Counting conjugacy classes in the unipotent radical of parabolic subgroups of $\GL_n(q)$
aa r X i v : . [ m a t h . G R ] J un COUNTING CONJUGACY CLASSES IN THE UNIPOTENT RADICALOF PARABOLIC SUBGROUPS OF GL n ( q ) SIMON M. GOODWIN AND GERHARD R ¨OHRLE
Abstract.
Let q be a power of a prime p . Let P be a parabolic subgroup of the generallinear group GL n ( q ) that is the stabilizer of a flag in F nq of length at most 5, and let U = O p ( P ). In this note we prove that, as a function of q , the number k ( U ) of conjugacyclasses of U is a polynomial in q with integer coefficients. Introduction
Let GL n ( q ) be the finite general linear group defined over the field F q of q elements, where q is a power of a prime p . A longstanding conjecture attributed to G. Higman (cf. [12])asserts that the number of conjugacy classes of a Sylow p -subgroup of GL n ( q ) is given bya polynomial in q with integer coefficients. This has been verified by computer calculationby A. Vera-L´opez and J. M. Arregi for n ≤
13, see [17]. There has been much interest inthis conjecture, for example from G. R. Robinson [15] and J. Thompson [16]. The reader isreferred to [1], [6], [9], [8], [10], and [11] for recent related results.In this note we consider the following question, which is precisely Higman’s conjecturewhen P = B is a Borel subgroup of GL n ( q ). Question 1.1.
Let P be a parabolic subgroup of GL n ( q ) and let U = O p ( P ) . As a functionof q , is the number k ( U ) of conjugacy classes of U a polynomial in q ? Here we recall that O p ( P ) is by definition the largest normal p -subgroup of P . In thispaper, we give an affirmative answer to Question 1.1 in the following cases. Theorem 1.2.
Let P be a parabolic subgroup of GL n ( q ) that is the stabilizer of a flag in F nq of length at most , and let U = O p ( P ) . Then, as a function of q , the number k ( U ) ofconjugacy classes of U is a polynomial in q with integer coefficients. We now explain the significance of the hypothesis imposed in Theorem 1.2. Let P be aparabolic subgroup of GL n (¯ F q ) and let U be the unipotent radical of P ; where ¯ F q denotes thealgebraic closure of F q . All instances when P acts on U with a finite number of orbits weredetermined in [13]; this is precisely the case when P is the stabilizer of a flag in ¯ F nq of lengthat most 5. So Theorem 1.2 deals with parabolic subgroups P of GL n ( q ) that correspond toparabolic subgroups P of GL n (¯ F q ) with a finite number of conjugacy classes in U . In suchcases, it is observed in [13, §
4, Rem. 4.13] that the parameterization of the P -conjugacyclasses in U is independent of q : this is the crucial point that we require for our proof ofTheorem 1.2.The proof of Theorem 1.2 involves a translation of the problem to a representation theoreticsetting. More precisely, we recall from [13, §
4] that the P -conjugacy classes in U correspond Mathematics Subject Classification . Primary 20G40 Secondary 20E45, 20D15. ijectively to the so called ∆-filtered modules of a certain quasi-hereditary algebra A t . Thisallows us to see that the parameterization of the P -orbits in U is independent of q and thatwe can choose a set R of representatives that are matrices with entries equal to 0 or 1. Theother key point is that the structures of the centralizers C P ( x ) and C U ( x ) for x ∈ R do notdepend on q ; this is covered in Propositions 2.5 and 2.8.We now discuss some natural generalizations of Theorem 1.2. First consider the case of anormal subgroup N of P with N ⊆ U . Still assuming that there is only a finite number of P -orbits in U , we readily derive from the proof of Theorem 1.2 that k ( U, N ), the number of U -conjugacy classes in N = N ∩ U , is given by a polynomial in q with integer coefficients. Itshould also be possible to prove that the