Projective spaces in flag varieties

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1 This article was downloaded by: [Universita Studi la Sapienza] On: 18 December 2011, At: 00:38 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: Projective spaces in flag varieties E. Strickland a a Dipartimento di Matematica, Universitá di Roma "Tor Vergata", Via della Ricerca Scientifica, Roma, 00133, Italy Available online: 23 Dec 2010 To cite this article: E. Strickland (1998): Projective spaces in flag varieties, Communications in Algebra, 26:5, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sublicensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 COMMUNICATIONS 1N ALGEBRA, 26(5), (1998) Projective Spaces in Flag Varieties 0. Introduction Dipartiniento tii Matcrnatica UniversitA di Roma "Tor Vergata", Via della Ricerca Scicntifica, 00133, Roma, Italy. Let G be a semisimple algebraic group over an algebraically closed field k of characteristic zero. Let B be the variety of Borel subgroups. Consider the the projective embedding U + B(HO(U, Lp)*), where Lp is the line bundle associated to the Steinberg weight p, which is the half sum of positive roots. In this paper we show than any positive dimensional projective space contained in the image of U is necessarily of dimension one. Furthermore we determine exactly these lines. Indeed we show that if P c B is such a line, then there exists a minimal parabolic subgroup P C G such that t is the set of Borel subgroups which are contained in P. In particular this implies that the possible homology classes of such lines correspond under the usual identification of the root lattice with H2(U, Z) to the set of simple roots. The result is obtained as an application of some properties of the intersection of Schubert cycles in the cohomology ring of U. 1 Copyright O 1998 by Marcel Dekker. Inc

3 STRICKLAND 1. Linear Spaces in U Given a semisimple sirnply cor~nt:c:ted algebraic group G we want to recall a few facts on the geometry of the projective variety, 8, of thr Borcl subgroups of G. Let 11s choose a maximal torus I' C G and a Bore1 suhgroup B > T. Let W = N(T) /T be the Weyl group. Let f = LicT be t,he corresponding Cartan C 1" be the root system associated to LieT anti cp+ be the set of positive roots corresponding to the choice of B and {a,,...,a,} the set of simple roots. Let us denote by B the unique Borel subgroup such that B fl B = T. Furthermore we shall denote by s, E W the simple refection corresponding to a root a E + and, for an element w E W, t(w) the length of w with respect to the generators given by the simple refections s, = sai. As usual we shall denote by wo the longest element in W. We shall also consider the weight lattice A = XS(T), which is the dual of the root lattice Q, and denote by {wl,...,w,) the set of fundamental weights defined by (w,,aj) = &,,. We set p = CaE9+ a = CI=l w,. Since all Borel subgroups in G are conjugate and a Borel subgroup equals its normalizer, we can identify L? with G/B. Once this has been done, we get an explicit cellular decompositiorl by locally closed affine spaces of U, whose cells, indexed by the elements w E W, are the B orbits C, = BrtwB/B, n,,, C N(T) being a representative of w. The C,,'s are called Schubert cells and their closures C, Schubert varieties. One has dim C, = e(w). It follows from this [F], that the Chow ring AS(8) has a basis given by the classes [c,], dual to the classes of the Schubert varieties and, if 6 coincides with the cohomology (doubling the degrees). We also have that if we consider the pairing on A8(U) given by (x, Y) = / xv tz for z, y E A*(U), then we haw that ([c,], [c,,,]) = 6,,,1,,,,,. This is easily seen by taking the cellular decomposition C, corresponding to B arid remarking that the intersection of C, and Ci, is t3mpt,y unless ww' = wo, while if ww' = wo, C, and C,, intersect transversally iri a single point.

