 1 y t ␯ 1 ␯ 1 y ␮1

q Ž y1 .

␮ 1 qdymq1

␮1 y ␮ 2

␮1 ␮2 y ␮3

␯1 y ␯2

␯1 ␯2 y ␯3

⭈⭈⭈ ⭈⭈⭈

␯2 y ␯3 ␯ 2 y ␮ 2 ⭈⭈⭈ .

We now put b s ␮ 1 , a i s ␮ iy1 y ␮ i , ri s ␯ i y ␮ i . After some algebraic manipulation the previous equation becomes

Ž y1.

1 ym q1 Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ cy js 2 ␮ j

t

s Ž 1 y t ␮c .

r Ý Ž y1.

1

Ž 2 y t ␮ cy 1 y t ␮ c .

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2yÝ js 2 Ž ␮ iqr i . c

Ý i r ism

=

b q r1 y 1 r1

q t ␮c Ž 1 y t a c .

a2 r2

a3 ⭈⭈⭈ r3

r Ý Ž y.

1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2yÝ js 2 Ž ␮ iqr i . c

Ýr ism

=

b q r1 y 1 r1

a2 r2

ac y 1 a3 ⭈⭈⭈ ⭈⭈⭈ . r3 rc y 1

Ž A.5.

44

SEVENHANT AND VAN DEN BERGH

Before we continue we derive an identity between Gaussian binomial coefficients using non-commutative generating functions. If x, y are variables satisfying yx s txy then it is well known that a

Ž x q y. s

Ý uG0

a x u y ayu . u

We will also apply this with negative exponents:

Ž x q y.

ya

s

Ý uG0

s

ya x u yyayu u

u Ý Ž y1. tya uqŽ uŽ uy1..r2

uG0

a q u y 1 x u yyayu . Ž A.6. u

Now we introduce variables Ž x i . i , Ž yi . i satisfying yi x j s tx j yi yi y j s y j yi xi x j s x j xi and for b g ⺞, Ž a i . i g ⺪, a i s 0 for i 4 0, we consider ⭈⭈⭈ Ž x n q yn .

an

a

⭈⭈⭈ Ž x 2 q y 2 . 2 Ž x 1 q y 1 .

yby1

.

Ž A.7.

This yields

Ž A.7. s

Ý Ž y1.

r 1 yŽ bq1. r yŽ r Ž r y1..r2 1 1 1

t

Ž ri .i

b q r1 r1

a2 r2

a3 ⭈⭈⭈ r3

y1yr 1 = ⭈⭈⭈ x 3r 3 y 3a 3yr 3 x 2r 2 y 2a 2yr 2 x 1r 1 yyb 1

s

Ý Ž y1.

r1 S r

t

Ž ri .i

b q r1 r1

a2 r2

a3 ⭈⭈⭈ = x 1r 1 x 2r 2 x 3r 3 ⭈⭈⭈ r3

y1yr 1 a 2yr 2 a 3yr 3 = yyb y2 y3 ⭈⭈⭈ , 1

where Sr s ybr1 y

r1Ž r1 q 1. 2

q

Ý Ž a i y ri . r j . 1Fj-i

Substituting x i s x, yi s y yields

Ž x q y. s

a 2 qa 3 q ⭈⭈⭈ yby1

Ý Ž y1. Ž ri .i

r1 S r

t

b q r1 r1

a2 r2

a3 ⭈⭈⭈ = x Ý r i yyby1yÝ r iqÝ i G 2 a i . r3

45

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

We consider the case b s Ý i G 2 a i and we look at the coefficient of x m yym y1. Using

Ž x q y.

y1

s

Ý Ž y1.

m yŽ mŽ mq1..r2

t

x m yymy1

m

we find r Ý Ž y1.

1

t Tr

Ý i r ism

b q r1 r1

a2 r2

a3 m ⭈⭈⭈ s Ž y1 . , r3

Ž A.8.

where Tr s Sr q

m Ž m q 1. 2

s

ri a j q

Ý j)iG2

Ý

ri Ž ri q 1 . 2

iG2

.

Let ␮ , ␯ , b, a2 , a3 , . . . be related as above. Then a straightforward computation yields that Tr s

Ý

␯ i Ž ␯ i q 1. 2

iG2

y

Ý

␮i Ž ␮i q 1. 2

iG2

so that we obtain the identity r Ý Ž y1.

1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2

Ý i r ism

b q r1 r1

m

s Ž y1 . t Ý i G 2 Ž ␮ i Ž ␮ iq1..r2 .

a2 r2

a3 ⭈⭈⭈ r3

Ž A.9.

