Christine Bessenrodt. Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, Hannover, Germany

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1 Séminire Lothringien de Comintoire 54A (2006), Article B54Ag ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS Christine Bessenrodt Fkultät für Mthemtik und Physik, Leiniz Universität Hnnover, Welfengrten 1, Hnnover, Germny Astrct. Finite directed grphs with reltions define finite-dimensionl lgers given s fctor lgers of the corresponding pth lgers. In joint work with T. Holm, we studied invrints of n ssocited generlized Crtn mtrix for certin types of lgers where the Crtn invrints siclly count nonzero pths; the min motivtion for this cme from the context of derived module ctegories nd derived invrints. For the so-clled (skewed) gentle lgers we computed explicit comintoril formule for the invrint fctors of the Crtn mtrices. Bsed on the tlk t the FPSAC 05 conference in Tormin, some of this work is surveyed here, nd some further results on lgers of different type re presented in the finl section. 1. Introduction Finite directed grphs ply n importnt rôle in the representtion theory of finite-dimensionl lgers, where they re clled quivers. For given quiver Q nd field K one otins n lger KQ y tking ll pths in Q, including the trivil pths of length 0 t ech vertex, s K-sis, nd defines multipliction s induced y conctention of pths; this is clled the pth lger to the quiver Q. The importnce of the pth lgers lies in fmous result y Griel tht ny finite-dimensionl K-lger over n lgericlly closed field K is Morit equivlent to fctor lger KQ/I, where I is n dmissile idel of KQ (i.e., contined in the squre of the idel of KQ generted y the rrows). Thus such fctor lgers KQ/I re centrl ojects of study in the representtion theory of lgers; this sitution is referred to s quiver with reltions. There re mny interesting representtion theoretic properties of such lgers for which one tries to find comintoril methods for computing them. It is of prticulr importnce to consider representtion theoretic prmeters of the lger which re invrints for pproprite equivlence clsses of lgers. In recent yers focus in the representtion theory of lgers hs een the investigtion

2 2 CHRISTINE BESSENRODT of derived equivlences of lgers; this is homologicl notion: two lgers re derived equivlent if their derived module ctegories re equivlent. A lot of progress hs recently een mde in this very ctive re. It is difficult prolem to find invrints of lgers preserved y derived equivlences; only few importnt representtion theoretic prmeters re known to e indeed invrints under derived equivlence, such s the numer of simple modules, the dimension of the center of the lger or the dimension of its Hochschild cohomology groups. Here, we will discuss invrints of the Crtn mtrix of finite-dimensionl lger A = KQ/I. Crtn mtrices contin crucil structurl informtion on the lger, s their entries re the multiplicities of simple A-modules s composition fctors of projective indecomposle A-modules. It is in generl difficult to compute the entries of the Crtn mtrix nd some fmous conjectures re relted to their properties. The min point to note here is tht the unimodulr equivlence clss of the Crtn mtrix of finite dimensionl lger is invrint under derived equivlence. From comintoril point of view it is importnt tht for finite-dimensionl lger A = KQ/I given y quiver with reltions, the entries of the Crtn mtrix cn e computed y considering pths in the quiver Q which re non-zero in the lger A. The lgers we study here re the (skewed-) gentle lgers which re defined comintorilly y conditions on the quiver nd reltions. Gentle lgers occur nturlly in mny plces in the representtion theory of finite dimensionl lgers, in prticulr in connection with derived ctegories. They mde their first ppernce in 1981 [1] when it ws shown tht the lgers which re derived equivlent to hereditry lgers of type A re precisely the gentle lgers whose underlying undirected grph is tree. The lgers which re derived equivlent to hereditry lgers of type à re certin gentle lgers whose underlying grph hs exctly one cycle [2]. Only recently it ws proved tht the clss of gentle lgers hs the remrkle property of eing closed under derived equivlence [10]. For more ckground on the lgeric context the reder is referred to [5]. In joint work with T. Holm [5] we hve studied the Crtn mtrices of (skewed- )gentle lgers nd some refinements nd generliztion of this notion; the strting point of our investigtion ws recent result y T. Holm [8] giving n explicit comintoril formul for the Crtn determinnts of gentle lgers. In [5], these results re improved to determintion of the invrint fctors of the Crtn mtrix C A of gentle lger A = KQ/I, nd the formule re lso extended to skewed-gentle lgers. Sections 2 nd 3 survey some of these results, nd (s in the tlk t FPSAC 05) we tke the opportunity to include n lterntive proof for the fct tht the numer of oriented cycles with full zero reltions in such quiver is ounded y the numer of vertices. The key to the refinement of the

