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Bruhat order for two subspaces and a flag
Evgeny Smirnov April 23, 2007
Abstract The classical Ehresmann­Bruhat order describes the possible degenerations of a pair of flags in a finite-dimensional vector space V ; or, equivalently, the closure of an orbit of the group GL(V ) acting on the direct product of two full flag varieties. We obtain a similar result for triples consisting of two subspaces and a partial flag in V ; this is equivalent to describing the closure of a GL(V )-orbit in the product of two Grassmannians and one flag variety. We give a rank criterion to check whether such a triple can be degenerated to another one, and we classify the minimal degenerations. Our methods involve only elementary linear algebra and combinatorics of graphs (originating in Auslander­Reiten quivers).

1

Intro duction

We will consider certain configurations of subspaces in an n-dimensional vector space V over an algebraically closed field K. These configurations (U, W, V· ) consist of two subspaces U and W of V of fixed dimensions k and l, and a partial flag V· = (Vd1 Vd2 · · · Vdm = V ), where dim Vdi = di . Our goal is to describe such configurations up to a linear change of coordinates in V and the ways how configurations degenerate. In other words, we consider the direct product X = Gr(k , V ) â Gr(l, V ) â Fld (V ) of two Grassmannians and a flag variety of type d = (d1 , . . . , dm ) in V , the group GL(V ) acting diagonally on this variety, and describe orbits of this action and the inclusion relations between their closures. One can easily show that the number of these orbits is finite. Such a product X of flag varieties is said to be a multiple flag variety of finite type. In the paper [MWZ] the authors list all such varieties and describe a way of indexing the orbits of the general linear group acting on them. They also obtain a necessary condition for the closure of a GL(V )-orbit on such a variety to contain another GL(V )-orbit. This condition comes from the results by C. Riedtmann [Ri] on degenerations of representations of quivers. It is not always clear whether this condition provides a criterion. As is mentioned in [MWZ], this is so in several cases, as follows from some general results on

1


quivers due to K. Bongartz ([B1, §4], [B2, §5.2]). One more case is treated in the paper [M] by P. Magyar, where a similar criterion is obtained for configurations of two flags and a line. Magyar's approach is elementary; it uses only combinatorics and linear algebra. The case X = Gr(k , V ) â Gr(l, V ) â Fld (V ) we are interested in is covered by the results of Bongartz. However, in this case we provide a simpler criterion for a configuration to degenerate to another one, in terms of dimensions of certain subspaces obtained from U , W , and V· by taking sums and intersections, and we give a completely elementary proof of this result. For this, we follow in general the approach of [M]. But the combinatorics we use for indexing the orbits in X is quite different. For a geometric study of orbit closures in X in the particular case d = (1, . . . , n) (that is, when Fld (V ) is the full flag variety; this case we call spherical ), we address the reader to our paper [Sm].

Structure of the pap er. This paper is organised as follows. In Section 2, we
recall some results from [MWZ] concerning classification of orbits in an arbitrary multiple flag variety of finite type. In Section 3, we introduce an indexing of orbits of GL(V ) in X by subsets of vertices of a certain quiver. Section 4 is devoted to defining three partial orders on this set of orbits: the first order is given by degenerations of orbits, the second one is given by conditions on dimensions of certain subspaces, and the definition of the third order is purely combinatorial, involving the description of orbits from Section 3. In Section 5, we discuss the relation of the third order with the "weak order" on spherical varieties in the spherical case. The principal result of this paper states that the first three orders are the same; this is proved in Section 6.

Acknowledgements. I am grateful to Grzegorz Zwara for extremely useful
discussions on Auslander­Reiten quivers, and to Andrei Zelevinsky for drawing my attention to the paper [M]. I also would like to thank Michel Brion for constant attention to this work.

2 Orbits and representations: a general approach
In this section, we consider the problem of classifying orbits of the general linear group in a multiple flag variety in its general setting, after [MWZ]. Let V be an n-dimensional vector space over a field K, which we suppose to be arbitrary throughout this and the next Section. Let Qp,q,r be the three-arm

2


star-like quiver of the following form: ·
/·

·

/·o ·o _dd dd dd d

· ·o

·o ·

·

·

with p + q + r - 2 vertices forming three arms of lengths p, q , and r, and with all arrows leading to the center. Let Rep(Qp,q,r ) denote the category of representations of this quiver. Magyar, Weyman, and Zelevinsky [MWZ] consider the full subcategory I njRep(Qp,q,r ) in Rep(Qp,q,r ) whose ob jects are those representations such that all the linear maps corresponding to the arrows are injections. The subcategory I njRep(Qp,q,r ) is closed under taking direct sums and subob jects (but not quotients!), so one can introduce the notions of decomposition into direct sums and indecomposable ob jects. The uniqueness of a decomposition into a sum of indecomposables is guaranteed by general results due to Kac [Ka]. In particular, the set of indecomposables Ind(I njRep(Qp,q,r )) forms a subset of Ind(Rep(Qp,q,r )), since it is closed under taking subob jects. Fix a dimension vector (a, b, c) = (a1 , . . . , ap ; b1 , . . . , bq ; c1 , . . . , cr ), where ap = bq = cr , and take a representation V = (V1 , . . . , Vp ; V1 , . . . , Vq ; V1 , . . . , Vr ) I njRep(Qp,q,r ) with dimension vector (a, b, c). This representation can be considered as a triple of partial flags in V = Vp = Vq = Vr with the given depths and dimension vectors, defined up to GL(V )-action. And, vice versa, any such triple of flags provides a representation from I njRep(Qp,q,r ). So, the orbits of the diagonal action of GL(V ) on the direct product of three partial flag varieties Fl(
a,b,c)

(V ) = Fla (V ) â Flb (V ) â Flc (V )

are in one-to-one correspondence with the elements of I njRep(Qp,q,r ) with dimension vector (a, b, c). In this category we have the uniqueness of a decomposition into a sum of indecomposables. We also have the following property: there exists at most one indecomposable ob ject with a given dimension vector. This means that the GL(V )orbits in Fl(a,b,c) (V ) correspond to the possible decompositions of the dimension vector (a, b, c): (a, b, c) = dimI , where I are indecomposable ob jects. So, if the number of GL(V )-orbits in Fl(a,b,c) (V ) is finite (in this case this multiple flag variety is said to be of finite type ), the classification of orbits is thus reduced to a purely combinatorial problem. So, knowing all the indecomposable ob jects in the category I njRep(Qp,q,r ) for a given quiver Qp,q,r allows us to describe the GL(V )-orbits in the multiple

3


flag variety Fl(a,b,c) (V ) for an arbitrary dimension vector (a, b, c). The complete list of all multiple flag varieties of finite type and indecomposable ob jects in the corresponding categories is given in [MWZ, Theorem 2.3]. In particular, this list includes quivers Qp,q,1 (type A) and Qp,2,2 (type D). The multiple flag varieties corresponding to these two series of quivers will be the main ob jects of our interest throughout this paper.

