C L A S S E S O F C O N J U G A T E E L E M E N T S O F T H E

U N I T R I A N G U L A R G R O U P

P. M. Gudivok, Yu. V. Kapitonova, S. S. Polyak,

V. P. Rud'ko, A. I. Tsitkin UDC 512.542+519.44

The paper examines the classes of conjugate elements of the unitriangular group. It is shown that for sufficiently barge n the problem of describing the classes of conjugate elements of the group G includes the problem of uni~- angular similarity of pairs of matrices. Effective algorithms are constructed for enumeration of classes of conjugate

elements and their elements. Representatives of the classes ofconjugate elements of the group G with n <- 9 are

obtained by computer.

Let K be a ring with unity and UT(n, K) the unitriangular group of degree n on the ring K, i.e., the group of all matri-

ces [I aij [1 (aij ~ K) of order n such that aij = 0 for i > j and ai i= 1 (i = 1 ..... n). The group LIT(n, K) plays a major role in the

general theory of linear groups. If K is a field of characteristic p > 0, then UT(n, K) is a Sylow p-subgroup of the complete

linear group GL(n, K) (see [1]). This assertion can be shown to hold also when K is an integral domain of characteristic p,

Levchuk [2] described the normal subgroups of the group UT(n, K) where K is a body. Polyak [3-5] obtahled exact bounds for

the orders of the classes of conjugate elements of the group UT(n, F), where F is a field of p elements, and const~cted an algo-

rithm to find representatives of these classes.

It is shown in [6-7] that when K is an infinite field, the triangular group T(n, K) has finitely many classes of conjugate

elements contained in UT(n, K) if and only if n < 6.

In this paper we investigate the classes of conjugate elements of the group UT(n, F). We show that for sufficiently large

n the problem of describing the classes of conjugate elements of the group UT(n, F) includes the problem of unitriangular

similarity of pairs of matrices of order [n/8] over the field F ([m] is the whole part of the rational number m). Effective algo-

rithms are constructed for enumerating: 1) the representatives of the classes of conjugate elements of h5e group UT(n, F), 2) all

the elements of a class of conjugate elements of the group UT(n, F) given its representative; 3) all the elements of a given order

of the group UT(IL b 0. Computer programs based on these algorithms were used to find and to count the repre~ntatives of the

classes of conjugate elements of the group UT(n, F) for n _< 9 (Tables 1-3).

COMPLEX/TY OF THE PROBLEM OF TESTING UNITPdANGULAR CONJUGATION OF MATRICES

Let R be an arbitrary ring with unity, T(n, R) the group of upper-triangular matrices of order n on the ring R. The pair

of matrices (A, 13) of order n on the ring R is called triangular (unitriangular) similar to the pair of matrices (A', B ~) if there

exists a matrix C E T(n, R) (C ~ UT(n, R)) such that C-1AC = A', C-1BC = 13'. According to the accepted terminology, an

algebraic problem is regarded as wild if its solution involves solving a problem with a pair of matrices, i.e., a problem of

classification up to similarity of all square matrices on some ring.

We introduce the following matrices on the ring R:

j 0 A '

Translated from Kibcrnetika, No. 1, pp. 40-48, January-February, 1990. Original article submitted April 11, 1989.

0011-4235/90/2601-0047512.50 o1990 Plenum Publishing Corporation 47

where

A =

x, y ~ R. Clearly, A(x, y) ~ UT(8, R).

::i 1iO io ~ 0 0

s (x, ~) = 0o0 x 0 0 0 0 0 ' 0 1 y

T H E O R E M 1. The matr ices A ( a l , a2) and A(fll, f12) (ai, fli ~ R, i = 1, 2) as e lements of the group T(8, R) are

conjugate in this group if and only if ya i = flit ( i = 1, 2) for some invertible element 7 from R.

Proof. Sufficiency is obvious. Let us prove necessity. Let C be a matrix from T(8, R) such that

where

Equality (1) can be rewritten in the form

CA (COl, ai) = A (131, [~z) C,

C, Co [, C 1 = C--- 0 C~

C 2

C 0 ~--

0 ?~2

0 o

0 o

742

%, vi~ 0 ?~2

0 0

0 0

?33 0

?s3 7s4 /

~ V" II

v;~ v; 7~3 V~4

0 v4~

ll C'o A C'S (~" ~') + C~ II = II 0 • + S (fh, #~) C~

O)

(2)

Hence we obtain

C1A = AC1,

C~A = hCv

ClS (~1, ~,~) + CoA = ACo + S (151, ~ ) C~.

