Affine Hecke algebras and the Schubert calculus

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1 Affine Hecke algebras and the Schubert calculus Stephen Griffeth Department of Mathematics University of Wisconsin Madison Madison WI USA Arun Ram Department of Mathematics University of Wisconsin Madison Madison WI USA Dedicated to Alain Lascoux 0. Introduction Using a combinatorial approach which avoids geometry this paper studies the ring structure of K T (G/B) the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data G B T is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G a Borel subgroup B and a maximal torus T and K T (G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. Because of the T-equivariance the ring K T (G/B) is an R-algebra where R is the representation ring of T. As explained by Grothendieck [Gd] the ring K T (G/B) has a natural R-basis {[O Xw ] w W } where W is the Weyl group and O Xw is the structure sheaf of the Schubert variety X w G/B. One of the main problems in the field is to understand the structure constants of the ring K T (G/B) with this basis that is the coeffients c z wv in the equations [O Xw ][O Xv ] = z W c z wv [O X z ]. (0.1) Our approach is to work completely combinatorially and define K T (G/B) as a quotient of the affine nil-hecke algebra. The fact that the combinatorial approach coincides with the geometric one is a consequence of the results of Kostant and Kumar [KK] and Demazure [D]. In the combinatorial literature the elements [O Xw ] are often called (double) Grothendieck polynomials. Let P be the weight lattice of G and for λ P let [X λ ] be the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. The theorem of Pittie [P] says that Research partially supported by the National Science Foundation (DMS ) and the National Security Agency (MDA ). Keywords: flag variety K-theory affine Hecke algebras Schubert varieties.

2 2 s. griffeth and a. ram the ring K T (G/B) is generated by the [X λ ] λ P. Steinberg [St] strengthened this result by displaying specific [X λ w ] w W which form an R-basis of K T (G/B). These results are often collectively known as the Pittie-Steinberg theorem. The theorems which we prove in Section 2 are simply different points of view on the Pittie- Steinberg theorem. Though we are not aware of any reference which states these theorems in the generality which we consider these theorems should be considered well known. Let s 1... s n be the simple reflections in W (determined by the data (G B T)) let w 0 be the longest element of W and let P + be the set of dominant weights in P. The Schubert varieties X w0 s i are the codimension one Schubert varieties in G/B. In section 3 we prove Pieri-Chevalley formulas for the products [X λ ][O Xw ] [X λ ][O Xw ] [X w 0λ ][O Xw ] and [O Xw0 s i ][O Xw ] (0.2) for λ P + w W and 1 i n. All of these Pieri-Chevalley formulas are given in terms of the combinatorics of the Littelmann path model [Li1-3]. The formula which we give for the first product in (0.2) is due to Pittie and Ram [PR1]. In this paper we provide more details of proof than appeared in [PR1]. The other formulas for the products in (0.2) follow by applying the duality theorem of Brion [Br Theorem 4] to the first formula. However here we give an independent combinatorial proof and deduce Brion s result as a consequence. The last formula is a consequence of the nice formula [O Xw0 s i ] = 1 e w 0ω i [X ω i ] (0.3) which is an easy consequence of the first two Pieri-Chevalley rules. It is not difficult to specialize product formulas for K T (G/B) to corresponding product formulas for K(G/B) H T (G/B) and H (G/B) (by using the Chern character and comparing lowest degree terms and ignoring the T-action). Thus the products which are computed in this paper also give results for ordinary Grothendieck polynomials double Schubert polynomials and ordinary Schubert polynomials. In section 4 we explain how to do these conversions. For most of these cases the specialized versions of our Pieri-Chevalley rules are already very well known (see for example [Ch]). In Section 5 we give explicitly (a) two different kinds of formulas for [O Xw ] in terms of X λ and (b) complete computations of the products in (0.1) for the rank two root systems. This data allows us to make a positivity conjecture for the coefficients c z wv in (0.1). This conjecture generalizes the theorems of Brion [Br formula before Theorem 1] and Graham [Gr Corollary 4.1] which treat the cases K(G/B) and HT (G/B) respectively. Acknowledgement. It is a pleasure to thank Alain Lascoux for setting the foundations of the subject of this paper. Our approach is heavily influenced by his teachings. In particular he has always promoted the study of the flag variety by divided difference operators (the affine or graded nil-hecke algebra) it is his work with Fulton in [FL] that provided the motivation for the Pieri- Chevalley rules as we present them and it his idea of transition (see for example the beautiful paper [La]) which allows us to obtain product formulas for Schubert classes in the form which we have given in Section 5 of this paper.

3 Fix the following data and notation: Schubert calculus 3 1. Preliminaries h is a real vector space of dimension n R is a reduced irreducible root system in h R + is a set of positive roots in R W is the Weyl group of R s 1... s n are the simple reflections in W m ij is the order of s i s j in W i j R(w) = {α R + wα R + } is the inversion set of w W l(w) = Card(R(w)) is the length of w W is the Bruhat-Chevalley order on W α 1... α n are the simple roots in R + ω 1... ω n are the fundamental weights P = n i=1 Zω i is the weight lattice P + = n i=1 Z 0ω i is the set of dominant integral weights. For a brief easy introduction to root systems with lots of pictures for visualization see [NR]. By [Bou VI 1 no. 6 Cor. 2 to Prop. 17] if w = s i1 s ip be a reduced word for w then R(w) = {α ip s ip α ip 1... s ip s i2 α i1 } (1.1) The affine nil-hecke algebra is the algebra H given by generators T 1... T n and X λ λ P with relations Ti 2 = T i T i T j T i = T j T i T j X λ X µ = X λ+µ (1.2) }{{}}{{} m ij factors m ij factors and Let T w = T i1 T ip for a reduced word w = s i1 s ip. Then are bases of H. Both the nil-hecke algebra X λ T i = T i X siλ + Xλ X s iλ 1 X α. (1.3) i {X λ T w w Wλ P } and {T w X λ w Wλ P } (1.4) H = Z-span{T w w W } and Z[X] = Z-span{X λ λ P } (1.5) are subalgebras of H. The action of W on Z[X] is given by defining wx λ = X wλ for w W λ P (1.6) and extending linearly. The proof of the following theorem is given in [R Theorem 1.13 and Theorem 1.17]. The first statement of the theorem is due to Bernstein Zelevinsky and Lusztig [Lu 8.1] and the second statement is due to Steinberg [St] and is known as the Pittie-Steinberg theorem.

