algorithmic testing for dense orbits of borel subgroups

Journal of Pure and Applied Algebra 197 (2005) 171–181www.elsevier.com/locate/jpaa

Algorithmic testing for dense orbits of Borelsubgroups

Simon GoodwinSchool of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK

Received 16 February 2004; received in revised form 4 August 2004Communicated by S. Donkin

Abstract

LetGbe a reductive algebraic group,Ba Borel subgroup ofGandU the unipotent radical ofB. Letu= Lie(U) be the Lie algebra ofU andn aB-submodule ofu. In this note we discuss the algorithmDenseOrbits of Borel Subgroups (DOOBS) which determines whetherBacts onnwith a dense orbit.We have programmedDOOBS in GAP4 and used it to classify all instances whenB acts onn witha dense orbit forG of semisimple rank at most 8 and chark zero or good forG. So in particular, wehave the classification forG of exceptional type.© 2004 Elsevier B.V. All rights reserved.

MSC:Primary: 14L30; secondary: 17B45; 20G15

1. Introduction

Let G be a reductive algebraic group over the algebraically closed fieldk. Let B bea Borel subgroup ofG andU the unipotent radical ofB. ThenB acts on the Lie algebrau=Lie(U) ofU via the adjoint action; this induces an action ofBon anyB-submodulen ofu. In this note we consider the question of whenB acts onn with a Zariski dense orbit. Wehave devised an algorithm Dense Orbits of Borel Subgroups(DOOBS)which answers thisquestion. In this note we describeDOOBS and discuss how it was used to determine allB-submodulesn such thatn is a prehomogeneous space forB, whenGhas semisimple rank

E-mail addresses:[email protected](S. Goodwin).URL: http://www.mat.bham.ac.uk/S.M.Goodwin

0022-4049/$ - see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jpaa.2004.08.038

http://www.elsevier.com/locate/jpaa

mailto:[email protected]

http://www.mat.bham.ac.uk/S.M.Goodwin

172 S. Goodwin / Journal of Pure and Applied Algebra 197 (2005) 171–181

at most 8 and chark is zero or good forG. This classification was obtained by programmingDOOBS in the computer algebra languageGAP4 [6].In 1987 Bürgstein and Hesselink wrote a computer program to analyse theB-orbits on

bothu andu∗ (see[5]). Using the ideas of this program, Jürgens andRöhrlewrote a programto determine when a parabolic subgroupPofG acts onP-submodules of the Lie algebra ofPu (the unipotent radical ofP) with a finite number of orbits (see[14]). Our algorithm wasinspired by these computer programs but employs quite a different approach.There hasbeena lot of other interest in the action of aBorel subgroupB (ormore generally

a parabolic subgroupP) of G on the Lie algebra of its unipotent radical. For example, allinstances whenP acts onpu = Lie(Pu) with a finite number of orbits are classified (see[11,14]). In fact there is a classification of all instances whenP acts on termsp(l)

u of thedescending central series ofpu with a finite number of orbits forG not of typeE7 or E8(see[3,4,10]).Richardson’s dense orbit theorem[16] says thatP always acts onpu with a Zariski open

orbit. Recently, there has been interest in the question of whenP acts onP-submodulesn of pu with a dense orbit. In[12,9,10] the casen = p

(l)u was considered. In particular,

the results of[9,10] give a classification of all instances whenB acts onb(l)u with a dense

orbit.We now give a short outline of the structure of this paper. We begin by introducing the

notation we require and proving some general results in Section 2. Then in Section 3 wedescribe the algorithmDOOBS and prove that it works when chark is zero or good forG.Next in Section 4, we explain howDOOBS was used to classify whenB acts onn with adense orbit forGof semisimple rank at most 8. In Section 5 we give two results which allowone to deduce further results from our classification and in Section 6 we discuss possiblegeneralisations ofDOOBS.As a general reference for the theory of algebraic groups we cite[1].

