Torus quotients of homogeneous spaces

Proc.lndian Acad. Sci. (Math. Sci.), Vol. 108, No.1, February 1998, pp. 1-12. © Printed in India

Torus quotients of homogeneous spaces

SSENTHAMARAIKANNAN SPIC Mathematical Institute, 92 G.N. Chetty Road, T. Nagar, Madras 600 017, India E-mail: [email protected]

MS received 24 February 1997; revised 21 August 1997

Abstract. We study torus quotients of principal homogeneous spaces. We classify the Grassmannians for which semi-stable = stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring of t invariants of the homogeneous co-ordinate ring of the Grassmannian G 2 (n odd) over the ring generated by RI , the first graded part of the ring of T invariants. .n

Keywords. Torus; Borel subgroups; Frobenius splitting; stable and semi-stable points.

1. Introduction

One of the classical problems in invariant theory is the study of binary quantics. The main object was to give an explicit description and study the geometric properties of SL 2 quotients of the projective space for a suitable choice of linearization.

The aim of this paper is to begin the study ofa natural generalization of this classical question.

Let k be an algebraically closed field. Let G be a semisimple algebraic group over k, T a maximal torus of G, B a Borel subgroup of G containing T, N the normalizer of Tin G, W= N IT, the Weyl group.

Consider the quotient variety N\\ G I B. In fact the aim is to study more generally the variety N\ \ G I Q, where Q is any parabolic subgroup of G containing B.

In the case when G = S Ln(k), the special linear group and Q is the maximal parabolic subgroup of SLn(k) associated to the simple root 1X2' one knows that GIQ is the Grassmannian G2 •n of two dimensional subspaces of an n dimensional vector space. One also has an isomorphism:

N\\(GIQ)SS(L) = N\\(G2•n)SS(L)'::+'SL2 \\(P(V))"s,

where V is the vector space of homogeneous polynomials of degree n in two variables and L is the line bundle associated to the fundamental weight tiJ2, and the scheme SL2 \\P(V)SS is precisely the space of binary quantics (for example, see the proof of Theorem [1] and the proof of Theorem [4] of [CSS III]. We will also give an outline of the proof in the proof of corollary [3.10]).

More generally one has an isomorphism:

T\\(GI P)SS(L) = T\\ G~~n(L) '::+'SLr \\(pr-l t, where G = SLn(k), P is the maximal parabolic subgroup associated to IXr' Gr.n is the Grassmannian of r dimensional subspaces of an n dimensional vector space and L the line bundle on G I P = Gr •n associated to tiJr •

2 S Senthamarai Kannan

In this paper, we prove the following results:

(a) The varieties T\\ G I Q and N \\ G I Q are Frobenius split and as an application the vanishing Theorems for higher cohomologies of these varieties.

(b) As a part of result (a), we prove the vanishing of the higher cohomology groups for the binary quantics.

(c) For the line bundle L on Gr •n associated to the fundamental weight 1lJr ,

(Gr.n)~(L) = (Gr.n)~(L) if any only if rand n are coprime. (d) Existence of smooth projective varieties as quotients of certain G I Q modulo

a maximal torus T(in the case of G = S Ln). ___ (e) For n odd, a partial result about Rl generation of the graded ring k[Gz.nY =

EBd;'O Rd'

The layout of this paper is as follows: Section 2 consists of notations, conventions and basic Theorems. In § (3), we prove the results (a), (b), (c) and (d), and in § (4), we prove the results (e) and (f).

2. Notations and conventions

Let k be an algebraically closed field. Let G be a semisimple algebraic group over k, Ta maximal torus of G and B a Borel subgroup of G containing T.

Let N be the normalizer of Tin G, W= N IT, the Weyl group. Let $ be the set of all roots. Let $+ and $- be the sets of positive and negative roots respectively. Let d = {(Xl' (X2"'" (Xl} be the set of all simple roots, where 1 is the rank of G. Let Si be the simple reflection with respect to (Xi'

Let 1lJi 'S be the fundamental weights. Let X ( T) and r (T) denote the sets of all characters and one parameter subgroups of T respectively. Denote the canonical bilinear form by

(,):X(T) x r(T)-+Z.

Let E = r(T)®R,

C(B) = {A.EE:«(X, A.) ~ 0 for all (XEd}.

