Representationsofquiverswithautomorphismsoverfinitefields.doc

representations of quivers with automorphisms over finite fieldsbangming dengschool of mathematical sciences, beijing normal university, beijing 100875, chinaabdukadir obulcollege of mathematics and system sciences, xinjiang university, urumqi, 830046, chinayali pangdepartment of mathematics, east china normal university, shanghai 200241, china.abstractlet fq be the finite field of q elements and k = fq be its algebraic closure. let q be a quiver with automorphism . in this survey we focus on the study of modules over the fq -algebra a(q, ; q) associated with the pair (q, ) in terms of f -stable representationsof q over k, and we also discuss several polynomials obtained by counting numbers ofindecomposable a(q, ; q)-modules. in case q is a tame quiver, we present the formula for the number of isoclasses of indecomposable a(q, ; q)-modules with a fixed dimension vector.keywords: quiver with automorphism, hereditary algebra, representationalgebras with frobenius morphisms were introduced in 3, 5 in order to deal with modules over finite dimensional algebras over a finite field fq . a typical exampleis the fq -algebra a(q, ; q) (or the tensor algebra of an fq -species) associated witha quiver q with automorphism . as shown in 3, 5, there is a one-to-one corre-spondence between the isoclasses of a(q, ; q)-modules and the isoclasses of f -stablerepresentations of q over the algebraic closure k = fq of fq .in the present paper we study a(q, ; q)-modules in terms of f -stable representa- tions of q over k. various polynomials for counting the numbers of indecomposable a(q, ; q)-modules with fixed dimension vectors are defined. we also introduce the notion of -absolutely indecomposable a(q, ; q)-modules which generalizes that of absolutely indecomposable ones. this notion seems to be suitable for the representa- tion theory of fq -species. in case q is a tame quiver, these polynomials are explicitly computed. in particular, the formula for the number of isoclasses of indecomposable a(q, ; q)-modules with a fixed dimension vector is presented. this is based on the classification of representations of q over k given in 9, 26, 8.0supported partially by the natural science foundation of china and the doctoral program ofhigher education.1272throughout the paper, fq denotes the finite field of q elements, and k is the algebraic closure fq of fq . for any r 1, let fqr be the unique extension field of fq of degree r contained in k.ijgiven a k-vector space v , an fq -linear isomorphism f : v v is called a frobenius map if the following conditions are satisfied: (a) f (v) = q f (v) for all v v and k; (b) for every v v , f n(v) = v for some n 0. in particular, if v is finite dimensional, the condition (b) is a consequence of the condition (a). for a matrix x = (xij ) kmn, we define x1 := (xq ) kmn.let f : k k be the field automorphism taking 7 q . for each k-vector spacev , let v (1) be the new vector space obtained from v by base change via f:v (1) = v f kfor each v v , we write v(1) = v 1. further, for a k-linear map : u v , the map (1) := f 1 : u (1) v (1) is again a k-linear map. let v : v v (1) be the fq -linear isomorphism sending v to v(1) .1. quivers with automorphisms and f -stable representationsin this section we recall from 3, 4, 5 some basic results on f -stable representations of a quiver q with automorphism . we then relate f -stable representations of q with a(q, ; q)-modules, where a(q, ; q) is the hereditary fq -algebra associated with the pair (q, ).let q = (q0 , q1 ) be a finite quiver with vertex set q0 and arrow set q1 . for each arrow in q1 , we denote by h and t the head and the tail of , respectively. let further be an automorphism of q, that is, is a permutation on q0 and on q1 satisfying (h) = h() and (t) = t() for any q1 . then also acts on the set of all paths of q