2. Preliminaries

LetRbe an algebraic group andVa rationalR-module. Letr=Lie(R) be the Lie algebraof R. ThenV is also a module forr. For x ∈ V , we writeR · x for theR-orbit of x inV andCR(x) for the stabiliser ofx in R. Similarly, we writer · x = {y · x : y ∈ r} andcr(x) = {y ∈ r : y · x = 0}. LetWbe anR-submodule ofV andSa closed normal subgroupof R. ThenV /W is a module forR and the action factors throughR/S if and only ifs · v − v ∈ W for all s ∈ S, v ∈ V .We recall thatV is said to be aprehomogeneous spacefor R, providedRacts onVwith

a dense orbit.The following lemma is elementary; the casewhereSis the trivial group is used repeatedly

in the sequel.

Lemma 2.1. Let R,S,V andW be as above and supposes ·v −v ∈ W for all s ∈ S, v ∈ V

(so that the action of R onV /W factors throughR/S). Suppose V is a prehomogeneousspace for R. ThenV /W is a prehomogeneous space forR/S.

S. Goodwin / Journal of Pure and Applied Algebra 197 (2005) 171–181 173

Proof. If x ∈ V is a representative of a denseR-orbit onV, thenx + W ∈ V /W is arepresentative of a denseR-orbit onV /W and thus a representative of a dense(R/S)-orbitonV /W . �

LetG be a reductive algebraic group over the algebraically closed fieldk. We denote thesemisimple rank ofGby ssrank(G) and writeg=Lie(G) for the Lie algebra ofG; likewisefor closed subgroups ofG. LetT be a maximal torus ofG and let� be the root system ofG with respect toT. A subgroup ofG is called (T-)regular if it is normalised byT. For aregular subgroupH ofGwe write�(H) =�(h) ⊆ � for the set of roots ofH with respecttoT. For a root� ∈ � we choose a generatore� of the corresponding root spaceg�.Let B be a Borel subgroup ofG containingT, let�+ = �(B) be the system of positive

roots determined byB and� = {�1, . . . , �r} the base determined by�+. We writeU forthe unipotent radical ofBandu for the Lie algebra ofU. Forx =∑

�∈�+ x�e� ∈ uwewritesupp(x) = {� ∈ �+ : x� �= 0} for thesupportof x. If chark = p >0, then we sayp is badfor G if there is some root� = ∑r

i=1 ai�i ∈ �+ such thatp dividesai for somei. We sayp is goodfor G if it is not bad forG. We recall the standard (strict) partial order≺ on�+is defined by:� ≺ � if � − � is a sum of positive roots.We recall that a subsetI of �+ is called anideal if � ∈ I , � ∈ �+ and� + � ∈ �+

implies� + � ∈ I . Given an idealI of �+ an element� ∈ I is called a generator if it is aminimal element ofI with respect to≺. We write�(I ) for the set of generators ofI; �(I )

forms an anti-chain in�+, that is�⊀� for all �, � ∈ �(I ). Further, the mapI → �(I ) is abijection between the set of all ideals of�+ and the set of anti-chains in�+. We refer thereader to[15, Sections 1, 2]for a more detailed account of ideals, anti-chains, etc.Let n be aB-submodule ofu. We recall thatn is determined by the ideal�(n) of �+.

The set of generators of�(n) is given by�(�(n))=�(n)\�([u,n]). Conversely, an idealI of�+ gives rise to theB-submodulenI = ⊕

�∈I g�.We now give some general results which we require in Section 3; the following two

Lemmas are[7, Lemma 3.1,3.2].

Lemma 2.2. Suppose the algebraic groupR =HN is the semi-direct product of the closedsubgroup H and the closed normal subgroup N. Let V be an R-module and letx ∈ V . Then

dim R · x = dim H · x + dim N · x − dim(H · x ∩ N · x).

In particular,H · x ∩ N · x is finite if and only ifdim R · x = dim H · x + dim N · x.

