Opposite relation on dual polar spaces and half-spin Grassmann spaces

OPPOSITE RELA TION ON DUAL POLAR SP A CES AND HALF-SPIN GRASSMANN SP A CES MARIUSZ KWIA TKO WSKI, MARK P ANKO V Abstract. W e c haracterize the collinearity (adjacency) relation o f half-spin Grassmann spaces in terms of the relation to b e opp osite i n the corresponding collinearity graphs. Our c haracterization is closely related with results given [1] and [2]. Also we sho w t hat this characte rization does not hold for dual polar spaces. De dic ate d to Pr of. Helmut Karzel on t he o c c a sion of his 80-th birthday 1. Introduction Let V be an n -dimensio nal v ector space ov er a divisio n ring. Denote by G k ( V ) the Gr assmannians consisting of all k -dimensional subspaces of V . Two dis tinct el- ement s o f G k ( V ) are ca lled adjac ent if their intersection b elongs to G k − 1 ( V ) (the la t- ter is equiv alent to the fact that the sum of these subspaces is ( k + 1)-dimensional). The cas es when k = 1 , n − 1 are no n-in teres ting, s ince any tw o distinct ele ments of G k ( V ) are adjacent if k = 1 or n − 1. F rom this momen t w e suppos e that 1 < k < n − 1. The Gr assmann gr aph is the graph whose vertex set is G k ( V ) and whose edges ar e pairs of adjace nt elements. By well-known Cho w’s theorem [4], every automorphism of this g raph is induced by a semilinea r iso morphism of V t o itself or to the dual vector space V ∗ , and the second p ossibility can b e realiz ed only in the ca se when n = 2 k . The Grassmann graph is connected a nd the dis tance b etw een S, U ∈ G k ( V ) is equal t o k − dim( S ∩ U ) = dim( S + U ) − k (the distance betw een t wo vertexes o f a connected gr aph is defined as the smallest nu mber i such that there is a pa th of leng th i co nnecting the vertexes); in pa rtic- ular, the diameter of the Gr assmann graph is finite. Two elements of G k ( V ) are called opp osite if the distance b etw een them is equal to the diameter. It follows from Blunc k–Havlicek’s res ults [2] (see also [5]) that t he adjacency relation can be characterized in terms of the rela tions to b e opp osite: distinct S 1 , S 2 ∈ G k ( V ) a re adjacent if and o nly if there e x ists S ∈ G k ( V ) \ { S 1 , S 2 } suc h that ev ery elemen t of G k ( V ) o ppo s ite to S is oppo site to S 1 or S 2 . In particular, this implies that every bijectiv e transforma tion of G k ( V ) pres e r ving the r elation to b e opp osite in both directions is an auto morphism of the Gras smann graph. In this note, we characterize the c o llinearity (adja c e ncy) relation of half-s pin Grassmann spa c es in terms of the relation to b e oppo s ite in the cor resp onding 2000 Mathematics Subje ct Classific ation. 51M35, 14M15. Key wor ds an d phr ases. dual polar space, half-spi n Grassmann spaces. 1 2 MARIUSZ KWIA TK OWSKI, MARK P ANK OV collinearity graph. Also w e give an example showing that this characterization do es not hold for dual p olar spaces. Usual and p olar Grass mann spaces (in particular, dual p olar space s and half- spin Grass ma nn spaces) ar e also known as the shadow spaces of buildings o f t yp e A n , C n , and D n . So our consider ations can b e also motiv ated b y Abra menk o– V an M aldeg hem’s r esult [1] concerning the adjacency and o ppos ite rela tions in the cham ber sets of spherical buildings. The present note is a part of Master thes is of the firs t author under superv i- sion of the s econd author. The authors thank Hendr ik V an Maldeghem for use ful discussion. 