3 ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS 3 comintoril nlysis of the quiver is to put weight on the pths ccording to their lengths insted of just counting them; in our context, this mkes good sense s the reltions on the quiver re homogeneous (in fct, they re even more specil). Tking n indeterminte q corresponding to the weight of n rrow, this gives us q-crtn mtrix C A (q) for the lger; this my lso e considered s so-clled filtered Crtn mtrix, counting the multiplicities of the simple modules in the rdicl lyers of the lger. Setting q = 1 gives the ordinry Crtn mtrix C A. We hve lredy pointed out tht the unimodulr equivlence clss of the Crtn mtrix of finite-dimensionl lger is invrint under derived equivlence. But unfortuntely, not even the determinnt of the q-crtn mtrix is in generl n invrint under derived equivlence. There re further finite-dimensionl lgers given y quivers nd reltions for which it is possile to determine the q-crtn mtrices nd otin nice formule for their invrints or t lest their determinnt. A further fmily of lgers ssocited with cyclic quivers is considered in the finl section 4; here, the q- Crtn mtrices re specil circulnt mtrices nd we compute their determinnts nd invrints explicitly. Finlly, in more recent joint work with T. Holm [6] we hve lso delt with clss of infinite-dimensionl lgers which we termed loclly gentle lgers; this clss is gin defined comintorilly y relxing the conditions on the gentle quiver. Also, in this context it is then well motivted to study more generl weights on the quiver. The comintoril point of view dds further interest to these quivers, s for exmple secnt configurtions in regulr 2n-polygons my e viewed s exmples of loclly gentle quivers. For detils the reder is referred to [6]. 2. Gentle lgers In this section, we wnt to descrie n extension nd refinement of the result on the determinnt of the Crtn mtrix of gentle lger from [8] otined in [5] (see [5] for detils). First we hve to give the definition of gentle lgers. They form n importnt suclss of the clss of specil iseril lgers which we now define. Let K e n lgericlly closed field. Let Q e quiver, i.e., finite directed grph, with set of vertices Q 0. Let I e n dmissile idel of the pth lger KQ, i.e., I J 2, where J is the idel of KQ generted y the rrows. For pth p in Q we denote y s(p) its strt vertex nd t(p) its end vertex. The pir (Q, I) is clled specil iseril if the following holds: (i) For ny vertex v Q 0 the set of lengths of the pths strting in v nd not eing in I is finite. (ii) Ech vertex v Q is the end point of t most two rrows nd the strting

