Algebra and Logic, Vol. 49, No. 5, 2010

PRIMITIVE PARABOLIC PERMUTATION

REPRESENTATIONS FOR FINITE SIMPLE

ORTHOGONAL GROUPS IN ODD DIMENSIONS

V. V. Korableva∗ UDC 512.542.5

Keywords: permutation representation, parabolic subgroup, finite simple group, classical group,isotropic subspace.

Ranks, degrees, subdegrees, and double stabilizers of permutation representations forfinite simple orthogonal groups in odd dimensions are defined on cosets with respect tomaximal parabolic subgroups.

INTRODUCTION

Sufficiently full information about a permutation representation of a finite group is given bythe following parameters: degree, rank, subdegrees, structure of a point stabilizer and of doublestabilizers. By now these parameters have been obtained for minimal faithful permutation repre-sentations of all finite simple groups of Lie type (see [1-6]). An important class of permutationrepresentations for finite groups of Lie type is formed by parabolic representations, that is, rep-resentations on cosets with respect to parabolic subgroups. Groups of Lie type are divided intoclassical groups (by which are meant those that have natural representations via automorphismgroups of vector spaces) and exceptional groups.

In [7-15], ranks, degrees, subdegrees, and double stabilizers of primitive parabolic permutationrepresentations were computed for all finite simple exceptional groups of Lie type and for finitespecial linear, unitary, and symplectic groups. Ranks of permutation representations for classicalgroups Bl(q), Cl(q), and Dl(q) on cosets with respect to maximal parabolic subgroups were deter-mined in [16]. In the present paper, we work to find parameters of primitive parabolic permutationrepresentations for finite orthogonal groups in odd dimensions over fields of odd characteristic. Ourargument is effected in terms of linear transformations and bilinear forms.

∗Supported by RFBR (project No. 10-01-00324).

Chelyabinsk State University, ul. Br. Kashirinykh 129, Chelyabinsk, 454021 Russia; vvk@gcsu.ru. Trans-lated from Algebra i Logika, Vol. 49, No. 5, pp. 615-629, September-October, 2010. Original article submit-ted September 28, 2009; revised February 1, 2010.

416 0002-5232/10/4905-0416 c© 2010 Springer Science+Business Media, Inc.

1. PRELIMINARIES

Let V be a vector space of dimension l over a finite field GF (q) and f a nondegenerate bilinearsymmetric form on V , i.e., a mapping from V × V to GF (q) such that f(αx + βy, z) = αf(x, z) +βf(y, z), {v ∈ V | f(v, x) = 0 for all x ∈ V } = {0}, and f(x, z) = f(z, x) for all α, β ∈ GF (q)and all x, y, z ∈ V . Denote by GL(V ) the group of all nondegenerate linear transformations ofV . A subgroup of GL(V ) consisting of elements ϕ for which f(xϕ, yϕ) = f(x, y) with all x and y

in V is called an isometry group of f and is denoted by I(f). Matrices of elements of GL(V ) inan arbitrary fixed basis for V form a group GLl(q) of all nondegenerate matrices of order l overGF (q), isomorphic to GL(V ). The order of GLl(q) is equal to q(l2−l)/2(ql− 1)(ql−1− 1) . . . (q− 1).

A quadratic form F on V is a mapping from V to GF (q) such that F (λx) = λ2F (x), whereλ ∈ GF (q), x ∈ V , and F (x+y) = F (x)+f(x, y)+F (y), where f is some symmetric bilinear form.Note that each symmetric bilinear form f is determined by a quadratic form F . On the otherhand, if GF (q) has odd characteristic, then 2 · F (x) = f(x, x) and a quadratic form F likewise isuniquely determined by a symmetric bilinear form f . The last claim is invalid for fields of evencharacteristic. A quadratic form F is nondegenerate if its corresponding symmetric bilinear formf is nondegenerate. Below the value f(x, y) of a form f on a pair (x, y) ∈ V × V is denoted by(x, y), and we assume that F is a nondegenerate form.

