Table Of Content

Conjugacy classes in parabolic subgroups of general linear groups 8 0 0 Anton Evseev∗ 2 n with an appendix by a Anton Evseev and George Wellen† J 1 2 ] R Abstract G We prove a formula connecting the number of unipotent conjugacy classes . in a maximalparabolicsubgroupof a finite generallinear groupwith the num- h bers of unipotent conjugacy classes in various parabolic subgroups in smaller t a dimensions. We generalise this formula and deduce a number of corollaries; in m particular, we express the number of conjugacy classes of unitriangular matri- [ ces over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers 1 v of unipotent conjugacy classes in parabolic subgroupsof smalldimensions may 8 be calculated. 7 1 3 1 Introduction . 1 0 LetqbeaprimepowerandF bethefinitefieldwithqelements. Ifkandmare q 8 nonnegative integers, let M (q) be the set of all m×k matrices over F . Let m,k q 0 l=(l ,...,l )beasequenceofnonnegativeintegers. WritingM (q)=M (q), : 1 s k k,k v let Ml(q) be the set of all matrices of the form i X M (q) M (q) ... M (q) r l1 l1,l2 l1,ls a  0... M.l2.(.q) ...... Ml2,...ls(q).    0 0 ... M (q)   ls    l l l Let P (q) be the group of all invertible matrices in M (q). The group P (q) is called a parabolic subgroup of the group GL (q), where m = l +···+l . m 1 s Alternatively, a parabolic subgroupmay be described as the stabiliser of a flag inthe vectorspace Fm. LetNl(q)be the setofallnilpotent matricesinMl(q). q The group Pl(q) acts on the set Ml(q) by conjugation: gx = gxg−1 (g ∈ l l P (q), x∈M (q)). We investigate the number of orbits of this action and also l l the numbers of orbits of the actions of P (q) on certain subsets of M (q). Throughout the paper, we denote by γ(G,X) the number of orbits of an action of a group G on a finite set X, where the action is understood. Let l l ρl(q)=γ(P (q),N (q)). ∗Selwyn College, Cambridge, CB3 9DQ, UK,[email protected] †Mathematical Institute,24-29 St Giles’, Oxford, OX13LB, UK,[email protected] 1 l Although the groups P (q) have received a lot of attention, not much is l l known about the numbers γ(P (q),P (q)) and ρl(q). In particular, it is not known whether the following is true. Conjecture 1. For every tuple l=(l1,...,ls) of nonnegative integers, ρl(q) is polynomial in q. (Here, and in what follows, we call a rational-valued function f(x) defined on a set D of integers polynomial if there exists a polynomial g with rational coefficientssuchthatf(x)=g(x)forallx∈D.) Write(1m)=(1,1,...,1). The m followingtwospecialcasesofConjecture1havereceivedconsiderableattention and will be of particular interest to us. | {z } Conjecture 2. For every positive integer m, ρ (q) is a polynomial in q with (1m) rational coefficients. Conjecture 3. For any positive integers k and m, ρ (q) is a polynomial in (k,m) q with rational coefficients. It is not difficult to show that Conjecture 2 is equivalent to the conjecture that the number of conjugacyclasses of the groupof upper unitriangular n×n matrices is a polynomial in q with rational coefficients (see [4, Section 4.2]). This last conjecture has been proved for n ≤ 13 by J.M. Arregi and A. Vera- Lo´pez [8, 9, 10]. Conjecture 3 has been proved for k ≤5 (or, alternatively, for m ≤ 5); indeed, S.H. Murray [7] proved that in those cases ρ (q) does not (k,m) depend on q. A partition is a (possibly, empty) non-increasing sequence of positive integers. Ifλ=(λ ,...,λ )isapartition,let|λ|=λ +···+λ ,andwritel(λ)=r. 1 r 1 r If k,m ∈ N, let P be the set of all partitions λ with |λ| = k, let Pm be the k