4 4 s. griffeth and a. ram Theorem 1.7. Define λ w = w 1 s i w<w ω i for w W. (1.8) The center of H is Z( H) = Z[X] W and each element f Z[X] has a unique expansion f = w W f w X λ w with f w Z[X] W. (1.9) Let ε i = 1 T i and let ε w = ε i1 ε ip for a reduced word w = s i1 s ip. Then ε w is well defined and independent of the reduced word for w since The second equality is a consequence of the formulas ε 2 i = ε i and ε i ε j ε i = ε j ε i ε j. (1.10) }{{}}{{} m ij factors m ij factors ε w = ( 1) l(v) T v and T w = v w v w( 1) l(v) ε v (1.11) which are straightforward to verify by induction on the length of w. 2. The ring K T (G/B) Let H and Z[X] be as in (1.5). The trivial representation of H is defined by the homomorphism 1:H Z given by 1(T i ) = 1. The first of the maps Z[X] HT w0 H H 1 f ft w0 f 1 is an H-module isomorphism if the action of H on Z[X] is given by The group algebra of P is T i f = Xα i f s i f X α i 1 for f Z[X]. (2.1) R = Z-span{e λ λ P } with e λ e µ = e λ+µ (2.2) for λµ P. Extend coefficients to R so that H R = R Z H and R[X] = R Z Z[X] are R-algebras. Define K T (G/B) to be the H R -module K T (G/B) = R-span{[O Xw ] w W } (2.3) so that the [O Xw ] w W are an R-basis of K T (G/B) with H R -action given by X λ [O X1 ] = e λ [O X1 ] and T i [O Xw ] = { [OXwsi ] if ws i > w [O Xw ] if ws i < w. (2.4)

5 If R is an R[X]-module via the R-algebra homomorphism given by Schubert calculus 5 e: R[X] R X λ e λ (2.5) then K T (G/B) = H R R[X] R as H R -modules. Let Q be the field of fractions of R and let Q be the algebraic closure of Q. For w W let b w in Q R K T (G/B) be determined by X λ b w = e wλ b w for λ P. (2.6) If the b w exist then they are a Q-basis of Q R K T (G/B) since they are eigenvectors with distinct eigenvalues. If τ i 1 i n are the operators on Q R K T (G/B) given by τ i = T i 1 1 X α i then b 1 = [O X1 ] and τ i b w = b wsi for ws i > w (2.7) because a direct computation with relation (1.3) gives that X λ τ i b w = τ i X s iλ b w = τ i e ws iλ b w = e ws iλ b wsi. Thus the b w w W exist and the form of the τ-operators shows that in fact they form a Q-basis of Q R K T (G/B) (it was not really necessary to extend coefficients all the way to Q). Equations (2.6) and (2.7) force τ i τ j τ i }{{} m ij factors = τ j τ i τ j }{{} and the equality τi 2 = m ij factors 1 (X α i 1)(X α i 1) is checked by direct computation using (1.3). Let τ w = τ i1 τ ip for a reduced word w = s i1 s ip. Then for w W b w = τ w 1b 1 [O Xw ] = T w 1[O X1 ] and we define [I Xw ] = ε w 1[O X1 ] (2.8) where ε w is as in (1.11). In terms of geometry [O Xw ] is the class of the structure sheaf of the Schubert variety X w in G/B and up to a sign [I Xw ] is class of the sheaf I Xw determined by the exact sequence 0 I Xw O Xw O Xw 0 where X w = v<w BvB (see [Ma Theorem 2.1(ii)] and [LS equation (4)]. We are not aware of a good geometric characterization of the basis {[X λ w ] w W } of K T (G/B) which appears in the following theorem. Theorem 2.9. Let λ w w W be as defined in Theorem 2.9 and let [X λ ] = X λ [O Xw0 ] = X λ T w0 [O X1 ] for λ P. Then the [X λ w ] w W form an R-basis of K T (G/B). Proof. Up to constant multiples [O Xw0 ] = T w0 [O X1 ] is determined by the property T i [O Xw0 ] = [O Xw0 ] for all 1 i n. (2.10) If constants c w Q are given by [O Xw0 ] = w W c w b w then comparing coefficients of b wsi for ws i > w on each side of (2.10) yields a recurrence relation for the c w ( ) 1 c w = c wsi 1 e wα for ws i i > w which implies c w0 v 1 = 1 1 e w 0α (2.11) α R(v)

6 6 s. griffeth and a. ram via (1.1) and the fact that c w0 = 1. Thus [X λ v ] = X λ v [O Xw0 ] = c w e wλ v b w w W and if C M and A are the W W matrices given by C = diag(c w ) M = (e wλ v ) and A = (a zw ) where b w = z W a zw [O Xz ] then the transition matrix between the X λ v and the [O Xz ] is the product ACM. By (2.8) and the definition of the τ i the matrix A has determinant 1. Using the method of Steinberg [St] and subtracting row e s αwλ v from row e wλ v in the matrix M allows one to conclude that det(m) is divisible by α R + (1 e α ) W /2 and identifying w W e wλ w = n i=1 s i w<w e ω i = (e ρ W /2 ) as the lowest degree term determines det(m) exactly. Thus det(acm) = 1 w W α R(w) ( 1 1 e α e ρ α R + ( 1 e α )) W /2 = (e ρ ) W /2. Since this is a unit in R the transition matrix between the [O Xw ] and the X λ v is invertible. Theorem The composite map is surjective with kernel Φ: R[X] H R T w0 H R K T (G/B) f ft w0 h h[o X1 ] ker Φ = f e(f) f R[X] W the ideal of the ring R[X] generated by the elements f e(f) for f R[X] W. Hence K T (G/B) = R[X] f e(f) f R[X] W has the structure of a ring. Proof. Since Φ(X λ ) = X λ T w0 [O X1 ] = X λ [O Xw0 ] it follows from Theorem 2.9 that Φ surjective. Thus K T (G/B) = R[X]/ ker Φ. Let I = f e(f) f R[X] W. If f R[X] W then for all λ P Φ(X λ (f e(f))) = X λ (f e(f))t w0 [O X1 ] = X λ T w0 (f e(f))[o X1 ] = X λ T w0 (e(f) e(f))[o X1 ] = 0 since f e(f) Z( H R ). Thus I ker Φ. The ring K T (G/B) = R[X]/ ker Φ is a free R-module of rank W and by Theorem 1.7 so is R[X]/I. Thus ker Φ = I.

7 If c µz wλ Recall that both Schubert calculus 7 3. Pieri-Chevalley formulas {X λ T w 1 λ Pw W } and {T z 1X µ µ Pz W } are bases of H. Z are the entries of the transition matrix between these two bases X λ T w 1 = z Wµ P then applying each side of (3.1) to [O X1 ] gives that [X λ ][O Xw ] = c µz wλ eµ [O Xz ] z Wµ P c µz wλ T z 1Xµ (3.1) in K T (G/B). This is the most general form of Pieri-Chevalley rule. The problem is to determine the coefficients c µz wλ. The path model A path in h is a piecewise linear map p:[01] h such that p(0) = 0. For each 1 i n there are root operators e i and f i (see [L3] Definitions 2.1 and 2.2) which act on the paths. If λ P + the path model for λ is T λ = {f i1 f i2 f il p λ } the set of all paths obtained by applying the root operators to p λ where p λ is the straight path from 0 to λ that is p λ (t) = tλ 0 t 1. Each path p in T λ is a concatenation of segments p = p a 1 w 1 λ pa 2 w 2 λ pa r w r λ with w 1 w 2 w r and a 1 +a 2 + +a r = 1 (3.2) where for v W and a (01] p a vλ is a piece of length a from the straight line path p vλ = vp λ. If W λ = Stab(λ) then the w j should be viewed as cosets in W/W λ and denotes the order on W/W λ inherited from the Bruhat-Chevalley order on W. The total length of p is the same as the total length of p λ which is assumed (or normalized) to be 1. For p T λ let r p(1) = a i w i λ be the endpoint of p i=1 ι(p) = w 1 the initial direction of p and φ(p) = w r the final direction of p. If h T λ is such that e i (h) = 0 then h is the head of its i-string S λ i (h) = {hf ih... f m i h} where m is the smallest positive integer such that fi m h 0 and f m+1 i h = 0. The full path model T λ is the union of its i-strings. The endpoints and the inital and final directions of the paths in the i-string Si λ (h) have the following properties: (f k i h)(1) = h(1) kα i for 0 k m either ι(h) = ι(f i h) = = ι(fi m h) < s i ι(h) or ι(h) < ι(f i h) = = ι(fi m h) = s i ι(h) and (3.3) either or s i φ(fi m h) < φ(h) = = φ(f m 1 i h) = φ(fi m h) s i φ(fi m h) = φ(h) = = φ(f m 1 i h) < φ(fi m h).