Remark 2.3. Suppose in Lemma 2.2 we do not assume thatN is normal inR. By lookingat the proof given in[7] we note that the result remains true in this more general situation.

Lemma 2.4. Let R be an algebraic group and let S be a maximal torus of R. Let V bean R-module and let�1, . . . , �j be linearly independent weights of V with respect to S. Letv1, . . . , vj beeigenvectors of Swithweights�1, . . . , �j respectively and letx=v1+· · ·+vj .ThenS · x = {t1v1+ · · · + tj vj : t1, . . . , tj ∈ k×}. In particular, dim S · x = j .

We now consider the unipotent varietyU ofGand the nilpotent varietyN of g. Supposethe derived subgroup ofG is simply connected and chark is zero or good forG. A slightly


strengthened version of a theorem of Springer (see[17, 13,�6.20]) says that there existsa G-equivariant isomorphism of varieties� : N → U. Such� is called aSpringerisomorphism.In the proof of Proposition 2.5 we frequently use the equivalent conditions for an orbit

map to be separable given by Borel[1, Proposition 6.7]; we do not make this reference inthe proof.

Proposition 2.5. Assumechark is zero or good for G.(i) Letu ∈ U . Then the orbit mapU → U · u is separable.(ii) Letx ∈ u. Then the orbit mapU → U · x is separable, so in particular

dim cu(x) = dim CU (x).

Proof. Assume the derived subgroup ofG is simply connected and let� : N → U bea Springer isomorphism. It is well-known that� mapsu to U. Let u ∈ U , we begin byshowing that�(cu(u)) = CU (u), wherecu(u) = {y ∈ u : u · y = y}.Let y ∈ cu(u), thenu ∈ CU (y) so u ∈ CU (�(y)), by G-equivariance and therefore

U-equivariance of�. Therefore,�(y) ∈ CU (u) and hence,�(cu(u)) ⊆ CU (u). A similarargument gives the reverse inclusion and therefore that�(cu(u)) = CU (u).In particular, dimcu(u) = dim CU (u), which implies that the orbit mapU → U · u is

separable, giving (i).Now letx ∈ u. The Springer isomorphism� transforms the orbit mapU → U · x to the

orbit mapU → U · �(x). The latter map is separable by (i) which implies that the formermap is also separable.SinceU is independent (up to isomorphism) of the isogeny class ofGwe can deduce the

result for arbitrary reductiveG. �

Remark 2.6. Let m be aB-submodule ofu and letx ∈ u. If chark is zero then allmorphisms are separable so the orbit mapU → U · (x + m) is separable. In fact usingrelative Springer isomorphismsone can show that this orbit map is separable if chark isgood forG (see[8, Corolary 4.4]). So in particular, we have

dim CU (x +m) = dim cu(x +m).

3. DOOBS

In this section we describe the algorithmDOOBS which determines whetherB acts onaB-submodulen of u with a dense orbit. We prove the algorithm works for chark zero orgood forG, so we make this assumption throughout this section. We begin by introducingsome notation.We fix an order for the roots�1, . . . , �m of �(n) (m = dim n) so that�j ≺ �i for each

i < j . We defineB-submodulesmi of n bymi = ⊕mj=i+1 g�j

for i = 0, . . . , m. Then wedefine the quotientsni = n/mi . DOOBS considers the action ofB on successivenis; ateach stage it finds a representativexi +mi (with supp(xi) linearly independent) of a denseB-orbit onni or decides thatni is not a prehomogeneous space forB.


We now give an outline of howDOOBS works. In this outline we do not justify why thealgorithm makes the decisions it does; this is covered in Theorem 3.1 below.0th step:DOOBS considers the action ofB on n0 = {0}. Trivially B acts onn0 with a

dense orbit, the algorithm chooses 0+m0 as a representative of a dense orbit and thereforesetsx0 = 0.ith step:DOOBS has chosen the representativexi−1+mi−1 of a denseB-orbit onni−1

with supp(xi−1) linearly independent. The algorithm considers the action ofB onni .