set of partitions λ such that λ ≤ m for all i, and let Pm = P ∩Pm. Let i k k p(k)=|P |. k Letm∈N,andletλ=(λ ,...,λ )beapartitionsuchthatλ ≤m. Define 1 r 1 δ(m,λ)=(λ ,λ −λ ,λ −λ ,...,λ −λ ,m−λ ). r r−1 r r−2 r−1 1 2 1 Let νm(q) = ρ (q). Thus, νm(q) is the number of Pl(q)-orbits of Nl(q) λ δ(m,λ) λ l where P (q) is the stabiliser of a flag Fλr ≤Fλr−1 ≤···≤Fλ1 ≤Fm. q q q q Weshallgeneralisethedefinitionofνm(q)asfollows. LetF bethealgebraic λ q closureofthefieldF ,andletF beasubsetofF . LetY =Y bethefamilyof q q F alllinearendomorphismsT :U →U (whereU isanarbitraryfinitedimensional vector space over F ) such that all eigenvalues of T over F are in F. We shall q q say that Y is the class of endomorphisms associated with F. We shall refer to elements of this class as Y-endomorphisms. Obviously, Y is preserved by conjugation. Note that the family of all nilpotent endomorphisms and the family of all invertible endomorphisms are both classes. If Y is a class and m ∈ Z , let c(m,Y) be the number of GL (F )-orbits on Y ∩M (F ). We ≥0 m q m q shall denote by N the class of all nilpotent endomorphisms. Note that c(m,N)=p(m), thenumberofpartitionsofm. Thisfollowsfromthefactthatnilpotentmatrices in Jordan canonical form form a complete set of representatives of GL (q)- m orbits on N (q), the set of nilpotent m×m matrices. m 2 LetY beaclass(ofendomorphisms),andletm∈N. Letλ=(λ ,...,λ )∈ 1 r Pm (that is, λ ≤m). Define 1 κm(Y)=γ(Pδ(m,λ)(q),Mδ(m,λ)(q)∩Y). λ One of the main results of this paper is as follows. Theorem 1.1. Let k and m be positive integers. Let q be a prime power and Y be a class of endomorphisms over F . Then q k κk+m(Y)= c(k−j,Y) κm(Y). (k) λ j=0 λ∈Pm X Xj In particular, k ρ (q)= p(k−j) νm(q). (k,m) λ j=0 λ∈Pm X Xj Remark. S.H. Murray [6] has proved a similar result stated in terms of irre- duciblerepresentationsofparabolicsubgroups. ThisresultimpliesTheorem1.1 in the case when Y is the class of all invertible matrices. The proof in [6] is differentfromtheonegivenhere. Unliketheproofinthispaper,theproofin[6] establishes not just a numerical equality, but also an explicit correspondence betweenrepresentations. Itisaninterestingquestionwhetherthemethodof[6] can be extended to prove an analogue of the more general Theorem 6.2. We will deduce the following two results from Theorem 1.1. Theorem 1.2. Let m ∈ N, and let n = m(m −1)/2. There exist integers a ,a ,...,a such that, for all prime powers q, 0 1 n n ρ (q)= a ρ (q). (1m) j (j,m) j=0 X Hence, Conjecture 3 implies Conjecture 2. If l=(l1,...,ls), write ρk,l(q) for ρ(k,l1,...,ls)(q). Theorem 1.3. Let k,m ∈ N, and let n = m(m−1)/2. There exist integers a ,...,a such that, for all tuples l=(l ,...,l ) of nonnegative integers with k0 kn 1 s l +···+l =m, 1 s n ρk,l(q)= akjρj,l(q). j=0 X UsingthemethodsdevelopedintheproofofTheorem1.1,onemaycompute ρl(q) for all tuples l = (l1,...,ls) with l1 +···+ls ≤ 6. In particular, the following holds. Proposition 1.4. Let l = (l ,...,l ) be a tuple of nonnegative integers with 1 s l1+···+ls ≤6. Thenρl(q)isapolynomialinq withpositiveintegercoefficients. Hence(byTheorem1.1)ρ (q)ispolynomialinqwheneverm≤6andk ∈N. (k,m) Thispaperisorganisedasfollows. InSection2wedescribeageneralframe- workofquiverrepresentationsandtheir endomorphisms,whichisusedtostate and prove the results. In Section 3 we show, in particular, how the numbers γ(Pl(q),Pl(q))andγ(Pl(q),Ml(q)) maybe expressedinterms ofρl′(qd)where d ∈ N and l′ is another tuple of nonnegative integers. This justifies our focus on the numbers ρl(q). 