An orthogonal group of a quadratic form F is I(F ) = {ϕ ∈ GL(V ) | F (xϕ) = F (x) for all x ∈V }. If GF (q) is a field of odd characteristic then I(F ) = I(f). If, however, GF (q) has evencharacteristic then (x, x) = 0 holds for any x ∈ V and f is a skew-symmetric form. ThusI(F ) < I(f) ∼= Sp(V ) with even q. Each nondegenerate skew-symmetric bilinear form should bedefined on a space of even dimension (see [17, Prop. 2.4.1]). Therefore, we have

LEMMA 1 [17, Prop. 2.5.1]. If the characteristic of a field GF (q) is equal to 2, then thedimension of a space V , on which a nondegenerate symmetric bilinear form is defined, is even.

LEMMA 2 [17, Prop. 2.3.5]. An odd-dimensional space V with a nondegenerate quadraticform F has a basis (e1, e2, . . . , em, f1, f2, . . . , fm, x) such that F (ei) = F (fi) = 0, (ei, fj) = δij ,and (ei, x) = (fi, x) = 0 for all 1 � i, j � m, with F (x) �= 0.

Lemma 2 implies that for a space V of odd dimension l over a field GF (q) and for all quadraticforms F on V , groups I(F ) are isomorphic. We denote any such group by O(V ) and a correspond-ing matrix group by Ol(q).

If we consider elements of an orthogonal group O(V ) (Ol(q)) whose determinant is equal to1 we arrive at a special orthogonal group SO(V ) (SOl(q), resp.) A projective special orthogonalgroup PSO(V ) (PSOl(q), resp.) is a factor group of SO(V ) (SOl(q)) with respect to its center.In general, PSO(V ) is not simple. But for finitely many exceptions, however, it contains a certainsubgroup that is simple. Such a simple subgroup is denoted by PΩ(V ), or PΩl(q) in the matrixoption.

We give an exact definition of PΩ(V ) following [17]. Let v ∈ V and F (v) �= 0. A transfor-

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mation rv in O(V ) defined by a formula xrv = x− (v, x) · v/F (v) has a determinant equal to −1,lies in O(V ), and is called a reflection generated by a vector v. In view of [17, Prop. 2.5.6], forq odd, every element of the group SO(V ) is a product of an even number of reflections. For aspace V in odd dimension, SO(V ) contains a unique subgroup of index 2, which we denote byΩ(V ) (see [17, Prop. 2.5.7]). Let q be odd, g ∈ SO(V ), and GF (q)∗ be the multiplicative group ofa field GF (q). Then the factor group GF (q)∗/(GF (q)∗)2 is of order 2, and g = rv1rv2 . . . rv2s

for some v1, v2, . . . , v2s ∈ V . A spinor norm Θ(g) of an element g is defined by a formulaΘ(g) = (v1, v1)(v2, v2) . . . (v2s, v2s) (mod (GF (q)∗)2). In this event Θ is a well-defined functionfrom SO(V ) into GF (q)∗/(GF (q)∗)2, and Ω(V ) = Ker(Θ) with odd q (see [17, Prop. 2.5.7]). Thefactor group of Ω(V ) with respect to its center is denoted by PΩ(V ). It is known that if V is anodd-dimensional space then PΩ(V ) = Ω(V ).

A subspace on which the restriction of a form f (F , resp.) is a zero form is said to be isotropicrelative to f (F ). For any subset J of V , we put J⊥ = {v ∈ V | (v, x) = 0 for all x ∈ J}. Elementsof a matrix (ϕij) for a transformation ϕ with respect to a basis (v1, v2, . . . , vl) of V are given by

the rule vϕi =

l∑j=1

ϕijvj , i = 1, 2, . . . , l. Let G = Ω(V ) and W be a nonzero isotropic subspace of V

relative to F . We know from [17, Sec. 3.1] that the stabilizer GW of the subspace W in the groupG is a maximal parabolic subgroup of G, and moreover, all maximal parabolic subgroups of G arerealized in just this way.

