Combinatorics of degeneracy loci

Table of contents :
1.1 Degeneracy lo c i ................................................................................................ 1
1.2 Description of the algorithm ......................................................................... 3
1.3 A conjecture for the coefficients c „ ( r) ........................................................ 6
2.1 Paths through the rank diagram .................................................................. 9
2.2 A criterion for factor sequences .................................................................. 12
2.3 An involution of Fom in ................................................................................... 19
2.4 The stronger conjecture ................................................................................ 24
2.5 Proof in a special c a s e ................................................................................... 27
3.1 Introduction ....................................................................................................... 31
3.2 Schubert polynomials ...................................................................................... 33
3.3 Stable Schubert polynomials ......................................................................... 35
3.4 Redundant rank conditions and products of perm utations ................... 39
3.5 Relations to a conjectured Littlewood-Richardson rule .......................... 42

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ABSTRACT Let X be a non-singular variety and



—>■••• —►£ „ a sequence of vector

bundles over X w ith maps between them. A set of rank conditions for this sequence is a collection r = (rt]) of non-negative integers, 0 < i < j < n. The associated quiver variety is the locus Qt (E .) = {x


X | rank(£j(x) —>E j(x)) < r^ Vi < j } .

W ith Fulton we recently found a formula for the cohomology class of £lr(E .) in the cohomology ring of X , when this locus is irreducible and of maximal codimension in X . This formula extends the Thom -Porteous formula, and it is general enough to give new expressions for ail types of Schubert polynomials. O ur formula writes the cohomology class of f2r (£ .) as a linear com bination w ith integer coefficients of products of Schur polynomials: [Qr (£ .)] = 5 ^ c „ ( r ) - £ 0) - . . . -s^n (£„ - £ n_ 0 . T he sum is over all sequences of partitions n = (n 1 , . . . , fin). The coefficients cM(r) are given by an explicit algorithm . Surprisingly, these coefficients ail seem to be non­ negative. We have conjectured a generalized Littlewood-Richardson rule saying th a t each coefficient is equal to th e num ber of sequences of sem istandard Young tableaux satisfying certain properties. In the first half of my thesis I will prove this conjecture in the special case where the sequence £ . contains four vector bundles. I will also pose a stronger bu t sim pler conjecture, and prove th a t it implies the generalized Littlewood-Richardson rule. In contrast to the Littlewood-Richardson rule, this stronger conjecture is easy to verify on a com puter, and this has been done in 500.000 randomly chosen examples w ith n
+lE t -> Ar‘>+l E j ) . Kj The later definition shows th a t Qr (E .) has a natural stru ctu re of subscheme of X . Let ru denote the rank of the bundle

We will dem and th a t the rank conditions

can occur, i.e. that there exists a sequence of vector spaces and linear m aps Vo -» Vi

Vn so that dim(V'i) = rti and rank (Vi —»> Vj) = rij. This is equivalent to

the conditions rX] < m in(rtJ _ 1; r l+ 1j ) for i < j , and rxj - r , j _ : - r ,+ lj- + r t + l j _i > for j — i >



Let E and F be vector bundles of ranks e and / over X and let / = (aL, . . . , Op) be a sequence of integers. We define th e double Schur polynom ial

S /(F

—E ) as follows.


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2 Let hk be the coefficient of the term of degree k in the formal power series expansion of the quotient of to tal C hem polynomials for the duals of E and F : V ^


. fc

q ( F v) ct (F v )

- Cl(E )t + c2(E)t'2 - - - - + ( - l ) ' c e(E )te i - Cl( F ) t + c2 ( F ) i 2 - . . . + ( - l ) / C/( F ) t / 1

Then the polynom ial s / ( F —E ) is the determ inant of the p x p m atrix whose (i, j ) th entry' is s/(F

E ) — det(/ia, +j_i)x

r ll r 0l



■■ ■

r 22


E}n Tn n


r l2 r n -2 ,n

r 02

T'On In this diagram we replace each sm all triangle of numbers riJ - 1

r i + 1J r ij

bv a rectangle Rij w ith r 1+lj —r tJ rows and rt^ _ i —t 13 columns.

r i J —1

r ij

These rectangles £ire then arranged in a rectangle diagram: Roi

R 12 £ 0 2

••• • • •

£71—1.71 £ r i — 2 ,7 1

Ron It turns out th a t the inform ation carried by the rank conditions is very well repre­ sented in this diagram . F irst, the expected codimension d(r) for th e locus Qr (£ .) is equal to the to tal num ber of boxes in the rectangle diagram. Furtherm ore, the con­ dition th at the rank conditions can occur is equivalent to saying th a t the rectangles get narrower when one travels south-w est, while they get shorter when one travels south-east. Finally, the element Pr depends only on the rectangle diagram . We will define Pr € A®" by induction on n. W hen n =


