EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TEN: MORE ON FLAG VARIETIES

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1 EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TEN: MORE ON FLAG VARIETIES WILLIAM FULTON NOTES BY DAVE ANDERSON 1 A. Molev has just given a simple, efficient, and positive formula for the structure constants c ν λµ for multiplication in H TGr(k,n), without puzzles [Mol07]: (1) c ν λµ = R (t l+t(α) c(α) t l+t(α) ρ(α)t(α) ). T α Here l = n k, as usual. The rest of the notation is described as follows: The outer sum is over all sequences R : µ = ρ (0) ρ (1) ρ (s) = ν, where s = ν µ, and ρ (i) is a partition obtained from ρ (i 1) by adding one box. Let r i be the row of the box added in ρ (i) ρ (i 1). The inner sum is over all reverse, barred, ν-bounded tableaux T on the shape λ. This means T is a filling of λ using entries from 1,...,k}, weakly decreasing along rows and strictly decreasing down columns. One also chooses s of the entries (or boxes of λ) to be barred ; these entries must be r 1,r 2,...,r s, occurring in this order when the columns of T are read bottom-to-top, left-to-right. Finally, the entries in the jth column of T must be less than or equal to the number of boxes in the jth column of ν (i.e., T(i,j) ν j ). The product is over the boxes α = (i,j) of λ containing an unbarred entry of T. Also, c(α) = j i is the content of α, and ρ(α) is the partition ρ (t), where t is the number of barred boxes occurring before α in the column reading order. Date: April 5,

2 2 10 MORE ON FLAG VARIETIES Example 1.1. For k = l = 3 and λ = µ = (2,1), ν = (3,1,1), there are two sequences R: R 1 : r 1 = 1, r 2 = 3 R 2 : r 1 = 3, r 2 = 1 There is only one tableau for the sequence R 1 : 3 1 t t = t 3 t 1. (ρ = (3,1,1)) For R 2, there are two tableaux: 3 1 t t = t 5 t 2 (ρ = (2,1)) 3 1 t t = t 6 t 4. (ρ = (2,1)) So the rule says c ν λµ = t 6 t 4 + t 5 t 2 + t 3 t 1. Part of the claim is that all terms are positive i.e., ρ(α) T(α) > c(α). The proof is almost the same as that of the original Molev-Sagan rule [Mol-Sag99] (remarkably, since that rule involved non-positive cancellation), together with a combinatorial argument showing that the ν-bounded tableaux pick out the positive (nonzero) terms. Question 1.2. Is there a bijection between the tableaux T in Molev s rule and the Knutson-Tao puzzles? Note the independence of k, and the simple dependence on l: Replacing k by k+h and l by l+m, the coefficient c ν λµ for multiplication in H T Gr(k+ h,n+h+m) is obtained from that for H T Gr(k,n) by replacing t i with t i+m. We ll see a generalization of this kind of stability below. Exercise 1.3. Prove this fact using puzzles: see what happens when you place a 0 at the beginning of each string, or a 1 at the end of each string. 2 In the last lecture, we saw that under the projection f : Fl(C n ) Gr(k,n), the inverse image of Ω λ (F ) is Ω w(λ) (F ), so f σ λ = σ w(λ). (Recall that if I(λ) = i 1 < < i k } and J(λ) = j 1 < < j l }, then w(λ) = j 1 j l i 1 i k..) Replacing k with k + h and l with l + m takes w(λ) to 1 2 m (j 1 + m) (j l + m) (i 1 + m) (i k + m) (n + m + 1) (n + m + h).