Consider a representation of a group G via permutations on a set Γ of left cosets of the groupG with respect to a subgroup GW , in which every element g of G is assigned a permutation thatmaps each coset xGW into gxGW . For this representation, the subgroup GW is the stabilizer ofthe point GW in Γ, and each point stabilizer is conjugate in G to GW . The number of orbits of thestabilizer GW on Γ is called the (permutation) rank of a permutation representation (G,Γ). Orbitsof the subgroup GW on Γ are suborbits of the group G. Cardinalities of these suborbits, calledsubdegrees of a permutation representation (G,Γ), can be computed as indices of double stabilizersGW ∩GWi in the group GW , where Wi is an isotropic subspace of V relative to F having the samedimension as W , and GWi is the stabilizer of the subspace Wi in the group G. In compliance withthis notation, GW is a trivial suborbit, whose subdegree is equal to 1.

If dim W = 1 then a permutation representation of a group G on Γ has minimal degree (see[2] for details).

An extension (split extension) of a group X by a group Y is denoted by X.Y (X : Y , resp.),and we write Za for a cyclic group of order a, a ∈ N. Denote by diag(a1, a2, . . . , al) a diago-nal matrix with an element ai sitting at an intersection of a row and column with number i,i = 1, 2, . . . , l, by Ej an identity matrix of order j, and by At the transpose of a matrix A. IfV =

⊕Vi is a decomposition of a space V into a direct sum of subspaces Vi, then the basis

(v11, . . . , v1m1 , v21, . . . , v2m2 , . . . ) for the space V such that (vi1, . . . , vimi) is a basis for a space Vi

is referred to as a basis consistent with that decomposition. If a superscript j in a productj∏1

is

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equal to zero then we assume that the product is equal to one.

2. ORTHOGONAL GROUPS Ωl(q), lq IS ODD

Thus V is a vector space of odd dimension l = 2m + 1 over a finite field of order q and oddcharacteristic. In this case a subspace of V that is isotropic relative to a nondegenerate quadraticform F is isotropic relative to its corresponding nondegenerate symmetric form f , and vice versa.Below, therefore, the form relative to which such subspaces are isotropic will not be specified. The

order of an orthogonal group O(V ) equals 2 · q(l−1)2/4(l−1)/2∏

i=1(q2i − 1). The order of its subgroup

G = Ω(V ) equals |O(V )|/4.LEMMA 3 [17, Prop. 4.1.12]. Let O = O(V ) and W be an isotropic subspace of V of

dimension k. Then:(1) there is a subspace Y of V such that V = (W ⊕ Y ) ⊥ (W ⊕ Y )⊥;(2) OW = C : S, where the group C acts identically on factors of the series 0 < W < W⊥ < V ,

S = K ×O(W ⊕ Y )⊥, and the group K is isomorphic to GL(W ).(3) the group C is a subgroup of Ω(V ).

LEMMA 4. Transformation matrices in the subgroup C of OW with respect to some basisconsistent with a decomposition V = (W ⊕ Y ) ⊥ (W ⊕ Y )⊥ have the form

⎛

⎜⎜⎜⎜⎜⎜⎝

Ek 0 0 0 0

A Ek U R Z

B 0 Em−k 0 0

Q 0 0 Em−k 0X 0 0 0 1

⎞

⎟⎟⎟⎟⎟⎟⎠,

where the following equalities hold for the matrices A, B, Q, R, U , Z, and X and for someλ ∈ GF (q)∗:

(A + URt)t + (A + URt) + λZZt = 0, U = −Qt, R = −Bt, λZ = −Xt.