• FirstDOOBS considers theB-orbit ofxi−1+mi . It calculates the dimension ofcu(xi−1+mi )—knowledge of the Chevalley commutator relations reduces this to a problem inlinear algebra (see the end of this section for more details). If this is equal to|supp(xi−1)|then the algorithm decides thatB · (xi−1+mi ) is dense inni and so setsxi = xi−1 andgoes to the(i + 1)th step.

• If the algorithm decides thatB · (xi−1 + mi ) is not dense inni , then it considers theB-orbit of xi−1 + e�i

+ mi . The set of roots supp(xi−1) ∪ {�i} is considered; if theseroots are linearly independent then the algorithm decides thatB · (xi−1 + e�i

+ mi )

is dense inni . The algorithm then setsxi = xi−1 + e�iand goes to the(i + 1)th

step.• If DOOBS decides that neitherB · (xi−1 +mi ) norB · (xi−1 + e�i

+mi ) is dense inni , then it decides thatB does not act onni (and therefore onn) with a dense orbit andstops.

(m + 1)th step:DOOBS has chosen a representative of a dense orbit onnm = n so itconcludes thatB does act onn with a dense orbit and finishes.In Theorem 3.1 below we justify thatDOOBS does correctly decide whetherB acts on

n with a dense orbit.

Theorem 3.1.DOOBS correctly decideswhetherBacts onnwith adenseorbit.Moreover,if B does act onn with a dense orbit thenDOOBS find a representative of this orbit.

Proof. We begin by introducing some notation which we require in the proof. When con-sidering the action ofB onni writeAH (y +mi ) = H · (y +mi ) ∩ (y + ke�i

+mi ) wherey ∈ n,H ∈ {B, T , U} andy + ke�i

+mi = {y + �e�i+mi ∈ ni : � ∈ k}.

Suppose thatB does act onn with a dense orbit. We work by induction and prove thefollowing for eachi = 0, . . . , m:

(IH1) A representativexi +mi of a denseB-orbit onni has been found.(IH2) xi is of the formxi = ∑

j∈J e�jwhereJ = {j : �j ∈ supp(xi)} and{�j : j ∈ J } is

linearly independent.(IH3) We have dim(U · (xi +mi )) = i − |J | and dim(T · (xi +mi )) = |J |.(IH4) For eachj = 1, . . . , i, AU (

∑k∈J,k<j e�k

+ mj ) has cardinality 1 or is equal tox + ke�j

+mj . Moreover,j ∈ J if and only if |AU (∑

k∈J,k<j e�k+mj )| = 1.

The theorem then follows from the casei = m.The base casei =0 is trivial so we assume (IH1)–(IH4) hold fori = l −1 andx =xl−1=∑j∈J e�j

whereJ = {j : �j ∈ supp(xl−1)}.


First, consider theU-orbitU · (x +ml ). By (IH3) for i = l − 1 and Lemma 2.2 we seethatU · (x + ml−1) ∩ T · (x + ml−1) is finite; from this and Lemma 2.4 it follows thatU · (x +ml ) ∩ T · (x +ml ) is finite. Clearly, dim(U · (x +ml )) is equal to eitherl − |J | orl − |J | − 1. Therefore, by Lemma 2.2 it follows thatx +ml is a representative of a denseB-orbit onnl if and only if dim CU (x +ml ) = |J |. By Remark 2.6 this occurs if and onlyif dim cu(x +ml ) = |J |. So we see that (IH1), (IH2) hold if dimcu(x +ml ) = |J |. Alsoclearly dim(T · (x +ml )) = |J | so (IH3) holds.Now considerAU (x + ml ). SinceU is unipotentU · (x + ml ) is closed innl . There-

fore, AU (x + ml ) is closed inx + ke�l+ ml . Thus, since as an algebraic varietyx +

ke�l+ ml is isomorphic tok, we have thatAU (y + ml ) is either finite or equal to

x + ke�l+ml .