3 EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY 3 Note that the last h entries are irrelevant, since they are larger than all the preceding entries. In general, the embedding S n S m+n (which lets S n act on the last n letters in an alphabet of size m + n) takes w to 1 m w, where 1 m w = 12 m (w 1 + m) (w n + m). Molev s stability generalizes as follows. For u,v,w S n, we have σ u σ v = c w uv σ w in H T Fl(Cn ), with c w uv Λ T = Z[t 1,...,t n ]. Proposition 2.1. c 1m w 1 m u,1 m v is obtained from cw uv by mapping t i to t i+m. We need an algebraic lemma: Lemma 2.2 ([Buch-Rim04], Cor. 4). For v S m+n, we have S v (z 1,...,z m,x 1,...,x n z 1,...,z m,y 1,...,y n ) Sw (x = 1,...,x n y 1,...,y n ) if v = 1 m w for some w S n ; 0 otherwise. The proposition follows, since we have S 1 m u(x t) S 1 m v(x t) = c w 1 m u,1 m v S w (x t). Set x i = t i for 1 i m in x = (x 1,...,x m+n ), and apply the lemma. The lemma can be proved geometrically: Proof. Recall from last lecture that S w is characterized by the fact that S w (x y) = [Ω w (ϕ)], for ϕ : E F a general map of flagged vector bundles. Take general line bundles L 1,...,L m, with z i = c 1 (L i ), and let H i = L 1 L i. Then we have a map id ϕ of flagged vector bundles H m E H m F, as in the following diagram: H 1 0 H m 0 H m E 1 H m E n id ϕ H 1 0 H m 0 H m F 1 H m F n. The locus Ω v (id ϕ) is empty unless v = 1 m w, since v(i) i for i m would force rk(h m H m ) < m. For v = 1 m w, the locus is the same as Ω w (ϕ), as can be seen from the diagram D(1 m w): m D(w).

4 4 10 MORE ON FLAG VARIETIES This stability corresponds to the embedding ι : Fl(n) Fl(m+n) which sends L 1 L n to C 1 C m C m L 1 C m L n. We have ι σw if v = 1 σ v = m w; and ι x i = ι t i = xi m if i > m; ti m if i > m; 0 otherwise. The other obvious embedding puts the fixed parts last: j : Fl(n) Fl(n + m) sends L to L 1 L n L n C L n C m = C n+m. The corresponding inclusion S n S n+m is the usual one, with v v. We have j σv if v S σ v = n S n+m ; and j x i = j t i = xi if i m; ti if i m; 0 otherwise. An important property of Schubert polynomials, visible from the second stability above, is that S w (x y) is independent of n, for w S n. Also, they multiply with the same structure constants as the Schubert classes σ w ; more precisely, for u,v S n we have (2) S u (x y) S v (x y) = c w uv (y)s w(x y), where the sum is over w S 2n 1. In fact, it suffices to consider w which are less than (2n 1)(2n 3) 3124 (2n 2) in Bruhat order (to be defined below), and such that w(n) < w(n + 1) <. The first condition must be satisfied, since all the monomials which appear on the LHS divide (x1 n 1 x n 1 ) 2. To see the second condition holds, recall that S w is symmetric in x k and x k+1 iff w(k) < w(k + 1); since x k does not appear S u or S v for k n, the LHS is certainly symmetric in x k and x k+1 for all k n. By the simple stability property, (2) specializes to the corresponding identity in HT Fl(N) for any N n, discarding those S w with w S N. Remark 2.3. If one uses an algebraic proof to see s λ (x t) = S w(λ) (x t), then the degeneracy locus formula for flags implies the formulas of Kempf- Laksov and Thom-Porteous.