in an obvious way.let a = kq denote the path algebra of q over k. the automorphism of qinduces a frobenius morphismsfq,;q : a a, x xsps x xq (ps),sswhere ps xsps is a k-linear combination of paths ps. the set of fixed pointsa(q, ; q) := a a | fq,;q (a) = abecomes an fq -algebra. by 4, th. a.3, a(q, ; q) is isomorphic to the tensor algebra of the fq -species associated with (q, ). hence, it is a hereditary fq -algebra.3an a-module m is called f -stable (with respect to fq,;q ) if there exists a frobe- nius map fm : m m satisfyingfm (am) = fq,;q (a)fm (m), for all a a and m m .thenm f = m fm = m m | fm (m) = mbecomes an a(q, ; q)-module. by 3, th. 3.2, the correspondence m 7 m f induces a bijection between the set of isoclasses of f -stable a-modules and the set of isoclasses of a(q, ; q)-modules. a nonzero a-module m is called an indecomposable f -stable a-module if it is f -stable and can not be decomposed into a direct sum of two nonzero f -stable submodules.now let m be an a-module defined by the k-algebra homomorphism : a endk (m ). the composition of the following maps1fq,;q a(1)a a a(1) endk (m )(1) endk (m (1) )defines an a-module structure on m (1) . we denote this module by m 1 and call it the frobenius twist of m (with respect to fq,;q ). if f : m n is an a-module homomorphism, then the k-linear map f (1) : m (1) n (1) becomes an a-module homomorphism m 1 n 1 which is denoted by f 1 . thus, we obtain the so-called frobenius twist functor:( )1 : mod a mod a, m m 1 .inductively, we can define the s-fold frobenius twist m s := (m s1 )1 of m and f s := (f s1 )1 for s 1, where m 0 = m and f 0 = f by convention. we call the a-module m f -periodic (with respect to fq,;q ) if m = m r for some r 1. the minimal positive integer r with m r = m is called f -period of m . by 4, prop. 2.7,every (finite dimensional) a-module m is f -periodic and, moreover, m is f -stable if and only if m = m 1 . the next result gives a way to construct indecomposablef -stable a-modules (see 3, th. 5.1).lemma 1.1. if m is an indecomposable a-module m with f -period r, thenmf = m m 1 m r1fis an indecomposable f -stable a-module. in other words, m fis an indecomposablea(q, ; q)-module. moreover, each indecomposable a(q, ; q)-module can be obtainedin this way.fif m is given as in the above lemma, we say that the a(q, ; q)-module m farisesfrom a-modules of f -period r. the lemma implies that there is a bijection betweenthe set of isoclasses of indecomposable f -stable a-modules and the set of isoclassesof indecomposable a(q, ; q)-modules.4the following lemma shows that the frobenius twist functor ( )1 preserves almostsplit sequences (see 3, lemma 7.3).lemma 1.2. let q be a quiver with automorphism . assume q does not containoriented cycle (i. e., a = kq is finite dimensional). then 0 n1 l m 01is an almost split sequence if and only if so is 0 n 1 l1 m 1 0. inparticular, if m is an indecomposable nonprojective a-module, then m 1 = ( m )1 ,where = a is the auslanderreiten translation of a.it is well known that representations of q over k can be identified with (left) modules over the path algebra a = kq of q. thus, a representation v = (vi, v) of q is called f -stable if v as an a-module is f -stable. it is easy to check that v = (vi, v) is f -stable if and only if there is a frobenius map f : iq0 vi iq0 vi such that f (vi) = v(i) for each i q0 , and the equality f v = v()f holds for each q1 , that is, the following diagram commutesvvt vhf fv() v(t)v(h)in general, let v = (vi, v) be a representation of q over k with dimension vector d = (di) nq0 , that is, dim vi = di for i q0 . by choosing bases of vi for i q0 , we simply identify vi with kdi . thus, for each q1 , v is written as a matrix in kdh dt . then the frobenius twist v 1 of v is isomorphic to w = (wi, w) defined byfor all i q0 and q1 .1wi = v 1 , w = v,(i) 1 ()remark 1.3. for each r 1, r is also an automorphism