Hall polynomials for symplectic groups

Table of contents :
CHAPTER PAGE
1 PRELIMINARIES AND MAIN THEOREMS 1
1.1 Introduction 1
1.2 Notation and Definitions 3
1.3 Unipotent classes in G F 5
1.4 The quadric Q 6
1.5 The main theorems 10
1.5.1 Theorem 1.1 10
1.5.2 Theorem 1.2 15
1.5.3 Theorem 1.3 22
1.5.4 Theorem 1.4 28
2 THE CASE OF ONE BLOCK 30
2.1 General remarks 30
2.2 A = l d 32
2.3 X = 2 d 32
2.3.1 Discussion of a Frobenius map 34
2.4 X = r d ,r>2 37
2.5 A closed formula for polynomials g^(q) 38
3 THE GENERAL CASE 41
3.1 Example 41
3.2 Some more notation 43
3.3 Compositions in 1-belt 47
3.4 Compositions in 2-belt 55
3.4.1 Discussion of a Frobenius map 56
3.5 Compositions in &-belt 62
3.6 The general case 71
3.6.1 Compositions in (l,0)-belt 71
3.6.2 Compositions in (2, l)-belt 72
3.6.3 Compositions in (k, k — ^)-belt 74
CITED LITERATURE 77
VITA 78

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Hall polynomials for symplectic groups Zabric, Eva, Ph.D. University of Illinois at Chicago, 1992

UMI

300 N. ZeebRd. Ann Arbor, MI 48106

HALL POLYNOMIALS FOR SYMPLECTIC GROUPS

BY EVA ZABRIC B.A. University of Ljubljana, Slovenia,1983 M.S. University of Illinois at Chicago, Chicago, 1986

THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Chicago, 1992

Chicago, Illinois

THE UNIVERSITY OF ILLINOIS AT CHICAGO Graduate College CERTIFICATE OF APPROVAL October 8. 1992

I hereby recommend that the thesis prepared under my supervision by

Eva Zabric Hall Polynomials for Symplectic Groups

entitled

be accepted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy \_t vu v/a/Ojtt/vu Advjper (ChairpersBji of Defense Committee)

I concur with this recommendation

rU^—-

La 7 fUk)- Then . A- = ( / 1 - I , / 2 - I , - , ^ - I ) • A D fj, if lj > rrii for all i = 1,k. • For A a partition let A* be the partition corresponding to the diagram of A with rows and columns exchanged. So if A = r"1 r£ 2 ,r\ > r2 > ... > r3 then A* = (ni +

+ ... +

«s)ra(«i + n2 + ... + • We will denote by (f) the smallest integer greater or equal to f and by [f] the greatest integer smaller or equal to f. • gjl is the number of A-fixed maximal totally isotropic subspaces of type fj, in a space of type A

Unipotent classes in G F

1.3

Let G be the symplectic group Sp(2m,F), P a maximal parabolic subgroup of G , and X = G/P. Let XUiW be the variety defined in the introduction. Let u be a unipotent element of G and F :G

G a Frobenius map. In each i^-stable unipotent class CU there are certain distinguished

unipotent elements in CUF\GF, called split, such that F acts on A(U) = CG{U)/CQ{U) trivially. Any two such elements are conjugate in G F . Here CG{U) is the centralizer of u in G and CQ{U) the connected component of 1 in CG{U). If u is split, then the G F -conjugacy classes in C u correspond bijectively to the conjugacy classes in A(u) as follows. For each A € A(m), there

6

exists g € G such that g - 1 F(g) = [a], where [a] is a representative of a in CQ(U), by Lang's theorem. Then the gug-1 give representatives of the (?F-conjugacy classes of Cu. Since [a] centralizes u we have an action of A(u) on XUtW, and can therefore define a map F[a] on Xu-L / < v > is a space of type r d ~ 2 (r — l)2, or [v, t>i] ^ 0 and < v > J - / < v > is a space of type r d - 1 ( r — 2) under the action that A induces on < v > ± / < v > . The quadric Q is a variety of one dimensional subspaces < v > of kerA, such that [v, v\\ = 0, i.e., Ya=i ai^i = 0. The number of irrational points of Q was calculated in [10] by giving kerA orthogonal structure defined by (J4r_1e,-, Ar~1ej) = Sijcti. In an orthogonal space we can find a basis of F-fixed vectors w,-, i = 1,2,..., tZ, such that 1. If d = 2m + 1, (w{, w is in the quadric if (v,v) = 0, i.e.,

v=

and

d 1 2 liWi i=1

8

1. m

Uld—i+l + a^m+1

2

=

0

t=l

when d = 2m + 1. The number of F,-rational points in Q (as calculated in [10]) in this case is Q2m+i{q) = (1 + Q + ... + q 2 m ~ l ).