Recall in the case of G = S Ln, G I Pr is the Grassmannian Gr.n of r dimensional subspaces of an n dimensional vector space, where Pr is the maximal parabolic subgroup associated to the simple root ar • Define the character ei of Tby ei(t) = ti for t = diag (tl' t2 ••• tn)ET.

Let Gr •n denote the cone over Gr •n with respect to the line bundle given by the character 1lJr • Note that the action of an element t in Ton the vector ei, /\ ei,/\ ••• /\ ei,ENV (V being the standard G module) ~is given by t.e;. /\ ... /\ ei, = (I1~= 1 ti,)·e;. /\ ... /\ ei,' hence on the co-ordinate ring k [Gr •n] is given by t'Pi,i" i, = (I1j= 1 tij )-l'Pi,i, i, where Pi i "i is the Plucker co-ordinate associated to (iI' i2 ,.··, ir) with il < i2 < ... < ir ~ n. L~t k[Gr.nY denote the T invariants of the co-ordinate ring k[Gr.n].

Note also that if Ul' jz,'" ,jr) is not in ascending order,

PM,' 'j. = sgn ("C)-( Pt(h) , , 't(j,»)

where "C is the permutation such that

1 ~ "CUd < "C(2) < ... < "CUr) ~ n.

Torus quotients of homogeneous spaces 3

The action of w, an element of Won P;, ... i, is given by w·p;, ... i, = PW-'(i,) ... w-'(i,)'

Denote r·PI.2 ..... r by p(r). For an r tuple j1 = (iI' i2,···, ir ), denote Pi,. h ... .,i, by PI"

Now, coming back to the general semisimple algebraic group G, let W o denote the longest element of the Weyl group of G, p denote the half sum of all positive roots. The coset w Bin G / B is denote by e(w).

Let U denote the unipotent radical of B. When the characteristic p of the field k is positive we have the Frobenius map F:(!Jx ~ (!Jx defined by F(f) = fP.

Denote the global sections of X with respect to the line bundle L on X by HO(X, L), and denote by p(id) a lowest weight vector of the G module HO(G/ B, L;) where LA is the line bundle associated to the dominant weight A.

Let Ga•1I

denote the unipotent group associated to the root oc. It is known that U = nlI>o Ga•

lI• Let XII be the co-ordinate function of the affine line Ga•

lI• Let X(r) denote

the Schubert variety defined by r in W. We use the notations for semistable points, stable points, numerical functions, ... as

in [GIT]. We recall some important results of [CSS I] here which will be used in § 3.

Lemma 2.1. [CSS 1] Let G be a semisimple algebraic group. T a maximal torus of G. B a Borel subgroup of G containing T, and C(B) be as in this section. (a) Let Lbe the line bundle defined by the character XEX(T). Then if xEG/B is represented by bwB. bEB. WE W represented by an element of N in the Bruhat decomposi-

tion of G and A is a 1 - PS of T which lies in C(B), we have

j1L(X, A) = - < w(X), A).

(b) Given any set S offinite number ofnon-triviaI1-PS A ofT, there is an ample line bundle Lon G/ B such that

j1L(X, A) # 0

for all xEG / B, AES.

3. Torus quotients

In this section we shall study the quotients of the homogeneous spaces under torus actions. In particular we give some natural criteria under which semi-stable points are actually stable. We also prove the Frobenius splitting property for these quotients.

PROPOSITION 3.1

Let G be a simply connected semisimple algebraic group. T a maximal torus of G. B a Borel subgroup of G containing T. Set X = G / B. Then there is a G linearized very ample line bundle L on X such that

X::;(L) = X~(L).

Proof Choose a set S of finite number of non-trivial 1 - PS A of C(B), that generates

C(B) as a R;.o monoid. By Proposition [2.1 (b)], there is a very ample line bundle L on X such that

j1L(X, A) # 0 for AES, xEG / B.


For AEC(B), write A = L).'ESasAs,as~ O. By Lemma [2.1 (a)J,

IlL(x,i)::-- \W(X), ~asAs )

= - ( ~as<w(x), As> )

(**)

Now, let xEX~(L) be arbitrary. By Theorem [2.1J of [GITJ, IlL(X, A) ~ 0 for alll-PS A of T i.e, Il( WX, WAW- 1

) = Il(x, A) ~ 0 for alll-PS A of T. So, Il( WX, A) ~ 0 for all WE W, AEC(B). Therefore by (*), we have ILL ( WX, As) > 0 for all wEWand SES.