O)

Comparing the elements of the matrices C1A and AC 1 in places (1, 2), (1, 3), (1, 4), (2, 4), and (3, 4), we find that ?~1 =

?~2 = Ys3 =- 744 = Y1, 723 = 0, Y~4 = "1'i~ + 3'is. Similarly the equality C2A = AC_. 2 gives ?~1 -- ?~2 = ?ss = 7~4 = Y2, ?24 = 71'2 + 3'~'3, 7~s = 0. Thus, if the matrices C 1 and C 2 satisfy the first two equalities in (3), then they are written in the form

C 1 =

C 2

2

V~

0

o ?;~ + % ,

71 7'34

0 7~

vi'~ vh o ~;'~ + 'e~

0 72

48

Now, equating the elements of the matrices ClS(0:l, 0:2) q- C0A and AC 0 -k S(ffl, ff2)C.2 ill places (3, 1), (4, 2), (2, 1), and (4, 3),

we obtain 741 ~ - 0 , Yl = Y2 = "~, YG~I ~ ~1"~, '~G~t = ~ Y , Q.E.D.

COROLLARY 1. Let R = K s be a total matrix ring of degree s over the field K. The p rob lem of description of all

classes of conjugate elements in the groups T(n, Ks) (s _> 1) for n _> 8 is a wild problem.

COROLLARY 2. Let R be a field. The problem of describing all classes of conjugate elements in the group T(n, R) (in

the group UT(n, R)) includes the problem of classification, up to triangular (unitriangular) similarity, of all pairs of matrices of

order In/8] over the field R.

Thus, the description of all classes of conjugate elements of the group LIT(n, R) for an arbitrary n includes the sotut~on

of the hard prob lem of classification of all pairs of square matrices over the field R relative to the unitr iangular similarity

transformation over this field.

NORMAL FORMS

Let F be a field of p elements. Denote by Gn the group UT(n, 12) = UT(n, p). If n = 1, then G 1 = {1}o Let n > 1 and

1 __ s < n, t = n - s. Denote by F(s, t) the addition group of rectangular s x t matrices over the field F. Each matrix from G n is

uniquely representable in the form

1!: '

where X s ~_ Go, Yt ~ Gt, and Z ~ F(s, t). Identify such a matrix with the triple (X s, Z, Y~). By the law of matrix multiplication, we obtain

(x~, z, z,) (x;, z ' , Y;) = (x~x;, x~z' + zY;, r ,Y ) . (4)

Let

H~ = ((Xs, O, E,)I Xs E C,},

H~,t = ((E~, Z, Et) l Z r F (s, t)}, (5)

where E t is the identity matrix of order t. Then Hs,t, H s are normal subgroups in Gn, and H s ~- O s. Define the mappings ~0: On ">

Go, Po: Gn '> Ot, setting % (X,, Z, Y~) = X~, p~ (X~, Z, g~) = Y~ (X~ E Q, Yr E G~,, Z C F (s, 0)- From (4) it follows that if the

elements a, b of the group O n are conjugate in this group, then for any s (1 _< s < n), z-sO), ~s(b) are conjugate in G 0 arid ps(a),

ps(b) are conjugate in G t.

T H E O R E M 2. For each na tu ra l n > 1, we can select in the g roup G n a set W n of r ep resen ta t ives of all classes of

conjugate elements such that ~n_l(Wn) = Wn_ 1.

Proof. Let a ~ Gn_l , then (a, 0, 1) E G n and there is a matrix (b, x, 1) (b E G~_I, x ~ F(n - 1, 1)) from W~ such that (b,

x, 1) and (a, 0, 1) are conjugate in G n. Hence a and b ar e conjugate in Gn_i, i.e., a is conjugate with b = rn_l(b, x, I) in Gn_l~ Q.E.D.

COROLLARY 3. For each a ~ Gn_ 1 denote by Wn(a ) the set of all pairw/se aonconjugate elements in G n of the form (a,

x, 1) (x E F(n - 1, 1)). Then W n = LI Wn(a ). aEWn. 1

Assume that the set W n is selected in the group G n so that Theorem 2 is satisfied. The matrices from the set W n are

called normal forms of unitriangular matrices of order n. Note that W 1 = {1}. In what follows, we describe a computer algo- rithm to find these normal forms.

Let xij be the coordinate function: if a E G, , then x~j(a) is the element in row i and column j in a. Clearly, xij(a ) = 0 for i > j and )qi(a) = 1.

For a, b E O n, define the matrices Xl(a , b) and X2(a, b):

I X~8 Y~ ... Y~.,,~ Y~,,_~

Xl (a, b) = [i O X~, ... Y~.,,-2 F2,n-, ,

/[11; " "0.'.." ' Xn-s ~)n-'2,~, (6)

49

where x ~ (a) 0 . . 0

X l ~ = ] x~'i+~ (a) x~+x,j+~ (a) . . .0

I �9 �9 �9 �9 i ~ . , . �9 ~ . , ,

LlXi.,(a) x i + i . n ( a ) . . . Xn--t.n

Y i j = "'.. ;

0 - - xi~ (b)

and

X 2 (a, b) = (X 1 (a, b), Z (a, b)),

where

(A T is the transpose of A).

Let

z T (a, b) = (ZT . . . . . Z.T_2),

Z7 = (x~,,+, (a - - b) . . . . . x~, (a - - b))

rl (a, b) = r (X, (a, b)),

r 2 (a, b) = r (X 2 (a, b)),

(7)

(8)

where r(Xi(a, b)) is the rank of the matrix Xi(a, b) (i = 1, 2).