(corresponding to

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a sequence of two vector bundles), the rectangle diagram has only one rectangle R = R qi. In this case we set Pr = s R e A® 1 where R is identified w ith the partition for which it is the Young diagram . This case recovers the Giambelli-Thom -Porteous formula. If n > 2 we let f denote the bottom n rows of the rank diagram . Then f is a valid set of rank conditions, so by induction we can assume th at ( 1.2 )

is a well defined element of A®” l . Now PT is obtained from Pf by replacing each basis element sfll ® • • • 0 slin_l in ( 1 .2 ) with the sum

This sum is over all partitions eri,. . . , cr„_t and r l , . . . , r n_! such th a t a x has fewer rows than R t - ij and the Littlewood-Richardson coefficient c£ r . is non-zero. A diagram consisting of a rectangle R i - ij w ith (the Young diagram of) a p artitio n a , attached to its right side, and

attached beneath should be interpreted as the sequence of

integers giving the num ber of boxes in each row of this diagram. It can happen th a t the rectangle

is empty, since the num ber of rows or

columns can be zero. If the number of rows is zero, then cr, is required to be empty, and the diagram is the Young diagram of t^ i . If the number of columns is zero, then the algorithm requires th a t the length of Oi is a t most equal to the num ber of rows ra —

of R i-i,i, and the diagram consists of Oi in the top

rows amd r,_!

below this, possibly w ith some zero-length rows in between.

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A conjecture for the coefficients c^(r)

Finally we will describe the conjectured formula for the coefficients cM(r). We will need the notions of (sem istandard) Young tableaux and m ultiplication of tableaux, see for example [6 ]. A tableau diagram, for a set of rank conditions is a filling of all the boxes in the corresponding rectangle diagram with integers, such th at each rectangle Rij becomes a tableau Ty. Furtherm ore, it is required th at the entries of each tableau TtJ are strictly larger than the entries in tableaux above Ty in the diagram, w ithin 45 degree angles. These are the tableaux Tkl w ith i < k < I < j and (k ,l) # (i, j) . A factor sequence for a tableau diagram with n rows is a sequence of tableaux (W*!,. . . , Wn). which is obtained as follows: If n = 1 then the only factor sequence is the sequence (Toi) containing the only tableau in the diagram. W hen n > 2, a factor sequence is obtained by first constructing a factor sequence ( L \ , . . . . Un- 1) for the bottom n — 1 rows of the tableau diagram , and choosing arbitrary factorizations of the tableaux in this sequence: l \ = P t ■Qi ■

Then the sequence ( W \ , . . . , Wn ) = (Toi • P i , Q i • T12 ■P2 , . ■■, Q n —i • T „_lin)

is the factor sequence for the whole tableau diagram. The conjecture from [1], which is the theme of C hapter


can now be stated as follows:

C o n je c tu re 1.1. The coefficient cM(r) is equal to the number o f different factor se­ quences ( W i,. . . , W'n) fo r any fixed tableau diagram fo r the rank conditions r, such that Wi has shape iix fo r each i. This conjecture first of all implies th a t the coefficients cM(r) are non-negative and that they are independent of the side lengths of em pty rectangles in the rectangle

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7 diagram . In addition it implies th a t the number of factor sequences does only depend on the rectangle diagram and not on the choice of a filling of its boxes w ith integers. E x a m p le 1.2. Suppose we are given a sequence of four vector bundles and the fol­ lowing rank conditions: So 1

—^ E i

—^ S 2

—► S 3



4 1




1 0

These rank conditions then give the following rectangle diagram:


□ □ From the bottom row of this diagram we get

= sn ■ T hen using the algorithm we obtain Pf = Sp ® Sp


1 ® Sj

and Pr = s p

denote the transitive closure of the relation [=. This notation depends on the choice of T, as well as the numbers a and


if T is empty.

L e m m a 2 .4 . Let W be a tableau containing T in its upper-left com er. Suppose that the entries o f T are smaller than all other entries in W . I f W = Q - T ’ P i s a simple factorization of W with respect to the rectangle {b)a, and if W = X ■T ■Y is any factorization, then (X, Y ) —* ( Q, P) . Proof. Let X Y' = X

= X Q • X be the vertical cut through X after column b. and put

Y . Then let Y ' = Y ' • Y0' be the horizontal cut through Y 1 after row a, and

put X " = Xo • Y'. We claim th at the pair (X ", F0') fits around T . Using Lemma 2.2 and th a t the entries of T are smaller than all other entries, it is enough to prove th a t the b + j th entry in the top row of X " is strictly larger than the j th entry in the bottom row of Y q. This will follow if the b + j th entry in the top row of X " is larger th an or equal to the j th entry in the top row of Y ' . Since X " = X 0 • Y ' and X 0 has a t most b columns, this follows from an easy induction on the number of rows of Y '. It follows from the claim th a t W = X " ■T ■F0' is the canonical factorization of W , and therefore we have (X, Y ) \= (X 0, Y ') |= (X ", Yq) [= (Q, P ) as required. Notice th a t if W = X • T ■Y is a simple factorization and (X, Y )

(X ', Y' ), then

W — X ' - T - Y ' must also be a simple factorization. It follows th a t Lemma 2.4 would be false w ithout the requirement th at W = Q • T ■P is simple.