3 Combinatorial enumeration of ob jects with a sp ecific dimension vector
Consider the Auslander­Reiten quiver (AR-quiver) for the category Rep(Q). Its vertices correspond to indecomposable ob jects, and arrows represent "minimal" morphisms between indecomposables -- i.e., morphisms f: I I that cannot be presented as a composition of two morphisms f = g h: I I I , where I , I and I are pairwise non-isomorphic indecomposables. Having the AR-quiver for Rep(Q), consider its subquiver defined as follows: we take all vertices that correspond to indecomposable ob jects from I njRep(Q) and all arrows between these vertices. This is the Auslander­Reiten quiver for the category I njRep(Q). We will refer to the latter quiver (not to the former) as to the AR-quiver for the quiver Q; it will be denoted by AR(Q). For background on Auslander­Reiten quivers, see the book [ARS]. Now let us pass to the explicit study of cases A and D.
h g

3.1

Case A: two flags

Let Q equal Qp,q,1 . That is, Q is a linear quiver with p + q - 1 vertices and arrows p oriented as follows: · / · · / · o · · o · All the indecomposable injective representations of this quiver are one-dimensional. They are as follows: Iij = 0 0
/K

K

/Ko

K

Ko

0

0,

where the first nonzero space has number i, the last -- the number p + q - j , and i [1, p], j [1, q ]. So, there are pq non-isomorphic indecomposable ob jects. The AR-quiver for such a quiver is a rectangle of size (p â q ). Let us draw the example where p = 4, q = 3:

4


I43 c c

I42 c cc ? cc
cc c

I41 c cc ? cc

I33 c c

I32 c cc ? cc
cc c

I31 c cc ? cc

I23 c c

I22 c cc ? cc
cc c

I21 c cc ? cc

(A)

I13

I12 ?

I11 ?

Given an ob ject F I njRep(Q), we will say that an indecomposable ob ject I occurs in F , if it occurs with nonzero multiplicity in the decomposition of F into indecomposables. Prop osition 1. Let F be an object in I njRep(Qp,q,1 ) corresponding to a configuration of two flags, such that dimF = (a1 , . . . , ap ; b1 , . . . , bq ), ap = bq = n, and let F= Iij be its decomposition into a sum of indecomposable objects. Then there are n summands. On each path formed by the elements Ii with i fixed, there are exactly ai - ai-1 indecomposable objects, counted with multiplicities, occuring in F . On each path formed by the elements Ij with j fixed, there are exactly bj - bj -1 indecomposable objects occuring in F . (We set formal ly a0 = b0 = 0). Proof. Since all the indecomposable summands are one-dimensional, there are exactly n of them. As we have seen before, dimIij = (0, . . . , 0, 1, . . . , 1, . . . , 1, 0, . . . , 0,).
i-1 entry j -1 entry

The resulting dimension is the sum of dimensions of the indecomposable ob jects occuring in F : dimF = dimIij . Denote the dimension vector of a representation by (a , b ) = (a1 , . . . , ap ; b1 , . . . , bq ). For a given i, the ob jects Iij are characterized by the equality ai = ai-1 + 1. For all other indecomposable ob jects, ai = ai-1 . This means that there are exactly ai - ai-1 ob jects of the form Iij occuring in F . The fact that F contains exactly bj - bj -1 summands of the form Iij for a given j is proved similarly. Corollary 2. Consider the particular case p = q = n, (a, b) = (1, 2 . . . , n; 1, 2 . . . , n). Then for any two summands Iij and Ii j occuring in F , we have i = i and j = j . So, objects with such dimension vector are in one-to-one correspondence with the

5


configurations of n rooks not attacking each other on the chessboard of size n â n, i.e., with the permutations of the set of n elements. In particular, there are n! such non-isomorphic objects. We will see in Section 5 that this description coincides with the well-known indexing of B -orbits in a full flag variety by permutations.

3.2

Case D: two subspaces and a flag

Now let Q be the quiver Dp+2 with all arrows mapping to the center. Having a representation
z zz zz z }zz / Kap bhh hh hh hh

Kb

Ka

1

/ Ka

2

/ ...

Kc we denote its dimension vector by (a1 , . . . , ap ; b; c). Here is the complete list of indecomposable ob jects in I njRep(Q), taken from [MWZ, Theorem 2.3]. There are four one-dimensional series, which we present in the table below together with their dimension vectors: Ii+ Ii- Ii I0i (0, (0, (0, (0, . . . . . . . . . . . . , , , , 0, 0, 0, 0, 1 1 1 1 , , , , . . . . . . . . . . . . , , , , 1; 1; 1; 1; 1; 0; 0; 1; 0) 1) 0) 1)

(all the maps between one-dimensional spaces are nonzero, the dimension jumps at the i-th step, i [1, p]), and one family of the following form:
| }||

K

0

/ ...

/0

/K

/ ...

/K

/ K2

/ ...

/ K2 aff f

K where all the images of the three maps K K2 are distinct (this guarantees indecomposability), and the dimension within the longest arm jumps at the i-th and the j -th steps, i < j . Denote these ob jects by Iij .

6


From the definition of AR(Q) we obtain the following example, where p = 5:
F 2H EQ F 3H E 4Q F 1H QQQ QQQ HHH HHH HHH I - Q I + Q I - HH I + HH I - HH x; 5 ppQQQ x; 4 ppQQQ y< 3 ggHH {= 2 ggHH {= 1 ggHH pp pp g g g { { xx ! ! { ! { # yy # xx I34 q I23 i I5 r I45 r I12 i : < < < I01 i i q rr r$ v: i" i" q# r y y y vvv yy yy yy vv $ I4 r : I35 rr ; I24 ii < I13 ii < I02 i i r$ r r$ w y y " vvv ww yy yy "
+ I5

I

-

I

+

I

-

I

+

(D)

I3 r r$ r

I2 r r$ r

: I25 qq q # vvv

I1

; I15 ii i w ww "

< I14 ii i y " yy



I05

< I04 y yy

< I03 y yy

Indeed, knowing the AR-quiver for Rep(Dp+2 ) with arrows oriented to the center, we restrict ourselves to its vertices corresponding to indecomposable ob jects from I njRep(Dp+2 ). Construction of the AR-quiver for Rep(Q) with Q arbitrary is discussed, for instance, in [ARS, Chap. VII] Notation. The two subsets of vertices of the two top rows connected by the dashed and the dotted line, formed by the ob jects of the form Ii+ and Ii- , are called zigzags. Subsets of vertices of the following form, represented by white circles on the figure below, are said to be roads : · · · o oG HHH G HHH o o oG HHH G HHH HH · oH · HH · HH H ? d H ? d H ? d H ? d H ~d~d~d~d ~~ dH ~~ dH ~~ dH ~~ dH · · · ·d d ~ ? dd ~ ? dd ~ ? dd ~ ? dd ~ ? d ~~ d ~~ d ~~ d ~~ d ~~ d · · · d ~ ? dd ~ ? dd ~ ? dd ~ ? d ~~ d ~~ d ~~ d ~~ · · ·d d ~ ? dd ~ ? dd ~ ? d ~~ d ~~ d ~~
G·H Ho o HHH H ~? · ddH ~ d ~

·d d d

·

?·d ? ~d~ ~~ d ~~

·

They are formed by the ob jects Ii , . . . , Ii,i+1 , Ii+ , Ii- , Ii-1,i , . . . , I0i for a given i. Each road starts on the left edge of the AR-quiver, at an ob ject Ii , goes up, then passes through the "mountain range" formed by two upper rows, bifurcates there and then goes down to the right edge, ending at the ob ject I0i . This road is said to be the i-th one. So, there are exactly 2 different zigzags and p different roads. Prop osition 3. Let F be an object in I njRep(Qp,2,2 ), such that dimF = (a1 , a2 , . . . , ap ; k ; l), and let F = I be its decomposition into a sum of indecomposables. Then:

7


(i) For the i-th road in AR(Qp,2,2 ) there are exactly ai - ai-1 objects occuring in F situated on this road (as before, a0 is set to be equal to 0); (ii) The total number of I of the form Iij , 1 i < j n, and Ii+ , equals k ; (iii) The total number of I of the form Iij , 1 i < j n, and Ii- , equals l. Proof. Fix a road; let Ii be its first element. From the description of indecomposable ob jects given on Page 6, it follows that the dimension vectors (a ; b ; c ) of the indecomposable ob jects situated on this road are characterized by the equality ai = ai-1 + 1. For all other elements, ai = ai-1 . So, F contains exactly ai - ai-1 indecomposable ob jects with dimension jump on the i-th step. This proves the first part of the proposition. (ii) and (iii) are proved similarly. So, an ob ject with dimension vector (a1 , . . . , ap ; k ; l) gives us a set of vertices in AR(Dp+2 ), satisfying the properties (i)­(iii). Obviously, the converse is also true: each set of vertices determines an ob ject, namely, the direct sum of the corresponding indecomposables, and the properties (i)­(iii) guarantee that the dimension vector of this ob ject equals (a1 , . . . , ap ; k ; l).