2. Definitions and resul ts Recall that a p artial line ar sp ac e is a pair Π = ( P , L ), where P is a set of p oints and L is a family of line such that each line contains at lea st t wo po in ts, every p oint is o n a certain line, and for a n y dis tinct p oints there is at most o ne line containing them (points connected by a line are calle d c ol line ar ). F or every par tial linear spa ce Π = ( P, L ) there is the asso ciated c ol line arity gr aph whose p oint set is P and whose vertexes ar e pair s of collinear p oints. F ollowing [3] w e define a p ol ar sp ac e of finite rank as a partial linear s pace Π = ( P, L ) sa tisfying the axio ms: (1) on every line there are a t le ast 3 p oints, (2) a p oint is collinear with all p o in ts o f a line or with precisely one p oint o f a line (Buekenhout–Shult’s axio m), (3) for each p ∈ P there is a p oint non-colline a r with p (our p olar space is non-degenera te), (4) every flag consisting of singular subspa ces is finite. Then all maximal singular s ubs paces a re pro j ective s pa ces of same finite dimension n , and the num b er n + 1 is known as the r ank of our p olar space. The collinearity relation will b e deno ted by ⊥ : we wr ite p ⊥ q if p, q ∈ P are co llinear and p 6⊥ q ov erwise. Similarly , X ⊥ Y means that every po in t of X is co llinear with all p o in ts of Y . W e denote by X ⊥ the set of a ll p oint s p ∈ P satisfying p ⊥ X . A subset F = { p 1 . . . p 2 n +2 } (recall tha t the rank of Π is equal to n + 1) is called a fr ame of Π if for every p i ∈ F there is precisely one p oint p j ∈ F , j 6 = i such that p i 6⊥ p j . In what fo llows we will use the following well-know fac t: for every t wo singular subspac es there is a frame who s e p oint s span b oth these subspaces. F or every natur al n ≥ 2 ther e a re precisely the following tw o t yp es of rank n po lar spaces: ( C n ) every ( n − 2)-dimensional s ingular subspace is con tained in at least three maximal sing ular subspace s, ( D n ) every ( n − 2 )-dimensional s ingular subspace is contained in precis e ly t wo maximal sing ular subspace s, we say that a rank n p olar spac e is o f type C n or D n if the cor resp onding case is realized. Let Π = ( P, L ) b e a p ola r s pace o f rank n ≥ 3. Denote b y G k (Π) the Grassmania n consisting o f all k -dimensional s ingular subspa ces. A subset of G n − 1 (Π) is called a line if it consists of all maximal sing ula r subspaces containing certain M ∈ G n − 2 (Π). The dual p olar sp ac e G n − 1 (Π) is the partial linear space whose p oints are elements OPPOSITE RELA TION ON DUAL POLAR SP ACES AND HALF-SPIN GRASSMANN SP A CES 3 of G n − 1 (Π) and who se lines are defined ab ov e. If our p olar spa ce is of t yp e D n the dual p ola r space is trivial: every line c onsists of pr ecisely tw o p oint s. W e s ay that t wo e le ments S, U ∈ G n − 1 (Π) are opp osite and write S o p U if the distance b et ween them in the co llinearity gra ph of G n − 1 (Π) is ma ximal; this is equiv alent to the fact that S and U are disjoint. Now supp ose that our p olar space is of type D n , n ≥ 3. Then the Grass mannian G n − 1 (Π) ca n b e pres en ted as the sum o f tw o disjoint s ubsets O δ (Π) , δ ∈ { + , −} such