THEOREM 3 [5]. The matrices a and b from the group G n are conjugate in this group if and only if rl(a, b) = rz(a, b)

and xi,i+t(a ) = xi,i+l(b ) (i = 1, ..., n - 1).

THEOREM 4 [3]. The class of conjugate elements of the group G n that contains the matrix a is of order p rl(a'a).

For each matrix a E Gn_ l , deno te by Y(a) the set o f all matr ices f rom G n of the form (a, x, 1) = (a, x), where

x E F(n - 1, 1) = F(n - 1). The s-th component of the column vector x E F(n - 1) is called inessential if there exists k (s <

k <- n) such that Xsk(a ) ;~ 0, but Xik(a ) = 0 for all i = 1 .... , k - 1.

LEMMA 1. Let il, ..., i s be the indices of some linearly independent rows of the matrix a - En_ 1 (a E Gn_l). Then any

matrix from Y(a) is conjugate with a matrix (a, y) ~ Y(a), y E F(n - 1) such that all the components in y with indices i 1 ..... i s

are zero. The conjugation matrix has the form (En_l, z) (z E F(n - 1)) .

Proof. Let x E F(n - 1), ( a , x ) E Y(a). For z E F(n - 1)we have (En_t, z) -1 (a, x ) ( E n - 1 , z) = (a, ( a - En-1)z +

-t- x)). Assume that the vector y = (a - En_l)Z + x has the form as in the lemma. Then for the components of the unknown

vector z we obtain a system of s linear equations whose left-hand sides (without the free terms) are linearly independent. Such

a system is always solvable. Q.E.D.

COROLLARY 4. Any matrix from Y(a) is conjugate with a matrix (a, x) (x ~ F(n - 1)) such that all the inessential

components in x are zero.

Proof The rows of the matrix a - En_ 1 corresponding to the inessential components of the vector x ~ F(n - 1) are

linearly independent. Q.E.D.

Let Aj be the subset in Y(a) of all matrices that differ from one another only by the values of the coordinate function

xjn. A matrix from Aj is denoted A(a) if the value of xjn on this matrix is a E F.

LEMMA 2. The matrices A(a) and A03 ) (a,/5 E F, a ;~/5) are conjugate in G n if and only if the matrices A(0) and A(1)

are conjugate.

Proof. Use Theorem 3. First note that for b E G n the matrix Xl(b, c) is independent of the last column of the matrix c.

This means that XI(A(a ), A(fl)) is independent of/5. If for some a,/5 ~ F (a r t ) the matrices A(a) and A(fl) are conjugate,

then by Theorem 3 r2 (A (r A (8)) = rl (A (a), A (~)) = rl (A (a), A (-r fora l l7 ~ F. If7 ~ a, then the column Z (A(a ) ,

A(?)) contains only one nonzero element equal a - ? (see (7) and the notation for Zi). Therefore

r2 (A Ca), A (V)) = r~ (A (~), A (~)) ([~ r ~) .

Thus, rl(A(a), A(7)) = r2(A(a ), A(?)), i.e., the matrices A(a) and A(?) are conjugate. Q.E.D.

50

ALGORITHM FOR THE DESCRIPTION OF NORMAL FORMS

The algorithm relies on Theorems 3 and 4 and Lemmas 1 and 2. For a given natural number n, the algorithm produces

all the normal forms W k (k = 1, 2, ..., n) -- the sets of representatives of all classes of conjugate elements of the group G k =

UT(k, p). The algorithm runs in n steps.

Step 1.

Step 2.

�9 '~ = {1}.

Assume that all the s e t s W 1 . . . . , ~ J n - 1 have been determined. Then

W~ = U uz, (a), (9) aEWn~ 1

]t a 7 Jl (x E F(n - 1)) that are pairwise nonconjugate in Gn. The descrip tion of where Wn(a ) is the set of all matrices of the form 0

the sets Wn(a ) is thus the main aspect of the algorithm.

Let the matrix a E Wn_ 1 be given. The indices of all the rows, except the last one, tl~at do not pass through some tL~ed

linearly independent subsystem of rows of the matrix a - En_I, e.g., the subsystem of inessential rows (see Corollary 3), will be called the digits of the matrix a.

Let s = s(a) be the number of digits in ao

1. The case s = 0. The matrix a has the form

1.0 it l 0r

0 1

where a i are nonzero elements from F (i = 1 .... , n - 2). The set Wn(a ) consists of matrices of tile form

o 1 I(~EF).