4 4 CHRISTINE BESSENRODT point of t most two rrows. (iii) For every rrow α there is t most one rrow β with t(α) = s(β) nd αβ I, nd there is t most one rrow γ with t(γ) = s(α) nd γα I. A specil iseril pir (Q, I) is clled gentle, if furthermore: (iv) There is generting set of I (s idel) consisting of pths of lengths 2. (v) For ny rrow α there is t most one rrow β with t(α) = s(β) nd αβ I, nd there is t most one rrow γ with t(γ) = s(α) nd γα I. A K-lger A is clled gentle (resp. specil iseril) if it is Morit equivlent to n lger KQ/I, for (Q, I) gentle (resp. specil iseril). As we ssume tht the set of vertices Q 0 of Q is finite, condition (i) implies tht the lger KQ/I is finite-dimensionl. As mentioned in the introduction, in [6] the notion of gentle quivers is generlized to loclly gentle quivers; for these quivers, condition (i) need not e stisfied, nd thus the corresponding lgers cn e infinite-dimensionl. By condition (iv), the reltions for gentle quiver re homogeneous, so the length of non-zero pth in KQ/I is well-defined. Then we define the q-crtn mtrix of A = KQ/I s C A (q) = (c ij (q)) i,j Q0, where for i, j Q 0 c ij (q) = n 0 n (i, j)q n with n (i, j) the numer of pths from i to j which re non-zero in A (nd different in A). Exmple. In the following picture, the dotted lines (or rcs) etween two rrows or in loop correspond to the generting reltions for the quiver, i.e., they indicte tht composing the corresponding rrows is zero in the lger A = KQ/I Here is the q-crtn mtrix for this gentle quiver with reltions: 1 + q 5 q + q 2 + q 7 q + q 4 + q 6 + q 9 q 3 + q 4 + q 8 q 2 + q 3 q q 6 q 2 + q 5 + q 8 q + q 3 + q 7 q + q 2 C A (q) = 0 q 1 + q 3 q 2 0 q q 3 q + q 2 + q q 4 0 q 2 + q 3 q 4 + q 5 q 3 + q 4 + q 6 + q 7 q + q 2 + q 5 + q q 3

5 ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS 5 The following property does not only hold for gentle lgers ut for those where we hve dropped the finl condition (v) in the definition of gentle pirs; in [5], it is useful reduction tool in the proof of the min result. Lemm 2.1. Let A = KQ/I e specil iseril lger, where I is generted y pths of length 2. Let α e n rrow in Q, not loop, such tht there is no rrow β with s(α) = t(β) nd βα I, or there is no rrow γ with t(α) = s(γ) nd αγ I. Let Q e the quiver otined from Q y removing the rrow α, I the corresponding reltion idel nd A = KQ /I. Then the q-crtn mtrices C A (q) nd C A (q) re unimodulrly equivlent (over Z[q]). For the comintorilly defined gentle lgers we cn provide n explicit comintoril description for very nice norml form of its q-crtn mtrix; elow, we sy tht mtrix C is unimodulrly equivlent (over Z[q]) to mtrix D, if there re mtrices S, T with entries in Z[q] nd determinnt 1 such tht SCT = D. Theorem 2.2. [5] Let A = KQ/I e gentle lger, defined y gentle pir (Q, I). Denote y c k the numer of oriented k-cycles in Q with full zero reltions. Then the q-crtn mtrix C A (q) is unimodulrly equivlent (over Z[q]) to digonl mtrix with entries (1 ( q) k ), with multiplicity c k, k 1, nd ll further digonl entries eing 1. This result hs some immedite nice consequences. Corollry 2.3. Let A = KQ/I e gentle lger, nd denote y c k the numer of oriented k-cycles in Q with full zero reltions. Then the q-crtn mtrix C A (q) hs determinnt det C A (q) = k 1(1 ( q) k ) c k. Exmple. In the exmple given efore, the q-crtn mtrix hs determinnt det C A (q) = 1 + q + q 3 q 5 q 7 q 8 = (1 + q)(1 + q 3 )(1 q 4 ). Indeed, the quiver hs loop t vertex 5 (which is 1-cycle with zero reltion), 3-cycle with full zero reltions from vertex 1 to 2 to 4 nd ck to 1, nd 4-cycle with full zero reltions running over 2,5,4,3 nd ck to 2. For n lger A = KQ/I s ove, let ec(a) nd oc(a) e the numer of oriented cycles in Q with full zero reltions of even nd odd length, respectively. Setting q = 1, the corollry ove immeditely implies the min result from [8] which ws in fct the strting point of our investigtions in [5]. Corollry 2.4. Let A = KQ/I e gentle lger. Then for the determinnt of its ordinry Crtn mtrix C A the following holds. { 0 if ec(a) > 0 det C A = 2 oc(a) else