4

Three orders

Throughout this section, Q is either the quiver Ap+q-1 = Qp,q,1 or the quiver Dp+2 = Qp,2,2 . Recall that throughout the rest of this paper, the ground field K is supposed to be algebraically closed. In this section we present three different ways to turn the set of ob jects F I njRep(Q) with a given dimension vector into a partially ordered set (or shortly poset ). We will show that these three orders are the same in the next section.

4.1

Degeneration order

The first definition uses the bijection between ob jects with dimension vector (a, b, c) and orbits in the corresponding multiple flag variety Fl(a,b,c) (V ). Given an ob ject F , we denote the corresponding orbit by OF . Definition. We say that F is less or equal than F w.r.t. the degeneration order, if there is an inclusion of the corresponding orbit closures (in the Zariski topology): FF
deg



¯ OF OF .

4.2

Rank order

Another partial order is defined by means of dimensions of the homomorphism spaces between ob jects in the category I njRep(Q). For short, for two elements F, G I njRep(Q) we denote the dimension dim Hom(F, G) by F, G .

8


Definition. F is less or equal than F w.r.t. the rank order (notation: F F ), if for each indecomposable ob ject I I njRep(Q) I, F I, F . (NB: the inequality is reversed!) In our cases (Ap+q-1 and Dp+2 ) we shall give a simple geometric interpretation of the numbers I , F . In general, this interpretation also exists (see [MWZ, Prop. 4.1]), but it is not evident at all. Prop osition 4. 1. Let Q and V· = (Vb1 · · · space V . Then for the the fol lowing equalities equal Qp,q,1 , and let V· = (Va1 · · · Vap = V ) Vbq = V ) be two flags of the same depth in a vector object F corresponding to the configuration (V· , V· ) hold: Iij , F = dim Vai V
bj

rk

for each i [1, p], j [1, q ]. (A description of the Iij is given on Page 4.) 2. Let Q equal Qp,2,2 , and let V· = (Va1 · · · Vap = V ), U and W be a flag and two subspaces in V . Then for the object F corresponding to the configuration (U, W, V· ) the fol lowing equalities hold:

Ii , F Ii+ Ii- ,F ,F

= dim V

a

i

= ai ; (1)
a

= dim Vai U ; = dim Vai W ; = dim Vai U W ; = dim V
a
j

I0i , F Iij , F

U W + dim Vai ((V

j

U ) + (V

a

j

W )).

Proof. A first observation: these formulas are additive under taking direct sums of ob jects and componentwise direct sums of corresponding configurations of subspaces. Next, the bracket ·, · is bilinear, so I, F F = I, F + I, F .

Thus, it only suffices to prove these formulas for an indecomposable F . And this is done by a direct verification. Definition. The numbers I , F are called rank numbers.

4.3

Move order

In the previous section we have obtained a combinatorial description of ob jects in I njRep(Q) with a given dimension vector. Ob jects are encoded by subsets of vertices of a certain quiver, satisfying a number of properties.

9


To introduce the third partial order, we define some operations, called elementary moves, that bring these subsets of vertices into other ones. As usual, we begin with type A. In this case the definition of elementary move is quite simple. Take the decomposition of F into indecomposables: F = I . Suppose that among these I 's there are two ob jects Iij and Ii j occuring in F (probably with multiplicities), such that i > i and j > j . Let us also suppose that there is no other Ii j , such that i > i > i and j > j > j . Graphically, this can be reformulated as follows: there is no other vertex occuring in F and situated in the following rectangle: ?·c c · ? c c c
c c c ? c ?·

·

If this is the case, this rectangle is called admissible. Having this, we construct an ob ject F by replacing this pair of indecomposables Iij Ii j with the pair Iij Ii j . This means that the multiplicities multF I of occurences of indecomposable ob jects I in F are obtained from multF I according to the following rule: multF Iij multF Ii
j j

= multF Iij - 1; = multF Ii
j

- 1;

multF Ii

= multF Ii j + 1; = multF Iij - 1; otherwise.

multF Iij

multF I = multF I Ii

j

Informally, can be described as flipping the rectangle, whose "corners" Iij and occuring in F are replaced by Ii j and Iij : · ? c c c c · ?
c c c
ij ? · Iij ? c c cc ? - · c · c c ?

I

Ii


j

· ? ·

c c c

c c c · ?

Iij

Let F be obtained from F by an elementary move. We denote this as follows: F F. Now we are ready to give the definition of the third order. Definition. An ob ject F is said to be less or equal than an ob ject F w.r.t. the move order, if there exists a sequence of ob jects F0 , F1 , . . . , Fs , such that F =F
0

F

1

···

Fs = F .

10


This is denoted as follows: F F . Remark. Of course, each element is less or equal than itself. This corresponds to the empty sequence. So, given two vertices of the AR-quiver, we have at most one possibility to perform an elementary move affecting them. As a result of this move, this pair of vertices is replaced with another pair. In type D everything is more complicated. As above, elementary moves consist in replacing a pair of marked vertices, but now they can be replaced by one, two or three other vertices. Moreover, the choice of an initial pair does not uniquely define the move any more; there may be up to three different possibilities. To begin with, we introduce some convention that allows us to make the description of elementary moves less bulky. Let us add a "fake vertex" in the missing lowest corner, and the corresponding fake indecomposable ob ject I0 , equal to zero. So, the resulting quiver will be as follows:
G·H G·H HHH HHH H H · dH · dH ~~? ddH ~~? ddH ~ ~ · · ·d d ~ ? dd ~ ? d ~~ d ~~ ·d · d ~ ? dd d ~~ d · ·d d ~? d ~~ F· · | = | | f f f! · |= | | f f f! · ·d d d | |= | II II f III f f! · |= | | f f f! · |= | | f f f! >· G·H G·H HHH HHH H H · dH · dH ~~? ddH ~~? ddH ~ ~ dd ~? · dd ~? · d~d~ ~ ~ ? · dd ~? · ~d~ ~ ~~ dd ~? · d~ ~ · ~? ~ ~

mv

·

Now let us describe the moves explicitly. Our general strategy will be as follows: first, we define regions, which are analogues of rectangles in the case An . A region is a triple (A, Init A, Term A), where A is a subquiver in our ARquiver of a certain form, described below. Each A has exactly one source (vertex of incoming degree 0) and one sink (vertex of outcoming degree 0). These two vertices are called initial vertices ; we denote this two-elementary set by Init A. There are also at least one and at most three vertices marked as terminal ones, denoted Term A (they will be defined below in an ad hoc way). Remark. The uniqueness of a source and a sink implies, in particular, that A is connected and that there exists an (oriented) path joining the initial vertices. Now let us describe regions explicitly. We distinguish between the following six cases, denoted I.a)-I.e) and II. The cases I.a)­I.e) are characterized by the following property: A consists of those vertices that are situated on the paths joining the source of A with its sink. I.a) The initial vertices of a region of type I.a) are of the form I1 = Ii j , I2 = Iij , where i < i < j < j . In this case we define an admissible region A of

11


type I.a) as follows: A = {I | i i , j j }. It is a rectangle with corners in I1 and I2 . We define the terminal vertices as the two other corners of this rectangle, Iij and Ii j : A region of this type is shown on the figure. The initial vertices are outlined by squares, the terminal ones -- by circles.
89:; ?>=< Ii jc ? c c · c c ? c c c · ? ? c c · c ? cc c c ·c c c ? 89:; ?>=< Iij

Ii

· ?
j

c c

·c c · Iij ?