8 8 s. griffeth and a. ram The first property is [L2] Lemma 2.1a the second is is [L1] Lemma 5.3 and the last is a result of applying [L2] Lemma 2.1e to [L1] Lemma 5.3. All of these facts are really coming from the explicit form of the action of the root operators on the paths in T λ which is given in [L1] Proposition 4.2. Let λ P + w W and z W/W λ and let p T λ be such that ι(p) ww λ and φ(p) z. Write p in the form (3.2) and let w 1... w r z be the maximal (in Bruhat order) coset representatives of the cosets w 1... w r z such that w w 1 w 2 w r z. (3.4) Theorem 3.5. Recall the notation ε v from (1.11). Let λ P + and let W λ = Stab(λ). Let w W. Then in the affine nil-hecke algebra H X λ T w 1 = p(1) and X λ ε w 1 = ( 1) l(w)+l(z) ε z 1X p(1) T φ(p) 1X p T λ ι(p) ww λ where if W λ {1} then T φ(p) 1 = T w 1 r p T λ ι(p)=w z W/W λ z φ(p) and ε z 1 = ε z 1 with w r and z as in (3.4). Proof. (a) The proof is by induction on l(w). Let w = s i v where s i v > v. Define T λ w = {p T λ ι(p) ww λ }. Assume w = s i v > v. Then the facts in (3.3) imply that (1) T λ w is a union of the strings S i(h) such that h T λ v and (2) If h T λ v then either S i(h) T λ v or S i(h) T λ v = {h}. Using the facts in (3.3) a direct computation with the relation (1.3) establishes that if h T v λ then T φ(p) 1X η(1) = T φ(h) 1X h(1) T i and Thus p S i (h) p S i (h) X λ T w 1 = X λ T v 1T i = = h T λ v e i (h)=0 = h T λ w e i (h)=0 = h T λ w e i (h)=0 = p T λ w { T φ(p) 1X η(1) T φ(h) 1X h(1) T i if S i (h) T v λ = T φ(h) 1X h(1) T i if S i (h) T v λ = {h}. p T λ v S i (h) T λ p S i (h) v S i (h) T λ v S i (h) T λ v T φ(p) 1X p(1). T φ(p) 1X p(1) T i T φ(p) 1X p(1) + T φ(h) 1X h(1) T i + T φ(h) 1X h(1) T i + (by induction) S i (h) T λ v ={h} T φ(h) 1X h(1) S i (h) T λ v ={h} T φ(h) 1X h(1) S i (h) T λ v ={h} p S i (h) T i T φ(p) 1X p(1) T i

9 (b) The proof is similar to case (a). For w W let Schubert calculus 9 T λ =w = {p T λ ι(p) = ww λ }. Assume w = s i v > v. Then the facts in (3.3) imply that (1) T λ =w is a union of the strings S i(h) such that h T λ =h and (2) If h T λ =v then either S i(h) T λ =v or S i(h) T λ =v = {h}. Let E φ(p) = z W/W λ z φ(p) ( 1) l(z) ε z 1. (3.6) Using (3.3) a direct computation with the relation (1.3) establishes that if h T=v λ with e ih = 0 then E φ(p) X p(1) T i = 0 and E φ(h) X h(1) T i = E φ(p) X p(1). p S i (h) p S i (h) {h} Thus X λ ε w 1 = X λ ε v 1ε i = ( 1) l(v) E φ(p) X p(1) T i = ( 1) l(v) = ( 1) l(v) 0 = ( 1) l(w) p T λ =v S i (h) T=v λ p S i (h) p T λ =w E φ(p) X p(1) + S i (h) T=v λ ={h} p S i (h) {h} E φ(p) X p(1). E φ(h) X h(1) T i S i (h) T=v λ ={h} E φ(p) X p(1) Corollary 3.7. Let λµ P + and let w W. Then in the affine nil-hecke algebra H X λ T w 1 = X w 0µ T w 1 = p T w 0 λ φ(p)=ww 0 p T µ φ(p)=ww 0 z W/W w0 λ zw 0 ι(p) z W/Wµ zw 0 φ(p) ( 1) l(w)+l(z) T z 1X p(1) ( 1) l(w)+l(z) T z 1X p(1). and Proof. The second identity is a restatement of the first with a change of variable µ = w 0 λ. The first identity is obtained by applying the algebra involution H H T w ε w and the bijection X λ X λ T λ T w 0λ p p

10 10 s. griffeth and a. ram where p is the same path as p except translated so that its endpoint is at the origin. Representation theoretically this bijection corresponds to the fact that L(λ) = L( w0 λ) if L(λ) is the simple G-module of highest weight λ. Note that p (1) = p(1) ι(p ) = φ(p)w 0 and φ(p ) = ι(p)w 0. Applying the identities from Theorem 3.5 and Corollary 3.7 to [O X1 ] yields the following product formulas in K T (G/B). In particular this gives a combinatorial proof of the (T-equivariant extension) of the duality theorem of Brion [Br Theorem 4]. For λ P and w W let [X λ ] = X λ [O Xw0 ] = X λ T w0 [O X1 ] and let c z λw be given by [X λ ][O Xw ] = z W c z λw [O X z ] (3.8) Corollary 3.9. Let λ P + w W and W λ = Stab(λ). Then with notation as in (3.8) c z λw = e p T λ ww λ ι(p) φ(p)=zw λ p(1) c z w 0 λw = ( 1)l(w)+l(z) c ww 0 λzw 0 and c z λw = ( 1)l(w)+l(z) c ww 0 w 0 λzw 0. Proposition For 1 i n [O Xw0 s i ] = 1 e w 0ω i [X ω i ]. Proof. We shall show that X ω i [O Xw0 ] = e w 0ω i ([O Xw0 ] [O Xw0 s i ]) (3.11) and the result will follow by solving for [O Xsi w 0 ]. Let ω j = w 0 ω i. By Corollary 3.9 c z ω i w 0 = ( 1) l(w 0)+l(z) c 1 ω j zw 0 = ( 1) l(w 0)+l(z) p T ω j zw 0 ι(p) φ(p)=1 e p(1). The straight line path to ω j p ωj has ι zw0 (p ωj ) = φ zw0 (ω j ) and is the unique path in T ω j which may have final direction 1. Suppose φ zw0 (p ωj ) = 1. Then since s j is the only simple reflection which is not in Stab(ω j ) it must be that zw 0 s k for all k j. Thus zw 0 = 1 or zw 0 = s j and so c z ω i w 0 0 only if z = w 0 or z = s j w 0 = w 0 s i. Now (3.11) follows since p ωj has endpoint ω j = w 0 ω i. Corollary Let c z wv be as in (3.8). Then for 1 i n c w w 0 s i w = (e (wω i w 0 ω i ) 1) and c z w 0 s i w = ( 1) l(w)+l(z)+1 e p T w 0 ω i zw 0 ι(p) φ(p)=ww 0 w 0 ω i +p(1) for z w. Proof. This follows from Proposition 3.10 and Corollary 3.9 and the fact that in the case when z = w there is a unique path p with ww 0 = ι(p) = φ(p) = ww 0 and endpoint p(1) = ww 0 ( w 0 ω i ) = wω i.