Proof. Given the basis of V specified in Lemma 2, we construct a new basis consistent withV = (W ⊕ Y ) ⊥ (W ⊕ Y )⊥. Let W be a subspace spanned by vectors e1, e2, . . . , ek and Y

a subspace spanned by f1, f2, . . . , fk. Then the subspace (W ⊕ Y )⊥ is a linear hull of vectorsek+1, ek+2, . . . , em, fk+1, fk+2, . . . , fm, x. Note that the subspace W⊥ is a linear hull of vectorse1, e2, . . . , em, fk+1, fk+2, . . . , fm, x. A transformation matrix for ϕ in C with respect to the result-ing basis (e1, e2, . . . , ek, f1, f2, . . . , fk, ek+1, ek+2, . . . , em, fk+1, fk+2, . . . , fm, x) is shaped like in theformulation of the lemma, where A = (aij)k,k, B = (bij)m−k,k, Q = (qij)m−k,k, U = (uij)k,m−k,R = (rij)k,m−k, X = (xi)1,k, and Z = (zj)k,1.

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Every transformation matrix in C preserves a nondegenerate symmetric form f . Let (x, x) =λ �= 0. For i = 1, 2, . . . , k and t = 1, 2, . . . ,m− k, we have

eϕi = ei, fϕ

i = fi +k∑

d=1

aided +m−k∑

d=1

uidek+d +m−k∑

d=1

ridfk+d + zix,

eϕk+t = ek+t +

k∑

d=1

btded, fϕk+t = fk+t +

k∑

d=1

qtded, xϕ = x +k∑

d=1

xded.

We compute the value of a form f on basis vectors. We have

0 = (fϕi , eϕ

k+t) = bti + rit, 0 = (fϕi , fϕ

k+t) = qti + uit, 0 = (fϕi , xϕ) = xi + λzi,

and so R = −Bt, U = −Qt, and λZ = −Xt. Furthermore,

0 = (fϕi , fϕ

j ) = aij + aji +m−k∑

d=1

uidrjd +m−k∑

d=1

ujdrid + λzizj .

Consequently, (A+URt)t +(A+URt)+λZZt = 0. The values of f on the remaining basis vectorsproduce trivial relations. The lemma is proved.

LEMMA 5. Let G = Ω(V ), V be a vector space of odd dimension l over a finite field GF (q)of odd characteristic, and W an isotropic subspace of V of dimension k. Then GW = C : L,where the group C acts identically on factors of the series 0 < W < W⊥ < V and the group L

is isomorphic to a subgroup SLk(q). Z q−12

of index 2 in GLk(q) for l − 2k = 1, and to a subgroup(SLk(q). Z q−1

2× Ωl−2k(q)). Z2 for l − 2k > 1.

Proof. As in the proof of Lemma 4, we choose a basis of V consistent with V = (W ⊕ Y ) ⊥(W ⊕ Y )⊥. Let groups O, C, S, and K be as in Lemma 3. Then OW = C : S, GW = C : (S ∩G),and K � SO(V ). To be more specific (see [17, Prop. 4.1.9]), transformations in the subgroup K

have matrices of the form ⎛

⎜⎝A 0 00 A−1t 00 0 El−2k

⎞

⎟⎠ ,

where A−1t = (A−1)t, A ∈ GLk(q), and S ∩ SO(V ) = K × SO(W ⊕ Y )⊥ ∼= GLk(q)× SOl−2k(q).If l = 2k + 1, then SO(W ⊕Y )⊥ = 1, and so S ∩G = K ∩G. The group K ∩G contains those

elements for which detA is a square in GF (q)∗ (see [17, Prop. 4.1.9]). Matrices for such elementshave the form ⎛

⎜⎝A0 0 00 A−1t

0 00 0 1

⎞

⎟⎠ diag(μ2α, 1, . . . , 1, μ−2α, 1, . . . , 1),

where μ is a generating element of the multiplicative group GF (q)∗, A0 ∈ SLk(q), α ∈{1, 2. . . . , (q − 1)/2}, and nonidentity elements in the diagonal matrix occupy the first and k + 1places.