number k ( U, N ) is a polynomial in q with just theassumption that there are finitely many P -orbits in N . For example, for N = U ( l ) the l thmember of the descending central series of U , there is a classification of all instances when P acts on U ( l ) with a finite number of orbits, see [2]. In such situations a generalization ofthe proof of Theorem 1.2 would require detailed knowledge of the P -conjugacy classes in N .Further, it is natural to consider the generalization of Question 1.1, where GL n ( q ) isreplaced by any finite reductive group G ; and also to consider the number k ( P, U ) of P -conjugacy classes in U rather than k ( U ). (In order to avoid degeneracies in the Chevalleycommutator relations, it is sensible to only consider these generalizations when q is a powerof a good prime for G .)At present there are no known examples, where k ( U ) is not given by a polynomial in q ,and there are many cases not covered by Theorem 1.2, where k ( U ) is given by a polynomialin q , see for example [10] and [17]. However, it is not necessarily the case that k ( P, U ) isa polynomial in q : indeed in [7, Exmp. 4.6], it shown that in case G is of type G , and P = B is a Borel subgroup of G , the number k ( B, U ) is given by two different polynomialsdepending on the residue of q modulo 3.Let P be a parabolic subgroup of a reductive algebraic group G defined over F q , andsuppose that P has finitely many conjugacy classes in U ; let P and U be the groups of F q -rational points of P and U respectively. Given the discussion after Theorem 1.2 a naturalgeneralization to consider is whether the number k ( U ) of conjugacy classes of U is a poly-nomial in q . Our proof of Theorem 1.2 is dependent on the detailed information about the P -conjugacy classes in U . For this reason the argument does not adapt to the case, where G is any finite reductive group. The main difficulty is that is it is not clear whether theparameterization of P -orbits in U and the structure of centralizers depends on the charac-teristic of the underlying ground field. Another problem is that centralizers C P ( u ) for u ∈ U need not be connected, so determining the P -classes in U from the P -classes in U may benon-trivial. 2. Translation to representation theory
In this section we recall the relationship between adjoint orbits of parabolic subgroupsand modules for a certain quasi-hereditary algebra that was established in [13, § et K be any field, and let n, t ∈ Z ≥ . Let d = ( d , . . . , d t ) ∈ Z t ≥ with d i ≤ d i +1 and d t = n . We define the parabolic subgroup P ( d ) = P K ( d ) of GL n ( K ) to be the stabilizer ofthe flag 0 ⊆ K d ⊆ K d ⊆ . . . ⊆ K d t in K n ; any parabolic subgroup of GL n ( K ) is conjugateto P ( d ) for some d . We write U ( d ) = U K ( d ) = { u ∈ GL n ( K ) | ( u − V i ⊆ V i − for each i } for the unipotent radical of P ( d ), and u ( d ) = u K ( d ) = { x ∈ M n ( K ) | xV i ⊆ V i − for each i } for the Lie algebra of U ( d ). Then P ( d ) acts on u ( d ) via the adjoint action: g · x = gxg − for g ∈ P ( d ) and x ∈ u ( d ). For x ∈ u ( d ), we write P · x for the adjoint P -orbit of x and C P ( x ) for the centralizer of x in P ; we define U · x and C U ( x ) analogously.Though we are primarily interested in the conjugacy classes of U ( d ) and the P ( d )-conjugacy classes in U ( d ), it is more convenient to consider the adjoint P ( d )-orbits in u ( d ).The map x x is a P ( d )-equivariant isomorphism between u ( d ) and U ( d ), which meansthat the adjoint P ( d )-orbits in u ( d ) are in bijective correspondence with the P ( d )-conjugacyclasses in U ( d ); this allows us to work with the adjoint orbits.The quiver Q t is defined to have vertex set { , . . . , t } ; there are arrows α i : i → i + 1 and β i : i + 1 → i for i = 1 , . . . , t −