2. The case s > 1. Let i I .. . . . i s be the indices of the digits of a. For each k-dimertsional vector 7 = (Yl, ~ Yk) (Yi ~ F)

from the space F(I , k) = F k (k = 1, ..., s) denote by Y(y) the (n - 2)-dimensiona! column vector from F(n - 2) wimse compo-

nents i 1 . . . . . i k are respectively equal to 71 ..... Yk and the remaining components are zero. For ~ ~ F and ~, E F k denote by aa(7 ) the matrix

a= (~,) = 0 ~ "

By Theorem 3, for a ~ 13 (a, fl E F) the matrices aa0, ) and a3(y ) are not conjugate in G n for all 7 and y'.

Let Yak(a) be the set of all matrices from Wn(a ) of the form aa(y), where 7 E F (a ~ F), then

w . (at = U Y'~ (a) ~EF

5t

Let us describe a method to find the sets YaS(a). Introduce the sets

r~ = {'e E F ~ I a~ (V) E Y~ (a)}.

For the matrix aa(7) let n(7) = rl(aa(7), aa(7)) (see (3)-(8)). Let 1 ~_ k <_ s - 1. The vector 7 ~ Fk is called augmentable if the

matrix aa((7, 1)) (where (7, 1) = (71, ..., 7k, 1) for ? = (71,-", 7k)) is not conjugate with any matrix from the set Yak(a).

Augmentable Vector Rule. Let ?o E F k (1 <_ k < s). Find n((70, 1)) and the subset F k' in F k that consists of all 7 E F k

such that nff) = n((~0 , 1)). For each 7 E Fk', find

m (v) = r~ (as (% as ((V0, 1))).

ff for all ? E F k' we have m(7) ~ n((70, 1)), then 70 is an augmentable vector. We apply this rule to find the set YaS(a) in (s +

1) steps.

Step 1. Input: the matrix and the first digit i 1, Output: the set F 1 and the sequence N 1 = {n(7 ) 17 E F1}.

Let 7o = 0, 71 = 1. Find n(70) and n(Ta). If n(70) ~ n(71), then F 1 = {/3 1 fl E F}. If n(70) = n(71), then find m I =

r2(a~(0), an( l)) . If m 1 = n(70), then set F 1 = ~ [/3 ~ F}. If m 1 = n(70) = n(71), then F 1 = {0}. Order the se t r I and take

N1 = (n(7) 17 ~ r l ) . Step k + 1 (1 < k < s). Input: matrix a, digit ik+ 1, the sets r k and N k from step k. Output: the sets Fk+ 1 and N~+ v

Using the augmentable vector rule, find the set Fk" of all augmentable vectors in F k. Construct the set Fk+l: 1) replace

each vec tor 7 = (71 . . . . . 71~) f rom rl( with the vec tor (7, 0) = (71 . . . . . 71r 0) and include the resul t ing (k + 1)-dimensional

vectors in Fk+l; 2) for each vector 7 = (71,-.., 7Q from Fk" (i.e., for each augmentable vector) write ] F] - 1 (k + 1)-dimen-

sional vectors (71 .. . . . 7k, (~) (3 ~ F - {0}) and also include them in Fk+ 1. Order the resulting set Fk+ 1. Find the numbers n(7,

(~) (7 E Fk", 6 E F, ~ ~ 0, 3 ~ 1). Construct the sequence Nk+ 1 = (u(7) [7 ~ r~+~}.

Step s + 1. The input is the output of step s. Construct the matrices

These matrices form the set YaS(a).

= , (v E r,) .

1

This concludes the examination of the case of a matrix a ~ Wn_ 1 with a positive number of digits.

In order to obtain the set Wn(a), describe the set YaS(a) for each a ~ F. If the sets Wn(a ) (a E Wn_I) have been

described, this gives a description of the set W n of all normal forms of the group G n (see (9)).

The algorithm described above was used to develop a program for the ES-1033 computer that finds all the normal forms

of the group UT(n, p). This program produced all the normal forms of the group UT(n, 2) for n ___ 9.

It is shown in [3] that the order of any class of conjugate elements of the group UT(n, p) does not exceed p(n-D(n-2)/Z

and for any integer a (0 <_ a ___ (n - 1)(n - 2)/2) there exists a class of conjugate elements of order pa of the group UT(n, p).

For the group UT(n, 2), a computer program was developed that counts the number t(m, n) of the classes of conjugate elements

of order 2 m of this group. The computer results are presented in Tables 1-3. For each n (2 __. n <-- 9), the eCtry (m, s) in these tables is the number of all classes of conjugate elements of the group

LIT(n, 2) which are of order 2 m and contain the matrix with s ones above the main diagonal produced by the algorithm.

We see from these tables that the group UT(3, 2) has 5 classes of conjugate elements, UT(4, 2) has 16 classes, UT(5, 2)

has 61 classes, UT(6, 2) 275 classes, UT(7, 2) 1430 classes, UT(8, 2) 8506 classes, and UT(9, 2) 57,205 classes of conjugate

elements. For n <_ 8 these results wTere reported in 1985 at the 18th All-Union Algebraic Conference [8] and in 1987 this case

was also considered in [9]. The results on normal forms of matrices for the group UT(n, 2) (n _< 9) give a description of the normal forms of the

matrices of the group UT(n, p) for any prime p. Before presenting this description, we introduce some conventions.