4 PROJECTIVE SPACES IN FLAG VARIETIES 1653 As far as A' is concerned, one can give another description. Namely, given a character X E X*(T), we can extend, using t,he natural homomorphism B + T, X to a character of B, and thus consider the corresponding one dimensional B module kx. We now define the line bundle LA on B = G/B by G XB kx. By taking the fust Chern class c(lx) we get a isomorphism between A and A1(B) which takes the fundamental weights w, to the classes [??,,.i]. Remark that, using the above considerations, we deduce that, if we identify (B) with the root lattice & by associating to a simple root cr, the class ~dim5i [c.;], we get that for z E A1(B) and y E AdimB'(B), (z, y) = (2,y). Before we proceed, we need the following simple fact: Proposition 1.1. Let X C B be a subvariety. Let [XI E A*(B) denote the corre sponding dual class. Then [XI is a linear combination of the Schuhert classes [Cw] with non negative integer coefficients. Proof. We can also clearly assume that X is of pure dimension d. The only thing we need to prove is that, when we express [XI as a linear combination of Schubert classes, the coefficients are non negative. Indeed by a result of Kleiman (see [HI p. 268), since B is a homogeneous space, we can find an element g E G such that for every Schubert cell C,, of dimension complementary to that of X, X n c, is contained in C, and consists of a finite number of simple points. This clearly implies that, if we set n, = IX n??,i, [XI = Cl(tu,=d nwow1[cw], hence the claim. O then Corollary 1.2. Let a,,..., at be Schubert classes, then the product class a,... at is a hear combination with non negative integer coefficients of Schubert classes. Proof. By an easy induction, we can assume that t = 2. Let al = [c,, 1, a2 = [C,",]. Then again, by the result of Kleiman quoted above, we can find g E G such that alaz = [C,, n g??,,] and the corollary follows from Proposition 1.1. U Recall now that to any simple root a, we can associate a minin~al parabolic group B c P, c G such that the natural map

5 is a PIfibration whose fiber over [I] is exactly the Schubert variety [?,;I. Also given a weight X E A, the corresponding line bundle LA is very aniple if arid only if (A, a;) > 0 for all i = 1,..., r. Furthemiore LA is the pull hack of a line bundle on GIP,, which by abuse of notation we shall denote by the samc name, if and only if (X,ai) = 0, and, if this is the case, LA is very ample on GIP, if and only if (A, u,) > 0 for all j # i. Let 11s now consider the weight p = 4 xaeo+ Q = C:=, wi. By the above discussion the line bundle Lp is very ample on B while the line bundle Lpi, with pi = p w, is very ample on GIP,. Consider now l? embedded in P(HO(D, Lp)*) and Ci,. We have the following Theorem 1.3. Let S be a positive dimensional linear subspace in P(HO(f?, L,)*). Assume S C B. Then (1) dim S=l (2) There is a Bore1 subgroup B c G and a simple re1 ection s, ; such that Sis the Schbert variety F,,reiative to B i.e. S = FIB, P being the unique minimal parabolic subgroup containing B and associated to s,. Proof. Set t = dims and N = = dimb. Denote by c, the Chern class of the line bundle L,,. so that for any X = XI niw,, c(lx) = xi nict. By definition we have that Now, by Proposition 1.1 and Corollary 1.2, we immediately deduce t,hat if M is any It follows, ~lsing the expansion of (c, c,)" that. that there exists a unique z such

6 PROJECTIVE SPACES IN FLAG VARIETIES while, for any other degree t monomid in cl...., c,, In particular we deduce that Consider now the projection a,b t GIP,. As we have already remarked, the line bundle L,...,, implies that dim a,(s) < t. is very ample on GIP,, so that Now xi is a PIfibration, so dim xi(s) < t 1. Moreover xl'(xi(s)) is an irre ducible variety containing S and of the same dimension of S. Hence S = x;' (x;(s)). We have thus shown that our S is the total space of the P1fibration S + xi(s). If we consider the line bundle L on S obtained by restricting Lpr we deduce that S equals P(a,,(L)). Thus HZ(S) = H2(x,(S)) HO(ri(S)). If t > 1, then rkh2(xi(s)) 2 1, so we would deduce that rkh2(s) > 2, a contradiction. Therefore the first part of our Theorem follows. As for the second part, notice that we have proved that S is a fiber of the PI fibration a,. Thus, if we take a Bore1 subgroup B E S, we deduce that S = C,,. [F] W. Fulton,, lnterseclron Theory, Springer Verlag,, [HI R. Hartshorne,, Algebraac Geometry, Springer Vedag, Received: March 1997 Revised: July 1997