aa r X i v : . [ m a t h . G R ] F e b The reverse decomposition of unipotents for bivectors
ROMAN LUBKOV
Abstract.
For the second fundamental representation of the general linear group over a commutative ring R we construct straightforward and uniform polynomial expressions of elementary generators as productsof elementary conjugates of an arbitrary matrix and its inverse. Towards the solution we get stabilizationtheorems for any column of a matrix from GL( n )( R ) or from the exterior square of GL n ( R ) , n > . Introduction
Alexei Stepanov proposed the decomposition of unipotens for GL n ( R ) in 1987 [10]. Almost at onceNikolai Vavilov generalized this result to other split classical groups [16], and in 1990 Nikolai Vavilov,Alexei Stepanov, and Eugene Plotkin developed the method for exceptional groups [23]. Since the 1990sthe decomposition of unipotents was the focus of a number of authors, see [5, 11, 12, 17, 20] for furtherreferences.Let Φ be an irreducible root system of rank > , R be an arbitrary commutative ring with , and G (Φ , R ) be the simply connected Chevalley group of type Φ over R . Fix a split maximal torus T (Φ , R ) of G (Φ , R ) and the corresponding elementary generators x α ( ξ ) , where α ∈ Φ , ξ ∈ R . Let E (Φ , R ) be theelementary subgroup spanned by all these elementary generators.The decomposition of unipotents can be viewed as an effective version of the normality of elementarysubgroups, i. e., that E (Φ , R ) is normal in G (Φ , R ) . In particular, for simply laced root systems admittingmicroweights the method provides straightforward formulae for gx α ( ξ ) g − as a product of elementarymatrices, where g is an arbitrary matrix in G (Φ , R ) . Let gx α ( ξ ) g − be a product of L elementarygenerators. Then L equals l + 1) l for A l , · l · l − for D l , · · for E , and · · for E ,see [1, 23, 24].The reverse decomposition of unipotents was proposed by Raimund Preusser in 2017. For classicalgroups the method provides similar explicit short polynomial expressions of the elementary generatorsof g E (Φ ,R ) in terms of the elementary conjugates of g and g − , see [7, 8, 9]. Soon Nikolai Vavilov andZuhong Zhang generalized the idea to exceptional groups and developed it for relative elementary matrices x ∈ E n ( J ) or x ∈ E n ( R, J ) , where J is an ideal of R [18, 19].In the present paper, we produce a variation of the reverse decomposition of unipotents for the exteriorsquare of an elementary group. For a commutative ring R by V R n we denote the second exterior powerof the free module R n . We consider the natural transformation, V : GL n → GL( n ) which extends the action of the group GL n ( R ) on the module V R n .An elementary group E n ( R ) is a subgroup of the group of points GL n ( R ) , so its exterior square V E n ( R ) is a well-defined subgroup of the group of points V GL n ( R ) . Recall that the Pl¨ucker relations consist ofvanishing certain homogeneous quadratic polynomials f I,J ∈ R [ x H ] H ∈ V [ n ] of Grassmann coordinates x H .For m –exterior power there are short and long Pl¨ucker relations, but for m = 2 all relations are short andhave the following form f I,J = X j ∈ J \ I ± x I ∪{ j } x J \{ j } , Mathematics Subject Classification.
Key words and phrases. general linear group, elementary group, fundamental representation, decomposition of unipotents,reverse decomposition of unipotents, sandwich classification.This publication is supported by RFBR, project number 19-31-90072. where I ∈ V [ n ] and J ∈ V [ n ] , see [22]. The second exterior power of GL n ( R ) [for n = 4 , n > ] is astabilizer of the Pl¨ucker ideal Pl¨u( n, R ) P R [ x I : I ∈ V { , . . . , n } ] , where Pl¨u( n, R ) is generated by the standard short Pl¨ucker relations.Further, the upper level of a matrix g ∈ V GL n ( R ) is the smallest ideal A = lev( g ) P R suchthat g belongs to the full preimage of the center of V GL n ( R/A ) under the reduction homomorphism V GL n ( R ) −→ V GL n ( R/A ) . The upper level is generated by the (cid:0) n (cid:1) − (cid:0) n (cid:1) off-diagonal entries and bythe (cid:0) n (cid:1) − pair-wise differences of its diagonal entries. Overall we have (cid:0) n (cid:1) − elements.Notice that the group V GL n ( R ) has an invariant form f for even n and an ideal of forms K for odd n . Thus, V GL n ( R ) can be regarded as a stabilizer of f or K in the sense of semi-invariance, see [3]. Westress that V (cid:0) GL n ( R ) (cid:1) do not equal V GL n ( R ) . The first group is the image of the general linear groupunder the Cauchy–Binet homomorphism, whereas the second one is the group of R -points of the image ofthe group scheme GL n under the natural transformation. We refer the reader to [22], where this point isdiscussed in detail.Note also that the exterior square of the general linear group for n = 3 is isomorphic to GL ( R ) . Thiscase has been completely solved by Raimund Preusser in [9]. So we work only in the case n > .In these terms the reverse decomposition of unipotens for the second exterior powers looks as follows. Theorem 1.