·

I.b) The initial vertices of regions of this type are of form I1 = Ii j , I2 = Iij , such that 0 i < j i < j . For each such pair of vertices, there are two regions of type I.b), defined as follows:
+ A+ = A- = {I | i i , j j } {I , I -

|j i}

Each such region has three terminal vertices, defined by
+ Term A+ = {Iij , Ij , Ii- }; - Term A- = {Iij , Ij , Ii+ }.

These two regions are shown on the figures below. A+ :
F·E EEE 89:; E - ?>=
·E I

· üC üü ü VV VV A· ñ ñ ñ ·d d

·

+ 89:; ?>=< Ij G EE EE E EE · EE EE aa E }> YY EE YYE aaE }} } · · ï? ggg ÿ@ ï g! ÿ ÿ ï ÿ ï ;·e ff f! xxxx ee w; · tttt y y< · ww %y ;· f f f! x x xx 89:; ?>=< Iij

` ` `

Iij }> }

12


A- :

89:; ?>=< Ii+ F DD DD = · Y DD D D | | | YYYD üüü | ü Ii j · hh ÿ@ VVVV ÿ h" ÿ ÿ ·e A ee ñ ñ ñ ·d d

I.c) For regions of this type, the initial vertices are of the form I1 = Ii j , I2 = Ii± , such that i < i < j . In this case, we define A to be
+ A = {I | i i , j } {I , I -

G·D EE DDD E - 89:; I · ` EEE ?>= j Y D YD Y ` E }} @· ? · gg ï gg ÿ ÿ ï ÿ ï ï !ÿ · ·f ff xxx; eee !x w; · tttt y y< · ww %y · ff ;· f! x x xx 89:; ?>=< Iij

·E I

` ` `

Iij }> }

| i i } {Ii± },

and Term A = {Ii± , Iij }. ·I ·C ·C Ii I G HI H CCC II CCC CC CC ?>=< II + 89:; I ?Ii i iII }> · XXXCC òB · XXXCC ii } òò " } ·e ; · ff A· A·Y ee xxx Yñ f! ññ Y ññ ñ x · · pp ·Y }> pp |||= Y ññA Yñ }} # · ii · 9· ~ i" tttt qqq# ~ > · pp =· p | p# | | 89:; ?>=< Iij
+

Ii

z= zz
j

hh h!

± I.d) The initial vertices are of the form I1 = Ij , I2 = Iij , and i < j < j . Then ± +- A = {I | i , j < j } {I , I | j j } {Ij }, ± and Term A = {Ij , Iij }. + Ij

EE EE EE EE

· ·C ·F I IF F EE CCC FF EEE FF + C ?>=< E 89:; óB · WWCCC óB · ddFFF z< Ij b EEE dd z b óó W óó b z z ·Y = · ppp >· A·Y YñY Y ññ Y ||| p# }}} ·Y · · Y ñ A fff xxx; eee Y ññ !x ·d · · d www; tttt y y< %y · ff ;· f! x x xx 89:; ?>=< Iij

hh h!

Ii j z z= z

13


I.e) The initial vertices are of the form Ii± and Ii (signs are different), i < i . Then +- A = {I | i < i } {I , I | i < < i } {Ii± , Ii }. Then there is a unique terminal vertex: Term A = {Iii }. Ii+ · H II II · II > ~ ~ ·d d
II F II II pppI x; # xx ; · ppp xxx # · gg {= g! { { 89:; ?>=< Iii

·F

·

FF · dFFF d · ~> ~

F

Ii-

I I. In this case, the initial vertices are of the form Iij and Ii j , where i < j < i < j . The corresponding subquiver A is given by
+ A = {I | i i , j j } {I , I -

| j i },

Ii j and Iij are its terminal vertices:
E ffP ||= ! | Ii j ?·b d b d ïï bb d ï ï ·b bb ïï? ï 89:; ?>=< Ii j E = z zz

·P PP · fPPP

·F

FF · YFF YF Y · æC ææ æ ·V VV V

One can think of the obtained set of vertices as a "folded rectangle", with corners in the initial and the terminal vertices. After having defined regions, we can go further and pass to the definition of the move order. For the following definition, we fix an ob ject F I njRep(Qp,2,2 ). Definition. A region A is called admissible w.r.t. an ob ject F , if for both initial vertices of A, the corresponding indecomposable ob jects occur in F with nonzero multiplicities. An admissible region A is called minimal, if any non-initial vertex from A occurs in F with multiplicity 0. As in the case A, elementary moves that can be performed with an ob ject F correspond to the minimal admissible regions: Definition. We say that F is obtained from F by an elementary move (notation: F F , if there is a minimal admissible region A w.r.t. F , such that multF I = multF I - 1 multF I = multF I for I Init A; otherwise.

· E PP · PPP P |= fffP | | ! b ? b b ïï b ï ï · ï? b b ï bb ï ï ·b @ bb b ïï ïï 89:; ?>=< Iij

·` ` ` ·

Iij ð@ ð ð

multF I = multF I + 1 for I Term A;

14


This means that, as a result of an elementary move, a pair of indecomposable ob jects is replaced by one, two or three other indecomposable ob jects. Now the move order is defined as follows: F is said to be less or equal than F (notation: F F ), if F is obtained from F by a sequence of elementary moves.
mv

5 The spherical case, B -orbits in Gr(k , V ) â Gr(l, V ), and weak order
Throughout this section, we let the dimension vector a be (1, 2, . . . , n), so Fla (V ) equals the full flag variety Fl(V ). Instead of studying orbits of GL(V ) acting on X = Gr(k , V ) â Gr(l, V ) â Fl(V ), one can consider the stabilizer B GL(V ) of a complete flag V· Fl(V ) (so that B is a Borel subgroup of GL(V )), and the orbits of B acting diagonally on Y = Gr(k , V ) â Gr(l, V ). There is an evident bijection between these two sets of orbits, that also respects the degeneration order. So, Y is a GL(V )-variety containing finitely many B -orbits. For an orbit O in X , denote by OY the corresponding orbit in Y . Consider an arbitrary GL(V )-variety Z with a finite number of B -orbits (for an arbitrary connected reductive algebraic group G, such varieties are called spherical ). The set of its orbits admits, along with the usual degeneration order given by ¯ O1 O2 O1 O2 , another partial order structure, called the weak order. It was first introduced in [RS] for symmetric spaces, and in [Kn] for spherical varieties. For its definition, we shall use the minimal parabolic subgroups in GL(V ), that is, minimal subgroups containing B . There are n - 1 of them; they are of the form Pi = StabGL( where V· is the partial flag standard flag V· by omitting It is interesting to know another orbit closure by the
(i) V) deg

V

(i) ·

,

V1 · · · Vi-1 Vi+1 Vn = V , obtained from the the i-th term. when the closure of an orbit in Y is obtained from action of a minimal parabolic subgroup: O Y = Pi · O Y . (2)