11 Schubert calculus Converting to H T (G/B) The graded nil-hecke algebra is the algebra H gr given by generators t 1... t n and x λ λ P with relations t 2 i = 0 t it j t i = t j t i t j x λ+µ = x λ + x µ and x λ t i = t i x si λ + λα i. (4.1) }{{}}{{} m ij factors m ij factors The subalgebra of H gr generated by the x λ is the polynomial ring Z[x 1... x n ] where x i = x ωi and W acts on Z[x 1... x n ] by wx λ = x wλ and w(fg) = (wf)(wg) for w W λ P fg Z[x 1... x n ]. Then the last formula in (4.1) generalizes to ft i = t i (s i f) + f s if α i for f Z[x 1... x n ]. Let t w = t i1 t ip for a reduced word w = s i1 s ip and let ZW be the subalgebra of H gr spanned by the t w w W. Then {x m 1 1 x m n n t w w W m i Z 0 } and {t w x m 1 1 x m n n w W m i Z 0 } are bases of H gr. Let S = Z[y 1... y n ] and extend coefficients to S so that H grs = S Z H gr and S[x 1... x n ] = S Z Z[x 1... x n ] are S-algebras. Define H T (G/B) to be the H grs module H T(G/B) = S-span{[X w ] w W } (4.2) so that the [X w ] w W are an S-basis of K T (G/B) with H grs -action given by x i [X 1 ] = y i [X 1 ] and t i [X w ] = { [Xwsi ] if ws i > w 0 if ws i < w (4.3) Let y be the S-algebra homomorphism given by y: S[x 1... x n ] S x i y i so that H T (G/B) = H grs S[x1...x n ] y as H grs -modules Then using analogous methods to the K T (G/B) case proves the following theorem which gives the ring structure of H T(G/B) (see also the proof of [KR Prop. 2.9] for the same argument with (non-nil) graded Hecke algebras). Theorem 4.4. The composite map Φ: S[x 1... x n ] H grs t w0 H grs H T (G/B) f ft w0 h h[x 1 ] is surjective with kernel ker Φ = f y(f) f S[x 1... x n ] W

12 12 s. griffeth and a. ram the ideal of the ring S [ x 1... x n ] generated by the elements f y(f) for f S[x 1... x n ] W. Hence H T(G/B) = Z[y 1... y n x 1... x n ] f y(f) f S[x 1... x n ] W has the structure of a ring. As a vector space H gr = Z[x 1... x n ] ZW gr. Let Ĥgr = Q[[x 1... x n ]] QW gr with multiplication determined by the relations in (4.1). Then Ĥgr is a completion of H gr (this simply allows us to write infinite sums) and the elements of Ĥgr given by ch(x λ ) = r 0 1 x r! xr αi λ and ch(t i ) = t i 1 ch(x α (4.5) i ) satisfy the relations of H and thus ch extends to a ring homomorphism ch: H Ĥgr. It is this fact that really makes possible the transfer from K-theory to cohomomology possible. Though is it not difficult to check that the elements in (3.5) satisfy the defining relations of H it is helpful to realize that these formulas come from geometry. As explained in [PR2] the action of T i on K T (G/B) and the action of t i on HT (G/B) are respectively the push-pull operators π i (π i)! and πi (π i) where if P i is a minimal parabolic subgroup of G then π i :G/P i G/B is the natural surjection. Then the first formula in (3.5) is the definition of the Chern character and the second formula is the Grothedieck-Riemann-Roch theorem applied to the map π i. The factor α i /(1 ch(x α i )) is the Todd class of the bundle of tangents along the fibers of π i (see [Hz page 91]). Then Ĥ T (G/B) Q = Q[[y 1... y n ]] Z[y1...y n ] HT (G/B) is the appropriate completion of HT (G/B) to use to transfer the ring homomorphism ch: H R Ĥgr to a ring homomorphism ch:k T (G/B) Ĥ T (G/B) Q by setting ch(h[o X1 ]) = ch(h)[x 1 ] for h H R. (4.6) The ring Ĥ T (G/B) Q is a graded ring with deg(y i ) = 1 and deg([x w ]) = l(w 0 ) l(w) (4.7) and for w W ch([o Xw ]) = [X w ] + higher degree terms. (4.8) In summary if e i = e ω i X i = X ω i y i = y ωi x i = x ωi R[X] = Z[e ± e±1 n X± X±1 n ] Z[X] = Z[X 1 ±1... X±1 n ] and Ŝ[x 1... x n ] = Q[[y 1... y n ]][x 1... x n ] then there is a commutative diagram of ring homomorphisms K T (G/B) = K(G/B) = R[X] f e(f) f R[X] W e i =1 Z[X] f f(1) f Z[X] W ch H T (G/B) Q = ch H (G/B) Q = Ŝ[x 1... x n ] f y(f) f Ŝ[x 1... x n ] W y i =0 Q[x 1... x n ] f f(0) f Q[x 1... x n ] W.

13 Schubert calculus Rank two and a positivity conjecture In this section we will give explicit formulas for the rank two root systems. The data supports the following positivity conjecture which generalizes the theorems of Brion [Br formula before Theorem 1] and Graham [Gr Corollary 4.1]. Conjecture 5.1. For β R + let y β = e β and a β = e β 1 and let d(w) = l(w 0 ) l(w) for w W. Let c z wv be the structure constants of K T(G/B) with respect to the basis {[O Xw ] w W } as defined in (0.1). Then c z wv = ( 1)d(w)+d(v) d(z) f(αy) where f(αy) Z 0 [α β y β β R + ] that is f(αy) is a polynomial in the variables α β and y β β R + which has nonnegative integral coefficients. In the following for brevity use the following notations: in K T (G/B) [w] = [O Xw ] α rs = e (rα 1+sα 2 ) 1 and y rs = e (rα 1+sα 2 ) in K(G/B) [w] = [O Xw ] α rs = 0 and y rs = 1 in H T (G/B) [w] = [X w] α rs = rα 1 + sα 2 and y rs = 1 in H (G/B) [w] = [X w ] α rs = 0 and y rs = 1 and in H T (G/B) and in H (G/B) the terms in { } brackets do not appear. Type A 2. For the root system R of type A 2 α 1 = ω 1 + 2ω 2 λ 1 = ρ λ s1 = ω 2 = 1 3 α α 2 λ s2 s 1 = s 2 ω 2 = 1 3 α α 2 α 2 = 2ω 1 ω 2 λ w0 = 0 λ s2 = ω 1 = 2 3 α α 2 λ s1 s 2 = s 1 ω 1 = 1 3 α α 2. Formulas for the Schubert classes in terms of homogeneous line bundles can be given by and [s 1 s 2 s 1 ] = 1 [1] = (1 e s 1ω 1 )[s 1 ] = (1 e s 2ω 2 )[s 2 ] [s 2 s 1 ] = 1 e ω 1 [s 1 s 2 ] = 1 e ω 2 [s 1 ] = (1 e s 2ω 2 )[s 2 s 1 ] [s 2 ] = (1 e s 1ω 1 )[s 1 s 2 ] [s 1 s 2 s 1 ] = 1 [s 1 s 2 ] = 1 e ω 2 [s 2 s 1 ] = 1 e ω 1 [s 1 ] = 1 e ω 2 X s 1ω 1 e ω 2 + e 2ω 2 [s 2 ] = 1 e ω 1 X s 2ω 2 e ω 1 + e 2ω 1 [1] = 1 e ω 2 X s 1ω 1 e ω 1 X s 2ω 2 + e 2ω 1 + e 2ω 2 e ρ X ρ. The multiplication of the Schubert classes is given by [1] 2 = α 10 α 01 α 11 [1] [1][s 1 ] = α 01 α 11 [1] [1][s 2 ] = α 10 α 11 [1] [1][s 1 s 2 ] = α 11 [1] [1][s 2 s 1 ] = α 11 [1] [s 1 ] 2 = α 01 α 11 [s 1 ] [s 1 ][s 2 ] = α 11 [1] [s 1 ][s 1 s 2 ] = y 01 [1] α 01 [s 1 ] [s 1 ][s 2 s 1 ] = α 11 [s 1 ] [s 2 ] 2 = α 01 α 11 [s 2 ] [s 2 ][s 1 s 2 ] = α 11 [s 2 ] [s 2 ][s 2 s 1 ] = y 10 [1] α 10 [s 2 ]