Let R be a commutative ring, n > , and g ∈ V GL n ( R ) . Then for any ξ ∈ lev( g ) , k = l n , the transvection V t k,l ( ξ ) is a product of (cid:0)(cid:0) n (cid:1) − (cid:1) elementary conjugates of g and g − . A key step in the proofs of all such results is the stabilization theorems for a column of an arbitrarymatrix g ∈ G (Φ , R ) . Therefore, we have to show that an analogous result is true in the case of V GL n ( R ) .The following result is not astounding, because many experts believed in this fact in the early 2000s. Theorem 2.
Let w be any column of a matrix in V GL n ( R ) , n > . Set T := V t , ( w ) V t , ( − w ) V t , ( w ) . Then T · w = w . We could use this result for the deal, but the rank n should be at least . Besides, the vector w isnot arbitrary — we require that the Pl¨ucker relations hold. So we have a natural question: is there atransvection T in V E n ( R ) such that T stabilizes an arbitrary column of a matrix in GL( n )( R )) ? Thefollowing result gives a positive answer. Theorem 3.
Let w be an arbitrary vector in R ( n ) , n > . Set T ∗ ,j := Q s = j V t s,j (sgn( s, j ) w sj ) , j ∈{ , . . . , n } , sgn( s, j ) are the structure constants for V E n ( R ) . Then T ∗ ,j · w = w . Let us emphasize the hidden features of the theorem. In addition to the proof of the reverse decom-position of unipotents with a smaller constraint we can use it for other structure problems, e. g., for thestandard description of overgroups. Ilia Nekrasov and the author also work in this direction to completea solution of this general problem, see [3].The present paper is organized as follows. In Section 1 we recall the basic definitions pertaining to ele-mentary groups and the second fundamental representation. Stabilization theorems 2 and 3 are discussedin Section 2. The core of the paper is Section 3, which is devoted directly to the proof of Theorem 1.
Acknowledgment.
I am very grateful to my teachers Nikolai Vavilov and Alexei Stepanov for theirconstant support during the writing of the present paper. I also thank Ilia Nekrasov for thoughtful readingof the original manuscript and the referee for useful notes in a preliminary version of this paper.1.