3 Section 4 contains a few standard results used later. In Section 5 we prove resultsthatserveasthemaintoolsallowingusto reduceproblemssuchasthat of counting ρ (q) to other problems in a smaller dimension. In Section 6 (k,m) we use those tools to prove a generalisation of Theorem 1.1 stated in terms of quiver representations (Theorem 6.2). In Section 7 we invert the formulae in Theorems 1.1 and 6.2 and deduce Theorems 1.2 and 1.3. This relies on combinatorial results proved jointly with G.Wellen in the Appendix. I amverygratefulto GeorgeWellen for his partin this work. InSection 8 we describe a method for computing ρl(q) when m=l1+···+ l is small. For this purpose, we generalise our problem to that of counting s conjugacy classes of groups associated with preordered sets. Finally, Section 9 investigatesthesymmetryaffordedbyconsideringadualquiverrepresentation. In particular, we show that ρ (q)=ρ (q). (l1,...,ls) (ls,...,l1) Acknowledgments. Most of this paper is a part of my D.Phil. thesis. I amverygratefultomysupervisor,MarcusduSautoy,andtoGeorgeWellenfor hispartinthiswork. Iwouldalsoliketothankmythesisexaminers,DanSegal andGerhardRo¨hrle,forspotting a numberoferrorsandforhelpful comments. Notation and definitions • γ(G,X) is the number of G-orbits on a finite set X where the action of a group G on X is understood; • [k,n]={k,k+1,k+2,...,n} where k ≤n are integers; • |X| is the cardinality of a set X; • A is a partition of a set X if A is a family of disjoint sets whose union is X; two elements x and y of X are said to be A-equivalent if there exists A∈A such that x,y ∈A; • δ =0 if i6=j, and δ =1; ij ii • A⊔B is the disjointunionofsets AandB (formallydefined asA×{0}∪ B×{1}); • I =I is the identity element of GL(V); V • I is the identity k×k matrix over an appropriate field. k • Suppose U and V are vector spaces, X ∈End(U) and Y ∈End(V); then I(X,Y) denotes the vector space of all linear maps T :U →V such that TX =YT; • N (K) is the set of all nilpotent matrices in M (K); n n,n • End(V;U ,...,U ) = {f ∈ End(V) : f(U ) ⊆ U ∀i} where U are sub- 1 k i i i spaces of V; • P(V;U ,...,U ):=End(V;U ,...,U )∩GL(V); 1 k 1 k • By convention,the set of 0×k matrices contains just one element (which is nilpotent if k =0), and the group GL (K) is trivial; 0 • IfI andJ arefinitesets,thenM (K)isthesetofallmatricesoverafield I,J K whose rows are indexed by the elements of I and whose columns are indexed by elements of J; we refer to these as I ×J matrices; note that, if A ∈ MI,J(K) and B ∈ MJ,J′(K), then the product AB ∈ MI,J′(K) is well defined; 4 • At is the transpose of a matrix A; • Tr(A) is the trace of a square matrix A; • IfagroupGactsonasetX,twoelementsofX aresaidtobeG-conjugate if they are in the same G-orbit; • All rings are understood to have an identity element; • IfRisaring,thenRop istheringwiththesameunderlyingabeliangroup and with the multiplication (r,s)7→sr; 2 Quiver representations and automorphisms We recall the standard definitions related to quivers (see [1], for example). A quiver is a pair (E ,E ) of finite sets together with maps σ : E → E and 0 1 1 0 τ : E → E . Elements of E may be thought of as nodes; then each element 1 0 0 e∈E may be represented as an arrow from σ(e) to τ(e). Let K be a field. A 1 representation of a quiver (E ,E ) over K is a pair (U,α) such that 0 1 (i) U=(U ) is a tuple of vector spaces over K; a a∈E0 (ii) α=(α ) where α ∈Hom(U ,U ). e e∈E1 e σ(e) τ(e) If(U,α)isarepresentationofaquiver(E ,E ),weshallrefertothequadruple 0 1 Q=(E ,E ,U,α) as a quiver representation. A quiver representation may be 0 1 thought of as a collection of vector spaces together with linear maps between some of those spaces. If (U,α) and (U′,α′) are two representations of a quiver (E ,E ), a mor- 0 1 phism between those representations is a tuple (X ) such that a a∈E0 (i) X ∈Hom(U ,U′) for all a∈E ; a a a 0 (ii) α′X =X α for all e∈E . e σ(e) τ(e) e 1 This defines the category of representations of (E ,E ). 