By (**), we have

ILL ( WX, A) = Is a.IlL( WX, As) for AE C(B).

Hence for any nontrivial 1 - PSA lying in C(B), WE W,

Il( WX, A) > O.

Since for any 1 - PSA of T, there is a wEWsuch that WAW- 1 EC(B), we have IlL(X, A) > 0 for all nontrivial 1- PSA of T. Hence we have

X~(L) = X~(L). Lemma 3.2. Let G be a semisimpie algebraic group of adjoint type such that none of its simple factor is IPSL2 (k). Let T be a maximal torus of G, B a Borel subgroup of G containing T. Let X = G / B. Then the set of points of X whose isotropy in Tis non trivial has codimension at least two.

Proof Let D be an arbitrary prime divisior of X.

Case 1: D n U'e(wo) is non empty. For any root rx, set

Z(X,,) = {x = u'e(wo)E U e(wo)/ X,,(u) = O}.

Subcase 1: Dn U'e(wo) # Z(X,,) for al1rxE~. Choose a point xED() U'e(wo) such that X«(x) # 0 for all rxE~. Then the isotropy Tx is contained in the intersection ()aEA ker(rx) = (1). Hence the isotropy Tx = (1) for some XED n U·e(wo).

Subcase 2: D()U·e(wo) = Z(X,,) for some iE{1,2, ... ,I}. Choose a point XED n U'e( wo), X = u'e( wo), UE U such that X,,(u) # 0 for all rx i= rx i • It is easy to see that for any group G as above, and for any i, either rx i - 1 + rx i or rx i + rx i + 1 is a root of G. So, Tx = T" C ()"#' ker(rx), n,,#. ker(rx) C ()j;tl ker(rx) () ker(rxi _ 1 + rx i ) = (1), if rx i - 1 + rx i

is a root. Hence, Tx = (1). The argument is similar if rxi + rx i + 1 is a root.

Case 2: D () U'e(wo) is empty. In this case D is a Schubert divisor = X(-r), where ,=WO·S i . Set

S(,) = {rxE<I>+ /,(rx)E<I>-} = <1>+ - {rxi}'

Then the unipotent group U (,) = II"""i.i%E<l>+ G a," is isomorphic to the big cell U'e( ,) of X (,). Choose a point UEU(,) such that X,,(u) # 0 for all rx i= rx i • Now, if x = u'e(,), then

Tx = T" = (1), by proof as in the previous case.


Theorem 3.3. Let X = Gr •n. L= 4, the line bundle on X given by the weight tiJr. Let Tbe a maximal torus of G = S Ln(k). Then

X~(L) = X~(L)

if and only ifr and n are coprime.

Proof <=: Assume that rand n are coprime. It is known that X = G I P, P the maximal parabolic

subgroup associated to the root rxr . It is easy to see that C(B) is generated as a R",o monoid by AI"" ,AI where < rxj , Ai > = n·bj •i, bj•i is the Kroneker number, n = I + l.

Now, let XE BwP be an arbitrary point. Let n:G / B ~ G I P be the canonical projection. Choose a point YEBwB such that n(y) = x. By Proposition [2.1] of [CSS], we have

jl"'(L)(y, A) = jlL(X, A). (*)

By Lemma [2.1 (a)], we have

jl"'(L) (y, A) = - < w(tiJr), A>

for AEC(B).

Claim. jlL(X, As)"# 0 for all XEX, and sE{1,2, ... , I}. From (*) and (**), we have

jlL(X, As) = - < w( tiJr), As> = - / tiJr - L mirxi, AS)' \ mi~O

Case 1: r ~ s. Since

we have

Hence,

1 (r-l I ) tiJr=-1 1 .Lj(l-r+1)rxj + ~r(l-j+1)rxj

+ )=1 )=r

(tiJr, As> = \ I ~ 1 (r(l- s + l) rxs)' As) = I ~ 1 (r(l- s + l)n = r(n - s)).

jlL(X, As) = - \ tiJr - itl mirxi, As) = - (r(n - s) - nms)"# 0

as rand n are copime.

