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 ...
CLASSES OF CONJUGATE ELEMENTS OF THE UNITRIANGULAR GROUP
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 j and aii= 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 algorithm 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 algorithms 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 b0. 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:
j0
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 =
::i~1iOio 0
(x, ~) =
s
0
0o0 x 0 0
0 0 0 0 1 y
'
x, y ~ R. Clearly, A(x, y) ~ UT(8, R). T H E O R E M 1. T h e matrices A ( a l , a2) and A(fll, f12) (ai, fli ~ R, i = 1, 2) as elements 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
CA (COl, ai) = A (131, [~z)C,
O)
where
%, vi~ v;~ v; C, Co [, C---
0
0
C1=
C~
0
?~2 7~3 V~4 0
0
0
0
?~2
0
o
?33
0
o
0
C2
C 0 ~--
?s3 742 ~
v4~
0
7s4 / V" II
Equality (1) can be rewritten in the form
ll C'oA C'S (~" ~') + C~ II = II 0
•
+ S (fh, #~)C~
(2)
Hence we obtain C1A = AC1, C~A = hCv
O)
ClS (~1, ~,~) + CoA = ACo + S (151, ~ ) C~. 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
o
?;~ + %
71
7'34
0
7~
C1 =
vi'~ vh
2
V~
o
~;'~ +'e~
0
0
72
C2
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 r o b l e m 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 problem of classification of all pairs of square matrices over the field R relative to the unitriangular 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. F o r each n a t u r a l n > 1, we can select in the g r o u p G n a set W n of r e p r e s e n t a t i v e s 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 algorithm 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 Xl (a, b) = [iO /[11;
Y~ ... X~, ...
Y~.,,~ Y~.,,-2
" "0.'.." ' Xn-s
Y~,,_~ F2,n-,
~)n-'2,~,
,
(6) 49
where
x ~ (a)
0
X l ~ = ] x~'i+~ (a)
I LlXi.,(a) 9
9
9
. .0
x~+x,j+~ (a) . . .0 9
i
~
.
,
.
xi+i.n(a)...
Yij =
9
~
.
,
Xn--t.n
"'.. 0
,
; - - xi~ (b)
and
X 2 (a, b) = (X 1 (a, b), Z (a, b)),
(7)
where z T (a, b) = (ZT . . . . . Z7 = (x~,,+, (a - - b) . . . . .
Z.T_2), x~, (a - - b))
(AT is the transpose of A). Let rl (a, b) = r (X, (a, b)), (8)
r 2 (a, b) = r (X 2 (a, b)), 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). F o r each matrix a E G n _ l , d e n o t e by Y(a) the set o f all matrices from 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