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If l − 2k > 1, then K and SO(W ⊕ Y )⊥ contain elements whose spinor norm is not a squarein GF (q)∗, whereas the spinor norm of a product of such two elements is a square in GF (q)∗.We specify such elements. Consider, for instance, reflections re1+f1, re1+βf1, rek+1+fk+1

, andrek+1+βfk+1

generated by respective vectors e1 +f1, e1 +βf1, ek+1 +fk+1, and ek+1 +βfk+1, whereβ ∈ GF (q)∗, but β �∈ (GF (q)∗)2. Put h1 = re1+f1 · re1+βf1 and h2 = rek+1+fk+1

· rek+1+βfk+1. Then

h1 ∈ K ≤ SO(V ) and h2 ∈ SO(W ⊕Y )⊥ ≤ SO(V ). Computing the spinor norm of these elementsyields Θ(h1) = (e1 + f1, e1 + f1)(e1 + βf1, e1 + βf1) = 2 · 2β and Θ(h2) = (ek+1 + fk+1, ek+1 +fk+1)(ek+1+βfk+1, ek+1+βfk+1) = 2·2β. Therefore, h1, h2 �∈ G, but Θ(h1·h2) = 16β2 ∈ (GF (q)∗)2

and h1 · h2 ∈ G.Thus L ≡ S ∩G = ((K ∩G)× Ω(W ⊕ Y )⊥). Z2

∼= (SLk(q). Z q−12× Ωl−2k(q)). Z2. In this way,

elements of the subgroup L have matrices like a · b · cd, where a is a matrix of the form⎛

⎜⎝A0 0 00 A−1t

0 00 0 B

⎞

⎟⎠ ,

with A0 ∈ SLk(q), B ∈ Ωl−2k(q), b = diag(μ2α, 1, . . . , 1, μ−2α, 1, . . . , 1), c = diag(β−1, 1, . . . , 1,β, 1, . . . , 1, β−1, 1, . . . , 1, β, 1, . . . , 1), and d ∈ {0, 1}. In the matrix b, nonidentity elements capturethe first and k + 1 places, and α ∈ {1, 2. . . . , (q − 1)/2}, while nonidentity elements in c occupythe first, k + 1, 2k + 1, and m + k + 1 places. The lemma is proved.

LEMMA 6. Let G = Ω(V ), V be a vector space of odd dimension l over a finite field GF (q)of odd characteristic, and W an isotropic subspace of dimension k in V . Then the order of asubgroup GW is equal to

12q(l−1)2/4

k∏

s=1

(qs − 1)(l−2k−1)/2∏

s=1

(q2s − 1).

Proof. The order of a subgroup L in the group GW can be readily found by using Lemma 5.The order of a subgroup C in the stabilizer GW was pointed out in [17, Prop. 4.1.20(II)]. Thelemma is proved.

LEMMA 7. Let V be a vector space of odd dimension l over a finite field GF (q) of oddcharacteristic and W an isotropic subspace of dimension k in V . A permutation representationfor a group G = Ω(V ) on left cosets with respect to a maximal parabolic subgroup GW has degree

(ql−1 − 1)(ql−3 − 1) . . . (ql−2k+1 − 1)(qk − 1)(qk−1 − 1) . . . (q − 1)

.

Proof. The required statement follows immediately from Lemma 6, since the degree of ourrepresentation equals the index |G : GW |.