Case 2: r > s

jlL ( x, As) = - \ tiJr - ~ mirxi, As)

= -( <tiJr, As>- ~minbis) = (-1/1 + l)(s(l- r + l)(rxs' As> - msn

2)

= (-1/1 + l)(s(l- r + l)n - msn2

)

= -((n-r)s-nms)"#O

since n - rand n are coprime.

(**)


Hence, for AEC(B), write A = :EaiAi' ai ~ O.

J.lL(x, A) = - ( w( tDr ), ~ aiAi)

= - Las<w(tDr ), As>

Let XEX"; be arbitrary. Therefore, J.lL(x,A)~O for all AEr(T) by Theorem [2.1J of [GIT].

Hence

J.l(wx, A) = J.l(x, w- 1 AW) ~ 0

for all AEC(B) and WEW. SO, J.l(W-1X,As»0 for all sE{1,2, ... ,1} and wEW(since J.l(w-1x, As) # 0 for all s and WE W).

By (***), we have

J.lL(W-1X, A) = L asJ.l(w-1x, As) > 0 AsES

for all nontrivial 1 - PS AEC(B) and WE W.

Since for any AEr(T), there is a wEWsuch that WAW-1EC(B). Therefore we have J.lL(x, A) > 0 for all non trivial 1 - PSA of T. Hence by Theorem [2.1J of [GITJ, xEX~(L).

Hence we have

X";(L) = XHL).

:~ Conversely suppose that rand n are not coprime. So write n = q-d; r = m·d with d> 1. For kE{1, 2, ... , r} write k = (j - 1)m + i, where jE{l, ... ,d} and iE{1, 2, ... ,m}. Choose WE Wsuch that w-1(k) = jq - m + i where k = (j - l)m + i, kE{l, 2, ... ,r}. Let XEBwP be such that p(r)(x) #0 for all, ~ w. For sE{1,2, .. . ,q}, define

J.ls = ([sJ, .. . ,[s + m -1J,q + [sJ, .. . ,q + [s + m -1J, . .. ,(d -1)q

+ [sJ, ... ,(d - 1)q + [s + m - 1J),

where [aJ is defined to be the unique integer in {1, 2, ... ,q - 1} such that [aJ = a mod q for a # q and [qJ = q.

Therefore the weight of p is equal to

[

d s+m-l Jl. ] - .L .L eU-l)q+[i] .

J=l ,=s

Hence the weight of the section a = TIs p Ils of the line bundle L i8)q is equal to

m[ - (.I ± e(j-l)Q+S)] = - (m) (.f 13,) = O. J=ls=l ,=1

Therefore a is a T invariant section of L ®q such that a( x) # O. Hence x is semistable.

Torus quotients of homogeneous spaces

Claim 2: x is not stable. For, IlL(:X, Aq) = - < w(tiJ,), Aq). Now,

w(tiJ,) = wCt1 ci - (rjl + 1) Ct1 Ci) ).

So,

since < L:7= 1 ci' Aq) = O. i.e,

<w(tiJ,), Aq) = \ Jl itl Cjq-m+i,Aq)

= m(n - q) - (d - 1)mq = 0,

since n = qd, i.e, IlL(X, Aq) = O. Hence x is not stable.

COROLLARY 3.4

7

Let G = S L .. T a maximal torus of G, B a Borel subgroup of G containing T. Let Q = nUEJ) Pj be a parabolic subgroup of G containing B such that there is an rEJ with rand n being coprime. Set Z = GjQ. Then there is a very ample G linearized line bundle N on Z such that Z~(N) = Z~(N).

Proof Let X = G,.n where rEJ be such that rand n are coprime. Let L be the line bundle on X associated to the weight tiJ,. Let M be the line bundle on Z associated to the weight LiEJ-{'} mi· Let N = aL+ bM be a line bundle on Z so that "bja is sufficiently small" as in Proposition [5.1] of [CSS I]. Then by Proposition [5.1] of [CSS I],

Z~(N) = ZHN).

Lemma 3.5. Let G =SL .. T, B, Q, Z and N be as in Corollary (3.4). Then T\\(Z~(N)) is smooth.

Proof By Corollary (3.4), there is a very ample G linearized line bundle N on Z such that Z~(N) = ZHN).

Claim. For ZE ZS, the isotropy of Z in Tis the center of the group G.