Next considerAB(x +ml ). SinceB · (x +ml ) is dense innl , it is open innl . Therefore,AB(x +ml ) is open inx + ke�l

+ml and thus is infinite.Let b = ut ∈ B be such thatb · (x +ml ) ∈ x + ke�l

+ml . We have thatt · (x +ml ) =∑j∈J �j (t)e�j

+ml . If �j (t) �= 1 for somej ∈ J we get a contradiction from (IH4) fori = l − 1. For letj be minimal subject to�j (t) �= 1, then we haveu · ((

∑k∈J,k<j e�k

) +�j (t)e�j

+mj ) = ∑k∈J,k<j e�k

+ e�j+mj . Therefore, sinceu · (e�j

+mj ) = e�j+mj ,

we haveu · (∑

k∈J,k<j e�k+mj ) = (

∑k∈J,k<j e�k

) + (1−�j (t))e�j+mj . It follows that

AU (x +ml )=AB(x +ml ) is infinite so thatAU (x +ml )=x +ke�l+ml .This gives (IH4)

for i = l and so completes the induction in the case dimcu(x +ml ) = |J |.Now suppose that dimcu(x +ml )=|J |+1 and lety +ml be a representative of a dense

B-orbit onnl . Then by Lemma 2.1,y +ml−1 is a representative of a denseB-orbit onnl−1.Therefore,y +ml−1 isB-conjugate tox +ml−1 and so wemay assume thaty is of the formy = x + �e�l

where� ∈ k\{0}.Our arguments above show thatAU (y +ml ) is either finite or equal tox + ke�l

+ml .SinceB · (x + ml ) is not dense innl (so y + ml is not in the sameB-orbit asx + ml)it follows thatAU (y + ml ) must be finite. As above we also have thatAB(y + ml ) isinfinite.Let b = ut ∈ B whereu ∈ U, t ∈ T be such thatb · (y + ml ) ∈ x + ke�l

+ ml . Wehave thatt · (y +ml ) = ∑

j∈J �j (t)e�j+ ��l (t)e�l

+ml . If �j (t) �= 1 for somej ∈ J weget a contradiction from (IH4) fori = l − 1 as above. Then since|AU (y +ml )| is finite, itfollows that�l (t) �= 1 for somet ∈ T , which implies that{�j : j ∈ J } ∪ {�l} is linearlyindependent. Then by Lemma 2.4 we see thatx + (k\{0})e�l

+ml ⊆ AT (y +ml ) so wecan assume� = 1 and also that dim(T · y) = |J | + 1. If |AU (y +m)| �= 1 then one can seethatx +ml ∈ AB(y +ml ) which is not possible. The above arguments give (IH1)–(IH4)in the casecu(x +ml ) = |J | + 1.This completes the induction.�

Remark 3.2. The proof above shows thatDOOBS would work for any value of chark ifdim CU (xi + mi+1) was calculated instead of dimcu(xi + mi+1). Also the algorithm isstill valid for other values of chark provided all the orbit mapsU → U · (xi +mi+1) areseparable.Let dimp cu(xi +mi+1) denote the dimension ofcu(xi +mi+1)when chark =p�0. By

considering the method used for calculating dimcu(xi +mi+1) outlined at the end of this


section, we see that dimp cu(xi +mi+1)�dim0 cu(xi +mi+1) for anyp. It follows that allorbit mapsU → U ·(xi +mi+1) are separable if dimp cu(xi +mi+1)=dim0 cu(xi +mi+1)for eachi. If this is the case for somep >0, the result given byDOOBS is valid forchark = p.

We now give the following corollary of the proof of Theorem 3.1.

Corollary 3.3. Supposen is a prehomogeneous space for B. Then there is a linearly inde-pendent subset ⊆ �(n) such thatx = ∑

�∈ e� is a representative of the dense B-orbitonn.Moreover we have

(i) dim U · x = dim n− ||;(ii) dim T · x = ||;(iii) U · x ∩ T · x is finite.