THEOREM. Let V be a vector space of odd dimension l over a finite field GF (q) of odd char-acteristic, W an isotropic subspace of dimension k in V , and G = Ω(V ). Then there are isotropic

421

subspaces Wi,k−i−j,j of dimension k in V , where 0 � i � k, 0 � i + j � k, 0 � j � (l − 1)/2 − k

for 4k � l − 1, and 0 � j � k for 4k � l − 1, such that GW ∩GWi,k−i−j,j= Ci,k−i−j,j : Li,k−i−j,j,

where |Ci,k−i−j,j| = q(i+j)(l−k)−(i+j)(i+j+1)/2−j2, Li,k−i−j,j

∼= Ni,k−i−j,j : (Mi,k−i−j,j × Gj). Z2,|Ni,k−i−j,j| = q(k−i)(i+j)−j2

, Mi,k−i−j,j is a subgroup of index 2 in GLi(q)×GLj(q)×GLk−i−j(q),and Gj is isomorphic to the stabilizer of an isotropic subspace of dimension j in Ωl−2k(q).

Proof. Let m = (l − 1)/2. In accordance with Lemma 2, in the space V we fix a basis(e1, e2, . . . , ek, f1, f2, . . . , fk, ek+1, ek+2, . . . , em, fk+1, fk+2, . . . , fm, x) consistent with V = (W ⊕Y ) ⊥ (W ⊕ Y )⊥.

For any k � m, consider the stabilizer GWi,k−i−j,jof the fixed isotropic space

Wi,k−i−j,j = 〈e1, e2, . . . , ei, fi+j+1, fi+j+2, . . . , fk, em−j+1, em−j+2, . . . , em〉,

where i and j are as in the formulation of the theorem. Our goal is to look into the structure ofthe intersection GW ∩GWi,k−i−j,j

of stabilizers of subspaces W and Wi,k−i−j,j. Notice that

GW ∩GWi,k−i−j,j= (C : L) ∩GWi,k−i−j,j

= (C ∩GWi,k−i−j,j) : (L ∩GWi,k−i−j,j

),

where C and L are defined as in Lemma 5. Now we clarify how matrices in the subgroupCi,k−i−j,j ≡ C ∩ GWi,k−i−j,j

are structured. Matrices in C were described in Lemma 4. Thoseof the matrices that stabilize the subspace Wi,k−i−j,j have the form indicated in Table 1.

Table 1. Matrix in Ci,k−i−j,j

e1...eiei+1...ei+jei+j+1...ek f1...fk ek+1...em−jem−j+1...em fk+1...fm−jfm−j+1...fm x

e1...ek

Ek 0 0 0

f1...fi

∗ ∗ ∗

fi+1...fi+j

∗ ∗

Ek

∗ ∗

∗ 0 ∗

fi+j+1...fk

∗ 0 0 ∗ 0

ek+1...em−j

∗ ∗

em−j+1...em

∗ 0

0 0 Em−k 0

fk+1...fm−j

∗ 0

fm−j+1...fm

∗

0 0 Em−k 0

x ∗ 0 0 1

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There, the symbol ∗ is used to mark rectangular matrices with arbitrary elements in the fieldGF (q), whose dimensions are readily guessed. Following the argument used in the proof ofLemma 4 in order to compute the number of elements in the subgroup Ci,k−i−j,j, we obtain|Ci,k−i−j,j| = q(i+j)(l−k)−(i+j)(i+j+1)/2−j2

.Our next goal is to clarify the form into which matrices in the subgroup Li,k−i−j,j ≡ L ∩

GWi,k−i−j,jare shaped. The proof of Lemma 5 implies that matrices in L with respect to the

basis chosen have the form a · b · cd, where a, b, and cd are the same as in that lemma. From thematrices in L we choose those that stabilize the subspace 〈em−j+1, em−j+2, . . . , em〉 and centralizea subspace spanned by e1, e2, . . . , ei and a subspace spanned by fi+j+1, fi+j+2, . . . , fk. For thematrix A0 to centralize a subspace 〈e1, e2, . . . , ei〉, A0 should be of the form