Note that in the proof of this identity we have not used that ␮ is a partition. In fact ␮ can be an arbitrary sequence of integers, zero in high degree and positive in degree 1 Žthe restriction ␮ 1 G 0 comes from the fact that ŽA.6. is not valid for negative a.. Below we will use ŽA.9. for ␮ which are not necessarily partitions. We will use ŽA.9. to simplify the right-hand side of ŽA.5.. Let’s first work on the second term. Put

␮i s ri s

½ ½

␮i y 1 ␮i ri y 1 ri

i-c i G c, isc i / c.

46

SEVENHANT AND VAN DEN BERGH

We make the companion definitions a i s ␮ iy1 y ␮ i s

½

ai ai y 1

i/c i s c,

b s ␮ 1 s b y 1. Making these substitutions, the second term of the right-hand side of ŽA.5. becomes t ␮c Ž 1 y t a c .

r Ž y1. 1 t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2

Ý

Ýr ismy1

b q r1 r1

a1 r1

a2 ⭈⭈⭈ . r2

Ž A.10. Using the identity ŽA.9., ŽA.10. may be rewritten as

Ž y1.

my 1 ␮ c

t Ž 1 y t a c . t Ý i G 2 Ž ␮ i Ž ␮ iq1..r2 ,

which is equal to

Ž y1.

my 1 ␮ c

t Ž 1 y t a c . t Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ js 2 ␮ j . cy 1

Substracting this from the left-hand side of ŽA.5., and dividing by 1 y t ␮ c , we are left with proving m

Ž y1. t Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ js 2 ␮ j Ž 1 y t ␮ c . s

c

r Ý Ž y1.

1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2yÝ js 2 Ž ␮ jqr j . c

Ýr ism

=

b q r1 y 1 r1

a2 r2

a3 ⭈⭈⭈ . r3

Ž A.11.

We now work on the right-hand side of ŽA.11.. We first make the change of variables

␮i s

½

␮i y 1 ␮i

i-c i G c,

and companion definitions a i s ␮ iy1 y ␮ i s b s ␮ 1 s b y 1.

½

ai ai y 1

i/c i s c,

47

ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

So the right-hand side of ŽA.11. becomes r Ý Ž y.

1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2yŽ ␮ cqr c .

Ýr ism

b q r1 r1

a2 a q1 ⭈⭈⭈ c ⭈⭈⭈ . r2 rc

Now we use the identity ac ac q 1 a s q t rc c rc rc rc y 1 and we obtain that the right-hand side of ŽA.11. is equal to r Ý Ž y.

1

b q r1 r1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2yŽ ␮ cqr c .

Ýr ism

q

r Ý Ž y.

1

ac a2 ⭈⭈⭈ ⭈⭈⭈ r2 rc y 1

b q r1 r1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2y ␮ c

Ýr ism

a2 a ⭈⭈⭈ c ⭈⭈⭈ . r2 rc

Using ŽA.9. the second part of the previous formula is seen to be equal to m

m

Ž y1. ty ␮ cqÝ i G 2 Ž ␮ i Ž ␮ iq1..r2 s Ž y1. t Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ js 2 ␮ j . c

Substracting this from the left-hand side of ŽA.11. we are now left with proving

Ž y.

1 mq 1 Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ cy js 2 ␮ j

t

s

r Ý Ž y.

1

t Ý i G 2 ŽŽ ␮ iqr i .Ž ␮ iqr iq1..r2yŽ ␮ cqr c .

Ýr ism

=

ac a2 ⭈⭈⭈ ⭈⭈⭈ . r2 rc y 1

b q r1 r1

Ž A.12.

We now put ri s

½

ri rc y 1

i/c i s c.

Then the right-hand side of ŽA.12. becomes r Ý Ž y1.

rismy1

1

t Ý i G 2 ŽŽ ␮ qr i .Ž ␮ qr iq1..r2

b q r1 r1

a2 ⭈⭈⭈ , r2

48

SEVENHANT AND VAN DEN BERGH

which by ŽA.9. is equal to

Ž y1.

my 1 Ý i G 2 Ž ␮ i Ž ␮ iq1..r2

t

s Ž y1 .

1 my1 Ý i G 2 Ž ␮ i Ž ␮ iq1..r2yÝ cy js 2 ␮ j

t

.

This is indeed equal to the left-hand side of ŽA.12. and hence we are done.

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ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

49

18. C. M. Ringel, Hall algebras and quantum groups, In¨ ent. Math. 101 Ž1990., 583᎐592. 19. B. Sevenhant, ‘‘Some More Evidence for a Conjecture of Kac,’’ in preparation. 20. B. Sevenhant and M. Van den Bergh, ‘‘A Relation Between a Conjecture of Kac and the Structure of the Hall Algebra,’’ to appear. 21. R. P. Stanley, ‘‘On Counting Inequivalent Representations over GFŽ q .,’’ notes, 1981, unpublished. 22. A. V. Zelevinski, ‘‘Representations of Finite Classical Groups,’’ Lecture Notes in Mathematics, Vol. 869, Springer-Verlag, Berlin, 1981.