Let A E UT(n, 2). Denote by A(p) the set of all matrices from UT(n, p) which are obtained from the matrix A by the

following transformations: the zero elements in A are replaced with zeros of the field F = 2~ p, the ones on the main diagonal

in A are replaced with the identity element of the field F, and other ones in A are replaced with any nonzero elements of the

field F. The matrix A from UT(n, 2) is called a schema for the group UT(n, p) if all the matrices of the set A(p) are pairwise

nonconjugate in UT(n, p). Note that the set A(p) contains { A(p) I = (P - 1) s elements, where s is the number of ones above

the main diagonal of the matrix A.

52

T A B L E 1

0 1 2 3 4 5 6 7 8 9

10

1 l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 i 2

1 1 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 2 t 0 0 3 4 1 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 ! 2 3 4

I l 0 0 0 0 2 1 0 0 0 3 3 0 0 0 4 9 3 0 0 0 9 9 2 0 0 3 6 2 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o I ~ 3 ~ ,5 6

1 1 0 0 0 0 0 0 2 t 0 0 0 0 0 3 3 0 0 0 0 0 4 7 I 0 0 0 0 5 16 8 1 0 0 0 0 19 22 6 0 0 0 0 15 31 15 2 0 0 0 4 26 22 7 t 0 0 0 14 17 5 0 0 0 0 3 7 3 0 0 0 0 0 2 I 0

I l I 3 1 I 5 7 2 1 10 25 20 E ! 15 65 105 I0 il~ I

T A B L E 2

m 0 I 2 3 4 5 6 7 0 1 ~ 3 4 5 6 7 8 9

4 5 6 7 8 9

10 11 12 13 14 i5 16 17 18

2O 21

- - 7 - -

1 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 3 3 0 0 0 0 0 0 4 7 l 0 0 O O 0 5 13 4 0 0 0 0 0 6 26 17 2 0 0 0 0 0 32 43 12 0 0 0 0 0 31 70 35 4 0 0 0 0 22 92 76 18 0 0 0 0 5 78 115 59 12 1 0 0 0 52 I07 69 18 2 0 0 0 24 81 67 20 2 0 0 0 4 43 51 21 3 0 0 0 0 I 6 23 8 0 0 0 0 0 3 8 4 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0

I 21 I40 385 490 30I 84 8

1 1 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 4 7 1 0 0 0 0 0 0 0 5 13 4 0 0 0 0 0 0 0 6 22 l l 0 0 0 0 0 0 0 7 39 33 4 0 0 0 0 0 0 0 49 75 22 l 0 0 0 0 0 0 5t 126 65 9 0 0 0 0 0 0 45 18i 148 38 2 0 0 0 0 0 30 215 267 106 ll 0 O 0 0 0 6 201 392 239 50 3 1 0 0 0 0 148 442 386 136 19 2 0 0 0 0 84 402 489 257 68 12 1 0 0 0 36 303 476 313 104 2t 2 0 0 0 5 185 391 288 87 10 0 0 0 0 0 84 257 242 94 16 1 0 0 0 0 27 134 151 63 9 0 0 0 0 0 4 57 79 38 6 0 0 0 0 0 0 !8 28 tI 0 0 0 0 0 0 0 3 9 5 0 0 0 0 0 0 0 0 2 I 0 0

[ ~ 266 1120 2345 26tvt 1565 493 77

Proposition 1. L e t n < 9 a n d le t W a b e t h e se t o f a l l n o r m a l f o r m s o f m a t r i c e s o f t h e g r o u p U T ( n , 2) g e n e r a t e d by

c o m p u t e r by the p r o p o s e d a lgor i thm. T h e n for any p, each ma t r ix A E W n is a s chema for t he g roup U T ( n , p) a n d t he u n i o n

U A ( p ) may be used as t he set o f all n o r m a l forms o f the mat r ices o f the g roup U T ( n , p). AEW n

P r o o f By L e m m a 2, a l l m a t r i c e s A E W n a r e s c h e m a s fo r t h e g r o u p U T ( n , p) . L e t area0) b e t h e e n t r 3, in r ow m a n d

c o l u m n s o f t he t ab le for n, t h e n t he n u m b e r o f classes o f c o n j u g a t e e l emen t s o f t he g roup U T ( n , p) w h o s e o rde r equa l s pm ~

no t less t h a n

It is easy to show t h a t X m pmf(m, n) = I UT(n, p) [, w h e r e ] U T ( n , p) ] is t he o r d e r o f t he g roup U T ( n , p). Tiffs p roves P ropos i .

t ion 1.

Remark 1. F o r any n a n d any in t ege r p, t h e se t W n o f n o r m a l fo rms o f mat r ices o f the g roup U T ( n , 2) p r o d u c e d by ou r

a l g o r i t h m is p r o b a b l y t h e se t o f s c h e m a s fo r t h e g r o u p U T ( n , p) , a n d t h e u n i o n U A ( p ) is p robab ly t h e se t o f al l n o r m a l AEWn

forms o f th is group.