that the dista nce d ( S, U ) = n − 1 − dim( S ∩ U ) (in the collinearity g raph of G n − 1 (Π)) is ev en if S, U belongs to the same O δ (Π) and o dd other wise. These subsets ar e known as the half-spin Gr assmannians o f Π. A subset of O δ (Π) is ca lled a line if it co nsists of all elements of O δ (Π) containing certain M ∈ G n − 3 (Π). W e get a partial linear s pace which will de no ted by O δ (Π). In the case when n = 3, any t wo distinct elements of O δ (Π) are connected by a line (their intersection is a sing le p oint) and O δ (Π) is a 3 -dimensional pr o jective space. If n = 4 then O δ (Π) is a p olar s pace of type D 4 . As ab ov e, tw o elements S, U ∈ O δ (Π) are said to b e opp osite , S op U , if the distance be t ween them in the co lline a rity gr aph of O δ (Π) is max imal. If n is even then this is equiv alent to the fact that S and U are disjoint. In the cas e when n is o dd, we hav e S op U if and only if the intersection of S a nd U is a single p oint. Theorem 1. If Π is of t yp e D n , n ≥ 4 t hen t he fol lowing c onditions ar e e quivalent (1) S 1 , S 2 ∈ O δ (Π) ar e c ol line ar p oints of O δ (Π) , (2) ther e exists S ∈ O δ (Π) \ { S 1 , S 2 } such that U op S implies that U op S 1 or U op S 2 . Corollary . Every bije ctive tr ansformation of O δ (Π) pr eserving the r elation to b e opp osite is a c ol line ation of O δ (Π) . In Section 4 we show that Theo rem 1 do es not ho ld for dual p olar spac e s a sso- ciated with sesquilinear for ms. 3. Proof of Theorem 1 In this pro of we will distinguish the following tw o cases : (I) n is even, then S op U is equiv alent to the fa c t that S ∩ U = ∅ , (II) n is o dd, then S op U if and o nly if S ∩ U is a sing le p oint. (1) = ⇒ (2). Show that every p oint S 6 = S 1 , S 2 on the line joining S 1 with S 2 is as r equired (this line consis ts of all element s of O δ (Π) containing S 1 ∩ S 2 ). Supp ose that U op S , but U is no t opp osite to b oth S 1 and S 2 . Case (I). In this c a se, U intersects S 1 and S 2 by subspa ces whos e dimensions are not les s than 1. W e take lines L i ⊂ U ∩ S i , i = 1 , 2. These lines do not intersect S 1 ∩ S 2 . Hence S i is spanned by S 1 ∩ S 2 and the line L i . The latter mea ns that L 1 6⊥ L 2 which co n tradict the fact that o ur lines are contained in U . Case (I I). Acco rding our as s umption, the dimensions of U ∩ S 1 and U ∩ S 2 are not less than 2. Let P i be a plane co n tained in U ∩ S i , i = 1 , 2 . The planes P 1 , P 2 bo th hav e a non- empt y intersections with S 1 ∩ S 2 (beca use S 1 ∩ S 2 is ( n − 3)-dimensional). Since U op S , these intersections b oth are 0- dimens io nal. This implies the existence 4 MARIUSZ KWIA TK OWSKI, MARK P ANK OV of lines L i ⊂ P i , i = 1 , 2 disjoint w ith S 1 ∩ S 2 . As in the pr e v ious ca se, L 1 6⊥ L 2 which is imp ossible. Therefore, in the b oth c a ses we hav e U op S i for at least one i ∈ { 1 , 2 } . (2) = ⇒ (1). W e prov e this implica tion in s everal steps. First we establish that for every distinct c ol line ar p oints p i ∈ S i ( i = 1 , 2) the line p 1 p 2 interse cts S . Pr o o f in the c ase (I) . Supp ose that the line p 1 p 2 is disjo in t with S . There exists a maximal singular subspace U containing p 1 p 2 and o ppo s ite to S (we can take any frame of Π whose p oint s span S and the line p 1 p 2 , the maximal singula r subspace spanned by points of the frame and o ppo s ite to S is as req uir ed). B y our hypo thesis, U is opp osite to S 1 or S 2 ; this means that p 1 or p 2 is not in U whic h contradicts to the fact that line p 1 p 2 is in U .  