(we suppose that OY = OY ; in this case dim OY = dim OY + 1). The following proposition shows that this relation corresponds to elementary moves with certain properties. Prop osition 5. The equality (2) holds iff for the objects F and F , corresponding to OY and OY , F F,

15


and, moreover, the corresponding elementary move is of type I.a), I.c), I.d), I.e), or II, and the source and the sink of the corresponding admissible region belong to neighbor roads. The proof of this proposition will be given at the end of Subsection 6.1. Now let us pass to the definition of the weak order. It is similar to the move order, but its "elementary moves" are given by the relation (2). Namely, OY is said to be less or equal than OY , if there exists a sequence (Pi1 , . . . , Pir ) of minimal parabolic subgroups (possibly with repetitions), such that ¯ ¯ OY = Pir . . . Pi1 OY . We denote this as follows: O
Y

OY .
deg

Obviously, if OY OY , then OY OY (this explains the term "weak"). However, for an arbitrary spherical variety Z , the converse is not true. For example, the degeneration order admits a unique maximal element, namely, the open B -orbit, and the weak order admits a maximal element for each G-orbit on Z : the maximal elements for the weak order are those B -orbits that are open in the corresponding G-orbit1 . In particular, Y = Gr(k , V ) â Gr(l, V ) is not GL(V )-homogeneous, so in this case the weak order is strictly weaker than the degeneration one. In our paper [Sm], we describe the weak order on the set of B -orbits in Y and then use this description for constructing desingularizations of their closures.

6

The main result
deg rk mv

Theorem 6. Let Q equal Qp,2,2 . Then for al l F, F I njRep(Q), such that dimF = dimF , FF F F F F.

So, al l the three orders are the same. This is proved in [M] for Q = Qp,q,1 . We follow the same strategy and split the proof into three lemmas, corresponding to [M, Lemmas 5,6,7]. Lemma 7. F F = F F . This will be proved in 6.1 by constructing an explicit degeneration of the larger of the corresponding orbits to the smaller one. Lemma 8. F F = F F . This is a particular case of [Ri, Prop. 2.1]. However, in 6.2 we present an elementary geometric proof of this result.
In general, this is also false for G-homogeneous varieties; an example is provided, for instance, by a full flag variety Fl(V ), where dim V 3.
1

mv

deg

deg

rk

16


Lemma 9. F F = F F . ~ This will be proved in 6.3 as follows: given F F , we find an ob ject F , such rk mv ~ that F F F .
rk

rk

mv

6.1

Move order implies degeneration order

First let us recall the description of "standard" representatives in GL(V )-orbits, taken from [MWZ, Def. 2.8, Prop. 2.9]. As usual, this is described on orbits OI corresponding to indecomposable ob jects I , and then extended via taking direct sums. Let (U, W, V· ) be a triple corresponding to an indecomposable ob ject. This means that V = Vap is of dimension 1 or 2. If dim V = 1, each of U and W is either equal to V or to zero. If I = Iij , 0 < i < j < , then dim V = 2. Let (ei , ej ) be an ordered basis of V , such that Vi = · · · = Vj -1 = ei . Then the triple (U, W, V· ) with U = ej , W = ei + ej is called the standard representative of the orbit OIij . Later on, we will deal with certain deformations of bases in our subspaces. For this, the following notational convention will be useful. Introduce two more "vectors": e0 and e . Set formally e0 = 0 and each linear combination of vectors involving e be also equal to 0. Note that with this convention, the definition of standard representatives for Iij , 0 < i < j < , is extended to the cases of I0i and Ii , so later we will consider these three cases simultaneously. Now we pass to the proof of Lemma 7. Proof of Lemma 7. The main idea is as follows: for any two ob jects F and F , such that F F , we take a specific representative (U, W, V· ) of the orbit OF and present a one-parameter family (U ( ), W ( ), V· ( )) of subspace configurations ( runs over the ground field), such that (U (0), W (0), V· (0)) = (U, W, V· ), and (U ( ), W ( ), V· ( )) OF when = 0. Since F is obtained from F by replacing exactly two indecomposable summands with some other ob ject (consisting of one, two or three indecomposables), and all the other summands in F remain unchanged, we can assume that F consists only of these two ob jects. It turns out to be convenient to take the representative (U, W, V· ) in its standard form, as indicated in the beginning of this subsection. Now consider all the cases listed in Section 4.3. We will consider an initial pair of ob jects depending on numbers i, j, i , j [0, n] {}, where n = dim V ; when we need to speak about linear combinations of vectors involving e0 or e , we follow the convention from the beginning of this subsection. By V· we always denote the flag whose components are spanned by basis vectors {e1 , . . . , en }, such that dim Va- /Va-1 = 1 iff {i, j, i , j }, and 0 otherwise. This flag will always be invariant along the curves we are going to construct: V· ( ) = V· . I.a) F = Iij Ii j , F = Ii j Iij , where i < i < j < j . (U, W ) = ( ej , e
j

, ei + ej , ei + ej ),

17


(U ( ), W ( )) = ( ej , e

j

, ei + ej , ei + ej + ej ).

The triple (U ( ), W ( ), V· ) for each nonzero corresponds to the ob ject F = Ii j Iij , as may be seen by calculating its rank numbers, or by the decomposition of this configuration into a direct sum of two indecomposables. Note that this deformation also works for the case when i = 0 or/and j = . - + I.b) F = Iij Ii j , F = Ii j Ii+ Ij or F = Ii j Ii- Ij where i < j i < j . In the first case the initial configuration (U, W ) = ( ej , ej , ei + ej , ei + ej ), is deformed to (U ( ), W ( )) = ( ej + ei , ej , ei + ej , ei + ej ). and in the second one -- to (U ( ), W ( )) = ( ej , ej , ei + ej + ei , ei + ej ). I.c) F = Iij Ii+ , F = Ii+ Ii j , where i < i < j . (U, W ) = ( ei , ej , ei + ej ), (U ( ), W ( )) = ( ei + ei , ej , ei + ej ). Ii- Similarly, if F = Iij Ii- for i < i < j , this ob ject is transformed to F = Ii j : for the representative (U, W ) = ( ej , ei , ei + ej ) there is a curve (U ( ), W ( )) = ( ej , ei + ei , ei + ej ), having the configuration type F . + + I.d) F = Ii j Ij for i < j < j , and F = Ij Ii j . Similarly, (U, W ) = ( ej , ej , ei + ej ), and (U ( ), W ( )) = ( ej , ej , ei + ej + ej ). For F = Ii
j - - Ij for i < j < j , and F = Ij Ii j , we have

(U, W ) = ( ej , ei + ej , ej ), (U ( ), W ( )) = ( ej + ej , ei + ej , ej ). I.e) F = Ii+ Ii- for i < i, F = Ii i . (U, W ) = ( ei , ei ),

