aa r X i v : . [ m a t h . R T ] F e b Orbits of Z ◦ (2 .O +8 (2) . in Dimension 8 Frank LübeckFebruary 12, 2021
Abstract
Groups of structure .O +8 (2) have an irreducible representation ofdegree which can be realized over Z and any prime field F p . This rep-resentation extends to a group of structure .O +8 (2) . . Any subgroup Z ≤ F × p acts by scalar multiplication on this module over F p .In this short note we determine for which primes p > and which Z the central products Z ◦ (2 .O +8 (2) and Z ◦ (2 .O +8 (2) . have a regularorbit on the -dimensional F p -module.This work was triggered by an omission in the paper [KP01] byKöhler and Pahlings, a paper which is used in various places in workon the k ( GV ) -problem. In [KP01] Köhler and Pahlings investigated the following problem:Let p be a prime, G be a finite group whose order is not divisible by p and V be a finite dimensional faithful F p G -module of dimension n . Further-more, assume that G has a quasisimple normal subgroup E which also actsirreducibly on V . Does G have a regular orbit on V ?In most cases the answer to this question is yes, but there is a list ofexceptions. The main result of [KP01, Theorem 2.2] is the table of theseexceptions.For a fixed quasisimple group E and F p E -module V the possible groups G (as groups of endomorphisms of V ) are generated by a subgroup of Aut ( E ) and a subgroup Z ≤ F × p of scalar matrices, see [KP01, Section 3].From now we consider the specific case E = 2 .O +8 (2) and its irreduciblerepresentation of degree n = dim( V ) = 8 , hence p > . In this case Aut ( E ) .O +8 (2) . and the representation extends in two ways to thisgroup, see [CCN +
85, p.85]. The authors of [KP01] determine for which p thegroups of form Z ◦ E have a regular orbit on V by an elaborate computationwith the table of marks of O +8 (2) . But they forgot (in statement and proof)to handle the groups of form Z ◦ E. . In this short note we will close thisgap. We will also recover (with a slight correction) their result for the groups Z ◦ E with an easier argument. .O +8 (2) . and its -dimensionalrepresentations over F p There are two isomorphism types of groups with structure .O +8 (2) . whichare isoclinic, see [CCN +
85, Ch.6, Sec.7]. For one type the (Brauer)-charactersof the -dimensional representations have values in the rational integers andfor the other type the character values generate Z [ i ] ( i = − ). In the lattercase the -dimensional representations can only be realized over F p if the fieldcontains primitive fourth roots of unity, that is if p ≡ mod . If Z ≤ F × p , | Z | = 4 , and G is one group of type .O +8 (2) . we find the isoclinic group assubgroup of index in Z ◦ G (exchange the generators x of G which are notin the derived subgroup G ′ with i · x ).Now let W be the Weyl group of type E . It has the structure .O +8 (2) . ,see [CCN +
85, p.85]. Since Weyl groups have rational character values it isclear which one of the isoclinic groups this is. We denote ˜ W the isoclinicgroup.Now we can state our result. Theorem 2.1.
A group of type Z ◦ (2 .O +8 (2)) has no regular orbit on its -dimensional irreducible F p G -module if and only if p ≤ or p = 31 and | Z | > .A group G of type Z ◦ W has no regular orbit on its -dimensional irre-ducible F p G -modules, if and only if p ≤ or p = 31 and | Z | > .In case p ≡ mod a group G of type Z ◦ ˜ W has no regular orbit onits -dimensional irreducible F p G -modules if and only if one of the followingholds • p < , • p = 29 and | | Z | , p = 31 and | Z | > . From one of the irreducible representations of .O +8 (2) . of dimension weget the other one by tensoring with the -dimensional representation withkernel .O +8 (2) . So the action of elements outside the derived subgroup onlydiffers by scalar multiplication with − . Since the central element, whichis contained in the derived subgroup, also acts by the scalar − , the orbitson V are the same for both module structures. Therefore, it is enough toconsider one of the -dimensional modules.We now consider the Weyl group W of type E . It can be describedas follows as subgroup of GL ( Z ) , and this way we get for any prime p an -dimensional representation of W over F p by reducing the matrix entriesmodulo p .Let Y = Z and X be the dual Z -lattice. We describe a root datum oftype E . For this we take the standard basis vectors of Y as set of simplecoroots α ∨ j , ≤ j ≤ , and as corresponding simple roots α j ∈ X the rows ofthe following matrix (written with respect to the Z -basis of X which is dualto the simple coroots, the elements of this basis are also called fundamentalweights): − − − − − − − − − − − − −
10 0 0 0 0 0 − For ≤ j ≤ we can use α j and α ∨ j to define the following reflection on X : s j : X → X, x x − α ∨ j ( x ) · α j . The group generated by these reflections W = h s j | ≤ j ≤ i is theWeyl group of type E , it is a Coxeter group with the s j , ≤ j ≤ , asset of Coxeter generators. The orbit α W is called the root system of type E , it contains roots. The dual construction on Y yields the correspond-ing coroots and the highest coroot (the one with is componentwise ≥ the3oordinates of all other coroots) is α ∨ = (cid:0) (cid:1) . The corresponding root is α = (cid:0) (cid:1) and this de-fines a reflection s : X → X as above.For any prime p > we are interested in the orbits of the action of W on X modulo pX (identifying F p = Z /p Z and F p = X/pX ).The subgroup of bijections X → X , generated by W and the translationsby elements of pX is called the affine Weyl group W p of type E , see [Hum90,4.3, 4.8] (we use that the Z -span of the roots is all of X , and we scale thetranslations in the reference by a factor p ).The action of W p on X can be R -linearly extended to the vector space X ⊗ R . We consider the following subset of X ⊗ R which is called the bottomalcove: A := { x ∈ X ⊗ R | α ∨ j ( x ) > for ≤ j ≤ , α ∨ ( x ) < p } , and will use the following theorem, see [Hum90, 4.8]. Theorem 3.1.