Pr o o f in the c ase (I I) . The intersection of S 1 and S 2 is not empty . If the line p 1 p 2 int ers e cts S 1 ∩ S 2 then p 2 ∈ S 1 and p 1 ∈ S 2 ; th us there are the following t wo po ssibilities: p 1 p 2 ⊂ S 1 ∩ S 2 or p 1 p 2 ∩ ( S 1 ∩ S 2 ) = ∅ . In each o f these cas es, we can choose a p oint p ∈ S 1 ∩ S 2 which is not on the line p 1 p 2 (in the first case, the dimensio n of S 1 ∩ S 2 is not less than 2). Consider the plane P spanned by p, p 1 , p 2 . Assume that P intersects S pr ecisely by a certain p oint . Using the existence of a fr ame who s e p oint s span P and S , we construct a maximal singula r subspace U opp osite to S and containing P . Then U is opp osite to a t las t one of S 1 , S 2 which contradicts the fact that the lines pp 1 and pp 2 are co n tained in U . Now suppo se that P ∩ S = ∅ . W e cho ose a p oint q from S ∩ P ⊥ (this is p o ssible since n is no t les s than 4) and extend P ∪ { q } to a maximal singula r subspa ce U opp osite to S (using a fr ame who se p oints span P ∪ { q } and S ). The dimension o f each S i ∩ U is no t less than 1 which is imp ossible. Therefore, dim( P ∩ S ) ≥ 1 a nd P ∩ S contains a line; this line intersects p 1 p 2 (since P is a pla ne).  Our next step is the equalities dim( S ∩ S i ) = n − 3 , i = 1 , 2 . Pr o o f. Le t us take a p oint p ∈ S 2 \ S . Then S 1 ∩ p ⊥ is a hyperplane of S 1 or it coincides with S 1 . Consider a line L ⊂ S 1 ∩ p ⊥ . Let u, v b e dis tinct po in ts on this line. The lines up a nd v p intersect S by p oints u ′ and v ′ , r espe c tively . Since p 6∈ S , we ha ve p 6 = u ′ , v ′ and the p oints u ′ , v ′ are dis tinct. The lines L and u ′ v ′ bo th are contained in the plane L ∪ { p } , thus they have a non-empty intersection. The inclusion u ′ v ′ ⊂ S guarantees that L intersects S . So, every line L ⊂ S 1 ∩ p ⊥ has a non-empty int ers ection with S . Thus S intersects S 1 ∩ p ⊥ at lea st by a hyperplane. The dimension of S 1 ∩ p ⊥ is no t less than n − 2 and we g et dim( S ∩ S 1 ∩ p ⊥ ) ≥ n − 3 which implies that S ∩ S 1 is ( n − 3)- dimensional. Similar ly , we s how that the dimension of S ∩ S 2 is eq ual to n − 3.  Now establish the equality dim S 1 ∩ S 2 = n − 3 which completes o ur pro of. OPPOSITE RELA TION ON DUAL POLAR SP ACES AND HALF-SPIN GRASSMANN SP A CES 5 Pr o o f. Define a U := ( S ∩ S 1 ) ∩ ( S ∩ S 2 ) . Since S ∩ S 1 and S ∩ S 2 are ( n − 3)-dimensiona l subspaces of S , one of the fo llowing po ssibilities is realiz e d: (1) S ∩ S 1 = S ∩ S 2 and U is ( n − 3)-dimensional, (2) dim U = n − 4, (3) dim U = n − 5. Since U is contained in S 1 ∩ S 2 , the dimension of S 1 ∩ S 2 is equal to n − 3 in the first and s econd cases . Let U b e a n ( n − 5)-dimensiona l subspace. If U do es not co incide with S 1 ∩ S 2 then S 1 ∩ S 2 is ( n − 3 )-dimensional. Now s upp ose that U = S 1 ∩ S 2 . W e take any line L ⊂ S 1 \ S and co nsider the singular subspace L ⊥ ∩ S 2 ; its dimension is no t less than n − 3. Mor e ov er, this subspace do es not contain S ∩ S 2 . Indeed, S is spa nned by S ∩ S 1 and S ∩ S 2 , and the inclusio n S ∩ S 2 ⊂ L ⊥ ∩ S 2 implies that L ⊥ S ; the latter is imp ossible, since S is a ma ximal singular subspace and L 6⊂ S . Therefore, ther e is a po in t p ∈ S 2 \ S satisfying p ⊥ L . As ab ov e, we show that the int ers e ction of S with the plane L ∪ { p } contains a line. This line intersects L which contradicts L ⊂ S 1 \ S . This means that the third case can not b e realized.  