Notation
One can find many details relating to Chevalley groups over rings and further references in [12]. Herewe only fix the main notation.First, let G be a group. By a commutator of two elements we always mean the left-normed commutator [ x, y ] = xyx − y − , where x, y ∈ G . By x y = xyx − and y x = x − yx we denote the left and the right he reverse decomposition of unipotents for bivectors 3 conjugates of y by x , respectively. In the sequel, we will also use the notation y ± x for the the rightconjugates of y by x or x − .Let R be an associative ring with 1. By default, it is assumed to be commutative. By GL n ( R ) wedenote the general linear group, while SL n ( R ) is the special linear group of degree n over R . Let g = ( g i,j ) , i, j n be an invertible matrix in GL n ( R ) . Entries of the inverse matrix g − are denoted by g ′ i,j ,where i, j n .By t i,j ( ξ ) we denote an elementary transvection, i. e., a matrix of the form t i,j ( ξ ) = e + ξe i,j , i = j n , ξ ∈ R . Here e denotes the identity matrix and e i,j denotes a standard matrix unit, i. e., a matrixthat has 1 at the position ( i, j ) and zeros elsewhere. Hereinafter, we use (without any references) thestandard relations among elementary transvections such as(1) additivity: t i,j ( ξ ) t i,j ( ζ ) = t i,j ( ξ + ζ ) . (2) the Chevalley commutator formula: [ t i,j ( ξ ) , t h,k ( ζ )] = e, if j = h, i = k,t i,k ( ξζ ) , if j = h, i = k,t h,j ( − ζξ ) , if j = h, i = k. By R n we denote the free R –module. It contains columns with components in R . Vectors e , . . . , e n denote the standard basis of R n . Let P m be a parabolic subgroup of the coordinate subspace h e , . . . , e m i .It equals the stabilizer Stab( h e , . . . , e m i ) . Further, let U m be the subgroup of P m , generated by t i,j ( ξ ) ,where i m , m + 1 j n , ξ ∈ R . It is called the uniponet radical of P m . Clearly, U m is normaland abelian.Now, let I be an ideal in R . Denote by E n ( I ) the subgroup of GL n ( R ) generated by all elementarytransvections of level I : t i,j ( ξ ) , i = j n, ξ ∈ I . In the primary case I = R , the group E n ( R ) is calledthe [absolute] elementary group. It is well known (due to Andrei Suslin [13]) that the elementary group isnormal in the general linear group GL n ( R ) for n > . Further, the relative elementary subgroup E n ( R, I ) of level I is defined as the normal closure of E n ( I ) in the absolute elementary subgroup E n ( R ) : E n ( R, I ) = h t i,j ( ξ ) , i = j n, ξ ∈ I i E n ( R ) . Let [ n ] be the set { , , . . . , n } and let I = { i , i } , i = i be a subset of [ n ] . Define an exterior squareof [ n ] . Elements of this set are ordered subsets I ⊆ [ n ] of cardinality without repeating entries: V [ n ] := { ( i , i ) | i j ∈ [ n ] , i = i } . Usually we record i j in ascending order, i < i . Let sgn( i , i ) := +1 , if i < i and sgn( i , i ) := − , if i > i .Let I, J be two elements of V [ n ] . We define a height of the pair ( I, J ) as the cardinality of theintersection I ∩ J : ht( I, J ) := | I ∩ J | . This combinatorial characteristic plays the same role as the distancefunction d ( λ, µ ) for roots λ and µ in a weight graph in root system terms.Let R be a commutative ring, n > , and N be the binomial coefficient (cid:0) n (cid:1) . We consider the standardaction of the group GL n ( R ) on the R –module R n . Let us define the exterior square V ( R n ) of the free R –module R n as follows. The standard basis of V ( R n ) consists of exterior products e i ∧ e j , i < j n .However, we also define e i ∧ e j for arbitrary pair i, j with e i ∧ e j = − e j ∧ e i . The standard action of thegroup GL n ( R ) on the module V ( R n ) is defined on the basis elements as follows g ( e i ∧ e j ) := ( ge i ) ∧ ( ge j ) for any g ∈ GL n ( R ) and i = j n. This action is extended by linearity to the whole module V ( R n ) . We define a subgroup V (cid:0) GL n ( R ) (cid:1) ofthe general linear group GL N ( R ) as the image of GL n ( R ) under the action morphism.In other words, let us consider the Cauchy–Binet homomorphism V : GL n ( R ) −→ GL N ( R ) , mapping a matrix x ∈ GL n ( R ) to the matrix V ( x ) ∈ GL N ( R ) . Entries of V ( x ) ∈ GL N ( R ) are the secondorder minors of the matrix x . Then the group V (cid:0) GL n ( R ) (cid:1) is the image of the general linear group under ROMAN LUBKOV the Cauchy–Binet homomorphism. It is natural to index entries of the matrix V ( x ) by a pair of elementsof the set V [ n ] : (cid:0) V ( x ) (cid:1) I,J = (cid:0) V ( x ) (cid:1) ( i ,i ) , ( j ,j ) := M j ,j i ,i ( x ) = x i ,j · x i ,j − x i ,j · x i ,j , where M JI ( x ) is the determinant of a submatrix formed by rows from the set I and columns from the set J . Here and further we use the notation t I,J ( ξ ) for elementary transvections in the elementary group E N ( R ) . We write indices I, J without any brackets in ascending order, e. g., t , ( ξ ) is a transvectionwith the entry ξ at the intersection of the first row and the second column. Suppose x ∈ E n ( R ) ; thenthe exterior square of x can be presented as a product of elementary transvections t I,J ( ξ ) ∈ E N ( R ) with I, J ∈ V [ n ] and ξ ∈ R . The following result can be extracted from the very definition of V (cid:0) GL n ( R ) (cid:1) ,see [3, Proposition 2]. Proposition 1.