0 1 If Q = (E ,E ,U,α) is a quiver representation, write End(Q) for the ring 0 1 of all endomorphisms of Q and Aut(Q) for the group of all automorphisms of Q. LetN(Q)bethesetofallnilpotentelementsofEnd(Q). ThegroupAut(Q) acts naturally on the set End(Q) by conjugation: if X = (X ) ∈ End(Q) and a g∈Aut(Q), then g◦X=(g X g−1) . a a a a∈E0 AssumethatK =F ,whereq isaprimepower. Weshallbeconcernedwith q the number of orbits of this action and, more generally, with the number of orbits of actions of certain subgroups of Aut(Q) on certain subsets of End(Q), suchasN(Q)forexample. NotethatAut(Q)andN(Q)maybedefinedinterms of the ring structure on End(Q), so they are preserved by ring isomorphisms. Let θ(Q)=γ(Aut(Q),N(Q)). All the problems discussed in the introduction may be stated in terms of quiver representations. If l = (l ,l ,...,l ) is a tuple of nonnegative integers 1 2 s with m=l1+···+ls, let Rl =Rl(q) be the quiver representation Flq1(cid:31)(cid:127) //Fql1+l2(cid:31)(cid:127) //···(cid:31)(cid:127) // Fql1+···+ls−1(cid:31)(cid:127) //Fmq where all the arrows represent injective linear maps. More formally, Rl =([1,s],[1,s−1],U,α) 5 with σ(i)=i, τ(i) =i+1 for all i∈[1,s−1], Ui =Fql1+···+li and αi injective. Obviously, these conditions define Rl up to isomorphism of representations. Wemaychooseabasis{b ,...,b }ofU =Fmsothat,foreachi,theimage 1 m s q ofUi =Fql1+···+li underαs−1···αi+1αiisequaltothespanof{b1,...,bl1+···+li}. LetJl :End(Rl)→Mm,m(q)bethemapwhichassignstoeachX=(Xi)i∈[1,s] ∈ End(Rl) the matrix of Xs ∈ End(Fmq ) with respect to the basis {b1,...,bm}. Then the following result is obvious. Lemma 2.1. The map Jl is a ring isomorphism from End(Rl) onto Ml(q). Hence, ρl(q)=θ(Rl). Call a quiver representation Q′ = (E′,E′,U′,α′) an extension of a quiver 0 1 representation Q=(E ,E ,U,α) if 0 1 (i) E′ ⊇E ; 0 0 (ii) U′ =U for all a∈E ; a a 0 (iii) for every X = (Xa)a∈E0′ ∈ End(Q′), the tuple (Xa)a∈E0 belongs to End(Q). Let Q′ be an extension of Q. Define a map π = πQ′ : End(Q′) → End(Q) Q by (Xa)a∈E0′ 7→(Xa)a∈E0. If B ⊆ End(Q), let BQ′ = π−1(B). If G is a subgroup of Aut(Q), define GQ′ =π−1(G)∩Aut(Q′). (Thiswillcausenoambiguityifagroupisconsidered a distinct object from the set of its elements.) Let Q = (E ,E ,U,α) be a quiver representation, and let e ∈ E with 0 1 σ(e) = a, τ(e) = b. Let Y = ker(α ) ≤ U , Z = im(α ) ≤ U . Define an e a e b extension K(Q,e)=(E′,E′,U′,α′) as follows: 0 1 (i) E′ =E ⊔{c}; 0 0 (ii) U′ =Y (and U′ =U for x∈E ); c x x 0 (iii) E′ =E ⊔{e′} where σ(e′)=c, τ(e′)=a; 1 1 (iv) α′ =α for all f ∈E , and α′ is the inclusion map Y ֒→U . f f 1 e′ a Define another extension I(Q,e)=(E′′,E′′,U′′,α′′) as follows: 0 1 (i) E′′ =E ⊔{d}; 0 0 (ii) U′′ =Z; d (iii) E′′ = (E \ {e}) ⊔ {e♯,e′′}, where σ(e′′) = d, τ(e′′) = b, σ(e♯) = a, 1 1 τ(e♯)=d; (iv) α′′ = α for all f ∈ E \{e}, α′′ is the inclusion map Z ֒→ U , and α′′ f f 1 e′′ b e♯ is the map U →Z given by α′′ (v)=α (v) ∀v ∈V. a e♯ e Clearly,K(Q,e)is anextensionof Q. Also, Q′′ :=I(Q,e)is an extensionofQ: if X∈End(Q′′), then α X =X α because α =α′′ α′′ . e a b e e e′′ e♯ Lemma2.2. LetQ=(E ,E ,U,α)beaquiverrepresentation,andlete∈E . 