18


and (U ( ), W ( )) = ( ei , ei + ei ). The case F = Ii- Ii+ , F = Ii i for i < i is completely analogous. And here comes the last case: I I. F = Iij Ii j , where 0 i < j < i < j , and F = Ii i Ij j . Then (U, W ) = ( ej , ej , ei + ej , ei + ej ), and (U ( ), W ( )) = ( ej + ei , ej , ei + ej + ei , ei + ej ) So, for all the possible types of elementary moves we constructed curves that are contained in the closure of the "larger" orbit and that intersect the "smaller" orbit in exactly one point. This proves the lemma. Proof of Prop. 5. Each minimal parabolic subgroup may be presented as the closure of the product Pi = Ui- · B , where Ui- = {E + Ei+1,i | K} is a one-dimensional unipotent subgroup consisting of the matrices whose diagonal entries equal 1, and the only nonzero non-diagonal entry, situated in the i + 1-th line and i-th column, equals . For a pair of orbits OY and OY , such that OY = Pi OY , and a representative (U, W ) OY , the action of Ui- gives us the curve Ui- (U, W ) = {(U ( ), W ( )} OY . For a general , the point (U ( ), W ( )) belongs to the orbit OY . We see that, for the canonical representative (U, W, V· ) O X corresponding to OY Y , the curve (U ( ), W ( ), V· ) O is exactly the one that was constructed in the proof of Lemma 7. The corresponding region has its source and sink on the roads beginning at Ii+1, and Ii and is not of type I.b). Conversely, let F F . Suppose that the elementary move transferring F to F is not of type I.b), and that the source and the sink of the corresponding minimal admissible region belong to the roads beginning in Ir and Is respectively, s < r. Then the curve constructed in the proof of Lemma 7 is of the form U ( ) = Ars ( )U ; W ( ) = Ars ( )W ; V· ( ) = V· , where Ars ( ) = E + Ers is again a matrix with one nonzero nondiagonal entry. So, this action is given by the minimal parabolic subgroup Pi iff s = i and r = i+1.

19


6.2

Degeneration order implies rank order

Proof of Lemma 8. According to Proposition 4, it suffices to show that all the inequalities of the form dim Vai U dim Vai W dim Vai U W d; d; d; (3)

dim(((U Vaj ) + (W Vaj )) Vai ) + dim(U W Vaj ) d

define closed conditions on X = Gr(k , V ) â Gr(l, V ) â Fla (V ). For the first three families of inequalities this is clear -- these conditions define closed subvarieties in X cut out by vanishing of certain determinants in the homogeneous coordinates on X . Let us show this for the last family of inequalities. Fix i and j , i < j , and take a configuration of subspaces (U, W, V· ). Now define a linear map ij : (U Vaj ) â (W Vaj ) Vaj /Vai by (u, w) u + w mod Vai . The dimension of its kernel equals dim(((U Vaj ) + (W Vaj )) Vai ) + dim(U W Vaj ). Indeed, dim Ker(ij ) = dim(U Vaj ) + dim(W Vaj ) - rk ij = dim(U Vaj ) + dim(W Vaj ) - dim(((U Vaj ) + (W Vaj ))/Vai ) = dim(U Vaj ) + dim(W Vaj ) - dim((U Vaj ) + (W Vaj ))+ dim(((U Vaj ) + (W Vaj )) Vai ) = dim((U Vaj ) (W Vaj )) + dim(((U Vaj ) + (W Vaj )) Vai ) = dim(U W Vaj ) + dim(((U Vaj ) + (W Vaj )) Vai ). Now let us prove that the condition dim Ker ij d defines a closed condition on X . This will be done as follows. Consider the direct product Y of X and three copies of V = Vn : Y = Gr(k , V ) â Gr(l, V ) â Fla (V ) â V â V â V , and take the subset Zij Y formed by the sixtuples (U, W, V· , x, y , z ) Y satisfying the following conditions: x, y Vaj ; x U; y W; z Vai ; x + y = z (as vectors in V ).

20


- Obviously, Zij is closed in Y . Moreover, Ker ij ij 1 ((U, W, V· )), where is the pro jection Zij X . This means that the condition 3 is equivalent to the condition

ij

dim

-1 ij

((U, W, V· )) d,

and the latter condition is closed on X .

6.3

Rank order implies move order

Let us first establish two general facts about rank numbers. Prop osition 10. The set of rank numbers uniquely defines the corresponding object. Proof. Assume the contrary: let F and F correspond to the same set of rank numbers. This means that I , F = I , F for each indecomposable I . Since the direct sums of ob jects correspond to the sums of their rank numbers, one can consider that no indecomposable ob jects appear in F and F simultaneously. Now take two rightmost ob jects I and I (in the sense of AR-quiver of type D) occuring in F and F . Without loss of generality suppose that I is situated in the same column or to the right of I , and, consequently, (non-strictly) to the right of all indecomposable ob jects appearing in F . This means that I , F = 0. Similarly, I is situated non-strictly to the right of all the indecomposables from F , except for I itself. So I , F = I , I = 1, a contradiction. Prop osition 11. Let A be a region with initial vertices I1 (source) and I2 (sink), and J the sum of the indecomposable objects corresponding to the terminal vertices of A. Then for an arbitrary object F I1 , F + I2 , F J, F . Moreover, if A \ I2 contains no indecomposable subobject of F , the inequality is an equality. Proof. By bilinearity of ·, · , one can assume F to be indecomposable. So, suppose F = I. Let I and I be two neighbor indecomposable ob jects in a horizontal line (that is, Iij and Ii+1,j +1 , or Ii± and Ii+1 ). Also denote by J the sum of the ob jects corresponding to vertices situated on the paths from I to I (J may consist of at most three indecomposable ob jects). With (1) from Page 9, one can see that I , I + I , I J, I , (4)

and the inequality is strict iff I = I . Now, taking the sum of the inequalities (4) over all pairs (I , I ), where both I and I belong to A, we obtain the desired inequality. If all the inequalities (4) are equalities, the latter is equality as well.

21


Next, we need notions of the interior and the nucleus of a region. Definition. Let A be a region. The interior and the nucleus of A (denoted by Int A and Nuc A, respectively) are sets of indecomposable ob jects, defined as follows: Int A = {I |
I Term A

I, I I, I
I Term A

<
I Init A

I , I }; I , I - 2} Int A;
I Init A

Nuc A = {I |

=

A simple verification shows that Int A A and that the difference between I , F ~ and I , F does not exceed 1 for regions of type I.a)-e) and 2 for regions of type II. (So, the nucleus is nonempty only for regions of type II). On the figures below, for a region of each type its nucleus is marked with stars, and the interior is formed by the union of the nucleus with the set of black dots. As before, the initial and terminal vertices are outlined by squares and circles, respectively. I.a) '&%$ !"# ? c c · ? c ? c c ?·c ?·c c ?·c ?· c ?· c '&%$ !"# I.b)
'&%$ !"# G HHH H > dHH ~ d ~ · d d ~> ~ d d HH G ·H HH !"#HHH '&%$ dHH ~ > ddH d ~~ dH · · d~ ~> d ~ > ~ · · d~d ~> d ~ > d ~ · · d d ~> d ~> ~d~ d · d ~> d ~~ '&%$ !"# G ~> · ~ d d HH HH dHH d · ~> ~ d d G ~> · ~ d d · ~> ~

d d · ~> ~

I.c)

~> ~ d d

G H G· !"#HHH > '&%$d H ~> ~d ~~ dH ~ · d d ~> d ~d · · d~ ~> d ~ > ~ · d d ~> d ~d d · d ~> d ~~ '&%$ !"#

HH HH dHH d · ~> ~

G·

· ~> ~

22


I.d) H H
G· HH HH > ~ ~ d d HH HH dHH d · ~> ~ d d G ~> · ~ d d · ~> ~ d d G ·H !"#HHH > '&%$d H ~d ~~ dH · d d ~> ~ · d ~> d ~ ·d · ~> d ~ > ~ ~ d · d ~> d ~~ '&%$ !"# HH HH dHH d · ~> ~ d d HH HH dHH d · ~> ~

d d · ~> ~

I.e) H H
G· HH HH > ~ ~ d d

I I.