Proof. The existence ofx such thatB · x is dense inn and satisfying (i) and (ii) followsdirectly from the proof of Theorem 3.1. Condition (iii) then follows from Lemma 2.2.�

We have programmedDOOBS in the computer algebra languageGAP4 [6]. We nowbriefly explain how this was achieved. The program is available on the author’s websitehttp://web.mat.bham.ac.uk/S.M.Goodwin/DOOBS.html .The functions for Lie algebras inGAP4 are used to define the required mathematical

objects. Checking if a set of roots is linearly independent is easily achieved by calculatingthe rank of the matrix whose rows correspond to these roots. The method for calculatingthe dimension of centralisers inu is similar to that used in the functionDenseTest whichwas used in[10]. We now describe this method.Let n be aB-submodule ofu,m a submodule ofn and letx ∈ n. An arbitrary element

y ∈ u can bewritten asy=∑�∈�+ y�e�.Writex+m=∑

�∈�(u)\�(m) x�e�+m. Using theChevalleycommutator relations,wemaycalculate[y, x]+m=∑

�∈�(n)\�(m) z�(y�)e�+m.This is done using the functions for Lie algebras inGAP4. Thenz� is linear iny� for all �.Therefore, we see that dimcu(x +m) is equal to the dimension of the solution space of thesystem of linear equationsz� = 0, for � ∈ �(n)\�(m). LetE be the(dim n− dim m) ×dim u matrix corresponding to this system of equations.The algorithmDOOBS calculates dimcu(x + m) by first determining the matrixE

and then row reducing it to determine its rank. To reduce the required computer time thealgorithm does not recalculate and row reduce the matrixE at each step but uses the rowreducedmatrix from the previous step. The reduction of thematrixEensures that the entriesof the reduced matrix are integers and attempts to keep the modulus of the entries low.The algorithm also keeps track of the values ofp for which we know dimp cu(xi +

mi+1) = dim0 cu(xi + mi+1) for eachi (using the language of Remark 3.2). That is thevalues for which our row reduced matrix has the same rank when reduced modulop. UsingRemark 3.2 we see that the result given byDOOBS is the same for these characteristics asfor characteristic zero. We note that it is possible that this need not be the case even whenp is good forG—this never occurred in the calculations we carried out, which means thatRemark 2.6 is not necessary for the results we obtained.

http://web.mat.bham.ac.uk/S.M.Goodwin/DOOBS.html


4. Classification for ssrank(G)�8

We have used the version ofDOOBS programmed inGAP4 to classify all instanceswhenn is a prehomogeneous space forBwhen ssrank(G)�8. The results are available athttp://web.mat.bham.ac.uk/S.M.Goodwin/DOOBS.html .We wrote a program inGAP4 which computes, for a givenG, all B-submodules of

bu, then runsDOOBS on each submodule. Our program outputs two files: the first is aLATEX file which can be used to create a dvi file that can be read easily; the second is a textfile which one can read intoGAP4 and is then easy to search through. In all cases the onlycharacteristic restrictions given by the program were primes which are not good forG (seethe end of Section 3).Below we includeTable 1which is the beginning of the file created by the LATEX file

output forG of typeF4. To understand it we needTable 2which gives a numbering of thepositive roots of the root system of typeF4. The numbering is that used inGAP4 we usethe notation of[2, Planche VII]for the roots. The first column ofTable 1gives the numbersof the roots generatingn and is lexicographically ordered. The second column is blank ifB does not act onn with a dense orbit and contains the numbers of the roots in the supportof the representative found byDOOBS if B does act onn with a dense orbit. The thirdcolumn gives the values of chark for which the stated result may not be true (see the end ofSection 3).

5. Further results

In this section we give two results which allow one to deduce further results from ourclassification. These results give instances when we can determine if a parabolic subgroupP acts on aP-submodule ofpu= Lie(Pu) with a dense orbit, herePu denotes the unipotentradical ofP.Our first result is a particular application of Lemma 2.1.

