Case i: q ; 0. From the description of the injective hull R of the module M (Lemma 12) there

follows that in this case either V = Q or V = R. Case 2: q = 0. Then from Lemma 13 either

u = M, or V = R, or V = N. Thus, one has in all five indecomposable KT3-modules KT 3 such that

V'J = 0 follows from Lemma 9 (i). The theorem is proved.

Proof of Theorem 2. It follows from Lemma 2 and from Theorem 2 of [4].

.

2.

3.

4.

5.

LITERATURE CITED

A. Clifford and G. Preston, The Algebraic Theory of Semigroups, Vol. i, Am. Math. Soc., Providence (1961). C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York (1962). I. S. Ponizovskii, "On modules over semigroup rings of completely 0-simple semigroups," in: Modern Algebra [in Russian], Leningrad Univ., Leningrad (1981), pp. 85-94. I. S. Ponizovskii, "On algebras with a finite number of classes of indecomposable non- equivalent modules," Zap. Nauchn. gem. Leningr. Otd. Mat. Inst., 132, 122-128 (1983). I. S. Ponizovskii, "On matrix representations of semigroups," in: Eighteenth All-Union Algebra Conference. Abstracts, Kishinev (1985), p. 105.

BRUHAT DECOMPOSITION DECOMPOSITION OF ROOT SEMISIMPLE SUBGROUPS

IN THE SPECIAL LINEAR GROUP

A. V. Yakovlev UDC 519.46

It is proved that the root semisimple subgroups in the special linear group over a

field intersects at most four cosets in the Bruhat decomposition and, moreover, most

elements of the given subgroup lie in the same coset, with the remaining elements ly-

ing in distinct cosets.

In this paper we indicate which classes in the Bruhat decomposition can be intersected

by a given root semisimple subgroup in the special linear group over a field. We recall that

by a root semisimple element in the special linear group F = SL(n, K) of degree n over a field

K we mean an element that is conjugate in this group with the element d (~) = diag (s, s -I,

i, ..., i). By a root semisimple subgroup we mean a one-parameter subgroup ~={~$($)~-~, ~E K*}

where x e F. The necessary facts on the Bruhat decomposition can be found, for example, in

[ 3 ] , [ 9 ] .

It is known that, for the elements of the subgroup of one-dimensional transformations,

the element of the Weyl group, occurring in the Bruhat decomposition, does not depend on the

value of the nonunit eigenvalue of this transformation (see [4]). This fact has allowed us

to look from a more general point of view at the technique of the extraction of transvections,

applied in [i], [2] for solving problems of describing subgroups of the full linear group,

containing the group of diagonal matrices. The same result has been obtained for microweight

elements in the special linear group, i.e., for elements that are conjugate with diag(g .....

s, i , . . . , I ) [ 6 ] .

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 239-246, 1987.

3178 0090-4104/90/5203-3178512.50 �9 1990 Plenum Publishing Corporation

However, already from [5] it is clear that in the case of two-dimensional transforma-

tions, a given subgroup may intersect more than two classes. We have succeeded to establish

that, nevertheless, in this case a given root semisimple subgroup lies almost entirely in

one class and only for at most three values of the parameter, including also the identity

i, the element xdl2(g)x -I is in other classes. We describe explicitly all the possibilities

that occur here. We mention that from the obtained result there follows the possible form

of the Bruhat decomposition of long root semisimple subgroups in the symplectic group, having

the same form as the root semisimple subgroups in F.

A similar description presents interest in connection with the problem of the description

of the subgroups of the special linear and symplectic groups, containing the group of diagonal

matrices. For example, it becomes clear the nature of the outwardly artificial methods in

the computations that serve the purpose of discovering the transvections in a symplectic

group [7]. We mention one more result, closely related to the problem solved here. In [8]

it has been established how a given root subgroup intersects the normalizer of the maximal

torus, in our case, the group of monomial matrices.