~? ~ d d



GH G·H HHH HHH ~? dHH ~? · dHH d d ~ ~ d ?·d d ~ ~ d ~ ? ~ · · d~d ~? d ~ ? d ~ · · d d ~> d ~> ~d~ · d d ~> d ~~ '&%$ !"#

HH G HH dHH ~> · d ~ · d ~> d ~ d · d ~> d ~~ '&%$ !"#

G·



G·H HHH ~? · dH ~ d H d d ~ ? ~ d ~? d ~ · d d ~> ~ d d

Now let us pass to the proof of Lemma 9.
rk

G·H HHH ~? · dHH d ~ · d d ~ ? d ~d · d~ ~? d ~ ? ~ · d d ~> d ~d · d~ ~> d ~ > ~ · d d ~> d ~d d d ~> d ~~ '&%$ !"#

· d ~> d ~ · ·d d ~> ~ · ~> ~

·d d

·

Proof of Lemma 9. Let F and F be two ob jects, such that dimF = dimF and F F . We have I , F I , F for all indecomposables I . For the "fake vertex" I0 we set I0 , F = I0 , F = 0. We begin with the following definition, which will be the last one in this paper. Definition. A region B is said to be dominant w.r.t. F and F , if the following inequalities hold: I, F > I, F I Int B; I, F > I, F +1 I Nuc B.

(Of course, the second set of inequalities is trivial for regions of type I). The following technical lemma is essential for the sequel.

23


Lemma 12. With the notation as above, take a rightmost object I , such that the corresponding rank numbers for F and F differ: I , F > I , F . Then there exists a dominant region B with sink I and an indecomposable object J = I situated in B and occuring in F as a direct summand. Proof. Take a maximal dominant region B with sink I . Assume the contrary: no indecomposable summand of F other than I is situated in B. 1. First suppose that B is of type II, with sink I = Iij and source I = Ii j . We know that i < j < i < j . Since B is maximal dominant, there must exist two ob jects J1 and J2 with the property J1,2 , F = J1,2 , F , such that J1 {Ij | [j, i )} and J2 {I i | (i, j ]} {Ii


| (i , j )} {Ii± }

(otherwise B would be contained in a larger dominant region). According to the position of J2 , three cases can occur: 1a. J1 = Ij , J2 = I i , where [j, i ), (i, j ]. Consider also two ob jects Ii j and I . These four ob jects determine a region of type II: G ·H G· P C ·S ââ SSS HHH PPP â· · ·h > dHH > h PP ââ t: qqSS
? · ? ``` Id @ d ïï ï

WW óó ó J1 e @· e ðð ð

·Y · · Y ñ A YY ? Y ññ Y B· ` B·V ` ææ VV æææ ` æ · ·W J2 B

~ ~

d ~ ~

h" â tt t pp p #

q# = ·Y Y {{ Y I · dd ææB d {= { æ · hh }> h" }} }

·

·

Apply Prop. 11 twice to this region, taking into account that I Int B: Ii j , F = J1 , F + J2 , F - I , F < J1 , F + J2 , F - I , F Ii j , F ,

that gives us a contradiction. This means that this smaller region, and hence B, contain subob jects of F different from I . 1b. J1 = Ij , J2 = Ii , where [j, i ), (i , j ).

24


In this case, we consider the ob jects Ii
D ûû ûûw; ûû w w

j

and I :

·G ·G G G GGG GGG @ · `G @ · `G ð ` ð ` hQ ð ` G ð ` G ð "ð · · · > J2 gg ~? WWW óóB WWW óóB ~ ~ ~ ! ~ ~ ó ó I · · ·e dd óóB WWW óóB WWW ? ee d e z z= z ó ó · ·i ·X · Id XX òòB d ||> iii ||> | ò "| · J1 h ;· h h" wwww ·Q QQ Q · hhQ · and again apply the same Proposition: Ii j , F = J1 , F + J2 , F - I , F < J1 , F + J2 , F

WW W B· óó ó

- I , F

Ii j , F ,

obtaining a contradiction with our assumption. 1c. J1 = Ij , [j, i ), and J2 = Ii± . ± We consider the pair of ob jects (Ii j , I ) and again apply the same procedure (see figure below). ·D ·G J2F G H G GGG DDD FFF ±G D F G ? · c c I d A · Y cF F ~ > dGG ñññ YYDD ~ · ·h ·i · h d~ h" zz< i i" y y< d ~ > y z ·a ·g > · ee e {= g g ÿÿ@ aa { } }} { !ÿ · ·b · ·X XX òòB bb ïï? ò ï · · }> } }

< z zz =· { ee {{ e Id d ïï? ï

d d @· ÿ ÿÿ

J1 e e

·

2. The region B is of type I.a)­I.c). This means that its source I is of the form Iij . The maximality of B implies the existence of at least two ob jects J B, such that J, F = J, F . We distinguish between the following subcases: 2a. There are two such ob jects of the form J1 = Ii j and J2 = Iij , j (i, j ).

25


Then we can consider the ob jects Iij and Ii j :
E z< z z

·I II · eII e

Iij

· }> }
` ` `

·

} d d d A ññ ññ c c c

J2 h h }> h

" · {= { {{ ·c c c c =· {{ {{ ·h h h"

z= zz ii ii " · J1 e y< e yy y · rr r$

eI >· } }} d d d A· ññ ññ c c c · }> }

E < y y y y ii ii " z< z zz d d d

·R RR · i RR i ·

Ii

j

·

·G G GGG @ · `G i R ð ` G i" ð ð` · < ·` ` ð@ y y ` ðð yy hh h h ñ A · YY ñY "ñ · ·V ~? VVV ææC ~ ~ ææ A·Y hh ñ Y h ! ññ Y · < ·` ` ð@ y y ` ðð yy ii i i ð @ · `` ð ` "ð v: · d ~ > · d~ vv · rr ~ r$ ~ > ·

`` ` ñA ñ ñ VV VV A ñ ñ ñ `` ` @ ð ðð

·

C ææ ææ ·Y Y Y @ ð ðð

·Y Y Y

·V VV V ·

A· ñ ñ ñ

and apply Prop. 11 twice, writing Iij , F = J1 , F + J2 , F - Ii j , F < J1 , F This gives us a contradiction. 2b. J1 = Ii j , but for all vertices Iij , where i < j < j , the inequality Iij , F > Iij , F holds. Then, by maximality of B, there exist two vertices J2 = Ii± and J3 = Ii (with different signs), such that Ii± Term B, and J2 , F = J2 , F J3 , F = J3 , F + J2 , F - Ii j , F Iij , F .

Let us take for J3 the leftmost element of form I· situated in B and satisfying the latter equality. If i i , we can consider region C of type I.c) with Init C = {Iij , Ii± } and

26


Term C = {Ii j , Ii± }, see figure: ·D E ·G H GG DDD +G Ii GG B · XXDD c G òò XD y< c ò y y · · ttt y< d ~ > d~ y t% y · ;· xx eee ññA YY ñY xx · · pp Yñ pp }}}> YY ññA # ;· A·Y xe Y ñ x x ee ññ Y ·Y · j p p }}> Y ñ A p} Y ññ # · ii · ~d i" ~ > d ·d · d ~> ~ ·d d
G· G +GGG I G ? i cG G c ii <· i y " yy · }> eee }} ee >· e }}} · }> eee }} ee >· e }}} < · ii i y yy " · ii i y< y "y <· y yy

w; · ww · }> fff } ! Iij =· e e ||| ·f f!