0 1 1 Let Q′ = K(Q,e) and Q′′ =I(Q,e). Then the maps πQ′ :End(Q′) →End(Q) Q and πQ′′ :End(Q′′)→End(Q) are ring isomorphisms. Q Proof. As above, let a = σ(e), b = τ(e), Y = ker(α ), Z = im(α ). Let e e X = (X ) ∈ End(Q). Since α X = X α , the map X preserves Y and i i∈E0 e a b e a X preserves Z. Any element X′ ∈ πQ′ −1(X) satisfies α′ X′ = X′α′ . b Q e′ c c e′ (cid:16) (cid:17) 6 Since α′ is injective, it follows that πQ′ −1(X) = {X′}, where X′ = X | . e′ Q c a Y Similarly, πQ′′ −1(X) = {X′′}, whe(cid:16)re X(cid:17)′′ = X | . Hence, πQ′ and πQ′′ are Q d b Z Q Q bijections.(cid:16) (cid:17) 3 Reduction to nilpotent endomorphisms In this section we show that, if Y is a class of endomorphisms over F , the q problemofcounting orbitsofY-endomorphismsofaquiverrepresentationmay be reduced to that of counting orbits of nilpotent endomorphisms of various quiver representations. (A Y-endomorphism is an endomorphism X such that X ∈Y foralla.) Weusestandardmethodsrelatedtorationalcanonicalforms. a Theresultsofthissectionarenotusedelsewhereinthepaper,butprovidesome motivation for our later focus on nilpotent endomorphisms. Let U be a vector space over a field K. Let L be a field containing K. Suppose U′ is a vector space over L that becomes U if one restricts the scalars to K; that is, U′ = U as sets and multiplication by scalars from K in U is the same as in U′. We shall say that U′ is an L-expansion of U and U is the restriction of U′ to K. Let Q = (E ,E ,U,α) be a quiver representation 0 1 over K. Say that a quiver representation Q′ = (E ,E ,U′,α) over L is an 0 1 L-expansion of Q (and Q is the restriction of Q′ to K) if, for each a∈E , the 0 space U′ is an L-expansion of U . (Then α is L-linear for each e∈E .) a a e 1 Let X ∈ End(Q). If a ∈ E and f is a monic irreducible polynomial over 0 F , let q UX,f,a ={u∈Ua :f(Xa)ku=0 for some k ∈N}. Then UX,f,a = 0 for all but finitely many f. For all monic irreducible polynomials f ∈ Fq[T] and all e ∈ E1, we have αe(UX,f,σ(e)) ≤ UX,f,τ(e). Thus, UX,f :=(UX,f,a)a∈E0 induces a subrepresentation QX,f of Q. Lemma 3.1. Let Q be a quiver representation over F . Suppose X∈End(Q). q Then Q= QX,f f M where the sum is over all monic irreducible polynomials f over F . Moreover, q the isomorphism class of each component QX,f is an invariant of the Aut(Q)- orbit of X. Proof. The first statement follows from the fact that Ua = fUX,f,a for each a. The second statement is clear. L We now consider each quiver representation QX,f separately. Fix a monic irreduciblepolynomialf =f(T)overF . Letdbethedegreeoff. SupposeXis q anendomorphismofaquiverrepresentationQoverFq suchthatQ=QX,f. Let F [T] be the localisation of F [T] at the ideal (f); it consists of all fractions q (f) q g/h such that g,h ∈ F [T] and h is not divisible by f. Then X induces an q F [T] -module structure on each U : multiplication by T is given by the q (f) a action of X . Moreover,each α becomes an F [T] -module homomorphism. a e q (f) Let S be the completion of the discrete valuation ring F [T] . That is, f q (f) S is the inverselimitof the ringsF [T]/(fk), k =1,2,.... Any finiteF [T] - f q q (f) module is annihilated by fk for large enough k. Thus, a finite F [T] -module q (f) may be thought of as a (finite) S -module (and vice versa). f 7 Now, by [2, §9, Proposition 3], S is isomorphic to F [[Z]], the ring of f qd formalpowerseriesovera variableZ. Indeed, let r be the element ofS whose f projection onto F [T]/(fk) is Tqkd for each k ∈ N. Then F [r] ⊆ S is a field q q f isomorphictoF ,andeachelementofS mayberepresentedasapowerseries qd f in f with coefficients