Formulation of the Fundamental Result

THEOREM. Let x e SL(n, K). A given root semisimple subgroup U x intersects at most four

double classes BwB and, moreover, most of its elements xdl2(s)x -l lie in the same class (the

generic case), while the remaining classes contain each at most one element (the degenerate

case). All the p~ssibilities are given in Table 1 (we indicate only the elements of the Weyl

group, generating the classes). We use the following notations: wil,..o,i k is the matrix

of permutation o = (i I ..... ik), rearranging cyclically il ..... ik, i.e., the matrix whose

(i, j)-entry is the element 6i,o(j) ; e is the identity matrix. We mention that in the formu-

lation of the theorem the relations between the indices are not indicated since this would

lead to a considerable extension of the table and would diminish its clarity. All the occur

ing variants are explicitly indicated in the proof.

Bruhat Decomposition of Two-Dimensional Transformations

We precede the proof of the theorem by the investigation of a somewhat more general situa-

tion~ Let d = diag(~, D, 1 ..... I). We establish the manner in which an element of the

Weyl group appears in the Bruhat decomposition of the two-dimensional transformation y = xdx -I

as a function of s and ~. We use the following standard notations: e is the identity matrix,

eij is the matrix whose unique nonzero element is equal to 1 and is in the i-th row and j-th

column, and tij($) = e + geij. If x = d0u0w0v0 is the reduced Bruhat right decomposition

of the matrix x, x e GL(n, K), then the element of the Weyl group in the decomposition of

the transformation y will coincide with the permutation in the decomposition of the matrix

z = w0v0dvolwo I and we restrict ourselves to the consideration of the latter. We mention that

the matrix v0dv~ I differs from the identity matrix only by the elements of the first two rows.

We denote them by a~, ..., a n and bl, ..., bn, respectively. If the elements of the matrix -I !

v 0 are denoted by vij, then it can be easily seen that

3179

TABLE 1

No. generic case degenerate case

(*)

In turn, the matrix z differs from the identity matrix only by the elements of the rows with

indices w0(1) and w0(2).

i. The case ~(4)<u~Ci'.

We set tff~(~= p, %~(s . The matrix z = (zij) is

Here i, j, r, s = i, ..., n and r ~ p, q. Since the multiplication of the given matrix on

the left and on the right by an upper triangular matrix does not change the element of the

Weyl group, occurring in its Bruhat decomposition, we consider the matrix z I = ~ ~ to4~-a~>

~,4.<~,<@~ , �9 In zz, in the columns with indices p + i, ..., q - l in the p-th row and

in the columns with the indices q + i, .. ~ n in the p-th and q-th rows we have zeros, the

remaining elements being the same as in the matrix z.

Now it is necessary to distinguish several cases.

i.i. In the matrix zz we have ~4=..=~p-~.=~ ..... }~.4=0

of linear equations with respect to g and q:

[ - u ' ~.K

This is equivalent to the system

~/--9/.=0. In this case, for any e and q the Here nontrivial solutions exist only for ~k- ~

matrix w is the identity matrix.

1.2. Assume that in the q-th row of the matrix zl there exists an element ~r ~ 0 such

that ~I ..... ~-4=~ ..... )~-4=0 and r < p (this, just as in the case i.i, depends only on the

3180

matrix v0). We multiply z~ o n the left by ~p~(-#~ ~) �9 We obtain the matrix z=, in the p-th

row of which one has the elements Sk=Ak-#$~#k ~ k=~,..~. We have two possibilities:

1.2.1. Among ~, ..., yp_~ there exist nonzero ones. Let s be the smallest of indices

for which Ys ~ 0. Then from [5], taking into account that 7p, E ~ 0, we obtain the value

: ~ = ~ p �9 Condition ~s ~ 0, is equivalent to ~)~-~#~0 and, introducing the cor- i I I I

responding values from (*), we obtain ~kg~-~k~0, Thus, the fact that Ys r 0 depends

only on v0.

1.2.2. If, however, y~ ..... ~p are zero (this depends again only on v0), then from

[5] we obtain that w = Wrq and case 1.2 has been completely examined.