Proposition 5.1. Let P and Q be parabolic subgroups of G withP ⊆ Q. Let Q̂ = Q/Qu,and P̂ = P/Qu. Let n be a P-submodule ofpu and let n̂ = n/(n ∩ qu). Then ifn is aprehomogeneous space for P, n̂ is a prehomogeneous space forP̂ .

Proof. This follows from Lemma 2.1 with ‘R = P ’, ‘ S = Qu’, ‘ n= n’ and ‘m= n ∩ qu’.�

In the notation of Proposition 5.1 we can think ofQ̂ as a Levi subgroup ofQ. ThenP̂ isa parabolic subgroup of̂Q andn̂ is a P̂ -submodule of̂pu. SupposeP̂ is a Borel subgroupof Q̂ which does not act on̂n with a dense orbit. Then we can deduce from Proposition 5.1thatP does not act onn with a dense orbit.Our next result generalises a technique used in the proof of[10, Theorem 6.1]. In its

statementP� denotes the standard minimal parabolic subgroup ofG corresponding to thesimple root� and in the proofU−� denotes the root subgroup corresponding to−�.

http://web.mat.bham.ac.uk/S.M.Goodwin/DOOBS.html


Table 1DOOBS output forF4

�(I ) supp(x) Primes

1 1, 19 21, 2 21, 2, 31, 2, 3, 4 1, 2, 3, 4 2,31, 2, 41, 2, 7 1, 2, 7, 10 21, 2, 10 1, 2, 9, 10 21, 3 1, 3, 10, 16 21, 3, 4 1, 3, 4, 13 21, 3, 6 1, 3, 6, 10 21, 41, 61, 6, 71, 6, 10 1, 6, 9, 10 21, 7 1, 7, 10, 13 21, 9 1, 9, 13, 23 21, 9, 10 1, 9, 10 21, 10 1, 10, 24 21, 13 1, 13, 23 21, 16 1, 16, 22 222, 3 22, 3, 4 22, 4 2, 4, 10, 15 22, 4, 5 2, 4, 5, 10 22, 5 22, 5, 7 2, 5, 7, 20 22, 5, 10 2, 5, 8, 10 22, 7 2, 7, 15 22, 8 2, 8, 16, 22 22, 8, 10 2, 8, 10 22, 102, 12 2, 12, 16, 20 22, 15 2, 15, 163 23, 4 23, 6 3, 6, 8, 18 244, 55 5, 17 25, 6 5, 6, 13, 23 25, 6, 7 5, 6, 7, 13 25, 6, 10 5, 6, 10 25, 7 2, 35, 9 5, 9, 16, 20 25, 9, 10 8 2, 3


Table 2Numbering of roots ofF4

1 0001 4 0100 7 0110 10 0120 13 1120 16 1220 19 1231 22 12422 1000 5 0011 8 0111 11 1111 14 1121 17 1221 20 1222 23 13423 0010 6 1100 9 1110 12 0121 15 0122 18 1122 21 1232 24 2342

Proposition 5.2. Let P = P� be a minimal parabolic subgroup of G and letnI be a P-submodule ofpu, whereI = �(nI ) is an ideal of�+. Suppose there exists� ∈ �(I ) suchthat � + � is a generator ofJ , whereJ = I\{�}. Then P acts onnI with a dense orbit ifand only if B acts onnJ with a dense orbit.

Proof. LetK = I\{�, � + �}. We can consider the action ofP on nI /nK . It is clear thate�+� + nK is a representative of a denseP-orbit onnI /nK . Therefore, ifP acts onnI witha dense orbit then there is a representative of the formx = e�+� + x′ wherex′ ∈ nK . Letx be of the above form then one can see thatB · x ∩ U−� · x must be finite. So by a moregeneral version of Lemma 2.2(see Remark 2.3) we have

dim(P · x) = dim(B · x) + 1.