6 6 CHRISTINE BESSENRODT The Theorem lso gives the following corollry. Corollry 2.5. Let A = KQ/I e gentle lger. Then there re t most Q 0 oriented cycles with full zero reltions in the quiver Q. Insted of deriving this from the min result (s done in [5]), this my e proved directly. As n illustrtion, we give this lterntive proof here. Clerly the result holds when Q 0 = 1. So we ssume now tht Q 0 > 1, nd we prove the result y induction. We my remove ll rrows going in or out of vertex nd not hving zero reltion t this vertex, without losing the property of the quiver with reltions eing gentle nd without chnging the numer of vertices nd the numer of oriented cycles with full reltions. After this removl, ll vertices re of degree 0, 2 or 4, nd there is zero reltion t ll vertices of degree 2. By induction, we my ssume tht there is no vertex of degree 0 nd tht the quiver is connected. If ll vertices re of degree 4, then there re pths of ritrry lengths, contrdicting the property of eing gentle. Hence there is vertex of degree 2, which then elongs to unique oriented cycle with full zero reltions. Removing this vertex nd the rrows incident to it reduces the numer of vertices s well s the numer of oriented cycles with full zero reltions y 1 (note tht ny rrow in gentle quiver elongs to t most one oriented cycle with full zero reltions). Hence the result follows y induction. Remrk 2.6. Note tht in our context Q 0 is the numer l(a) of simple A- modules, which is lso invrint under derived equivlence. Hence this implies tht the Crtn determinnt of gentle lger A is t most 2 l(a). Recll tht the property of n lger eing gentle is invrint under derived equivlence [10]. Also, we hve pointed out erlier tht the invrint fctors of the ordinry Crtn mtrix C A = C A (1) re invrints of the derived equivlence clss of the lger A = KQ/I. Thus we now hve some esily computle invrints for gentle lgers to distinguish the derived equivlence clsses. Corollry 2.7. Let A = KQ/I nd A = KQ /I e gentle lgers s ove. If A is derived equivlent to A, then ec(a) = ec(a ) nd oc(a) = oc(a ). Our new invrints re quite powerful tool for distinguishing gentle lgers up to derived equivlence which cnnot e seprted y the more clssicl invrints. An illustrtion on how to use the invrints to tell non-equivlent gentle lgers prt is given in [5], where the 9 gentle lgers with two simples nd the 18 gentle lgers with three simples nd vnishing Crtn determinnt re discussed in detil.