14 14 s. griffeth and a. ram [s 1 s 2 ] 2 = y 01 [s 2 ] α 01 [s 1 s 2 ] [s 1 s 2 ][s 2 s 1 ] = { [1] } + [s 1 ] + [s 2 ] [s 2 s 1 ] 2 = y 10 [s 1 ] α 10 [s 2 s 1 ]. Type B 2. For the root system R of type B 2 α 1 = 2ω 1 ω 2 λ 1 = ρ = 2α α 2 λ s1 = ω 2 = α 1 + α 2 α 2 = 2ω 1 + 2ω 2 λ w0 = 0 λ s2 = ω 1 = α α 2 λ s2 s 1 = s 2 ω 2 = α 1 λ s1 s 2 s 1 = s 1 s 2 ω 2 = α 1 λ s1 s 2 = s 1 ω 1 = 1 2 α 2 λ s2 s 1 s 2 = s 2 s 1 ω 1 = 1 2 α 2. Formulas for the Schubert classes in terms of homogeneous line bundles can be given by and [s 1 s 2 s 1 s 2 ] = 1 [1] = (1 e s 1ω 1 )[s 1 ] = (1 e s 2ω 2 )[s 2 ] [s 1 s 2 s 1 ] = 1 e ω 2 [s 2 s 1 s 2 ] = 1 e ω 1 [s 2 s 1 ] = (1 e ω 1 X s 1ω 1 )[s 2 s 1 s 2 ] [s 1 s 2 ] = (1 e s 2s 1 ω 1 )[s 2 s 1 s 2 ] [s 1 ] = (1 e s 2ω 2 )[s 2 s 1 ] [s 2 ] = (1 e s 1ω 1 )[s 1 s 2 ] [s 1 s 2 s 1 s 2 ] = 1 [s 1 s 2 s 1 ] = 1 e ω 2 [s 2 s 1 s 2 ] = 1 e ω 1 [s 1 s 2 ] = (1 e ω 2 ) e ω 2 e ω 2 X s 2ω 2 + (e ρ + e s 1ρ ) [s 2 s 1 ] = 1 e ω 1 e ω 1 X s 1ω 1 + e 2ω 1 [s 1 ] = (1 e ω 2 ) + (e ρ + e s 1ρ )X s 1ω 1 + (e ρ + e s 1ρ ) e ω 2 X s 1s 2 ω 2 e ω 2 X s 2ω 2 (e 2ω 2 + e ω 2 ) [s 2 ] = (1 + e 2ω 1 ) + e 2ω 1 X s 2ω 2 + e 2ω 1 e ω 1 X s 2s 1 ω 1 e ω 1 X s 1ω 1 (e 3ω 1 + e ω 1 ) [1] = (1 + e 2ω 1 ) e ω 1 X s 2s 1 ω 1 + (e ρ + e s 1ρ )X s 1ω 1 (e 3ω 1 + e ω 1 ) The multiplication of the Schubert classes is given by e ω 2 X s 1s 2 ω 2 + e 2ω 1 X s 2ω 2 (e 2ω 2 + e ω 2 ) + e ρ X ρ. [1] 2 = α 10 α 01 α 11 α 21 [1] [1][s 1 ] = α 01 α 11 α 21 [1] [1][s 2 ] = α 10 α 11 α 21 [1] [1][s 1 s 2 ] = α 11 α 21 [1] [1][s 2 s 1 ] = α 11 α 21 [1] [1][s 1 s 2 s 1 ] = α 11 (1 + y 11 )[1] [1][s 2 s 1 s 2 ] = α 21 [1] [s 1 s 2 s 1 ] 2 = { y 11 [s 1 ] } + (y 01 + y 11 )[s 2 s 1 ] α 01 [s 1 s 2 s 1 ] [s 1 s 2 s 1 ][s 2 s 1 s 2 ] = { [1] [s 1 ] [s 2 ] } + [s 1 s 2 ] + [s 2 s 1 ] [s 2 s 1 s 2 ] 2 = y 10 [s 1 s 2 ] α 10 [s 2 s 1 s 2 ] [s 2 s 1 ] 2 = α 21 y 10 [s 1 ] + α 10 α 21 [s 2 s 1 ] [s 2 s 1 ][s 1 s 2 s 1 ] = y 21 [s 1 ] α 21 [s 2 s 1 ] [s 2 s 1 ][s 2 s 1 s 2 ] = { y 10 [1] } + y 10 [s 1 ] + y 10 [s 2 ] α 10 [s 2 s 1 ] [s 1 ] 2 = α 01 α 11 α 21 [s 1 ] [s 1 ][s 2 ] = α 11 α 21 [1] [s 1 ][s 1 s 2 ] = α 11 (y 01 + y 11 )[1] + α 01 α 11 [s 1 ] [s 1 ][s 2 s 1 ] = α 11 α 21 [s 1 ] [s 1 ][s 1 s 2 s 1 ] = α 11 (1 + y 11 )[s 1 ] [s 1 ][s 2 s 1 s 2 ] = y 11 [1] α 11 [s 1 ] [s 2 ] 2 = α 10 α 11 α 21 [s 2 ] [s 2 ][s 1 s 2 ] = α 11 α 21 [s 2 ] [s 2 ][s 2 s 1 ] = α 21 y 10 [1] + α 10 α 21 [s 2 ] [s 2 ][s 1 s 2 s 1 ] = y 21 [1] α 21 [s 2 ] [s 2 ][s 2 s 1 s 2 ] = α 21 [s 2 ]