Proof of Claim. Let ZE ZS. Then ~ is finite i.e, Tn gQg -1 is finite, where Z = gQ. Then Tn S is finite for every maximal torus S of gQg - 1.

Since G is reductive for any maximal torus S of G, S must be the centralizer C G(s) for some s in S. Therefore for any maximal torus S of gQg - 1, we have

TnS = {t = diag(t1, ... ,tn)ET. tlj-1 = 1 if sij # 0 or sij # O}

is finite (where s = (Si)ES is such that S = CG(s)).


Therefore, the roots

{ei-ej:sij#O or Sji#O}

generate the root lattice. Hence Tn S is the center of G. Since any semisimple element of gQg-l lies in some

maximal torus S of gQg - 1, we have ~ = Tn gQg - 1 = Z (G). Hence the claim. Now, if we set G = PSLn = G/Z(G), '1'= T/Z(G), B = B/Z(G), Q = Q/Z(G). Then Z = G/Q and Zf(N) = Z,. (N). Therefore the isotropy ~ of z in ZS is trivial. Hence by Proposition [0.9] of [GIT],

T\\ZSS(N) is smooth.

Lemma 3.6. Let r # 2 or n # 4. Let G = SLn, T a maximal torus ofG, B a Borel subgroup of G conraining T. Let P be the maximal parabolic subgroup of G associated to the simple root (1.r' Let X = Gr •n = G/P. Let L be the line bundle associated to thefundamentalweight tiJr . Then codim(X~(L) - X;.(L)) ~ 2.

Proof By Theorem (3.3), without loss of generality we can assume that 1 < r < n - 1 ( = I) (since if r = 1 or n - 1, then XSS(L) - XS(L) is empty set and therefore trivially the Lemma is true).

It is enough to prove that for any prime divisor D of X,D nXsis nonempty. LetD be any prime divisor of X. Set U = B'e(wo) = B'(woP) c: X, the big cell.

Case 1: Assume that W· U meets D for every WE W. Since D is irreducible and W is finite, the intersection

is nonempty. Now, choose a point XE(nwewwU)nD. Let A. be a nontrivial one

parameter subgroup of T. Choose a WE W such that W..1.W - 1 E C(B). Since WXEU,

Il(X, A.) = Il( WX, W..1.W- 1) = - < wo( tiJr ), W..1.W- 1) = < tiJn - r , W..1.W- 1).

Since W..1.W-1EC(B), write W..1.W- 1 = Las~oas..1.s. So,

s<lI-r s~n-r

= L as(sr) + L as(n - r) (n - s). 8<II-r s~n-r

Since A. is a nontrivial 1 - PS of T, there is a s E {1, 2, ... ,I} such that as> 0, hence Il(x, A.) > O. Hence by Theorem [2.1] of [GIT], XED nXs.

Case 2: Assume that there is a WE Wsuch that wU does not meet D. Therefore we have w-1D = X('r), where. = WO·sr • Choose XEX(.) such that

p(a)(x) # 0 for all a ~ •.


Claim: x is stable. By Lemma [2.1(a)],

Il(x, As) = - < r( tilr), As> = - < wo( tilr - ar), As> = < tiln - r - IXn - r, As>· (*)

Fors"#n-r,

Il(x, As) = < tiln - r , As> > 0

by proof as in Theorem [3.3]. Fors=n-r,

Il(x, A) = < tiln - r, As> - n(jn-r,s = (n - r)(n - s) - n = nr - r2 - n > 0

since 1 < r < n - 1 and r "# 2 or n "# 4. Therefore Il( x, A) > 0 for all 1-PS A of T lying in

the closure of the Weyl chamber C(B). Now, let A be a nontrivial 1-PSA of T. Choose a O'E W so that

.1.1 = 0'.1.0'-1 E C(B).

Since the weight of p(wo) = tiln - r , weight of p(r) = tiln - r -lXn - r, we have weight of p(O'- 1 r) "# weight of p(O'- 1wo)'

Therefore since the only extremal section vanishing at x is p(wo), we have either p(r)(O'x)"# 0 or p(wo)(O'x)"# O. So,

Il( x, A) = Il(O'x,O'AO' - 1) = Il(O'x, .1. 1) ~ - < r( tilr), .1.1 > = Il( x, .1.1)

which is positive Hence we have xEDnXs. Hence the Lemma.