Ii

d d ñA ñ ñ Y Y Y ñA ñ ñ Y Y Y ~> ~

ñA ñ ñ ·Y Y Y ñA ñ ñ

·Y Y Y

·Y Y Y ·

· ñA ñ ñ

·

·

Then we can again apply Prop. 11 and obtain Iij , F = Ii j , F + Ii± , F - Ii+ , F < Ii j , F + Ii± , F - Ii+ , F Iij , F .
ii

2c. If i > i , we consider the region C of type I.b), with Init C = {Iij , I and Term C = {Ii j , Ii± , Ii },
G· G GGG I + G ? i cG G c ii <· i y " yy · }> eee }} @· b b b ïï b ïï ·f |= f f | | ! · ii i y< y "y · ii i"

}

Iij

}

· }> ·

aa a

· ~> ~ Y Y Y B· ææ æ `` ` ·

·D G·H H HHH DDD I - HH · D D H > i eeH òòB XXXD ò }} qq w; · d ~ > · d~ q# ww = · ff A·Y | Y f! ññ Y ñ || ·V ? B· c c c VV ææ c æ · · {= ggg ð @ `` ð ` {{ !ð qq ; · d ~· q# www d ~ > w; · qqq ~ > · d ww #~d ·f = ·Y A· f! Yñ || Y ññ Ii j · ff ñ A ñ !ñ ·

ss ss $ ·i y< i i y i" y y · ff |> VVV f || · < Ii i pp ð @ p ðð x xx " ss s s u u: · $ uu :· uu uu

Again we apply Prop. 11 to this region twice, obtaining Iij , F = Ii j , F + Ii± , F + Ii , F - Ii < Ii j , F + Ii± ,F
i

,F - Ii
i

+ Ii , F

,F

< Iij , F .

27


3. The region B is of type I.d) or I.e). Let its source be situated at the vertex ± I = Ij . The maximality of B means that there exists at least one element Iij , such that Iij , F = Iij , F . Let Iij be the leftmost element with this property. We distinguish between the following subcases: ± ± ± 3a. There exists an element Ij , such that Ij , F = Ij , F , and i j . In this case, take a leftmost such element and consider region C of type I.d), defined ± ± by Init C = {Ij , Iij } and Term C = {Ij , Iij }. It does not contain ob jects occuring in F , so proceed as usual:
± ± Ij , F = Iij , F - Ij , F - Iij , F

< Iij , F a contradiction.
+ Ij

± - Ij , F

- Iij , F

± Ij , F ,

·G H GG GGG GG GG ñA · d GGG d ñ d ñ ·Y · Y ~> Y ~ ~ ·e e

· F DD FQ DD QQQ < · p Q = · Y DD p Y yyy p # Q || Y D y Iij · iii ff ñ A ñ i" x x; x !ñ · < · pp Y pp |||= YY y yy # Iij i ;·f A· i ñ i" xxxx ff! ññ · · tt tt% www; ·

+ Ij

Y Y Y · ñA ñ ñ

± ± ± 3b. For all elements Ij , such that i < j < j , the inequality Ij , F Ij , F is strict, and the element Ii belongs to B. Then we consider C of type I.e), with ± Init C = {Ii , Ij } and Term C = {Iij }, and apply the same method: ± Ij , F = Ii , F - Iij , F < Ii , F + Ij

- Iij , F

± Ij , F .

·E ·F ·C I H H FF FFF EEE CCC FF C - FF ? · cEEE Ii FFF B · X CC d F òò XXC F cc E ~ ? d ò ~ · · ·e > · ee e Y }> YY ñ A } ñ e }} } ñ Iij >·e A·Y e Y e }}} ee ññ Y ñ · ii ·d · ~ i" y y< d ~ > y · ii · ~ i" ~ > ·

Y Y Y · ñA ñ ñ

3c. Here comes the last possibility: the equality of rank numbers holds in Iij , ± but for all vertices I B, = j , the inequality
± ± I , F I , F

28


is strict, and the vertex Ii± does not belong to B. The latter means that B is of type I.d) (not I.e)). Denote its sink by Ii0 j0 . In this case, we claim that region C with Init C = {Ij,j +1 , Ii0 j0 } and Term C = {Ii0 j , Ij +1 } is dominant. Since B is dominant and by the hypothesis of Case 3c, we see that for each ~ Int C, I , F I , F + 1. I So, we have to show that for each vertex from Nuc C, that is, for each vertex of the form I , where j0 < j - 1, the inequality I , F I , F +1

is strict. Let us prove this. Suppose that there exists an ob ject I0 0 , where this inequality is an equality. Then we can apply Prop. 11, in a slightly different way than before: I0 j , F = I
0 0

, F + I0 j , F - Ii < I
0 0



0

,F - Ii

0

,F

+ 1 + I0 j , F

,F

I0 j , F

+ 1,

that yields a contradiction. Here is the corresponding figure: Ij üC âC GGG üü â ââ s9 · eeGG üüü t: ü eG ütttt ââss Ij,j +1 @ · qqq rrr ÿ qq ÿ r# ÿ ÿ # ·e t: ee t ttt · tt t$
+

Ij0

TT · tt TTT tttT $ ; w ww ww · tt ttt $ t: tt
,j +1

·T T

· G }> · } } ` ` ` ` }> } } ·e e

QQ QQ Q qq QQ qqQ # {= { {{ qq qq # w; ww Iij gg !

Ii0

F·E EEE · E { = YYEE {{ Y { c C· c c ææ c æ = ·Y Y {{ Y {{ ·g gg ñ A · g ññ ! · ?
j

ff f

Ii0 j0 y< yy

So, having obtained a dominant region of type II, we proceed as in the case 1. The lemma is proved. Having such a region B, let us take a minimal dominant region in it; that is, a dominant region C satisfying the following properties: 1. The sink of C equals I , and its source occurs in F as a direct summand; 2. C contains no subob jects of F other that its source and its sink (minimality). The properties 1 and 2 imply that such a region C is minimal admissible. So we ~ may perform the elementary move corresponding to C, thus obtaining an ob ject F

29


~ from F . The property of C to be dominant implies that I , F I , F for each ~ indecomposable ob ject I . So, we have found the desired ob ject F , such that F This concludes the proof of Lemma 9. ~ F F .
rk

References
[ARS] [B1] [B2] [Ka] [Kn] M. Auslander, I. Reiten, S. O. SmalÜ. Representation theory of Artin algebras. Cambridge University Press, 1995. K. Bongartz, On degenerations and extensions of finite dimensional modules, Adv. Math. 121 (1996), 245­287. K. Bongartz, Degenerations for representations of tame quivers, Ann. Sci. ´ Ec. Norm. Sup., (4) 28 (1995), 647­668. V. Kac. Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), pp. 57­92. F. Knop, On the set of orbits for a Borel subgroup. Comment. Math. Helv., 70 (2), pp. 285­309, 1995.

[MWZ] P. Magyar, J. Weyman, A. Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 171 (1999), 285­309. [M] [Ri] [RS] [Sm] P. Magyar, Bruhat order for two flags and a line, J. Algebraic Combin. 21 (2005), no. 1, 71­101. C. Riedtmann, Degenerations for representations of quivers with relations, ´ Ann. Sci. Ec. Norm. Sup. (4) 19 (1986), 275­301. R. W. Richardson, T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata, 35 (1990), 389­436. E. Smirnov, Desingularizations of Schubert varieties in double Grassmannians, arXiv:math.AG/0608554, to appear in Funct. Anal. Appl., 2007.

Independent University of Moscow, Bolshoi Vlasievskii per., 11, 119002 Moscow, Russia ` Institut Fourier, 100 rue des Maths, 38400 Saint-Martin d'Heres, France E-mail address: smirnoff@mccme.ru

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