15 Schubert calculus 15 [s 1 s 2 ] 2 = α 11 (y 01 + y 11 )[s 2 ] + α 01 α 11 [s 1 s 2 ] [s 1 s 2 ][s 2 s 1 ] = ({ α 11 } + y 21 )[1] α 11 [s 1 ] α 21 [s 2 ] [s 1 s 2 ][s 1 s 2 s 1 ] = { (y 01 + y 11 )[1] } + y 01 [s 1 ] + (y 11 + y 12 )[s 2 ] α 01 [s 1 s 2 ] [s 1 s 2 ][s 2 s 1 s 2 ] = y 11 [s 2 ] α 11 [s 1 s 2 ] [s 2 s 1 ] 2 = α 21 y 10 [s 1 ] + α 10 α 21 [s 2 s 1 ] [s 2 s 1 ][s 1 s 2 s 1 ] = y 21 [s 1 ] α 21 [s 2 s 1 ] [s 2 s 1 ][s 2 s 1 s 2 ] = { y 10 [1] } + y 10 [s 1 ] + y 10 [s 2 ] α 10 [s 2 s 1 ] Type G 2. For the root system R of type G 2 λ 1 = ρ = 5α + 3α 2 λ s1 s 2 s 1 = s 1 s 2 ω 2 = α 2 λ s1 = ω 2 = 3α 1 + 2α 2 λ s2 s 1 s 2 s 1 = s 2 s 1 s 2 ω 2 = α 2 λ s2 = ω 1 = 2α 1 + α 2 λ s1 s 2 s 1 s 2 = s 1 s 2 s 1 ω 1 = α 1 λ s2 s 1 = s 2 ω 2 = 3α 1 + α 2 λ s1 s 2 s 1 s 2 s 1 = s 1 s 2 s 1 s 2 ω 2 = 3α 1 α 2 λ s1 s 2 = s 1 ω 1 = α 1 + α 2 λ s2 s 1 s 2 s 1 s 2 = s 2 s 1 s 2 s 1 ω 1 = α 1 α 2 λ s2 s 1 s 2 = s 2 s 1 ω 1 = α 1 λ w0 = 0. Formulas for the Schubert classes in terms of homogeneous line bundles can be given by [s 1 s 2 s 1 s 2 s 1 s 2 ] = 1 [1] = (1 e s 1ω 1 )[s 1 ] = (1 e s 2ω 2 )[s 2 ] [s 1 s 2 s 1 s 2 s 1 ] = 1 e ω 2 [s 2 s 1 s 2 s 1 s 2 ] = 1 e ω 1 [s 2 s 1 s 2 s 1 ] = (1 e ω 1 X s 1ω 1 )[s 2 s 1 s 2 s 1 s 2 ] [s 1 s 2 s 1 s 2 ] = (1 e s 1ω 1 )[s 2 s 1 s 2 s 1 s 2 ] [s 1 s 2 s 1 ] = see below [s 2 s 1 s 2 ] = 1 e s 2s 2 ω X ω [s 1 1 s 2 s 1 s 2 ] [s 2 s 1 ] = (1 e ω 1 X s 1s 2 s 1 ω 1 )[s 2 s 1 s 2 ] [s 1 s 2 ] = (1 e s 2s 1 ω 1 )[s 1 s 2 ] [s 1 ] = (1 e s 2ω 2 )[s 2 s 1 ] [s 2 ] = (1 e s 1ω 1 )[s 1 s 2 ] [s 1 s 2 s 1 ] = (1 e α 2 )[s 2 s 1 s 2 s 1 ] + e α 2 (1 + e ω 1 )[s 2 s 1 ] 1 + e α 2

16 16 s. griffeth and a. ram and [w 0 ] = 1 [s 2 s 1 s 2 s 1 s 2 ] = 1 y 21 [s 1 s 2 s 1 s 2 s 1 ] = 1 y 32 [s 2 s 1 s 2 s 1 ] = 1 y 21 y 21 X s 1ω 1 + y 42 [s 1 s 2 s 1 s 2 ] = (1 y 32 ) + (y 22 + y 42 + y 43 + y 53 ) y 32 X s 1ω 1 y 32 X s 2s 1 ω 1 y 32 y 32 X s 2ω 2 [s 2 s 1 s 2 ] = (1 y 21 + y 42 ) + (y 42 y 21 y 52 y 53 y 63 ) + (y 42 y 21 )X s 1ω 1 + (y 42 y 21 )X s 2s 1 ω 1 + y 42 + y 42 X s 2ω 2 [s 1 s 2 s 1 ] = (1 2y 32 ) + (y 22 + y 42 + y 43 + y 53 ) + (y 22 + y 42 + y 43 + y 53 )X s 1ω 1 y 32 X s 2s 1 ω 1 y 32 X s 1s 2 s 1 ω 1 (y 32 + y 43 + y 53 ) y 32 X s 2ω 2 y 32 X s 1s 2 ω 2 [s 2 s 1 ] = (1 y y 42 ) + (y 42 y 21 y 52 y 53 y 63 ) + (y 42 y 21 y 32 y 53 y 63 )X s 1ω 1 + (y 42 y 21 )X s 2s 1 ω 1 + (y 42 y 21 )X s 1s 2 s 1 ω 1 + (y 42 + y 63 ) + y 42 X s 2ω 2 + y 42 X s 1s 2 ω 2 [s 1 s 2 ] = 1 y 11 y 21 y 32 y 43 y 53 + (y 22 + y 32 )(1 + y 10 + y 20 ) + (y 22 + y 32 + y 42 )X s 1ω 1 + (y 22 + y 32 + y 42 )X s 2s 1 ω 1 (y 32 + y 43 + y 53 ) (y 32 + y 43 + y 53 )X s 2ω 2 y 32 X s 1s 2 ω 2 y 32 X s 2s 1 s 2 ω 2 [s 2 ] = (1 + y 31 + y y 42 + y 63 ) (y 21 + y 52 + y 53 + y 84 ) (y 21 + y 52 + y 53 )X s 1ω 1 (y 21 + y 52 + y 53 )X s 2s 1 ω 1 y 21 X s 1s 2 s 1 ω 1 y 21 X s 2s 1 s 2 s 1 ω 1 + (y 42 + y 63 ) + (y 42 + y 63 )X s 2ω 2 + y 42 X s 1s 2 ω 2 + y 42 X s 2s 1 s 2 ω 2 [s 1 ] = 1 (y 11 + y 21 + y y y 53 ) + (y 22 + y 54 )(1 + y 10 + y 20 ) + (y 22 + y 54 )(1 + y 10 + y 20 )X s 1ω 1 + (y 22 + y 32 + y 42 )X s 2s 1 ω 1 + (y 22 + y 32 + y 42 )X s 1s 2 s 1 ω 1 (y 32 + y 43 + y 53 + y 64 ) (y 32 + y 43 + y 53 )X s 2ω 2 (y 32 + y 43 + y 53 )X s 1s 2 ω 2 y 32 X s 2s 1 s 2 ω 2 y 32 X s 1s 2 s 1 s 2 ω 2 [1] = (1 + y 31 + y 42 + y 63 y 53 y 43 ) y 21 (1 + y 32 ) 2 + y 22 (1 + y 10 + y 20 )(1 + y 21 + y 31 )X s 1ω 1 (y 21 + y 52 + y 53 )X s 2s 1 ω 1 + y 22 X s 1s 2 s 1 ω 1 y 21 X s 2s 1 s 2 s 1 ω 1 y 32 (1 + y 11 )(1 + y 21 ) + (y 42 + y 63 )X s 2ω 2 (y 32 + y 43 + y 53 )X s 1s 2 ω 2 + y 42 X s 2s 1 s 2 ω 2 y 32 X s 1s 2 s 1 s 2 ω 2 + y 53 X ρ. The multiplication of the Schubert classes is given by [1] 2 = α 10 α 01 α 11 α 21 α 31 α 32 [1] [1][s 1 ] = α 01 α 11 α 21 α 31 α 32 [1] [1][s 2 ] = α 10 α 11 α 21 α 31 α 32 [1] [1][s 1 s 2 ] = α 11 α 21 α 31 α 32 [1] [1][s 2 s 1 ] = α 11 α 21 α 31 α 32 [1] [1][s 1 s 2 s 1 ] = α 11 α 21 α 32 (1 + y 11 + y 21 )[1] [1][s 2 s 1 s 2 ] = α 21 α 31 α 32 [1] [1][s 1 s 2 s 1 s 2 ] = α 21 α 32 (1 + y 21 )[1] [1][s 2 s 1 s 2 s 1 ] = α 21 α 32 (1 + y 21 )[1] [1][s 1 s 2 s 1 s 2 s 1 ] = α 32 (1 + y 32 )[1] [1][s 2 s 1 s 2 s 1 s 2 ] = α 21 (1 + y 21 )[1]