Let t i,j ( ξ ) be an elementary transvection. For n > , matrix V t i,j ( ξ ) can be presented asthe following product: V t i,j ( ξ ) = i − Y k =1 t ki,kj ( ξ ) · j − Y l = i +1 t il,lj ( − ξ ) · n Y m = j +1 t im,jm ( ξ ) (1) for any i < j n . As an example, V t , ( ξ ) = t , ( − ξ ) t , ( ξ ) t , ( ξ ) for n = 5 .2. Stabilization theorems
Let G = G (Φ , R ) be the simply connected Chevalley group of type Φ over a commutative ring R , where Φ is an irreducible root system of rank > . E (Φ , R ) denotes the elementary subgroup spanned by allelementary generators x α ( ξ ) , α ∈ Φ , ξ ∈ R .A crucial move to use the decomposition of unipotents for solving problems in various fields is a trick tostabilize a column [row] of an arbitrary matrix g ∈ G (Φ , R ) . For the general linear group, the orthogonalgroup, or other Chevalley groups in the natural representations this trick can be found in the paper [12]by Alexei Stepanov and Nikolai Vavilov. Therefore, we present the similar maneuver for the general lineargroup in the bivector representation. In this section, we construct two special elements T ∗ ,j and T in V E n ( R ) stabilizing an arbitrary column of a matrix g ∈ GL N ( R ) or g ∈ V GL n ( R ) respectively. Theorem 3.
Let w be an arbitrary vector in R N , n > . Set T ∗ ,j := Q s = j V t s,j (sgn( s, j ) w sj ) , where j is in [ n ] . Then T ∗ ,j · w = w .Remark. Similarly, the element T i, ∗ := Q s = i V t i,s (sgn( i, s ) z is ) , i ∈ [ n ] , stabilizes an arbitrary row z ∈ N R ,i. e., z · T i, ∗ = z .To uncover the idea of the proof and forlmulae appearing in Thearem 3 the language of weight diagramsbe useful. Here we recall the necessary portion of it. The exterior square of an elementary group V E n ( R ) corresponds to the representation with the highest weight ̟ of Chevalley group of the type A n − . Theweight diagram in this case is a half of the square with ( n − vertices, see [6, Figure 5] and Figure 1.The action of the elementary group E N ( R ) on any vector w ∈ R N can be presented on the diagram. Anelementary transvection t I,J ( ξ ) is a root unipotent for E N ( R ) . So, t I,J ( ξ ) -action is an addition of J thcoordinate to I th coordinate of w with the multiplier ξ : t I,J ( ξ ) w = w + ξw J e I . For n = 5 the action t , ( ξ ) on the weight diagram looks as follows: he reverse decomposition of unipotents for bivectors 5 /.-,()*+ ❇❇❇❇❇❇❇❇ ⑤⑤⑤⑤⑤⑤⑤⑤ /.-,()*+ ❇❇❇❇❇❇❇❇ ⑤⑤⑤⑤⑤⑤⑤⑤ /.-,()*+ ⑤⑤⑤⑤⑤⑤⑤⑤ ❇❇❇❇❇❇❇❇ /.-,()*+ /.-,()*+ ξ ` ` s❴❑ ❇❇❇❇❇❇❇❇ /.-,()*+ ⑤⑤⑤⑤⑤⑤⑤⑤ ❇❇❇❇❇❇❇❇ /.-,()*+ ⑤⑤⑤⑤⑤⑤⑤⑤ /.-,()*+ /.-,()*+ /.-,()*+ Figure 1.