in F [r] (see loc. cit. for more detail). q Hence, Q gives rise to a finite F [[Z]]-module. This module corresponds qd to an F -expansion Q′ of Q and a nilpotent endomorphism of Q′ (given by qd multiplication by Z). Conversely, an F -expansion Q′ of Q together with a nilpotent endomor- qd phismofQ′ givesrisetoanF [[Z]]-modulestructureoneachU insuchaway qd a that all α are F [[Z]]-endomorphisms. Identifying F [[Z]] with S as above, e qd qd f we get an endomorphism X of Q such that QX,f = Q. We have proved the following result. Lemma 3.2. Let q be a prime power, and let f be a monic irreducible polynomial over F of degree d. Suppose Q is a quiver representation over F . Then q q the Aut(Q)-orbits of endomorphisms X of Q such that QX,f =Q are in a one- to-one correspondence with Aut(Q)-orbits of pairs (Q′,Z) such that Q′ is an F -expansion of Q and Z∈N(Q′). qd Let Y be a class of linear endomorphisms over F , as defined in Section 1. q Let End (Q)={X∈End(Q):X ∈Y for all a∈E }. Y a 0 Let F be a subset of F such that Y = Y . Call a polynomial f ∈ F [T] a q F q Y-polynomial ifalltherootsoff (overF )areinF. Letk (q)bethenumber q Y,d of monic irreducible Y-polynomials of degree d. Consider again an arbitrary quiver representation Q = (E ,E ,U,α) over 0 1 F . Lemmata 3.1 and 3.2 allow us to express γ(Aut(Q),End (Q)) in terms of q Y θ(Q′) where Q variesamongdirectsummands ofQ andQ′ is anexpansionof 0 0 0 Q . 0 Let X ∈ End (Q). By Lemma 3.1, there exists a finite set {f } of Y i i∈J irreducible Y-polynomials such that Q = j∈JQj where Qj = QX,fj. Here, J is a finite indexing set. Let ǫ(j) = degf . By Lemma 3.2, each Q together j j L with the restriction of X to Q corresponds to an F -expansion Q′ of Q j qǫ(j) j j together with a nilpotent endomorphism Z(j) ∈ N(Q′). Up to conjugation by j Aut(Q′), the endomorphism Z(j) may be chosenin θ(Q′) ways (by definition). j j Let A be the partition of J that consists of the sets {j ∈J :Q′ ≃Q′} j i where i varies among the elements of J. In particular, ǫ(i) = ǫ(j) whenever i and j are A-equivalent. Proposition 3.3. Let Q be a quiver representation over F . Then q ∞ 1 k (q)! Y,d γ(Aut(Q),End (Q)) = Y |A|! (k (q)−|ǫ−1(d)|)! (QXi)i∈JXA A∈A Xǫ dY=1 Y,d × Qθ(Q′ )|A|. A (Q′AX)A∈AAY∈A Here, the first sum is over all decompositions Q = Q of Q as a direct i∈J i sum of representations; two such decompositions are considered to be the same L if one may be obtained from the other by permuting the indices i and replacing 8 each Q with an isomorphic representation (we assume that J = [1,|J|]). The i secondsumis overallpartitions Aof J suchthatQ ≃Q whenever iandj are i j A-equivalent. The third sum is over all maps ǫ : J → N such that ǫ(i) = ǫ(j) wheneveriandj areA-equivalent. Thelastsumisoverallisomorphism classes of tuples (Q′ ) where Q′ is an F -expansion of Q (i is an arbitrary A A∈A A qǫ(i) i element of A) and the quiver representations Q′ (A ∈ A) are pairwise not A isomorphic. Proof. Suppose A, ǫ, Q′ =Q′ (i∈A), as above, are fixed. There are A i ∞ k (q)! Y,d (k (q)−|ǫ−1(d)|)! Y,d d=1 Y ways to assign a monic irreducible Y-polynomial f of degree ǫ(i) to eachi∈J i sothatallthosepolynomialsaredistinct. (Notethatallbutfinitelymanyterms in the product are equal to 1, so the product is well defined.) There are θ(Q′ )|A| A A∈A Y ways to choose, for each i ∈ J, an Aut(Q′)-orbit in N(Q′). By Lemmata 3.1 i i and 3.2, these assignments determine an Aut(Q)-orbit in End (Q), and all Y orbits occur inthis way. However,a permutationofthe indices within a subset A ∈ A yields the same