7 ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS 7 3. Skewed-gentle lgers Also skewed-gentle lgers re defined comintorilly. They were introduced in [7]; for the nottion nd definition we follow here mostly [4], ut we try to explin how the construction works rther thn stting the technicl definitions. We strt with gentle pir (Q, I). A set Sp of vertices of the quiver Q is n dmissile set of specil vertices if the quiver with reltions otined from Q y dding loops with squre zero t these vertices is gin gentle; we denote this gentle pir y (Q sp, I sp ). The triple (Q, Sp, I) is then clled skewed-gentle. We wnt to point out tht the dmissiility of the set Sp of specil vertices is oth locl s well s glol condition. Let v e vertex in the gentle quiver (Q, I); then we cn only dd loop t v if v is of degree 1 or 0 or if it is of degree 2 with non-loop zero reltion. Hence only vertices of this type re potentil specil vertices. But for the choice of n dmissile set of specil vertices we lso hve to tke cre of the glol condition tht fter dding ll loops, the pir (Q sp, I sp ) still does not hve pths of ritrry lengths. Given skewed-gentle triple (Q, Sp, I), we now construct new quiver with reltions ( ˆQ, Î) y douling the specil vertices, introducing rrows to nd from these vertices corresponding to the previous such rrows nd replcing previous zero reltion t the vertices y mesh reltion. More precisely, we proceed s follows. The non-specil vertices in Q re lso vertices in the new quiver; ny rrow etween non-specil vertices s well s corresponding reltions re lso kept. Any specil vertex v Sp is replced y two vertices v + nd v in the new quiver. An rrow in Q from non-specil vertex w to v (or from v to w) will e douled to rrows ± : w v ± (or ± : v ± w, resp.) in the new quiver; n rrow etween two specil vertices v, w will correspondingly give four rrows etween the pirs v ± nd w ±. We sy tht these new rrows lie over the rrow. Any reltion = 0 where t() = s() is non-specil gives corresponding zero reltion for pths of length 2 with the sme strt nd end points lying over. If v is specil vertex of degree 2 in Q, then the corresponding zero reltion t v, sy = 0 with t() = v = s(), is replced y mesh commuttion reltions sying tht ny two pths of length 2 lying over, hving the sme strt nd end points ut running over v + nd v, respectively, coincide in the fctor lger to the new quiver with reltions ( ˆQ, Î). We will spek of ( ˆQ, Î) s skewed-gentle quiver covering the gentle pir (Q, I). Note tht lso here the generting reltions re homogeneous. A K-lger is then clled skewed-gentle if it is Morit equivlent to fctor lger K ˆQ/Î, where ( ˆQ, Î) comes from skewed-gentle triple (Q, Sp, I) s descried ove.

8 8 CHRISTINE BESSENRODT Exmples. (1) We tke the gentle quiver Q s shown elow, with reltion idel I generted y nd. Then we cn tke Sp = {2}, i.e., only the vertex 2 is specified s specil vertex. This gives the skewed-gentle quiver ˆQ shown elow, with reltion idel Î generted y + +, ± ±, ±. Q Q^ + (2) We tke the gentle quiver Q s shown elow, with reltion idel generted y. This time we tke Sp = Q 0, i.e., ll vertices re chosen to e specil. This gives the skewed-gentle quiver ˆQ shown elow, where for simplicity ll rrows lying over or, respectively, re lso mrked or, respectively, ut for writing down the reltions generting the reltion idel Î we will put signs on, so tht e.g. + denotes the rrow going from 1 + to 2. In this nottion the generting reltions re given y , , , + +. Q Q^ Our result on gentle lgers generlizes nicely to skewed-gentle lgers (see [5] for detils) s follows. Theorem 3.1. [5] Let (Q, I) e gentle quiver, ( ˆQ, Î) covering skewed-gentle quiver. Let  = K ˆQ/Î e the corresponding skewed-gentle lger. Denote y c k the numer of oriented k-cycles in (Q, I) with full zero reltions. Then the q-crtn mtrix CÂ(q) is unimodulrly equivlent (over Z[q]) to digonl mtrix with entries 1 ( q) k, with multiplicity c k, k 1, nd ll further digonl entries eing 1. Remrk 3.2. Thus, the q-crtn mtrix C A (q) for the gentle lger A to (Q, I), nd the q-crtn mtrix CÂ(q) for skewed-gentle cover  re unimodulrly equivlent to digonl mtrices which only differ y dding s mny further 1 s on the digonl s there re specil vertices chosen in Q; in prticulr, with nottion s ove, det CÂ(q) = det C A (q) = k 1(1 ( q) k ) c k nd thus lso for the ordinry Crtn mtrices det C A = det CÂ.