53

TABLE 3

9

o I 2 3 4 5 6 7 8 9 I0 11~

1 1 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 0 0 0 0 0 0 5 13 4 0 0 0 0 O 0 0 0 0 6 22 11 0 0 0 0 0 0 0 0 0 7 34 25 1 O 0 0 0 0 0 0 0 8 56 60 9 0 0 0 0 0 0 0 0 0 70 123 39 2 0 0 0 0 0 0 0 0 76 205 110 15 0 0 0 0 0 0 0 0 73 302 249 60 2 0 0 0 0 0 0 0 61 389 467 174 16 0 0 0 0 0 0 0 39 451 765 398 59 0 0 0 0 0 0 0 7 430 1100 834 222 19 4 0 0 0 0 0 0 357 1313 1366 557 89 11 0 0 0 0 0 0 236 1377 1976 1158 295 37 2 1 0 0 0 0 122 1197 2308 1824 711 164 26 2 0 0 0 0 50 939 2345 2263 1039 262 42 5 0 0 0 0 6 632 2125 2548 1510 543 141 25 2 0 0 0 0 338 1631 2339 1518 510 98 8 0 0 0 0 0 134 1089 1969 1541 617 143 18 1 0 0 0 0 40 607 1363 1238 543 123 12 0 0 0 0 0 5 296 827 807 337 57 2 0 0 0 0 0 0 112 443 531 265 57 5 0 0 0 0 0 0 30 180 242 121 19 0 0 0 0 0 0 0 4 69 108 58 10 0 0 0 0 0 0 0 0 20 33 14 0 0 0 0 0 0 0 0 0 3 I0 6 0 0 0 0 0 0 0 0 0 0 2 I 0 0 0

l 36 462 2772 8715 15372 15862 9693 3493 718 78 3

Remark 2. Let r(n, p, m) be the number of all classes of conjugate elements of the group UT(n, p) of order pro. Then for

n _ 9

S

where amn (s) are the entries in places (m, s) in Tables 1-3 for n. Hence it fo l lows that the number d(n, p) of all classes of conjugate elements of the group UT(n, p) for n ___ 9 is d(n, p) = X ~ ( S ) ( p " 1) s, where co(s) is the entry in place (co, s) in the table

for n.

DESCRIPTION OF THE CONJUGATION CLASS OF THE UNITRIANGUI.AR GROUP

FROM ITS REPRESENTATIVE

Denote by gij the matrix from UT(n, p) in which the entry (i, j) (1 _< i < ] <__ n) is 1 and all other off-diagonal entries are

0. Order the set V = {gij [ 1 <_ i < i -< n} so that gij < grk if i < r or j > k for i = r. Let V = {gt . . . . . gt}, where the elements gi

(i = 1 . . . . . t) are arranged in this order. Note that gl = gln, gt = gn-l,n and t = n(n - 1)/2. THEOREM 5. Each element g of the group UT(n, p) is uniquely representable in the form g = gl)q...gt 'tt (~1 i E {0,1 ... . .

p - 1 } ) .

Proof. We will refer to this form of elements of the group UT(n, p) as canonical. The proof is by induction on n. For

n = 2' the the~ is ~176 Let n > 2" The matrices [i 110 A 0!! E UT(n' p) f~ a subgr~ H1 ~ the gr~ UT(n' p)" Let H

be the subgroup in UT(n, p) generated by the matrices gli (i = 2, ..., n). Then H is a normal subgroup in UT(n, p) and UT(n,

p) is a semidirect product of H and H 1. Since H ~ UT(n - 1, p), then the inductive hypothesis applies to the group H 1. Let

g E UT(n, p). Then g = lag', where h ~ H, g' E H~ and the matrix g' is representable in canonical form. Since gli (i = 2, ..., n)

form a basis in H, the matrix h is also representable in canonical form. Clearly, each element gli (i = 2, ..., n) is less than any

factor gr, k # 1 in g' E H 1. Taking h and g' in canonical form, we obtain that hg' = g is a canonical form of the e lement g.

Q.E.D.

54

This theorem is the basis of a simple algorithm, using the matrix multiplication rule, for the description of the class C(g)

of conjugate elements in UT(n, p) that contains the matrix g ~ UT(n, p). This algorithm consists of t steps. In each step, it

computes some subsets in C(g). If Ck(g ) is the subset of C(g) produced in step k (k < t), then the set computed in step (k + i)

coincides with

C~+, (g ) = {g~-2-,C~ (g ) g ~ + , I s = O, 1 . . . . . p - - ~}.

Step 1.

Step Z

C , ( e ) = { , ~ - , '++~+ . =

COROLLARY 5. The set Ct(g ) computed in the last step t = n(n - 1)/2 coincides with C(g).

Indeed, by Theorem 4, C(g) consists of all elements of the form

where 0 _< a i _ p -- 1 (i = 1 .... , t).