2.

m 2 } ] lild—t'+l

=

0

i=l

when d = 2m and ( , ) is split.

Q32m(q)= ( l + q + . . . + q m ~ 2 + 2q m ~ 1 + q m + ... + q 2 m ~ 2 ).

3. m—1 2

Itfd-i+l + lm ~ ^m+1 ~ 0

1=1 when d = 2m and ( , ) is nonsplit.

= (1 + ? + ».+ qm~3 + qm~2 + qm + qm+1 + - + q2m~2).

Remark 0.4.1: The quadric Q can be defined without the use of a particular basis. There a Levi subgroup L of CQ{U) and a maximal parabolic subgroup N of L° such that Q = L°/N

9

over F Q (see [10]). The connected component L° of L is needed here since CG(U) need not be connected. The action of F on Q is defined by the action of F on CG(U) and is independent of a particular basis chosen. Hence if we change the basis in our definition, the type of the quadric over FQ is not changed.

1.5

The main theorems From this point on A is some jF-fixed nilpotent element.

1.5.1

Theorem 1.1

The variety Xu>w we are studying is the variety of all A-fixed maximal totally isotropic subspaces of V, where V is a space of type A = pf1 pi?...Prr, a partition of 2m. Every A-fixed maximal totally isotropic subspace of V has type fi where fi is a partition of m, and we will denote it by In section 1.4 we will prove that every A-fixed maximal totally isotropic subspace of V is of the form V„ =< y > ©(< y >x nVM), where y is a suitable non-zero vector and

is an

A-fixed maximal totally isotropic subspace of V. Here we will study the space < y >± nVM in some detail. Let Vn be an A-fixed maximal isotropic subspace of V (dim V^ = m ) of type fi = r"1^2...^3. Then

is spanned by a set

where Uij form a diagram of the form r"1

basis of Vfj, extends to a hyperbolic basis

A diagram

of V, but not necessarily to a diagram basis of

V. Here we will introduce a further refinement of the index sets f1% defined in section 0.2. We will write Qk in place of of its subsets

from here on. Each

= ^fc,i U ilk,2 U ... U £lk,mk, a disjoint union

where the top nodes in the diagram of

in the columns indexed by £lk,l,

i.e., y T k ,j\j € ilk,l have hyperbolic pairs in the same row ( i.e., for all j € ft*,/ (rk,j)' = (L,l) for some I ). Note that L = r,- + 1 for some i = 1,2,..., s or L = 1.

11

We now define a subset S of {i/i.j} as follows: yij € S if and only if y^j £ and

Ayitj

=

yi-ij-

Each

yij

in S is of the form

yTk+i,j

yi-ij €

for some k .

We will say that y € V lies in the r-th row if and only if y = J2i,j ki,jyi,j, where k i j = 0 for i > r and not all krj = 0. Let

y - X ) k i,jyu + y ^ Vijts where y^ is in V^, so the span of y and

is ^4-stable.

Theorem 1.1 If 77 = min{7" : for some j ( r j ) = (i , j )', such that k i j

0} , then < y > ±

nVM is A-stable and of type

I* = r"1r22...4nj-1)(r/ - l)...r?a.

Proof: First we note that < y > ©V^ is A-stable. Therefore (< y > ©V^)1" = < y >- L DV^1 is ^4-stable. So (< y >-L nV^) n

is A-stable. Since

D V^, we finally see that < y >J- nVM

is an A-stable subspace of V.

• Let

U fii,2 U ... U fii.n be the index set for the 1-st block defined above. Sup­

pose y = Eyri+liJe5;i6ni,i

kn+ijyn+i,j

+ y„. Then for 2 =

J2i,j U,jyi,j

G

[y,z] =

1,1 ^n+i.i^j-i+ij)'- Consider the equation [y,z] = 0. This is a linear equation con­

12

necting the /(n+1 jy- Let U be the space spanned by the

jy, j G

The solutions

of this equation form a subspace of dimension one less than the dimension of U. Suppose that z eVfj. is such that [y,z] ^ 0. Then [Ay^A^z] ^ 0, so A~xz g Vlu, and 77 is one of ri,i = 1,2,

Further, if for all j G

(7*1 + 1, j)' = (ri1,m) for some m, the type of

< y >± DVfj, is (j! = r^r^ 2 ...r n ^ h

1 (r

Il

- l)...r?\

• We assume now that the statement holds for y = J2i~1

,• k r i +i,jy r i +i,j + Vn-

Suppose now that y = YA=1 £ienlfi k r i +i,jy r i +i,j + jfc = £;=i Siefii.i kn+ijVn+ij + Vn + HjeQlik kri+ljVn+ij = y* + y** We know the type of < y* >x nVM by induction hypothesis and the type of < y** >x DFm by the first part. We are looking for the type of < y* + y** >-L flV^, which is

{zeVri[z,v*] + [z,v**] = o}.