The weight diagram ( A , ̟ ) and the action of t , ( ξ ) By formula (1) an exterior transvection V t i,j ( ξ ) can be presented as a product of ( n − elementarytransvections. Thus T ∗ , is a product of ( n − · ( n − elementary transvections in E N ( R ) . So we presentthe action of T ∗ , on the column w ∈ R N on this diagram. For the case n = 5 , T ∗ , is the following productof the root unipotens: T ∗ , = x α + α + α + α (+ w ) · x α + α + α (+ w ) · x α + α (+ w ) · x α (+ w ) . In termsof elementary transvections T ∗ , equals t , ( − w ) t , ( − w ) t , ( − w ) · t , (+ w ) t , ( − w ) t , ( − w ) · t , (+ w ) t , (+ w ) t , ( − w ) · t , (+ w ) t , (+ w ) t , (+ w ) . We have shown on Figure 2 all these transvections. Different types of arrows correspond to differentuniponets. Note that the signs in the expression above appear due to the definition of exterior transvec-tions. Namely indices I = { i , i } , J = { j , j } are not necessarily in ascending order, so we need to cal-culate sgn( i i , j j ) = sgn( i , i ) · sgn( j , j ) . Hence, t I,J ( ξ ) = t σ ( I ) ,π ( J ) (sgn( σ ) sgn( π ) ξ ) where σ ( I ) , π ( J ) are in ascending order. • w (cid:11) (cid:11) w (cid:11) (cid:19) w (cid:8) (cid:16) ❳❳❳❳❨❨❩❩❬❬❪❪❫❫❵❵❛❛❝❝❡❡❣❣✐✐❧❧♥♥qqtt✇✇④④⑧⑧✄✄✝✝✠✠☞☞✍✍✏✏✒✒✔✔✕✕✗✗✘✘✙✙ • • w (cid:11) (cid:11) w (cid:11) (cid:19) − w (cid:1) (cid:1) ❚❲❨❭❫❛❞❣❥♠q✉①⑤✁ • • • • w (cid:11) (cid:11) − w _ g ✖✖✖✖✕✕✕✕✔✔✒✒✑✑✎✎✌✌✡✡✞✞✄✄⑧⑧③③✉✉qq♠♠❥❥❢❢❝❝❵❵❪❪❬❬❳❳❱❱❙❙◗◗❖❖▼▼▲▲❑❑❏❏■■❍❍●● − w W W ✍✌☛✡✟✆✄⑧✇♦❣❴❲❖●❄❀✾✻✹✸✶✵ • − w l t − w p x ❱❱❱❱❲❲❳❳❳❳❨❨❩❩❩❩❬❬❭❭❭❭❪❪❫❫❫❫❴❴❵❵❵❵❛❛❜❜❜❜❝❝❞❞❞❞❡❡❢❢❢❢❣❣❤❤❤❤ − w o o P◗❘❙❚❯❱❲❳❨❩❬❭❪❫ • •
Figure 2.
The transvection T ∗ , on the weight diagram ( A , ̟ ) : each arrow stands for oneelementary transvection t I,J ( ξ ) , one type of the arrows corresponds to one root unipotent x α ( w ) Proof of Theorem . The transvection T ∗ ,j acts on the vector w by adding the expression z ( p, q, j ) := sgn( pq, jq ) sgn( p, j ) w pj w jq + sgn( qp, jp ) sgn( q, j ) w qj w jp to (cid:0) n − (cid:1) entries w pq of w , pq ∈ V ([ n ] \ j ) , i. e., ( T ∗ ,j w ) pq = w pq + z ( p, q, j ) . It is necessary to analyze cases for the numbers p, q, j n . Below we give an analysis of these cases. By the above we mustonly prove that the expressions z ( p, q, j ) vanish in all cases. They equal: • ( − w pj w jq + (+1)(+1) w qj w jp = 0 for p < q < j ; ROMAN LUBKOV • (+1)(+1) w pj w jq + (+1)( − w qj w jp = 0 for p < j < q ; • (+1)( − w pj w jq + ( − − w qj w jp = 0 for j < p < q ; • ( − − w pj w jq + (+1)( − w qj w jp = 0 for j < q < p ; • (+1)( − w pj w jq + (+1)(+1) w qj w jp = 0 for q < j < p ; • (+1)(+1) w pj w jq + ( − w qj w jp = 0 for q < p < j . (cid:3) Now, let g be a matrix in V GL n ( R ) , i. e., columns g ∗ ,J satisfy the Pl¨ucker relations. Of course, V GL n ( R ) is a subgroup of GL N ( R ) , so we can use the elements T ∗ ,j to stabilize columns of V GL n ( R ) .But T ∗ ,j contains ( n − elementary exterior transvections. For the case of V GL n ( R ) we construct theelement in V E n ( R ) that contains three elementary exterior transvections for an arbitrary rank n .The idea of the construction is the following. Choose one vertex on the weight diagram and chooseexterior transvections so that the result of their action on this vertex is a short Pl¨ucker relation. At thesame time the result on other vertices is either another Pl¨ucker relation or a trivial Pl¨ucker relation. Since w is a column of a matrix in V GL n ( R ) , we see that all Pl¨ucker relations equal zero. However, for thismethod n should be at least . Theorem 2.