Theorem. 3.7. Let G be a simply connected semis imp Ie algebraic group. Let T be a maximal torus of G. Let B be a Borel sub group of G containing T. Let X = G/ B. Let L be any ample line bundle on X. Then the good quotient Y= T\\Xs.;(L) is Frobenius split.

Proof By Theorem [2] (p. 38) [M-R],X is Frobenius split. Let Vbea Tinvariant affine open subset of X. Let R be the coordinate ring of V. Let ¢: F * R ~ R be a R linear map such that ¢oF = idR•

Let n:R ~RTbe the Reynolds operator. Define ¢:F*RT -'>RTby ¢(f) = no¢(f) for [EF*RT.ltis easy to see that ¢ is RT linear and ¢o F = idRT. Hence Yis Frobenius split.

COROLLARY 3.8

Let G be a simply connected semisimple algebraic group. Let T, B, X, L, Y be as in previous Theorem. Let N be the normalizer of T in G. Let W be the Weyl group. Then if p the characteristic of the field is bigger than the order of the group W. then the good quotient Z = N\\ X';:(L) is Frobenius split.

Proof Since Z = W\\ Y~, Y is Frobenius split and W is linearly reductive (since p the characteristic of the field is bigger than the order of the Weyl group W) by proof as in previous Theorem Z is Frobenius split.

Theorem 3.9. Let G be a simply connected semisimple algebraic group. Let T, B be as in Theorem [3.7]. Let Q be any parabolic subgroup of G containing B. Let X = G/Q. L be


any very ample line. bundle on X. Then we have

1. The good quotient ¥= T\\XS;(L) is Frobenius split. 2. If p the characteristic of the field is bigger than the order of the Weyl group W, the

good quotient Z = N\\X";(L) is Frobenius split.

Proof It is known that GIQ is Frobenius split (for a proof, see [M-R]). So using the Reynolds operator (as in Theorem [3.7] and corollary [3.8]) we can prove (1) as well as (2).

COROLLARY 3.10

Let V be the vector space of homogeneous polynomials of degree n in two varibales. Then for the natural action ofSL2 on the projective space P(V) the good quotient SL 2 \\P(V)SS is Frobenius split.

Proof Let M(2, n) denote space of 2 x n matrices with entries in k. The morphism ¢:M(2, n)-+ Vdefined by

gives an isomorphism between the varieties (P(M(2, n)))"' II Nand P(V). On the other hand, the morphism X:M(2, n)-+A2(kn) defined by

gives an isomorphism between the varieties SL2 \\(P(M(2, n)))"' and the Grassmannian G2 •n. Hence the variety N\\ G'l.n is isomorphic to

SL 2 \\(P(M(2, n)))SSIIN)SS

via X and the latter is isomorphic to S L2 \\ (P (V))"' via <p. Thus we have established an isomorphism:

N\\(GIQ)""(L)'=;SL 2 \\P(V)"'

where G = SLn, Q is the maximal parabolic subgroup associated to !X2 and L is the line bundle on GIQ associated to the fundamental weight tlJ2 • From Theorem [3.9] it is easy to see that SL2 \\P(V)"" is Frobenius split.

COROLLARY 3.11

Let V be the vector space of homogeneous polynomials of degree n in two variables. Then for any ample line bundle L on the good quotient Z = SL2 \\P(V)SS the cohomologies Hi(Z, L) = o for i > O.

Proof It is known that for any normal Frobenius split projective variety X, and for any ample line bundle L, Hi(X, L) = 0 for all i > O. (For a proof, see [M-R].) Hence by Corollary (4.10), Hi(Z, L) = 0 for all i > O.


4. RI generation

In this section we prove a partial theorem of the RI generation property for the homogeneous coordinate ring of T\\ G~.n(L) for odd n, where Lis the line bundle on G2 •n associated to the fundamental weight 'IlJ2 •

Let G = SLn, T a maximal torus of G. Let B be a Borel subgroup of G containing T. Let P be the maximal parabolic subgroup associated to the simple root (X2' Let X = G2 •• = GIP, let L be the line bundle associated to the fundamental weight 'IlJ2 . Let X denote the cone over X with respect to the line bundle L.