orbit. Thus, each orbit giving rise to this particular A is obtained by (|A|!) such assignments. Hence, we must divide by this A∈A number to obtain the number of orbits. Q Now let l=(l ,...,l ) be a tuple of nonnegative integers, and consider the 1 s quiverRl(q). Ifl′ =(l1′,...,ls′)isanothersuchs-tuple,letl+l′ =(l1+l1′,...,ls+ l′); write l≤l′ if l ≤l′ foralli. Also, ifb∈Q, letbl=(bl ,...,bl ). Itis easy s i i 1 s to see that all decompositions of Rl into direct sums of subrepresentations are of the form n Rl = Rli i=1 M where l=l1+···+ln. Let d∈N. If not all l are divisible by d, then there is i no Fqd-expansion of Rl(q). If all li are divisible by d, then Rl(q) has a unique (up to the action of Aut(Rl(q))) Fqd-expansion, namely Rl/d(qd). Consider the set of tuples J =(l1,...,ln,d ,...,d ) such that each li is an 1 n s-tuple of nonnegative integers, not all equal to zero, d ∈N for each i and i n d li =l. i i=0 X Let D(l) be a complete set of representatives of the action of the symmetric group S on this set, where the action is by permuting the indices 1,...,n. n Less formally, D(l) is in a one-to-one correspondence with ways to decompose Rl(q) as a direct sum of quivers Rl′(q) and to expand each Rl′(q) to Rl′/d(qd) for some d. l l l l By Lemma 2.1, the numbers γ(P (q),P (q)) and γ(P (q),M (q)) may both beexpressedasγ(Aut(Rl(q)),EndY(Rl(q)))whereY istheclassofallinvertible endomorphisms or the class of all endomorphisms, respectively. If Y is one of those classes, then k (q) is polynomial in q. We deduce the following from Y,d Proposition 3.3. 9 Corollary 3.4. Let l = (l ,...,l ) be a tuple of nonnegative integers. Then 1 s there exist tuples (aJ)J∈D(l) and (bJ)J∈D(l) where aJ(T) and bJ(T) are polynomials with rational coefficients such that, for all prime powers q, n γ(Pl(q),Pl(q)) = aJ(q) ρli(qdi) and J∈XD(l) iY=1 n γ(Pl(q),Ml(q)) = bJ(q) ρli(qdi) J∈XD(l) iY=1 In each case, the first sum is over all elements J = (l1,...,ln,d ,...,d ) of 1 n D(l). 4 Preliminary results In this section we state severalstandardand straightforwardresults. We prove the last of these results; the first two are easy exercises. Lemma 4.1. Let G be a group acting on a set Y. Let N be a normal subgroup of G, and let S be the set of N-orbits on Y. The action of G on Y induces an action of G/N on S via gN◦Ny =Ngy (for all g ∈G, y ∈Y). Moreover, the G-orbits on Y are in a one-to-one correspondence with the G/N-orbits on S. Lemma 4.2. Let G be a group acting on finite sets X and Y. Suppose S is a subset of X ×Y preserved by G. Let R be a complete set of representatives of G-orbits on X (so each G-orbit on X contains exactly one element of R). Then γ(G,S)= γ(Stab (x),{y ∈Y :(x,y)∈S}). G x∈R X Recall that if V and W are vector spaces and X ∈ End(V), Y ∈ End(W), then I(X,Y)={T ∈Hom(V,W):TX =YT}. Lemma 4.3. Let V and W be finite dimensional vector spaces over a field K. Suppose X ∈End(V) and Y ∈End(W). Then dimI(X,Y)=dimI(Y,X). Proof. Let V∗ and W∗ be the dual spaces to V and W. Let X∗ ∈ End(V∗) and Y∗ ∈End(W∗) be the maps dual to X and Y: (X∗f)(v)=f(X(v)) for all f ∈V∗, v ∈V. Then dimI(Y,X)=dimI(X∗,Y∗): a linear bijection between I(Y,X) and I(X∗,Y∗) is given by assigning to each map its dual. Also, X∗ and X have the same rational canonical form, as do Y and Y∗. Thus, dimI(X,Y)=dimI(X∗,Y∗)=dimI(Y,X). 5 Action on the dual space Let V be a finite dimensional vector space over a finite field F . If w ∈V∗ and q v ∈ V, we shall write (w,v) for w(v). Suppose a group G acts on V by linear maps, so a representation of G on V is given: (g,v)7→gv. 10