Let w be any column of a matrix in V GL n ( R ) , n > . Set T := V t , ( w ) V t , ( − w ) V t , ( w ) . Then T · w = w .Proof. T is a product of unipotens x α (+ w ) x α + α ( − w ) x α + α + α (+ w ) , see figure 3. This productacts on the vector w by adding Pl¨ucker polynomials to ( n − entries of w . Namely, ( T w ) i = w i + f i ( w ) , where i ∈ [ n ] \ . Suppose that i ∈ { , , } ; then f i ( w ) is a trivial polynomial w A w B − w B w A =0 . Otherwise, i ∈ [ n ] \{ , , , } , f i ( w ) = w w i − w w i + w w i is a short Pl¨ucker polynomial. Sinceentries of the vector w satisfy the Pl¨ucker relations, we see that all polynomials f i ( w ) equal zero. (cid:3) • ❅❅❅❅❅❅❅⑦⑦⑦⑦⑦⑦⑦ (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? (cid:127)? • ❅❅❅❅❅❅❅⑦⑦⑦⑦⑦⑦⑦ w (cid:5) (cid:13) • ❅❅❅❅❅❅❅⑦⑦⑦⑦⑦⑦⑦ • − w (cid:3) (cid:3) ❵❞✐♦✉⑥✆ ❅❅❅❅❅❅❅⑦⑦⑦⑦⑦⑦⑦ • ⑦⑦⑦⑦⑦⑦⑦ ❅❅❅❅❅❅❅ • ⑦⑦⑦⑦⑦⑦⑦ ❅❅❅❅❅❅❅ w m m • • w (cid:127) (cid:127) ❅❅❅❅❅❅❅ • ⑦⑦⑦⑦⑦⑦⑦ ❅❅❅❅❅❅❅ • ⑦⑦⑦⑦⑦⑦⑦ ❅❅❅❅❅❅❅ w m m w [ c • ⑦⑦⑦⑦⑦⑦⑦ − w n n ✪✰✸❅▲❚❩ • w o w • • w m m − w f f ♦❥❞❴❩❚❖ • w l t − w n n ✪✰✸❅▲❚❩ Figure 3.
The transvection T on the weight diagram ( A , ̟ ) : the arrows of type “ → ”correspond to x α ( w ) , “ ” to x α + α ( − w ) , “ ⇒ ” to x α + α + α ( w ) Now we give a criterion that determine whether a matrix g ∈ GL N ( R ) belongs to the group V GL n ( R ) .Denote by a HA,C ( g ) the sum P B ⊔ D = H sgn( B, D ) g B,A g D,C , where H ∈ V [ n ] , and A, C ∈ V [ n ] . The nextresult can be found in [4, Theorem 3]. It is an analog of [14, Proposition 4], [15, Proposition 1], and [21,Theorem 5]. Proposition 2.
A matrix g ∈ GL N ( R ) belongs to V GL n ( R ) if and only if the following equations hold: • a HA,C ( g ) = 0 for any H ∈ V [ n ] and for any A, C ∈ V [ n ] such that A ∩ C = ∅ ; • sgn( A, C ) · a HA,C ( g ) = sgn( A ′ , C ′ ) · a HA ′ ,C ′ ( g ) for any H ∈ V [ n ] and for any A, C, A ′ , C ′ ∈ V [ n ] suchthat A ∪ C = A ′ ∪ B ′ and A ∩ C = ∅ . he reverse decomposition of unipotents for bivectors 7 In terms of a distance on a weight graph the first type is an analog of equations on a pair of adjacentcolumns , whereas the second type is an analog of equations on two pairs of nonadjacent columns . Lemma 1.