Let k[X] denote the co-ordinate ring of X. Let {Pij:i <j, i,jE{l, 2, ... , n}} denote the set of Plucker co-ordinates.

Recall from § 1, for t = diag(tl' t 2 , ••• , tn)E T, the action of ton Pij is given by

t· Pij = (titj) -I Pii'

An elementfEk [X] is Tinvariantifand only iff = L,aiMi,aiEk, each Miin the sum is a T invariant monomial in p,/s. (For a proof: Since the standard monomials [cf. [CSS II]] form a basis for the k vector space k [X] and they are all T weight vectors, any fEk[XY can be written as f = LxaxMx' axEk and Mx's are standard monomials in the Pi./S of T weight X, forcing that the weight X is zero whenever ax 1= 0.)

Now, for a monomial f = TI«jPu'J set m'j = mji for i > j. With these additional symbols {mij:i > j} it is easy to see that a monomial f = TIi<jPUV is T invariant if and only if Lk,.imik = Lk,.jmjk for every i,j E{1, ... , n}.

Write k[X] = A = EBd;.oAd, where Ad is the space of homogeneous elements of degree d in the Pij's. For n odd, it is easy to see that

AJ 1=0

if and only if n divides d. Since the action of T on X is linear with respect to L, the ring of T invariants R = k[XY is a graded ring, say R = EBd;.oRd•

Let S be the subring of R generated by the vector space Rl of R. Let M denote the monoid of n x n symmetric matrices with non negative integer

entries and whose diagonal is zero. Let

Let

M R ={(mi)EM:Lmij=2d forsome dEZ;.o, forall iE{1,2, ... n}}. j,oi

Ms = Z;.o span {(mij) EM: L mij = 2 for all i E {l, ... , n}}. j,oi

It is easy to see that the monomials {TIi <j P7'?: (mi)E M R} generate the vector space Rover k. Also it is easy to see that {TI«jPuv:(mi)EMs} generate the vector space S over k.

DEFINITION 4.1

A square matrix v = (mij) is called 'doubly stochastic' if L? = 1 mij = 1 for every j E {1,2, ... ,n}, LJ=I mij= 1 for every iE{1,2, ... ,n}, and mij's are non negative real numbers.

DEFINITION 4.2

A permutation matrix is a {O, I} square matrix with exactly one 1 in each row and each column.


Theorem 4.3. [cf [S] p·108]. A matrix v is a 'doubly stochastic' matrix if and only ifv is a convex combination of permutation matrices.

Theorem4.4. lfn is odd, the ring ofT invariants R is afinite module over S of degree 2r for some r.

Proof Since T is a reductive group, R is a finitely generated k algebra. Choose a finite set {f.:sE{1, 2, ... , N}} of monomials generating R.

Write f.= D;<jPij'J for sE{1, ... ,N}. Therefore the matrix v=(m;)EMR• So, 1/2d times () is a 'doubly stochastic' matrix for some positive integer d. By Theorem (5.4), (1/2d)·{) is a convex combination of permutation matrices. Hence () is a sum of permutation matrices say equal to 2:>1; (not neccessarily distinct).

Since () is symmetric,

2{) = () + {)I = 2:)0'; + aD E Ms· ;

Therefore f; E S. Hence the Theorem.

Acknowledgements

The author wishes to thank Prof. C S Seshadri for his encouragement and his help and to Profs T R Ramadas, V B Mehta, S Raghavan and Kashiwara and also to V Balaji, P A Vishwanath, P Sankaran and K N Raghavan for helpful discussions.

References

[M-R] Mehta V B and Ramanathan A, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. 122 (1985) 27-40

[GIT] Mumford D, Fogarty J and Kirwan F, Geometric Invariant Theory (third edition) (New York: Springer-Verlag, Berlin, Heidelberg)

[S] Schrijver A, Theory of Linear and Integer Programming (John Wiley and Sons Ltd.) (1986) [CSS I] Seshadri C S, Quotient spaces modulo reductive algebraic groups, Ann. Math. 95 (1972) 511-556

[CSS II] Seshadri C S, Introduction to the theory of standard monomials, Brandeis Lecture Notes 4, June 1985

[CSS III] Seshadri C S, Mumford's conjecture for GL (2) and applications. International Colloquim on Algebraic Geometry, Bombay, 16-23 January (1968)

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