17 Schubert calculus 17 [s 1 ] 2 = α 01 α 11 α 21 α 31 α 32 [s 1 ] [s 1 ][s 2 ] = α 11 α 21 α 31 α 32 [1] [s 1 ][s 1 s 2 ] = α 11 α 21 α 32 (y 01 + y 11 + y 21 )[1] + α 01 α 11 α 21 α 32 [s 1 ] [s 1 ][s 2 s 1 ] = α 11 α 21 α 31 α 32 [s 1 ] [s 1 ][s 1 s 2 s 1 ] = α 11 α 21 α 32 (1 + y 11 + y 21 )[s 1 ] [s 1 ][s 2 s 1 s 2 ] = α 21 α 32 (y 11 + y 21 )[1] α 11 α 21 α 32 [s 1 ] [s 1 ][s 1 s 2 s 1 s 2 ] = α 32 (y 22 + y 32 )[1] + α 11 α 32 (1 + y 11 )[s 1 ] [s 1 ][s 2 s 1 s 2 s 1 ] = α 21 α 32 (1 + y 21 )[s 1 ] [s 1 ][s 1 s 2 s 1 s 2 s 1 ] = α 32 (1 + y 32 )[s 1 ] [s 1 ][s 2 s 1 s 2 s 1 s 2 ] = y 32 [1] α 32 [s 1 ] [s 2 ] 2 = α 10 α 11 α 21 α 31 α 32 [s 2 ] [s 2 ][s 1 s 2 ] = α 11 α 21 α 31 α 32 [s 2 ] [s 2 ][s 2 s 1 ] = α 21 α 31 α 32 y 10 [1] + α 10 α 21 α 31 α 32 [s 2 ] [s 2 ][s 1 s 2 s 1 ] = α 21 α 32 (y 21 + y 31 )[1] α 21 α 31 α 32 [s 2 ] [s 2 ][s 2 s 1 s 2 ] = α 21 α 31 α 32 [s 2 ] [s 2 ][s 1 s 2 s 1 s 2 ] = α 21 α 32 (1 + y 21 )[s 2 ] [s 2 ][s 2 s 1 s 2 s 1 ] = α 21 (y 31 + y 52 )[1] + α 21 α 31 (1 + y 21 )[s 2 ] [s 2 ][s 1 s 2 s 1 s 2 s 1 ] = y 63 [1] α 21 (1 + y 21 + y 42 )[s 2 ] [s 2 ][s 2 s 1 s 2 s 1 s 2 ] = α 21 (1 + y 21 )[s 2 ] [s 1 s 2 ] 2 = α 11 α 21 α 32 (y 01 + y 11 + y 21 )[s 2 ] + α 01 α 11 α 21 α 32 [s 1 s 2 ] [s 1 s 2 ][s 2 s 1 ] = α 21 α 32 (y 11 + y 21 + α 31 )[1] α 11 α 21 α 32 [s 1 ] α 21 α 31 α 32 [s 2 ] [s 1 s 2 ][s 1 s 2 s 1 ] = α 32 (y 32 + y 42 { +α 11 (y y 11 + y 21 ) })[1] + α 11 α 32 (y 01 + y 11 )[s 1 ] + ( α 31 α 32 y 11 + α 11 α 32 (y 01 + y 11 + y 21 ) ) [s 2 ] α 01 α 11 α 32 [s 1 s 2 ] [s 1 s 2 ][s 2 s 1 s 2 ] = α 21 α 32 (y 11 + y 21 )[s 2 ] α 11 α 21 α 32 [s 1 s 2 ] [s 1 s 2 ][s 1 s 2 s 1 s 2 ] = α 32 (y 22 + y 32 )[s 2 ] + α 11 α 32 (1 + y 11 )[s 1 s 2 ] [s 1 s 2 ][s 2 s 1 s 2 s 1 ] = ( y 63 {+α 32 (y 11 + y 21 ) } ) [1] α 32 y 11 [s 1 ] ( ) α 32 (y 11 + y 21 ) + α 31 y 32 [s2 ] + α 11 α 32 [s 1 s 2 ] [s 1 s 2 ][s 1 s 2 s 1 s 2 s 1 ] = { (y 33 + y 43 + y 53 )[1] } + y 33 [s 1 ] + (y 33 + y 43 + y 53 )[s 2 ] α 11 (1 + y 11 + y 22 )[s 1 s 2 ] [s 1 s 2 ][s 2 s 1 s 2 s 1 s 2 ] = y 32 [s 2 ] α 32 [s 1 s 2 ]