Let R be a commutative ring, n > , and g ∈ V GL n ( R ) . Suppose that one column of g with index I = { i i } is trivial; then for any indices K ∈ V ([ n ] \ { i , i } ) and J ∈ V [ n ] such that ht( I, J ) = | I ∩ J | = 1 , we have g K,J = 0 .Proof.
Suppose that a matrix g belongs to V GL n ( R ) ; then by Proposition 2 for any indices A, C ∈ V [ n ] , ht( A, C ) = 1 and for any H ∈ V [ n ] , we have X B ⊔ D = H sgn( B, D ) g B,A g D,C = 0 . Let g has the trivial column A . It follows that the above sum equals g H \ A,C . (cid:3) Example.
Let g ∈ V GL ( R ) has the first trivial column, i. e., g sits in the maximal parabolic subgroup P . Then by the latter lemma g also sits in the submaximal parabolic subgroup P : ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Reverse decomposition of unipotens
In this section, we explain the main idea of the reverse decomposition of transvections for the bivectorrepresentation of GL . For this, we have to recall some notation.Let A P R be an ideal, and let R/A be the factor-ring of R modulo A . Denote by ρ A : R −→ R/A the canonical projection sending λ ∈ R to ¯ λ = λ + A ∈ R/A . Applying the projection to all entries of amatrix, we get the reduction homomorphism ρ A : V GL n ( R ) −→ V GL n ( R/A ) a a = ( a I,J ) The kernel of the homomorphism ρ A , V GL n ( R, A ) , is called the principal congruence subgroup of level A , whereas the full congruence subgroup C V GL n ( R, A ) is the full preimage of the center of V GL n ( R/A ) under ρ A . Let us remark that V GL n ( R, A ) C V GL n ( R, A ) and both groups V GL n ( R, A ) and C V GL n ( R, A ) are normal in V GL n ( R ) . Proposition 3.
Let R be a commutative ring, n > . Then for any ideal A P R the equality [C V GL n ( R, A ) , V E n ( R )] = V E n ( R, A ) holds This result is called the standard commutator formula . It has a lot of proofs in various contexts, seee. g., the paper [2].The upper level of a matrix g ∈ V GL n ( R ) is the smallest ideal I = lev( g ) P R such that g ∈ C V GL n ( R, I ) . As in the case of the general linear group, the upper level is generated by the off-diagonalentries g I,J , I = J and by the pair-wise differences of its diagonal entries g I,I − g J,J , I = J . Note that itsuffices to consider only the “fundamental” differences g I,I − g ˜ I, ˜ I , where ˜ I is the next index after I in V [ n ] in ascending order. Thus, the upper level lev( g ) is generated by (cid:0) n (cid:1) − elements. Lemma 2.
Let P ij := V t i,j (1) V t j,i ( − V t i,j (1) ∈ V E n ( R ) , where i = j n . Then for any index k = i, j and any ξ ∈ R : • P ki V t i,j ( ξ ) = V t k,j ( ξ ) ; • P kj V t i,j ( ξ ) = V t i,k ( ξ ) . ROMAN LUBKOV
The proof is straightforward.To formulate the main result of the present paper we introduce some notation. A matrix of the form g ± h is called an elementary [exterior] g -conjugate, where g ∈ V GL n ( R ) and h ∈ V E n ( R ) . Theorem 1.
Let R be a commutative ring, n > , and g ∈ V GL n ( R ) . Then for any ξ ∈ lev( g ) thetransvection V t k,l ( ξ ) is a product of (cid:0)(cid:0) n (cid:1) − (cid:1) elementary conjugates of g and g − . Namely, (1) V t k,l ( g I,J ) is a product of eight elementary exterior g -conjugates, where ht( I, J ) = 1 ; (2) V t k,l ( g I,J ) is a product of sixteen elementary exterior g -conjugates, where ht( I, J ) = 0 ; (3) V t k,l ( g I,I − g J,J ) is a product of elementary exterior g -conjugates, where ht( I, J ) = 1 ; (4) V t k,l ( g I,I − g J,J ) is a product of elementary exterior g -conjugates, where ht( I, J ) = 0 . Before we prove the theorem, we formulate a corollary. The latter result can be regarded as a strongerversion of the standard description of V E n ( R ) –normalized subgroups. Hence, we have one more veryshort proof of the Sandwich Classification Theorem for the exterior square of elementary groups. Theorem 4.