4. Example Let V b e a left vector space ov er a division ring R and Ω : V × V → R b e a no n- degenerate r eflexive sesquilinea r for m of Witt index n ≥ 3 . W e wr ite Π = ( P , L ) for the ass o ciated p olar space ( P and L a re the sets of 1-dimensio nal and 2-dimensional totally isotr opic s ubs paces, resp ectively) and supp ose that it is o f t yp e C n . Ev ery element of G n − 1 (Π) ca n be obtained from a cer tain maxima l singula r subs pa ce of the for m Ω. W e a ssert that the following conditions are no t equiv alen t (1) S 1 , S 2 ∈ G n − 1 (Π) are collinear p oints of G n − 1 (Π), (2) there exists S ∈ G n − 1 (Π) \ { S 1 , S 2 } such that U op S implies that U op S 1 or U op S 2 . It is not difficult to see tha t (1) implies (2) (a ny S ∈ G n − 1 (Π) \ { S 1 , S 2 } b elonging to the line joining S 1 with S 2 is as required). No w we show that (2) do es not imply (1). Let p 1 , . . . , p n , q 1 , . . . , q n be a frame of Π such that p i 6⊥ q i for each i . F or some vectors x 1 , . . . , x n , y 1 , . . . , y n ∈ V we hav e p i = h x i i , q i = h y i i and Ω( x i , y i ) = 1 . The maximal s ingular s ubspaces of Π ass o ciated with the maximal totally isotropic subspaces h x 1 , x 2 , x 3 , . . . , x n i , h y 1 , y 2 , x 3 , . . . , x n i , h x 1 + y 2 , x 2 − y 1 , x 3 , . . . , x n i will b e denoted by S 1 , S 2 , and S (resp ectively). Their intersection is the ( n − 3)- dimensional sing ular subspace N asso cia ted with h x 3 , . . . , x n i . 6 MARIUSZ KWIA TK OWSKI, MARK P ANK OV Now co nsider the line L joining h x 1 + y 2 i with h x 2 − y 1 i . Every p oint on this line is of type (1) h ( x 1 + y 2 ) + t ( x 2 − y 1 ) i , t ∈ R If p ∈ p 1 p 2 \ { p 1 p 2 } and q ∈ q 1 q 2 \ { q 1 q 2 } ar e collinear then p = h x 1 + ax 2 i and q = h y 2 − ay 1 i for a certain s c alar a ∈ R ; every p oint on the line p q is of type (2) h x 1 + ax 2 + s ( y 2 − ay 1 ) i , s ∈ R The lines L and p q have a no n-empt y intersection (b ecause (1) co incides with (2) if t = a and s = 1 ). Similarly , we establish that for a n y tw o co llinear p oints p ∈ S 1 \ N and q ∈ S 1 \ N the line p q intersects S \ N . There fo re, if U ∈ G n − 1 (Π) is opp osite to S then it is opp osite to S 1 or S 2 . How ev er, S 1 and S 2 are not collinear . References [1] Abramenko P ., V an Maldeghem H ., On opp osition in spheric al buildings and twin buildings , Ann. Combinatorics 4(2000), 125–137. [2] Blunck A., Hav li cek H., O n bije ctions that pr eserve c omplementarity of subsp ac es , Di screte Math. 301 (2005), 46–56. [3] Buekenh out F., Sh ult E., On the foundations of p olar ge ometry , Geom. Dedicata 3 (1974), 155–170. [4] Chow W. L., On the ge ometry of algebr a ic ho mo gene ous sp ac es , Ann. of Math. 50 (1949) 32–67. [5] Havlicek H., P anko v M ., T r ansformations on the pr o duct of Gr assmann sp ac es , Demonstratio Math. 38(2005), 675–688. Dep ar tment of Ma thema tics a nd Informa tion Technology, University of W armia and Mazur y, ˙ Zolnierska 14A, 10-561 Olsztyn, Poland E-mail addr ess : kfiecio@o2.p l, pankov@ma tman.uwm.edu. pl