1.3. If in the p-th row of the matrix z~ there is an element ~ ..... ~.~=# ..... ~_~=0~ #~0

for s > r (depends only on vo), then we have to consider three cases:

1.3.1 s < p. We multiply z I by ~ t~) ~ ~(-~4~) on the right and by its inverse

on the left. The result will be the matrix z s in which only the columns with the indices r,

s, p, q differ from the corresponding columns of the identity matrix. We write down those

elements of these columns which are in the rows r, s, p, q:

i 000ii 00 i As before, we bring the matrix z s to monomial form, multiplying it on the left and on the

right by appropriate upper triangular matrices: on the left by ~p(-f~)~5~#~ ~) and on the

right by ~p~ ~)'~(~-~)~)'~ ' and finally on the right by ~5~(-/$~)~-~(~-~)~) We have

obtained, to within a diagonal factor, the permutation matrix w = WrpWsq we are interested in,

and again its form does not depend on the values of E and n-

1.3.2. p < s < q (depends exclusively on v0). Conjugating the matrix z I in the same

way as in Seco 1.3.1, we obtain a matrix of the following form:

li 0 0 0 &p 0 0 0 4 0 0 0 4

It is easy to derive that for it we have w = Wrp = Wsq.

1.3.3. s = q. We conjugate z I with the aid of the matrix ~+~-~i.<~ -4~ %~i(.~ -4 ~i) ~ and then

we multiply by L~p (- ~ ) ' ~p~ (-~ &~) on the left and by %~ (-~ gp) on the right. After

this it is obvious that w = Wrp-

1.4. If, however, in the matrix z I we have ~ ..... ~p_~=}~=. =~4=0, ~s~0~ ~<s<~, then, after

the conjugation of z I with the aid of the matrix ~ ~i(~i) and multiplication by

3181

~(-#s~1~-~p~)~ '~ " -~ on the left and by tsq(--~ ~) on the right, it becomes clear that w = W s q .

Case i has been completely examined.

2. The case ~14)>~0(~.

Assume that, by the conjugation by the matrix w 0, the elements of the first row of the

matrix v0dv~ ~ go into the elements of the q-th row of the matrix z, while the elements of

the 2nd row go into the elements of the p-th row. The matrix z will be the following:

~pi=e~i , where d~=~,d~=0, Giq,0(k)=(~-4)ViK for k~4,~, r~l / / I

, , , , h e r e f o r

In the same way as in Case i, we shall assume that gp+~=,..=~4=&~+~ ,..=ib=~+~=..,=#~=O(if neces-

sary, we multiply the matrix z on the left by appropriate upper triangular matrices).

2.1. Let ,,l,~=...=,~pq=.~4=.,,=pp.~=O this, as before, depends only on the matrix v 0.

2.1.1. We assume that ~p=(~-~)~O . Then we multiply z on the right by ~J

and on the left by E ~pi(&pf/~i ) ; we obtain a matrix with a unique nonzero off-diagonal

element:

which, after multiplication by ~p~p{Ap)- on the left and by ~_#~#~],-4 on the right, yields

a matrix that differs from Wpq by a diagonal factor.

2.1.2. If, however, ~p = 0, then, depending on whether the equality ~p+4=...=#~-4 = 0 holds

or not, we obtain the permutation Wsq or e. However, the validity of the equality for ~ = I and also for v~ = 0 depends only on the matrix v 0.

2.2. We assume now that &~ .... =A~4=~=...=~=O~&~@O. The fact that the first r -- 1 ele-

ments of the p-th and q-th rows vanish simultaneously depends exclusively on the matrix v0;

i (g-~)i~(~q)~;=~0, here w0(k) = however, the condition ~0~ #~=0 (in other words, F~k~0, I -- ! _

r) will be satisfied either for ~k-9~-0 for all e and ~ but if ~90 then only for

6=(~k-9~k(~-~)).(~i) -4 Let s be the smallest index for which Ss ~ 0. As in [5], we shall

consider several cases.

2.2.1. s < p. The condition ~+4=. ~#s=0

appear in the following manner:

forms a homogeneous system, whose equations

/ / 1

If the pairs (~r~ are proportional for the considered j and vzj ~ 0, then the system _ I l / , ~.~

is consistent for ~-(q~i'~ii(4-{)'(9~i) ; otherwise, the consistency of the system depends only !

on the matrix v 0. In [5] it is shown that if )p=(~-~)~=0 then the element of the Weyl

q§ group we are interested in is %r ; otherwise w = Wrpsq. We note that for t

3182

condition $p = 0 is satisfied, by virtue of what has been said in 2.2, t ~! - - J (4-~))'($~f~ , i .e . , only for @~k-~4k.