Let H be a subgroup of V GL n ( R ) . Then H is normalized by V E n ( R ) if and only if V E n ( R, A ) H C V GL n ( R, A ) for some ideal A of R .Proof. Let H is normalized by V E n ( R ) . Set A := { ξ ∈ R | V t , ( ξ ) ∈ H } . It is obvious that V E n ( R, A ) H . It remains to check that if g ∈ H then g I,J , g I,I − g ˜ I, ˜ I ∈ A . But Theorem 1 states exactly this claim.Hence, H C V GL n ( R, A ) . The converse follows from Proposition 3. (cid:3) Now we must only check Theorem 1. A proof of statement (1) is a key step in our verification. Othercases follow from the fist one.
Proof.
We will use the first method of a column stabilization (without the Pl¨ucker relations) to cover thecase n = 4 . Let T = T ∗ , = Q s =1 V t s, ( g s, ) . By Proposition 3 the first column of T g equals the firstcolumn of the matrix g . Hence the first column of the matrix h := g − T g is standard, i. e., h lies in theparabolic subgroup P . By Lemma 1 h also lies in the submaximal parabolic subgroup K P , where K corresponds the latter (cid:0) n − (cid:1) rows.Next, note that for any j ∈ { , , , . . . , n } the exterior transvections V t ,j ( ξ ) and V t ,j ( ξ ) sit in theunipotent radical U of the parabolic subgroup K P . Using obvious formula [ xy, z ] x = [ y, z ] · [ z, x − ] , weget z := [ T − h, V t , (1)] T − = [ h, V t , (1)] · [ V t , (1) , T ] . Now the matrix z is a product of four elementary exterior conjugates of g and g − . The first commutator [ h, V t , (1)] belongs to the unipotent radical U , whereas the second one equals V t , ( g , ) . Since thetransvection V t , (1) also sits in U and the unipotent radical is abelian, we obtain [ V t , ( − , z ] = [ V t , ( − , u · V t , ( g , )] = V t , ( g , ) . Consequently the transvection V t , ( g , ) is a product of eight elementary exterior conjugates of g and g − . By Lemma 2 it follows that V t k,l ( g , ) is a product of eight elementary exterior g -conjugates.It remains to note that we can bring g I,J to position (13 , conjugating by monomial matrices from V E n ( R ) .Since n > there are two distinct indices h , h ∈ [ n ] \ { i, j } . Let us remark that the entry of g V t i,j ( − in the position ( ih , ih ) equals g jh ,ih + g ih ,ih . Applying (1) to g V t i,j ( − , we get that V t k,l ( g jh ,ih + g ih ,ih ) is a product of eight elementary exterior g -conjugates. Therefore, V t k,l ( g jh ,ih ) = V t k,l ( g jh ,ih + g ih ,ih ) V t k,l ( − g ih ,ih ) is a product of sixteen elementary exterior g -conjugates. he reverse decomposition of unipotents for bivectors 9 The third assertion follows from the above. Obviously, the entry of g V t i,j (1) in the position ( ih, jh ) equals g ih,ih − g jh,jh + g ih,jh − g jh,ih for any index h = i, j . Applying (1), we obtain that V t k,l ( g ih,ih − g jh,jh + g ih,jh − g jh,ih ) ∈ V E n ( R ) is a product of eight elementary exterior g -conjugates. Finally, V t k,l ( g ih,ih − g jh,jh ) = V t k,l ( g ih,ih − g jh,jh + g ih,jh − g jh,ih ) V t k,l ( g jh,ih − g ih,jh ) is a product of 24 elementary exterior g -conjugates.To finish the proof of the theorem it only remains to check the last assertion. There is an index K ∈ V [ n ] such that ht( I, K ) = ht(
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