Suppose that for all j e fti,* (ri + 1,j)' = (ri k ,l) for some /, then the set of all z such that [z, ?/*] = [z, y**] = 0 is a subspace < y* >-L D< y** y1 n V^, whose type is

1 (r

I -l)...(r I k )

nri*

\rIk -

Here 77 = m i n { r : for some I (r,l) = (r,+ 1, j)' such t h a t f c r , - + i , j ^ 0 , i = 1

,

2

1} .

13

Now we can write

= (< y* > x n< y ** >- L n V J ® < (y*)' > © < (y**)' > ,

where [y*,(j/*)'] = [y**, (j/**)'] = [(2/*)', (y**)'] = 0 and we can choose (y*)' to lie in the

77 —

th row and (?/**)' in rjk — th row of the original diagram. If 2 is such that

[z, 2/*] = -[z, y**] = k

^ 0,

then z = k((y*)' - (y**)') + w, w in < y* >L D< y** >x D

Now we can write

< y* + y** > x nv^ = (< y * > x n< y ** > x n v M )® < (y*)' -

>

which is a space of type

if TM = min{ri, ri k }. • Assume the theorem holds for y = J2i=l E^+i^eSjjen, kn+i,jyn+i}j + y„. Then V ~

5Z

yi,j£s

Vp

can be written as a sum y = V l + y 2 where y x = Ef=i Eyr(+lij6S;i€n, *VI+i,j2/n+i,j + 2//i and y2 = T,yr3+li]es-,jena krs+i,jyr3+i,j- Then by induction we know the type of < j/x >x nVM

14

and the type of < j/2 >x 0VM. We are looking for the type of < y\ + y 2 >-L

which is

the space {z G

: [z, yi] + [z, y 2 ] = 0}.

The set of all z such that [z, yi\ = [z, y 2 ] = 0 is a subspace < y\ >J- fl < y2 >x flV^, whose type is r?r?.-r^~\rh - 1 )...r^~\rh - l)...r-.

Where r/x and rj 2 are in place of 77 of the above argument for y\ and y2 respectively. From here the proof follows as in the first part. We write

= ( < 2/1 >- L n
x

nV M )®

< y x > © < y' 2 >

— th row, i = 1,2. Finally we get

x nv^ = (
J- n
x

ClV^)©

< y[ - y 2 >

which is a space of type n 1 _ i\ _n rrni rr n2 rr lM~ (rrT l )— r s 3 5 l 2 — IM \ iM

if rj M = m i n i r ^ i = 1,2}.

•

15

Corollary 1.1:

If also r j = max{ri;k r i + ij ^ 0 for some j} and [y, Ay] = 0 (that is, the

coefficients fcri+i.j satisfy the equation J2i=i,j ^i+i = 0)> then the space spanned by (< y >± nVM) and < y > is an A-fixed maximal isotropic space of type

r"1^2 ...(rj + l)rj J ~ 1 ...r^ I ~ 1 (ri — l)...r"a.

•

1.5.2

Theorem 1.2

Let V be of type A = r d ,

an A-fixed maximal isotropic subspace of type fj, = r"1 r% 2 ...r™ s ,

and let n D /i = r"1 ...r^ l k ~ 1 \rk — l)...r™3. Then Q ^ will denote the number of 1-dim A-fixed subspaces < v > of

such that v € kerA, V^f < v > has type // and < v > J - / < v > has

type r d ~ 2 {r — l)2, that is v is in the quadric of V . When a distinction is needed (A = (2m)2"), we will use Q s , for the split case and Q n s , for the nonsplit case. MM

MM

We will calculate Q,,,/ for all A.

1. A = (2m+ l)2n In this case all the choices for v are in the quadric, so we only need to count the choices with cotype /J! in

This result is already known;

«M,'

=

16

a Hall polynomial [5]. The formula for g*, (q) that can be found in [5] is

i>0

Here n(A) = X) (^ — 1)^»>

[?](g) = (1 -

so n (l) =

'

0, and

q n ) ( 1 " ^-(l -

?n"r+1)/(l -