(A1 0

∗ A2

),

where A1 ∈ GLi(q), A2 ∈ GLk−i(q), and det A1A2 = 1.On the other hand, for A−1t

0 to centralize a subspace 〈fi+j+1, fi+j+2, . . . , fk〉, the matrix M =A−1t

0 must be the following: ⎛

⎜⎝B1 ∗ ∗0 B2 ∗0 0 B3

⎞

⎟⎠ ,

where B1 ∈ GLi(q), B2 ∈ GLj(q), B3 ∈ GLk−i−j(q), detB1B2B3 = 1, and ∗ is used to markrectangular matrices with arbitrary elements in the field GF (q), whose dimensions are readilyguessed. The matrix B is chosen in the stabilizer of a space 〈em−j+1, em−j+2, . . . , em〉 (which isisotropic and has dimension j) in the group Ωl−2k(q). Such stabilizers were described in Lemma 5.

Thus the subgroup Li,k−i−j,j consists of matrices like⎛

⎜⎝M−1t 0 0

0 M 00 0 B

⎞

⎟⎠ · b · cd,

where b and cd are as in the proof of Lemma 5 and M and B are as above. We have thus establishedhow the group Li,k−i−j,j is structured, proving the theorem.

COROLLARY 1. In the notation of the theorem, the subdegrees

ni,k−i−j,j ≡ |GW : GW ∩GWi,k−i−j,j|

of a permutation representation for a group G on left cosets with respect to a subgroup GW areequal to

qa ·

k∏s=1

(qs − 1)(l−2k−1)/2∏

s=1(q2s − 1)

i∏s=1

(qs − 1)j∏

s=1(qs − 1)2

k−i−j∏s=1

(qs − 1)(l−2k−2j−1)/2∏

s=1(q2s − 1)

,

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where a = (k − i− j)(l − 2k) + (k − i− j)(k − i− j − 1)/2 + j2.Proof. The order of the group Ci,k−i−j,j was pointed out in the theorem, and so we need only

compute the order

|Li,k−i−j,j| = q(k−i)(i+j)−j2 · |GLi(q)| · |GLj(q)| · |GLk−i−j(q)| · |Gj |.

The order of a subgroup Gj in Li,k−i−j,j can be found by using Lemmas 5 and 6. The corollary isproved.

COROLLARY 2. Let V be a vector space of odd dimension l over a finite field GF (q) ofodd characteristic and W an isotropic subspace of dimension k in V . The rank of a permutationrepresentation of a group G = Ω(V ) on left cosets with respect to a maximal parabolic subgroupGW equals (l − 2k + 1)(6k − l + 5)/8, for 4k � l − 1, and (k + 1)(k + 2)/2 for 4k � l − 1.

Proof. Put W = Wk,0,0. Note that the inequalities 0 � i+ j � k hold as well as the equalities

n =k∑

i=0

(l−1)/2−k∑j=0

ni,k−i−j,j, for 4k � l − 1, and n =k∑

i=0

k∑j=0

ni,k−i−j,j for 4k � l − 1, where n is the

degree of the representation in question, computed in Lemma 7. In order to complete the proof,therefore, it suffices to compute the number of subspaces of the form Wi,k−i−j,j specified in thetheorem for each of the cases 4k � l − 1 and 4k � l − 1. The corollary is proved.

At this moment, we look at the results obtained above for orthogonal groups in odd dimensionsfrom the standpoint of groups of Lie type. It is known that an adjoint group Bm(q) (q is odd) ofLie type having the Dynkin diagram

is isomorphic to an orthogonal group Ω2m+1(q).

COROLLARY 3. Let Ps be a maximal parabolic subgroup of Bm(q), where q is odd, obtainedby removing the sth vertex from the Dynkin diagram of type Bm. Then the rank of a permutationrepresentation for Bm(q) on left cosets with respect to Ps equals s(2m− 3s + 5)/2, for s � m

2 + 1,and (m− s + 2)(m − s + 3)/2 for s � m

2 + 1.Proof. We need only observe that a number for the vertex s is equal to m − k + 1, where k

is the dimension of an isotropic subspace W and Ps is isomorphic to a group GW . The corollaryis proved.

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