2.2.2. s = p, according to [5], yields w = Wrp q and is encounterd for ~# 0

therefore, only for c and q that satisfy the relations from 2.2; moreover,

!

for ~: ~= (t~/k- ~ ~'k

and,

' = 0, i.e. because of the matrix v 0 or for c = q 2.2.3. p < s < q occurs either for vi2

in the situation described in case 2.2.1. The unknown here turns out to be ~=~p~

2.2.4. Finally, s = q can arise under the same conditions as in case 2.2.3; the element

of the Weyl group will be the transposition w = Wrp.

2.3. Assume that the matrix z satisfies the following conditions:

~,~=.,.=:c~..~=#4=...=j~_~=O , ),~-~0~ ,~<p.

The possible form of the elements of the Weyl group for this situation is found in Sec. 3

of [5]. We shall find out what determines the appearance of a certain permutation for these

elements.

We multiply z on the left by ~pr �9 In the p-th row of the obtained matrix we

have the elements ~=a~-#$~#~ and, moreover, ~i = .-- = Yr = 0. Assume, as usually, that s

is the smallest index for which u ~ 0. We have two possibilities:

2.3.1. s < p. The result in this case depends on the value of the element ~=gp-~pg~#~.

If yp ~ 0, then the element of the Weyl group is w = WrqWsp; if, however, yp = 0, then w =

Wrqsp. The equality yp = 0 can be written in the following manner:

~P =0 I (r~-4) 'O';k ~ !=0,

if k=~(~) This is equivalent to the equation ~(~q)~kC~-~-O~=O, We mention that

the left-hand side of this equation cannot be identically equal to zero since then it turns

out that also #~=0.

2.3.2. s = p. As before, the result depends on ~p. If yp ~ 0, then w = Wrq , otherwise

w = Wrp q. In conclusion we note that the vanishing of the elements yj, j = r + i, .... s - I,

is equivalent to the simultaneous vanishing of the determinants

% <t

and depends only on the matrix v 0.

Proof of the Theorem

We apply the just obtained results to the case q = c -z. For the sake of brevity, we

shall omit mentioning the obvious degeneration for c = i. We need the notation 6i=-~

Everywhere we shall assume that ~<s<5~<p<~<~<~.

3183

The case described at i.i is in the first row of Table i. From 1.2.1, where the generic

case is WrqWsp, we find ourselves in the sixth row, while from 1.2.2 (the generic case is

Wrq) is row 2. From 1.3.1, where WrpWsq degenerates to WrpWszq, we fall in row 6 and 1.3.2

gives again the result of the sixth row for WrpWsq. The cases 1.3.3 and 1.4 give Wrp and

Wsq , respectively, in the second row of the table.

We note that for w0(1) < w0(2) the degenerations occur only in the trivial case.

T The results of 2.1.1 and 2.1.2 for v12 ~ 0 yield the third row. In this case the degenera-

tion of Wsq to Wsz q occurs for ~ = ~-I = --i. If, however, vz2 = 0, then we obtain the result

indicated either in the second or in the first row.

Now we consider the variants, occurring in 2.2. If Vlk' = vz2' = 0, then, depending on

the matrix v0, we obtain the generic case WrpWsq with degeneration for ~ = s~ to WrpWsl q

(2.2.1) or to WrpW~q (2.2.3), or to Wrp (2.2.4). Thus, we obtain the rows 7 or 8. No other

generic case can occur in the case described in 2.2.3: WrpW~q with degenerator for WrpW~q

(2.2.3) or to Wrp (2.2.4), which again gives the results indicated in rows 7 and 8.

We proceed to the consideration of the situation from 2.3. The generic case WrpWsq

(2.3.1) may degenerate for ~=(g~#-~ to Wrqsp (2.3.1) while if ~=~-~ then we find our-

selves in 2.2.1, while the result, equal to Wrpsq, gives us the ninth row of the formulation.

This same generic case for ~=~=g~ =-~ gives us, according to 2.2.1, WrpWsq (row 7) or, if

there are sufficiently many zero elements in the q-th row of the matrix v0dv~ I, then from

2.2.3 we conclude that for the same value of g we have a degeneration to Wr~Wpq (row 7) and,

possibly, according to 2.2.4, to Wrp (row 8).