R e m a r k 3. Let gk = gij (1 --< i < j _< n), ,t ~ {0,1 . . . . . p - 1}. Then the matrix gij-aggij 'l is obtained by the following

transformations from the matrix g: multiply the i-th column of the matrix g by 2 and add the product to column j, then multiply

the j-th row by - 2 and add the product to row L All the operations are rood p.

The second algorithm for the description of C(g) is based on the defining relationships in the group UT(n, p). Ttiese

relationsbAps have the form

g~i = E, g ~ g ~ g h ~ = gi~g~k, g~ lg~gk~ = g~tg~j, (10)

and gij-lgkrgij = gkr if and only if i ~ r and j ,e k (1 _< ~ < j _< n; 1 _ k < r -< n).

The algorithm, as in the previous case, consists of t steps, and each step produces the canonical form of the elements of

the set Ck(g ) (1 _< k _< t). The input for the algorithm: the element g E UT(n, p) in canonical form, the ordered set V. In the

first step, Cl(g ) = g. Let 1 < k < t and Ck(g ) = {h 1 ..... ht}, where each element h i (i = 1 .... , / ) is in canoniea~ form..

f ~ - ~ h ~ I ~ = l , t; ~ = 0 , I, In step (k + 1), generate the canonical form of all the elements of Ck+t (g) = tg~+~ ~ u ~ . ~ . . . . . . . .

~.~ (L~ C { 1, /9 - - 1 }) be the canonical form of p - 1 } by the following rule. Let gk+l = ~ (1 --< c~ </5 __< n) and let h = gr ... gr . . . .

the element h; here gil < gi2 < ... < gis. In order to obtain h ' = g ~ - l h g ~ , replace each gik = grs with g a F l g r s g ~ by (10). All

the factors gik from h are repeated in h ' in the same order in which they occur in h. "New" factors o f the type ga/~' or ga,~, -~

may appear that violate the ordering. In order to obta in the canonical form of h ' , each "new" fac tor x = gr%' sl~ould be

successively interchanged from right to left with each factor y = gij such that y > x. This basic component of the algorithm is

justified by the following property of the "new" factors.

LEMMA 3. Let x be a "new" factor in h ' and let y be the factor in h' nearest to x from the left such that y > x. Then:

1) there are no factors from V between y and x, 2) yx = xy, 3) x is less than all the factors from h that ocz~r to its right in h'.

The proof of part 3 follows from (10). It is easy to show that the conditions o f the lemma are satisfied only if h =

h0hlh 2 (see Theorem 5), where

x, !xvC{i , , p - - 1 ) , s ~ < ~ , ~ < / ) .

Then

lh = xs xs - x 1 x i- g-d~ lg=~ ( g ~ g ~ ) ... ~g=fif~i) and x = g=? y = g ~

Clearly, yx = xy and part 1 holds. Q.E.D.

ORDERS OF THE ELEMENTS OF THE UN1TRIANGUIAR GROUP

Unitr iangular matrices arise in the description of a special class of signals, the so-called quick search sequences [10].

Each of these signals is described by a unitriangular matrix over the field Z2, whose rows are the indices of n Walsh functions

55

(for odd n) of various periods that are condensed by majority voting. Let the matrices A 1 and A2, and thus also the signals, be

linked by the unitriangular transformation B, i.e., A 2 = BA 1.

It is noted in [10] that the proper t ies of the pair of signals S 1 and S x depend only on the t ransformat ion B and the

transformation B -1 produces a pair of signals with the same properties as the pair of signals defined by the transformation B,

i.e., if B = B -1 a new pair of signals is not produced. It is thus relevant to consider the following problem: find all the matrices

B ~ UT(n, 2) with odd n such that B 2 = E (E is the identity matrix of order n).

In what follows, we solve the problem of computing the orders of the elements of the group UT(n, p) and the problem

of counting the elements of a fixed order of this group. As a corollary, we obtain a solution of the previous problem of finding

all the matrices of order 2 from UT(n, 2).

Fi rs t we descr ibe an exponen t i a t i on a lgor i thm for a given matr ix A E UT(n , p). Le t m be a na tu ra l number . Let

aij(m ) = xij(Am), where xij is the coordinate function, ai(m ) = (al , i+l(m) . . . . . ai , i+l(m)) T is the column vector above the i-th

diagonal element in A m, A i is the matrix of order i in the top left corner of the matrix A (1 < i < j __. n). Using the binomial

coefficients cmk computed in the field F, we define the matrices Mi(m, k) for A (i = 1, ..., n - 1, k = 0,1,...): if i = 1, then

Ml(m, k) = Cm k (for k > m we take Cm k = 0); if i > 1, then

M,(m,k) = tlM~-'~m' k) M'-' (m'le + l)a';-t (1)lt

m - - 1

LEMMA4. ~ Mi(j,k)= M~(m, le+ l). / '=0

ra--! m--I m--1

Proof. ~ M1 (j, k) = ~ C~. By induction on m we can easily show that ~ C~ = C~ +~, i.e., the lemma is true for i = ./=0 /=0 j=O

1. The proof is easily completed by induction on i.

THEOREM 6. For any natural m, we have the equality ai(m ) = Mi(m , 1)ai(1 ).