1. Below, in Figure 1, we give an example of a quiver Q t . t t t t t ✲ α ✲ α ✲ α ✲ α ✛ β ✛ β ✛ β ✛ β Figure 1.
The quiver Q Let I t = I t,K be the ideal of the path algebra K Q t of Q t generated by the relations:(2.1) β α = 0 and α i β i = β i +1 α i +1 for i = 1 , . . . , t − . The algebra A t = A t,K is defined to be the quotient K Q t /I t .Recall that an A t -module M is determined by a family of vector spaces M ( i ) over K for i = 1 , . . . , t such that M = L ti =1 M ( i ), and linear maps M ( α i ) : M ( i ) → M ( i + 1) and M ( β i ) : M ( i + 1) → M ( i ) for i = 1 , . . . , t − dim M ∈ Z t ≥ of an A t -module is defined by dim M = (dim M (1) , . . . , dim M ( t )).Let M t = M t,K be the category of A t -modules M such that M ( α i ) is injective for all i . Write M t ( d ) = M t,K ( d ) for the class of modules in M t with dimension vector d . It isshown in [13, §
4] that the orbits of P ( d ) in u ( d ) are in bijective correspondence with theisoclasses in M t ( d ). Moreover, there is a unique structure of a quasi-hereditary algebra on A t such that M t is the category of ∆ -filtered A t -modules, see [13, §
4] and [5, § K is infinite. Using the above correspondence from [13, §
4] and the results from [5], it was proved in [13, Thm. 4.1] that there is a finite number of P ( d )-orbits in u ( d ) if and only if t ≤
5. More precisely, this is deduced from the fact that A t has finite ∆-representation type if and only if t ≤
5, see [5, Prop. 7.2].Let t ≤
5. Because the results in [13, §
4] are proved for an arbitrary field, see [13, Rem.4.13], the parametrization of indecomposable ∆-filtered A t -modules does not depend onthe field K ; we explain this more explicitly below. Let { I , . . . , I m } be a complete set ofrepresentatives of isoclasses of indecomposable ∆-filtered A t -modules, and write d i for thedimension vector of I i . Let x i ∈ u ( d i ) be such that the P ( d i )-orbit of x i corresponds to theisoclass of I i . As discussed in [13, § x i to be a matrixwith entries 0 and 1; and these matrices do not depend on K . In particular, this impliesthat the modules I i are absolutely indecomposable. nother important consequence for us is the following lemma. Lemma 2.2.
Assume t ≤ . We may choose a set R of representatives of the adjoint P ( d ) -orbits in u ( d ) such that each element of R is a matrix with all entries equal to or .Moreover, the elements of R do not depend on the field K , i.e. the positions of entries equalto do not depend on K . We continue to assume that t ≤
5, and let d ∈ Z t ≥ . Let P = P ( d ), U = U ( d ) and x ∈ u = u ( d ). For the proof of Theorem 1.2 we require information about the structure ofthe centralizers C P ( x ) and C U ( x ), which is given by Propositions 2.5 and 2.8.Let M be a ∆-filtered A t -module (with dimension vector d ) whose isoclass corresponds tothe P -orbit of x . Extending the arguments of [13, § A t ( M ) of M is isomorphic to C P ( x ). Below we explain the structure of End A t ( M )and Aut A t ( M ), this uses standard arguments that we outline here for convenience. Weproceed to explain how C U ( x ) is related to End A t ( M ).As above, let { I , . . . , I m } be a complete set of representatives of isoclasses of indecom-posable ∆-filtered A t -modules. We may decompose M as a direct sum of indecomposablemodules(2.3) M ∼ = m M i =1 n i I i , where n i ∈ Z ≥ . Then End A t ( M ) ∼ = m M i,j =1 n i n j Hom A t ( I i , I j )as a vector space and composition is defined in the obvious way.We observed above that I i is absolutely indecomposable, which means that End A t ( I i ) is alocal ring, and that we have the decomposition End A t ( I i ) = K ⊕ m i , where K is acting byscalars and m i is the maximal ideal. Therefore, n i End A t ( I i ) ∼ = M n i ( K ) ⊕ M n i ( m i ) , where M n i ( K ) is a subalgebra and M n i ( m i ) is an ideal. In fact, we have that M n i ( m i ) is theJacobson radical of n i End A t ( I i ).Now one can see that the Jacobson radical of End A t ( M ) is J (End A t ( M )) ∼ = m M i =1 M n i ( m i ) ⊕ M i = j n i n j Hom A t ( I i , I j ) . Further, there is a complement to J (End A t ( M )) in End A t ( M ) denoted by C (End A t ( M ))with C (End A t ( M )) ∼ = m M i =1 M n i ( K ) . We are now in a position to describe the automorphism group Aut A t ( M ). We haveAut A t ( M ) ∼ = U ( C (End A t ( M ))) ⋉ (1 M + J (End A t ( M ))) , where U ( C (End A t ( M ))) denotes the group of units of C (End A t ( M )) and 1 M + J (End A t ( M ))is the unipotent group { M + φ | φ ∈ J (End A t ( M )) } . We have U ( C (End A t ( M ))) ∼ = mi =1 GL n i ( K ), and therefore Aut A t ( M ) ∼ = m Y i =1 GL n i ( K ) ⋉ N, where N is a split unipotent group over K . By saying N is a split unipotent group we meanthat N has a normal series with all quotients isomorphic to the additive group K . Thedimension of N is(2.4) δ := m X i =1 n i (dim End A t ( I i ) −
1) + X i = j n i n j dim Hom A t ( I i , I j ) . One can compute all Hom-groups Hom A t ( I i , I j ) from the underlying Auslander-Reiten quiv-ers of A t in [5, p221–222], see also [3, App. A]; the dimensions dim Hom A t ( I i , I j ) are inde-pendent of K . Therefore, the positive integer δ is also independent of K .We said above that Aut A t ( M ) of M is isomorphic to C P ( x ), so we have the followingproposition. Proposition 2.5.
The Levi decomposition of C P ( x ) is given by C P ( x ) ∼ = m Y i =1 GL n i ( K ) ⋉ N, where N , the unipotent radical of C P ( x ) , is a split unipotent group over K of dimension δ .Remark . It is natural to ask whether Proposition 2.5 holds without the restriction t ≤ t > K is assumed to be algebraically closed. It wouldbe interesting to know what happens in general, and also if Corollary 3.1 holds for t > C U ( x ). By a further extension of thearguments in [13, § C U ( x ) ∼ = 1 M + End ′A t ( M ) , where End ′A t ( M ) := { φ ∈ End A t ( M ) | φM ( l ) ⊆ M ( l −
1) for all l } , here we are identifying M ( l −
1) with its image in M ( l ) under M ( α l − ). We have thatEnd ′A t ( M ) is a nilpotent ideal of End A t ( M ). We defineHom ′A t ( I i , I j ) := { φ ∈ Hom A t ( I i , I j ) | φI i ( l ) ⊆ I j ( l −
1) for all l } . Then we have the isomorphismEnd ′A t ( M ) ∼ = m M i,j =1 n i n j Hom ′A t ( I i , I j ) . We write(2.7) δ ′ := dim End ′A t ( M ) = m X i,j =1 n i n j dim Hom ′A t ( I i , I j ) . From the Auslander-Reiten quivers of A t exhibited in [5, p221–222], one can computethe dimensions dim Hom ′A t ( I i , I j ). These integers are independent of K , so that δ ′ is also ndependent of K . The above discussion proves the following proposition, which concludesthis section. Proposition 2.8.
The centralizer C U ( x ) is a split unipotent group over K of dimension δ ′ . Proof of Theorem 1.2
Let q be a prime power and let K = F q be the field of q elements. Let t ≤ d ∈ Z t ≥ . Let P = P ( d ), U = U ( d ) and u = u ( d ) be as in the previous section, so P is aparabolic subgroup of GL n ( q ).The following corollary of Propositions 2.5 and 2.8 is a key step in our proof of Theorem1.2. It follows immediately from Propositions 2.5 and 2.8 along with the elementary factthat the order of a general linear group over F q is given by a polynomial in q . The positiveintegers in the statement are determined in (2.3), (2.4) and (2.7). Corollary 3.1.