Finally, from 2.3.2 the generic case, equal to Wrq, is possible, which, for g=(s

degenerates to Wrqp, while for g=g~@~ to Wrp q (2.2.2). This is the result of row 4. If, how-

ever, g=~:-~ then in this case the degeneration of the transposition Wrq will be either

Wr~Wpq (2.2.3) or Wrp (2.2.4). We have obtained rows 5 and 3 from the formulation of the

theorem and the proof is concluded.

LITERATURE CITED

i. Z.I. Borevich, "The description of the subgroups of the general linear group that con- tain the group of diagonal matrices," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 64,

12-29 (1976). 2. Z.I. Bor~vich and N. A. Vavilov, "Subgroups of the general linear group over a semilocal

ring that contain a group of diagonal matrices," Trudy Mat. Inst. Akad. Nauk SSSR, 148,

43-57 (1978). 3. N. Bourbaki, Groupes et Algebres de Lie, Chaps. 4-6, Hermann, Paris (1968). 4. N.A. Vavilov, "The Bruhat decomposition of one-dimensional transformations," Vestn.

Leningr. Univ. Ser. I, No. 3, 14-20 (1986). 5. N.A. Vavilov, "The Bruhat decomposition of two-dimensional transformations," Vestn.

Leningr. Univ. Ser. I, No. 2, 3-8 (1987). 6. N.A. Vavilov, "The Bruhat decomposition of weight elements in Chevalley groups," in:

Eighteenth All-Union Algebra Conference. Abstracts of Communications, Part I, Kishinev (1985), p. 75.

7. N.A. Vavilov and E. V. Dybkova, "Subgroups of the general symplectic group containing the group of diagonal matrices," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 103, 31-47

(1980).

3184

,

9 .

A. E. Zalesski, Semisimple root elements of algebraic groups. Preprint Inst. Mat. Akad Nauk BSSR, No. 13(93), Minsk (1980). J. E. Humphreys, Arithmetic Groups, Lecture Notes in Math., No. 789, Springer, Berlin (198o).

INVESTIGATION ON THE BESM-6 COMPUTER OF THE STRUCTURE OF THE

NILPOTENT SERIES FOR THE METABELIAN 2-GENERATED GROUP OF

EXPONENT 27

A . I. Skopin UDC 519.~5

A special symbolic form for the notation of the elements of a 2-generated metabelian

group is realized on a computer, with the aid of which for the exponent 27 one car-

ries out the search of the relations in the factors of the lower central series. The

first 36 factors are investigated.

This paper is based on the LOMI preprint [i], published by the author on the same topic

and containing many technical details. Here the technical details, referring mainly to the

description of the programming methods, have been omitted and only the algebraic aspect of

the problem has been retained.

We study the additively written group G, having the following definition:

~--<x,yl (02) 2--0, 27 ~=0>.

We consider the nilpotent series of this group,

and we wish to give an explicit description (in terms of generators and defining relations)

of the factors M n = Gn/G n+1 of this series. A similar investigation (but "by hand") has been

carried out earlier for the exponents 8 and 9 (see [2]).

It is known (see, for example, [3]) that if in the definition of the group G the condi-

tion 27G = 0 is omitted, then for the free metabelian group G, defined by the single condition

(G2) = = 0, with generators x, y, the double sequence of monomials

Vk =r~ y .~ , -k yk-~ , ~ = i , 2 . . . . ; k = i , . . . , ~ . - i ( i )

forms (in the expression of the naturally used additive terminology) a basis, which will be

called F. Hall's basis of monomials. However, by adding the condition 27G = 0, i.e., in the

group exponent 27, investigated by us, the sequence (I) is a system of generators of this

group and we have to find relations for these generators.

In the first section we describe the obtained results: we enumerate the relations in

the factors M n, n <_ 36. (In order not to complicate the writing, we preserve without modifica-

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. Vo A. Steklova AN SSSR, Vol. 160, pp. 247-256, 1987.

0090-4104/90/5203-3185512.50 �9 1990 Plenum Publishing Corporation 3185