Proof. It is easy to see that r/$--I

at(m) = ~ A~ai(1). ~=o (11)

By induction on i we can show that

y , = M,(m, 1), /=0

m--1 rn--i

Indeed, for i = t, we have A i = 1 from ~ , A{ = Mx(m, 1). Let i > 1 and M , - t (m, 1) = ~ A~. Then from (11) and j = o i = 0

Lemma 4 we obtain

Q.E.D.

rn--1 rn--I

/ '=0 =

= Ms (m, 1).

Thus, in order to compute A m for a given natural m and a given A E UT(n, p), it suffices to find all the matrices Mi(m,

k) (i = 1 .. . . . n - 1, k + i = n) and to apply Theorem 6. Let us now present an algori thm which, given n > 1 and m = pa (a E Z , a > 0) produces the subset ~ (n, m) in

UT(n, p) that consists of elements of order m. LEMMA 5. The maximum order of the elements of the group UT(n, p) (n > 1) is pk+l, where k is the greatest integer

such that Pk < n but pk+l _> n. The proof follows from the theory of representations of p-groups over a field of characteristic p.

The following lemma is obvious.

56

LEMMA 6. Let A E UT(n, p) and let X be an arbitrary n-dimensional vector over F. Assume that p~ is the order of the

element A in the group UT(n, p). Then the order of the element 0 of the group UT(n + 1, p) is pa or pa+l, according as pc~--i

the sum ~ A~X equals zero or not. ~'=0

Algorithm

For a given n > 1 and m = pa (a is a natural number) it is required to find the set of alt elements of order m in

UT(n, p).

1. If m/p > n, then the group UT(n, p) contains no elements of order m. If m/p _< n, then find alt the elements of order

m and m/p in the group UT(n - 1, p).

2. Let ~* 1 = ~ (n -- 1, m/p), ~ 2 = 9~ (n -- 1, m) and ~ 1 ;e O. For each matrix A from .~ 1 U ~ 2 compute Mn_l(m , 1)

(see Theorem 6). Construct the equation Mn_l(m , 1)X = 0, where X is a (n - 1)-dimensional column vector over the field F consisting of p elements.

Find ~I(A) -- the subset of all solutions of this equation in F(n - 1). Let

[d) 1 (A), if A E ~,, (D (A) = ~F (n-- l)\00x (A), i f AE~v

For each matrix A E 9~ 1 I.J 932 2, find ~(A). Form the set of matrices

~ (m,n ) = ~ U IIA x[l. AEg]21U~)j~e XE~(A) 0 It lr l

The result is the set ~3~ (m, n) of all elements of order m in the group LIT(n, p).

The authors acknowledge the programming assistance of A~ A. Sakach.

LITERATURE CITED

1~

2.

3.

4.

5.

6.

7.

8.

9.

10.

D. A. Suprunenko, Matrix Groups [in Russian], Nauka, Moscow (1972).

V. M. Levchuk, "Subgroups of the unitriangular group," Izv. Akad. Nauk SSSR, Set~ Mat., 38, No. 6, 1202-1220 (1974)o

S. S. Polyak, "On classes of conjugate elements of the group of triangular matrices," Dokl. Soobshch. Uzhgomd. Oos.

Univ., Ser. Fiz.-Mat. Nauk, No. 4, 92-95 (1961).

S. S. Potyak, ~On classes of conjugate elements of the group of triangular matrices," Nauk. Zap. Uzhgorod. Derzh. Univ., 49, 26-45 (1962).

S. S. Polyak, "On triangular similarity of triangular matrices," Ukr. Mat. Zh., 16, NOo 4, 539-544 (1964).

A~ E. Zalessldi, "A comment on the triangular linear group," Izv. Akad. Nauk BSSR, Set. Fiz.-Mat. Nauk, No. 2, 129-i31 (1968).

T. E. Azhiganich and M. I. Kabenyuk, "Conjugation classes of the triangular matrix group," 16th All-Union Algebraic Cons Abstracts of Papers[in Russian], Pa~t 1, LOMI, Leningrad (1981), p. 3.

S. S. Polyak and A. I. Tsitkin, "On classes of conjugate elements of the group UT(n, 2)," 18th All-Union Algeb~'a~c Conf. [in Russian], Inst. Mat., VTs Akad. Nauk MSSR, Kishinev (1985), p. 103.

H. Biirgstein and W. H. Hesseliak, "Algorithmic orbit classification for some Borel group actions, ~ Compos. Math., 6I, No. 1, 3-41 (1987).

V. V. Losev and !C V. Yakutik, "Quick search sequences for overlapped systems, ~ Izv. Vyssh. Uchebn. Zaved., Radio- 61ektron., 26, No. 11, 17-21 (1983).

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