Let x ∈ u . Then there are positive integers n , . . . , n m , δ and δ ′ independentof q such that | C P ( x ) | = m Y i =1 | GL n i ( q ) | · q δ , and | C U ( x ) | = q δ ′ . In particular, both | C P ( x ) | and | C U ( x ) | are polynomials in q with integer coefficients. We are now in a position to prove Theorem 1.2.
Proof of Theorem 1.2.
We have to prove that k ( U ) is given by a polynomial in q . As dis-cussed in the previous section k ( U ) = k ( U, u ), the number of adjoint U -orbits in u . We willprove that k ( U, u ) is a polynomial in q with integer coefficients.We may choose a set of representatives R of the adjoint P -orbits in u , as in Lemma 2.2and consider R to be independent of q . We have k ( U, u ) = X x ∈R k ( U, P · x ) , where k ( U, P · x ) is the number of U -orbits contained in P · x . For x ∈ u and g ∈ P , we have C U ( g · x ) = gC U ( x ) g − . Therefore, we get | U · x | = | U · ( g · x ) | and k ( U, P · x ) = | P · x | / | U · x | .It follows that k ( U, u ) = X x ∈R k ( U, P · x ) = X x ∈R | P · x || U · x | = | P || U | X x ∈R | C U ( x ) || C P ( x ) | = | L | X x ∈R | C U ( x ) || C P ( x ) | , where L is a Levi subgroup of P . Since | L | is a polynomial in q , it follows from Corollary3.1 and the fact that R is independent of q that k ( U, u ) = k ( U ) is a rational function in q .Since k ( U ) takes integer values for all prime powers, standard arguments show that k ( U ) isin fact a polynomial in q with rational coefficients, see for example [9, Lem. 2.11].Let P be the subgroup of GL n (¯ F q ) corresponding to P and let U be the unipotent radicalof P . The commuting variety of U is the closed subvariety of U × U defined by C ( U ) = { ( u, u ′ ) ∈ U × U | uu ′ = u ′ u } . etting C ( U ) = C ( U ) ∩ ( U × U ) and using the Burnside counting formula we get |C ( U ) | = X x ∈ U | C U ( x ) | = | U | · k ( U ) . Since | U | = q dim U and k ( U ) is a polynomial in q with rational coefficients, so is |C ( U ) | .Now using the Grothendieck trace formula applied to C ( U ) (see [4, Thm. 10.4]), standardarguments prove that the coefficients of this polynomial are integers, see for example [14,Prop. 6.1]. Thus, it follows that k ( U ) is a polynomial function in q with integer coefficients,as claimed. (cid:3) Remark . Let t ≤ d , d ′ ∈ Z t ≥ with d t = d ′ t = n . Suppose that P = P ( d )and Q = P ( d ′ ) are associated parabolic subgroups of GL n ( F q ), i.e. P and Q have Levisubgroups that are conjugate in GL n ( q ). This means that there is a σ ∈ S n such that d i − d i − = d ′ σ ( i ) − d ′ σ ( i ) − for all i = 1 , . . . , t , with the convention that d = d ′ = 0. Let U = U ( d ) and V = U ( d ′ ). A consequence of [13, Cor. 4.7] is that the number k ( P, U ) of P -conjugacy classes in U is the same as k ( Q, V ); the reader is referred to [9, Cor. 4.8] forsimilar phenomena. However, it is not always the case that the number of conjugacy classesof U is the same as the number of conjugacy classes of V . For example, take t = 3 andconsider the dimension vectors d = (2 , ,
4) and d ′ = (1 , , P ( d ) and P ( d ′ ) areassociated parabolic subgroups of GL ( q ). Let U = U ( d ) and V = U ( d ′ ). Then by directcalculation one can check that k ( U ) = ( q − + 6( q − + 5( q −
1) + 1 = ( q − + 4( q − + 6( q − + 5( q −
1) + 1 = k ( V ) . Acknowledgments : This research was funded in part by EPSRC grant EP/D502381/1.
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School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom
E-mail address : [email protected] URL : http://web.mat.bham.ac.uk/S.M.Goodwin Fakult¨at f¨ur Mathematik, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany
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