5 EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY 5 3 Recall that we write p v for the flag p v = e v(n) e v(n),e v(n 1) (which is in the Schubert variety Ω v (F )). For w S n, let σ w v be the image of σ w under the restriction map to H T (p v) = Λ. Proposition 3.1. σ w v = S w (t v(1),...,t v(n) t 1,...,t n ). Proof. Restricting to p v, the tautological quotient bundle Q p becomes C n / e v(n),...,e v(n+1 p) = e v(1),...,e v(p), so x i t v(i). Example 3.2. We have σ sk v = n i=1 (t v(i) t i ). Note that if u v, there is at least one k such that σ sk u σ sk v : for example, the minimal k such that u(k) v(k) works. As usual, σ w v = 0 unless p v Ω w, i.e., Ω v Ω w. This is one characterization of the Bruhat order on S n. There are many others: One writes w v if, equivalently, (i) Ω v Ω w. (ii) r w (q,p) r v (q,p) for all p and q. (iii) w 1,...,w k } v 1,...,v k } for all k, where the order on subsets is by sorting the elements, and comparing termwise. (iv) There is a chain w = w (0) w (1) w (s) = v, where each step is of the form u u t, with t = (i,j) the transposition exchanging entries in positions i and j, and l(u t) = l(u)+1. That is, u i < u j, and u k does not lie between u i and u j for all i < k < j. (v) There is an expression v = s i1 s il with l = l(v) such that w is given by a subsequence of length l(w). Write u k v if v = u t as in (iv) with t = (i,j) and i k < j. Proposition 3.3. We have σ w w = S w (t w(1),...,t w(n) t 1,...,t n ) = (t w(i) t w(j) ). i<j w(i)>w(j) There are algebraic proofs (cf. [Buch-Rim04]). Geometrically, it is similar to the Grassmann case: Look at the neighborhood U w of p w, and compute the tangent space to Ω o w as in the last lecture; the weights on the normal space to Ω w at p w will be the weights of T pw U w not in T pw Ω o w.

6 6 10 MORE ON FLAG VARIETIES Example 3.4. For w = , the normal space to Ω w is given by the s: The corresponding weights are t 3 t 2, t 4 t 1, t 4 t 3, t 4 t 2, t 6 t 3, t 6 t 2, t 6 t 5, and t 7 t 5. Proposition 3.5 (Equivariant Monk rule). We have σ sk σ w = w k w + σ w+ + (σ sk w )σ w. Proof. As for the Grassmannian case, the only possible σ v appearing on the RHS have v w and l(w) l(v) 1. (One sees this by Poincaré duality, intersecting with σ w0 v.) The sum in the first part of the RHS is the classical Monk rule; see [Ful97] for a proof. The second part is seen by restriction to p w, using the fact that σ w w 0 (and σ w + w = 0). Proposition 3.6. The polynomials c w uv satisfy and are uniquely determined by the following three properties: (i) c w ww = σ w w = (t w(i) t w(j) ); i<j w(i)>w(j) (ii) (σ sk u σ sk v )c u uv = u cuv ; and + v v k + (iii) (σ sk w σ sk u )c w uv = u k u + c w u + v w k w c w uv. Proof. The proof is essentially the same as in the Grassmannian case. To show that (iii) is satisfied, use the Monk rule and associativity: σ sk (σ u σ v ) = c w uv σ s k σ w = w cuvσ w + + c w uv(σ sk w )σ w, w w k + w

7 and EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY 7 (σ sk σ u ) σ v = σ u+ σ v + (σ sk u )σ u σ v u u k + = w cu + v σ w + (σ sk u ) c w uv σ w. u u k + w Equating the coefficients of σ w on the RHS s gives c w uv + (σ sk w )c w uv = w cu + v σ w + (σ sk u )c w uv, w w k u u k + which is (iii). Setting w = v in (iii) gives (ii ) (σ sk v σ sk u )c v uv = v cu + v, u u k + using the fact that c v uv = 0, since v v. Using commutativity (c w uv = cw vu ) and interchanging u and v turns (ii ) into (ii). The uniqueness statement is also almost the same as before. If u = v = w, c w uv is given by (i). If u = w, then c w uv is given by (ii), using induction on l(u) l(v): one starts with v = w 0 and uses the fact that one can always find a k such that σ sk u σ sk v. Finally, if u w, (iii) gives c w uv by induction on l(w) l(v). Remark 3.7. Conditions (i), (ii ), and (iii) also characterize c w uv. Remark 3.8. These conditions also determine the (unknown!) classical coefficients c w uv, as well as the Grassmannian coefficients cν λµ = cw(ν) w(λ) w(µ). References [Buch-Rim04] A. Buch and R. Rimányi, Specializations of Grothendieck polynomials, C. R. Acad. Sci. Paris, Ser. I 339 (2004), 1 4. [Ful97] W. Fulton, Young Tableaux, Cambridge Univ. Press, [Mol07] A. Molev, Littlewood-Richardson polynomials, math.ag/ [Mol-Sag99] A. Molev and B. Sagan, A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc. 351 (1999), no. 11,