The closure ¯ A of A is a fundamental domain for the actionof W p on X ⊗ R . If x ∈ ¯ A then the stabilizer of x in W is generated bythose s j , ≤ j ≤ , with α ∨ j ( x ) = 0 together with s in case α ∨ ( x ) = p .In particular, every W p -orbit on X ⊗ R has a unique representative x ∈ ¯ A and the orbit is regular if and only if x ∈ A . Restricting this to points x = ( x , . . . , x ) ∈ X we conclude: There existsa regular W -orbit on X modulo pX if and only if A contains a point withinteger coordinates. That is, all x j ∈ Z > (because α ∨ j ( x ) = x i ) for ≤ j ≤ and α ∨ ( x ) < p . Since the coordinates of α ∨ are all positive, such an x exists if and only if ρ := (cid:0) (cid:1) ∈ A if and only if α ∨ ( ρ ) = 29 < p .It is an easy programming exercise to enumerate for moderate p all in-tegral points x ∈ ¯ A , and to read off their stabilizers in W . For all p < these stabilizers all have order > . This shows that also the orbits of thederived subgroup W ′ = 2 .O +8 (2) of W are never regular.The case p = 29 . Here all integral x ∈ ¯ A with x = ρ have stabilizer oforder > . But ρ ∈ ¯ A and its stabilizer is generated by s and is of order . Since the reflections are a single conjugacy class of W and generate W wesee that s / ∈ W ′ . So, the orbit of ρ is (the only) regular W ′ -orbit.We have shown our Theorem 2.1 for G = W ′ and G = W .4 ction of scalars Now we consider the action of scalars. We will see that we find all informationwe need by considering the orbit of ρ ∈ X modulo pX .We want to know for all primes p all c ∈ Z modulo p such that there isan element w ∈ W with ρw ≡ cρ mod pX . The center of W ′ = 2 .O +8 (2) actsas scalar − . So, all W ′ -orbits on X will be closed under multiplication with − .Fixing w ∈ W and setting y = ( y , . . . , y ) := ρw we want to know forwhich p there is a c such that y ≡ cρ mod pX . This relation is equivalent tothe condition gcd ( y − c, . . . , y − c ) ≡ mod p for some c ∈ Z . We usegcd ( y − c, . . . , y − c ) = gcd ( y − c, y − y , . . . y − y ) | gcd ( y − y , . . . , y − y ) and compute the latter expression.The possible p are the prime divisors of gcd ( y − y , . . . , y − y ) . And itis clear that for each such prime there is some c (unique modulo p ) such that y − c is also divisible by p .Using a computer and GAP [GAP19] we computed the full W -orbit of ρ and the decribed gcd’s. For this we used the explicit construction of therepresentation given above. The computation took about 45 minutes on theauthors notebook.The prime divisors of these numbers are all ≤ . So, for p > the only F p - multiple of ρ occuring in the orbit of ρ modulo pX is − ρ , and thereforethe orbit of ρ is also regular for any central product Z ◦ W and so for Z ◦ W ′ , Z ≤ F × p .The case p = 31 . It is easy to see that ρ is the only integral point in A for p = 31 , so there is only one regular W -orbit modulo , namely theorbit of ρ . Therefore it is not surprising that this W -orbit contains all scalarmultiples of ρ . Looking closer at the set of w ∈ W which yield multiplesof ρ modulo in our gcd-computations we notice that they are all of evenlength, so that already the W ′ -orbit of ρ contains all multiples of ρ modulo . This proves the exceptions for p = 31 in Theorem 2.1.The case p = 29 . Our gcd-computations show that in the orbit of ρ modulo the only multiples of ρ are ± ρ . The orbit is not regular, and ρw = ρ modulo for w = 1 and w = s . Now we consider the isoclinicgroup ˜ W which is generated by the is j , ≤ j ≤ where i ∈ F is of order .The orbit of ρ under ˜ W contains powers of i scalar multiples of the vectors5n the orbit under W . The element s ∈ W is of length , so that thecorresponding product of generators of ˜ W will map ρ to iρ . This shows thatthe orbit of ρ under ˜ W contains all four multiples i k ρ , ≤ k ≤ . So ρ ˜ W istwice as long as ρW , hence a regular orbit.We have shown all statements in Theorem 2.1 concerning the cases with p = 29 . This finishes our proof. Acknowledgement.
I thank Melissa Lee for making me aware of thegap concerning O +8 (2) in the paper [KP01]; this caused a corresponding gapin early versions of her article [Lee20] (which hopefully can now be closed). References [CCN +
85] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, andR. A. Wilson.
Atlas of Finite Groups . Oxford University Press,Eynsham, 1985. Maximal subgroups and ordinary characters forsimple groups, With computational assistance from J. G. Thack-ray.[GAP19] GAP – Groups, Algorithms, and Programming, Version 4.10.2. , Jun 2019.[Hum90] J. E. Humphreys.
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Cambridge Studies in Advanced Mathematics . Cam-bridge University Press, Cambridge, 1990.[KP01] Ch. Köhler and H. Pahlings. Regular Orbits and the k ( GV ) -Problem. In Groups and Computation, III (Columbus, OH, 1999) ,volume 8 of
Ohio State Univ. Math. Res. Inst. Publ. , pages 209–228. de Gruyter, Berlin, 2001.[Lee20] Melissa Lee. Regular Orbits of Quasisimple Linear Groups I. https://arxiv.org/abs/1911.05785https://arxiv.org/abs/1911.05785