9 ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS 9 4. Cycles nd circulnts There re some more types of lgers for which one cn determine the unimodulr equivlence clss of their q-crtn mtrices. As n exmple, we consider q-crtn mtrices to cyclic quivers with reltions which occur in other interesting contexts. In this sitution, circulnts mke n ppernce, nd we first define the relevnt nottion. For x = (x 1,..., x n ) the circulnt mtrix to x is C = x 1 x 2... x n 1 x n x n x 1... x n 2 x n 1. x 2 x 3... x n x 1. Let ω e primitive n-th root of unity in C. Then C hs the eigenvlues n x k (ω j ) k 1, j = 1,..., n. k=1 Specil circulnts pper s q-crtn mtrices for cyclic quivers with monomil reltions nd commuttion reltions which re homogeneous of degree 2. Exmple. Here is the corresponding cyclic quiver for the cse of 6 vertices: where we tke s generting reltions: 2, 2, (t ech vertex of the quiver). The q-crtn mtrix for such cyclic quiver is just the circulnt to v = (1 + q 2, q, 0..., 0, q). We will descrie more precise result on these q-crtn mtrices elow, ut s it is esy to do we compute here the determinnt. Note tht s in the cse of gentle quivers we hve contriution 1 ( q) n for ech of the two oriented cycles with full zero reltions in the quiver. Theorem 4.1. Let q e n indeterminte, v = (1 + q 2, q, 0..., 0, q) (of length n), nd C(q) the circulnt to v. Then we hve det C(q) = (1 ( q) n ) 2.

10 10 CHRISTINE BESSENRODT Proof. By the ove, we hve n det C(q) = (1 + q 2 + qω j + q(ω j ) n 1 ) = = j=1 j=1 n (1 + q 2 + q(ω j + ω j )) n n. (q + ω j )(q + ω j ) = ( (q + ω j )) 2 Now ω j is zero of q n 1, if n is even nd zero of q n + 1, if n is odd, for ll j, hence we hve the ssertion. In fct, it is not hrd to trnsform the circulnt C(q) to v = (1 + q 2, q, 0..., 0, q) into etter form: Theorem 4.2. Assume n 3, nd let C(q) e s ove. (i) Over Z[q], we cn trnsform C(q) unimodulrly into the form E n ( n ( q) n q 2j 1) ( q) n 1 j=1 j=1 j= ( q) n where E n 2 is the identity mtrix of type n 2. (ii) Over Q[q], we hve the following unimodulr equivlences: n 1 For n odd, C(q) dig(1 n 2, ( q) j, q n+1 + q n + q + 1). (n 2)/2 For n even, C(q) dig(1 n 2, q 2j, q n+2 q n q 2 + 1). j=0 j=0 References [1] I. Assem, D. Hppel, Generlized tilted lgers of type A n, Comm. Alger 9 (1981), no. 20, [2] I. Assem, A. Skowroński, Iterted tilted lgers of type Ãn, Mth. Z. 195 (1987), [3] V. Bekkert, H. Merklen, Indecomposles in derived ctegories of gentle lgers, Algers nd Representtion Theory 6 (2003), no. 3, [4] V. Bekkert, E. N. Mrcos, H. A. Merklen, Indecomposles in derived ctegories of skewedgentle lgers, Comm. in Alger 31 (2003), no. 6, [5] C. Bessenrodt, T. Holm, q-crtn mtrices nd comintoril invrints of derived ctegories for skewed-gentle lgers, to pper in: Pcific J. Mth. [6] C. Bessenrodt, T. Holm, Weighted loclly gentle quivers nd Crtn mtrices, preprint [7] C. Geiss, J. A. de l Peñ, Auslnder Reiten components for clns, Boll. Soc. Mt. Mexicn 5 (1999), [8] T. Holm, Crtn determinnts for gentle lgers, Archiv Mth. 85 (2005),

11 ALGEBRA INVARIANTS FOR FINITE DIRECTED GRAPHS WITH RELATIONS 11 [9] J. Rickrd, Morit theory for derived ctegories, J. London Mth. Soc. (2) 39 (1989), [10] J. Schröer, A. Zimmermnn, Stle endomorphism lgers of modules over specil iseril lgers, Mth. Z. 244 (2003),