of q. thus, we have afrobenius morphism (with respect to fqr )rfq,r ;qr : a a, x xsps x xq r (ps),q,;qwhich clearly equals to f rsss. its fixed-point algebraa(q, r ; qr ) = a a | fq,r ;qr (a) = ais an fqr -algebra. it is easy to see that there is an fqr -algebra isomorphismqa(q, r ; qr ) = a(q, ; q) ffqr .5moreover, if an a-module m satisfies m r = m (with respect to fq,;q ), then m isf -stable (with respect to fq,r ;qr ).2. counting indecomposable a(q, ; q)-modulesin this section we study certain polynomials obtained by counting numbers of indecomposable a(q, ; q)-modules. we will also define -absolutely indecomposable a(q, ; q)-modules which generalize the notion of absolutely indecomposable modules over an fq -algebra.let q be a quiver with automorphism . associated with (q, ), there is a valued quiver = (q, ) = (i, 1 ) whose vertex set i = 0 (resp., arrow set 1 ) is the set of -orbits in q0 (resp., in q1 ), and whose valuation is defined as follows: we associate to each i 0 the positive integer i which is the number of vertices in the orbit i, and to each arrow : t h in the pair (d, d0 ) of the positive integers definedby(2.0.1)d = /h and d0= /t,where denotes the number of arrows in the orbit . if = id, the identity auto- morphism of q, then the valued quiver (q, ) coincides with q.to the valued quiver = (q, ) we attach a matrix cq, = (cij )i,ji defined by 2 2 x /i,if i = j,cij = x/i,if i = j,where the first sum is taken over all loops at i, and the second is taken over all arrows in between i and j. the matrix cq, is a (symmetrizable) borcherds cartan matrix in the sense of 2. if, moreover, = (q, ) contains no loops, i. e., cii = 2 for all i i , then cq, is a generalized cartan matrix. in case = id, we denote cq, by cq, which is clearly symmetric.in what follows, we assume that = (q, ) contains no oriented cycles (thus, no loops). let g = g(cq, ) be the kacmoody lie algebra associated with cq, ; see the definition in 20. it is known that g admits a root space decompositiong = g0 m g,where is the root system of g. the dimension of each root space g, , is called the multiplicity of the root , denoted mult for the notational simplicity, we write i for the simple root i and, thus, view as a subset of the free abelian group zi with basis i . to emphasize the role of the6pair (q, ), we write (q, ) for . it is known from 20 thatim(q, ) = +(q, ) (q, ),where +(q, ) = +(q, )ni and (q, ) = +(q, ). the roots in +(q, ) (resp., (q, ) are called positive (resp., negative) roots. let further re (q, ) andim (q, ) denote the set of real and imaginary roots, respectively. set+re(q, ) = +(q, ) re(q, ) and + (q, ) = +(q, ) im(q, ).in case = id, we write (q), + (q), and + (q) instead of (q, ), + (q, ),re im reimand + (q, ), respectively.the quiver automorphism extends linearly to a group automorphism on zq0defined by( x xaa) = x xa(a).aq0aq0let (zq0) denote the subset of -fixed points in zq0. this set can be identified with the group zi via the canonical bijection : (zq0) zi ;x xaa x yii,where yi := xa for a i.aq0iifor (q), let t 1 be the minimal integer satisfying t() = . we call t the -period of , denoted by p() = p (). we have the folding relation between the root systems (q) and (q, ) (see 32, prop. 2 and 16, prop. 4).lemma 2.1. let (q) and set := + () + + t1 () (zq0) ,where t = p (). then 7 () defines a surjective map (q) (q, ). more- over, if () is real, then is real and is unique up to -orbit.for a kq-module (resp., an a(q, ; q)-module) m , we denote by dim m nq0(resp., ni ) the dimension vector of m .now, for each ni , letmq, (, q) = # of isoclasses of a(q, ; q)-modules of dimension vector ,iq, (, q) = # of isoclasses of indecomposable a(q, ; q)-modules of dimension vector .the following result is proved in 18 for the case = id, in 14 for some special, and in 5, 9.19.2 for the general case.7lemma 2.2. both mq, (, q) and iq, (, q) are polynomials