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17 L e m m a 2 .5 . Let a > 0 be an integer, and let Y and S be tableaux with product A = Y ■S . Let A = A ■A 0 and Y = Y -Y Q be the horizontal cuts through A and Y after row a, and let Y = M ■N be any factorization. Then N -Yq ■S = A ' • .4o fo r some tableau A!, and M ■A' = A . Yn Y =

A = Y ■S =


Proof. The first statem ent follows from the observation th a t the bottom rows of Y can ’t influence the top p art of Y • S, which is a consequence of the row bumping Lemma 2.2 then shows th a t the factorization .4 = (M ■A ') • .40 is a

algorithm .

horizontal cut, so M • A' = A as required. Lem m a

2 .6

. Let


be a path through the rank diagram, and let ( . . . , A, B ■C , . . . ) be

a factor sequence fo r


such that the product B ■C is the label o f a down-going line

segment. Then ( . . . . A - B . C . . . . ) is also a fa cto r sequence fo r




Proof. We will first consider the case where th e line segment corresponding to .4 goes up. Let


be the path under


th a t cuts short this line segment and its successor. B C

.4 A B C Then by definition (

A ■B ■C , . . . ) is a factor sequence for 7 ', which m eans that

( . . . , .4 • B , C , . . . ) is a factor sequence for


. In general


lies over a p ath like the one

above, and the general case follows from this.

Similarly one can prove th a t if ( . . . , A • B , C , . . . ) is a factor sequence for a path, such th a t A ■B is the label of an up-going line segment, then ( . . . , A , B - C , . . . ) is also a factor sequence for this path.

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18 P r o p o s iti o n 2 .7 . Let




be paths related as in Case 2 o f Figure 2.2, and

let ( . . . , W . . . . ) be a factor sequence fo r


such that W is the label o f the displayed

horizontal line segment. W

I f W = Q ■T ■P is any simple factorization of W , then ( . . . , Q, P , . . . ) is a factor sequence fo r

7 '.

Proof. Since ( . . . , W , . . . ) is a factor sequence for


, there exists a factorization W —

X ■T ■Y such th a t ( . . . , X , Y , . . . ) is a factor sequence for

7 '.

By Lemma 2.4 we

have { X . Y ) —> (Q , P ). It is therefore enough to show th a t if { X . Y ) )= ( X ' . Y 1) then ( . . . . X ' , Y ' , . . . ) is a factor sequence for 7 '. Let a be the num ber of rows in (the rectangle corresponding to) T, and let Y = Y ■Yq be the horizontal cut through Y after the a th row. We will do the case where a factor of Y is moved to X , the other case is proved using a symmetric argum ent. We then have a factorization Y = M • N such th a t X ' = X ■M and Y ' = N ■lo- We can assume th a t the paths factor sequence for





go down after they meet, and th a t the original

is ( . . . , W, S , . . . ) . W

P u t A = Y - S . Then ( . . . , X , A , . . . ) is a factor sequence for the path with these labels in the picture. Now let T ' be the rectangular tableau associated to the lower triangle, and let A = U ■T' • V be the canonical factorization of A. Since this is a simple factorization we may assume by induction th a t ( . . . , X , U, V, . . . ) is a factor sequence. Using Lem m a 2.5 we deduce th a t N - Y q - S — U' - T ' - V for some tableau U' , such th a t M -U ' = U. Since ( . . . , X , M -U', V , . . . ) is a factor sequence, so is ( . . . , X - M , U', V, . . . ) by Lem m a 2 .6 . This means th a t ( . . . , X ■M, U' ■T' • V,

, X ' , Y ' • S , . . . ) is

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19 a factor sequence, which in turn implies th a t ( . . . , X ', Y ' , S , . . . ) is a factor sequence for Y as required.

The proof of Proposition 2.7 also gives the following: C o ro lla ry

2 .8

. Let ( . . . , X , Y , . . . ) be a factor sequence fo r the path 71 in the propo­

sition. I f (X , Y ) —> (X ', Y 1) then ( . . . , X

V ",. • •)

a factor sequence fo r y .

P roo/ 0/ Theorem 2.3. The “if” im plication follows from the definition. If the se­ quence ( W i , . . . , W n) is a factor sequence, then n applications of Proposition 2.7 shows that (Q0, Pi, Q lt P2, . . . , Qn- i >Pn) is a factor sequence for the path with these labels.

Pi • Qi

P 2 • £?2

It follows th a t Qq and P„ are empty, and { P i - Q i ,..., Pn- 1 -