From this it follows thatP · x is dense innI if and only ifB · x is dense innJ . �

Remark 5.3. There is a natural generalisation of Proposition 5.2 to the case whereB andP are arbitrary parabolic subgroups withB ⊆ P . We choose not to include this here as itsstatement and proof are quite technical.

6. Generalisations

We now discuss howDOOBS could be adapted to work in more general settings.First we consider simply weightedB-modules which are defined in[5]; these include

u and its dualu∗. Further, any quotient of a submodule ofu or u∗ is simply weighted.The strategy ofDOOBS works on any such modules, the algorithm would only need to bemodified to consider the different root space decompositions.Our algorithm stops if it determines thatB does not act onn with a dense orbit. It would

be possible to modifyDOOBS so that it instead determines a family ofB-orbits (whichwould be parameterised bykm for somem) which is dense inn. This may be useful inmaking estimates for the modality ofB on submodules ofu.Finally, we discuss how one could try to generaliseDOOBS to work for an arbitrary

parabolic subgroupP. Let n be aP-submodule ofpu. One would consider a compositionseries 0=m0 ⊆ · · · ⊆ ms =n ofP-submodules and the corresponding quotientsni =n/mi .Since the quotientsmi/mi−1 would not in general be 1-dimensional it would be moredifficult to determine a representative of a denseP-orbit onni from a representative of adenseP-orbit onni−1.


Acknowledgements

I would like to thank Gerhard Röhrle for many useful discussions on earlier versions ofthis paper. The calculations were carried out on a cluster of computers at the Universityof Birmingham, courtesy of Robert Wilson. I also acknowledge the financial support ofEPSRC.

References

[1] A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126, Springer, Berlin, 1991.[2] N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1975 (Chapitres 4–6).[3] T. Brüstle, L. Hille, Finite, tame and wild actions of parabolic subgroups in GL(V) on certain unipotentsubgroups, J. Algebra 226 (2000) 347–360.

[4] T. Brüstle, L. Hille, G. Röhrle, Finiteness results for parabolic group actions in classical groups, Arch. Math.76 (2) (2001) 81–87.

[5] H. Bürgstein, W.H. Hesselink, Algorithmic orbit classification for some Borel group actions, CompositeMath. 61 (1987) 3–41.

[6] The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.3, 2002. Available from:<http://www.gap-system.org >.

[7] S. Goodwin, Prehomogeneous spaces for the coadjoint action of a parabolic group, J. Algebra 279 (2) (2004)558–565.

[8] S. Goodwin, Relative Springer Isomorphisms, Preprint, The University of Birmingham, 2004.[9] S. Goodwin, G. Röhrle, Prehomogeneous spaces for parabolic group actions in classical groups, J. Algebra276 (1) (2004) 383–398.

[10] S. Goodwin, G. Röhrle, Finite orbit modules for parabolic subgroups of exceptional groups, Indag. Math. 15(2) (2004) 189–207.

[11] L. Hille, G. Röhrle, A classification of parabolic subgroups with a finite number of orbits on the unipotentradical, Transformation Groups 4 (1) (1999) 35–52.

[12] L. Hille, G. Röhrle, Variation on a theme of Richardson, Linear Algebra Appl. 365 (2003) 239–246.[13] J.E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys andMonographs,

vol. 43, American Mathematical Society, Providence, RI, 1995.[14] U. Jürgens, G. Röhrle,MOP—algorithmic modality analysis for parabolic group actions, Exp. Math. 11 (1)

(2002) 57–67.[15] D. Panyushev, Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra 274 (2) (2004)

822–846.[16] R.W. Richardson, Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London

Math. Soc. 6 (1974) 21–24.[17] T.A. Springer, The unipotent variety of a semisimple group, Algebraic Geometry (Internat. Colloq. Tata Inst.

Fund. Res., Bombay, 1968), Oxford University Press, London, 1969, pp. 373–391.

http://www.gap-system.org

algorithmic testing for dense orbits of borel subgroups

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