in q with rational coef- ficients and are independent of the orientation of q.the next result is known as kacs theorem. for its proof, see 18 for the case = id, and see 14, 6 for the general case.retheorem 2.3. let q be a prime power. for each ni , iq, (, q) = 0 if and only if +(q, ). moreover, if + (q, ), then iq, (, q) = 1.remark 2.4. the treatment for the case where (q, ) contains oriented cycles (in particular, loops) can be found in 33, 34.now, for each +(q, ) and each r 1, letiq, (, q; r) = # of isoclasses of indecomposable a(q, ; q)-modules of dimension vector arising from indecomposable representations of q over kof f -period r.clearly, for each fixed , only finitely many iq, (, q; r) can not be zero. more- over, iq, (, q) = pr1 iq, (, q; r), and iq, (, q; 1) is the number of isoclasses ofabsolutely indecomposable a(q, ; q)-modules of dimension vector . (an indecom- posable a(q, ; q)-module x is called absolutely indecomposable if x fq k is an indecomposable kq-module.)for a root = pii xii +(q, ), the number pii xi is called the height of .by induction on the heights of roots, lemma 2.2 gives the following result.proposition 2.5. for each +(q, ) and each r 1, iq, (, q; r) is a polynomial in q with rational coefficients and is independent of the orientation of q.let us recall that the classical mobius function : n+ = n0 1, 0, 1 is defined by 1,if n = 1,(n) = (1)t,if n = p1 pt with p1, . . . , pt distinct primes, 0,if n is not square-free.it is known that for r 1,(2.5.1)x (s) = x ( r ) = sr,1 .s|rs|r8we also need the mobius inversion formula (see 22, th. 3.24) which states the following: let and be two functions: n+ c. then(n) = x (d) for all n n+d|n(2.5.2) (n) = x ( n )(d) for all n n+ . dd|nnow, for each +(q) and r 1, define hq, (, q; r) (resp., kq, (, q; r) to be the number of isoclasses of indecomposable kq-modules of dimension vector satisfying m r = m (resp., having f -period r). by the definition, a necessarycondition for hq, (, q; r) = 0 or kq, (, q; r) = 0 is that r () = . clearly, wehavex kq, (, q; s) = hq, (, q; r).s|rapplying the mobius inversion formula givesrkq, (, q; r) = x (s|rs )hq, (, q; s).then, for each +(q, ),11riq, (, q; r) = x kq, (, q; r) = xr rx (s )hq, (, q; s),1where the sum is over all (q) satisfyings|r( + () + + r1() = .in conclusion, we obtain the following formula(2.5.3)iq, (, q) = x xrx ( r s)hq, (, q; s).r1 s|rfurthermore, by remark 1.3, if hq, (, q; s) = 0 (in particular, s() = ), thensshq, (, q; s) = iq,s (b (), qs; 1),bwhere s: (zq0 ) z0(q, s) is induced by the automorphism s and s() b+ (q, s). here 0(q, s) is the vertex set of the valued quiver (q, s) associated with the pair (q, s). in other words, the formula (2.5.3) relates the numbers of indecomposable modules over a family of algebras a(q, s; qs).we now consider the special case = id. for each + (q), we write iq(, q) for iq, (, q), and also write aq(, q) for iq, (, q; 1), which is the number of isoclasses of absolutely indecomposable representations of q over fq of dimension vector . in this case, kq, (, q; r) = riq, (, q; r) is the number of isoclasses of indecomposable representations m of q over k which have dimension vector and whose minimal9field of definition is fqr (that is, r is the minimal positive integer such that m is defined over fqr ). these polynomials have been studied in 13. moreover, in this case, hq, (, q; s) = aq(, qs). thus, if +(q) is indivisible, then (2.5.3) implies that for each m 1, the equality(2.5.4)iq(m, q) = xrr|mx ( r ss|rm)aq( r1, qs)holds. this formula was obtained by kac 19, 1.14.let us return to the general case. let o() denote the order of . an indecompos- able a(q, ; q)-module is called -absolutely indecomposable if m fq k decomposes into a direct sum of r indecomposable summands with r|o() . for each + (q, ), let aq, (, q) be the number of isoclasses of -absolutely indecomposable a(q, ; q)- modules of d