18 18 s. griffeth and a. ram [s 2 s 1 ] 2 = α 21 α 31 α 32 y 10 [s 1 ] + α 10 α 21 α 31 α 32 [s 2 s 1 ] [s 2 s 1 ][s 1 s 2 s 1 ] = α 21 α 31 (y 21 + y 31 )[s 1 ] α 21 α 31 α 32 [s 2 s 1 ] [s 2 s 1 ][s 2 s 1 s 2 ] = α 21 (y 51 + y 52 { +α 31 y 10 })[1] + α 21 (α 10 y 31 + α 32 y 10 )[s 1 ] + α 21 α 31 (y 10 + y 21 )[s 2 ] α 10 α 21 α 31 [s 2 s 1 ] [s 2 s 1 ][s 1 s 2 s 1 s 2 ] = ( y 62 { +α 31 (y 21 + y 31 ) } ) [1] ( α 31 y 21 + α 10 (y 31 + y 41 ) ) [s 1 ] ( ) α 31 y 21 + α 32 y 31 [s2 ] + α 21 α 31 [s 2 s 1 ] [s 2 s 1 ][s 2 s 1 s 2 s 1 ] = α 21 (y 31 + y 52 )[s 1 ] + α 21 α 31 (1 + y 21 )[s 2 s 1 ] [s 2 s 1 ][s 1 s 2 s 1 s 2 s 1 ] = y 63 [s 1 ] α 21 (1 + y 21 + y 42 )[s 2 s 1 ] [s 2 s 1 ][s 2 s 1 s 2 s 1 s 2 ] = { y 31 [1] } + y 31 [s 1 ] + y 31 [s 2 ] α 31 [s 2 s 1 ] [s 1 s 2 s 1 ] 2 = α 32 (y 32 + y 42 { +α 11 (y 11 + y 21 ) })[s 1 ] + ( α 11 α 32 (y 01 + y 11 + y 21 ) + α 31 α 32 y 11 ) [s2 s 1 ] α 01 α 11 α 32 [s 1 s 2 s 1 ] [s 1 s 2 s 1 ][s 2 s 1 s 2 ] = ( 1{ +α 11 (y 11 + y 22 + y 33 + y 31 + y 42 ) + α 31 (y 21 + y 32 ) + α 32 y 21 } ) [1] ( α 11 (y 21 + α 32 ) + α 10 (y 31 + y 41 + y 32 + y 42 ) ) [s 1 ] (α 31 (y 21 + y 32 ) + α 11 (y 21 + y 32 + y 31 + α 42 )[s 2 ] + α 11 α 32 [s 1 s 2 ] + α 21 α 31 [s 2 s 1 ] [s 1 s 2 s 1 ][s 1 s 2 s 1 s 2 ] = { (y y 43 + y 53 + α 11 (y 01 + y 11 ) + α 21 (y 11 + y 21 ))[1] } + ( y 33 + y 43 { +α 11 (y 01 + y 11 ) + α 21 (y 11 + y 21 ) } ) [s 1 ] ( (y33 + y 43 + y 53 ){ +α 11 (y 01 + y 11 ) + α 21 (y 11 + y 21 ) } ) [s 2 ] α 11 (y 01 + y 11 + y 22 )[s 1 s 2 ] ( α 11 (y 01 + y 11 ) + α 21 (y 11 + y 21 ) ) [s 2 s 1 ] + α 01 α 11 [s 1 s 2 s 1 ] [s 1 s 2 s 1 ][s 2 s 1 s 2 s 1 ] = (y 62 { +α 32 y 21 })[s 1 ] ( α 31 y 32 + α 32 (y 11 + y 21 ) ) [s 2 s 1 ] + α 11 α 32 [s 1 s 2 s 1 ] [s 1 s 2 s 1 ][s 1 s 2 s 1 s 2 s 1 ] = { (y 43 + y 53 )[s 1 ] } + (y 33 + y 43 + y 53 )[s 2 s 1 ] α 11 (1 + y 11 + y 22 )[s 1 s 2 s 1 ] [s 1 s 2 s 1 ][s 2 s 1 s 2 s 1 s 2 ] = { (y 11 + y 21 )[1] (y 11 + y 21 )[s 1 ] (y 11 + y 21 )[s 2 ] } + y 11 [s 1 s 2 ] + (y 11 + y 21 )[s 2 s 1 ] α 11 [s 1 s 2 s 1 ] [s 2 s 1 s 2 ] 2 = α 21 (y 21 + y 42 )[s 2 ] + ( ) α 11 α 21 y 31 + α 21 α 31 y 10 [s1 s 2 ] α 10 α 21 α 31 [s 2 s 1 s 2 ] [s 2 s 1 s 2 ][s 1 s 2 s 1 s 2 ] = y 53 [s 2 ] ( ) α 21 y 31 + α 11 α 21 α 32 y 21 [s1 s 2 ] + α 21 α 31 [s 2 s 1 s 2 ] [s 2 s 1 s 2 ][s 2 s 1 s 2 s 1 ] = { ( ) y 51 + y 52 + α 31 y 10 [1] } + (y41 { +α 31 y 10 })[s 1 ] + (y 42 + y 52 { +α 31 y 10 })[s 2 ] (α 11 y 31 + α 31 y 10 )[s 1 s 2 ] α 31 y 10 [s 2 s 1 ] + α 10 α 31 [s 2 s 1 s 2 ] [s 2 s 1 s 2 ][s 1 s 2 s 1 s 2 s 1 ] = { (y 31 + y 32 + y 42 )[1] (y 31 + y 32 )[s 1 ] (y 31 + y 32 + y 42 )[s 2 ] } + (y 31 + y 32 )[s 1 s 2 ] + y 31 [s 2 s 1 ] α 31 [s 2 s 1 s 2 ] [s 2 s 1 s 2 ][s 2 s 1 s 2 s 1 s 2 ] = y 31 [s 1 s 2 ] α 31 [s 2 s 1 s 2 ]

19 Schubert calculus 19 [s 1 s 2 s 1 s 2 ] 2 = { y 43 [s 2 ] } + (y 32 + y 42 { +α 01 y 21 + α 32 y 11 })[s 1 s 2 ] ( α 01 (y 11 + y 21 ) + α 31 (y 01 + y 11 ) ) [s 2 s 1 s 2 ] + α 01 α 11 [s 1 s 2 s 1 s 2 ] [s 1 s 2 s 1 s 2 ][s 2 s 1 s 2 s 1 ] = { (y 21 + y 31 + y 32 + y 42 + α 11 )[1] (y 21 + y 31 + y 32 + α 11 )[s 1 ] (y 21 + y 31 + y 32 + y 42 + α 11 )[s 2 ] } + (y 31 + y 42 {+α 11 })[s 1 s 2 ] + (y 21 + y 31 { +α 11 })[s 2 s 1 ] α 11 [s 1 s 2 s 1 ] α 31 [s 2 s 1 s 2 ] [s 1 s 2 s 1 s 2 ][s 1 s 2 s 1 s 2 s 1 ] = { (y 01 + y 11 + y 21 + y 22 + y 32 )[1] + (y 01 + y 11 + y 21 + y 22 )[s 1 ] + (y 01 + y 11 + y 21 + y 22 + y 32 )[s 2 ] (y 01 + y 11 + y 21 + y 22 )[s 1 s 2 ] (y 01 + y 11 + y 21 )[s 2 s 1 ] } + y 01 [s 1 s 2 s 1 ] + (y 01 + y 11 + y 21 )[s 2 s 1 s 2 ] α 01 [s 1 s 2 s 1 s 2 ] [s 1 s 2 s 1 s 2 ][s 2 s 1 s 2 s 1 s 2 ] = { y 21 [s 1 s 2 ] } + (y 11 + y 21 )[s 2 s 1 s 2 ] α 11 [s 1 s 2 s 1 s 2 ] [s 2 s 1 s 2 s 1 ] 2 = { y 52 [s 1 ] + (y 42 + y 52 )[s 2 s 1 ] } (α 11 y 31 + α 31 y 10 )[s 1 s 2 s 1 ] + α 10 α 31 [s 2 s 1 s 2 s 1 ] [s 2 s 1 s 2 s 1 ][s 1 s 2 s 1 s 2 s 1 ] = { y 42 [s 1 ] (y 31 + y 41 )[s 2 s 1 ] } + (y 31 + y 32 )[s 1 s 2 s 1 ] α 31 [s 2 s 1 s 2 s 1 ] [s 2 s 1 s 2 s 1 ][s 2 s 1 s 2 s 1 s 2 ] = { y 10 [1] + y 10 [s 1 ] + y 10 [s 2 ] y 10 [s 1 s 2 ] y 10 [s 2 s 1 ] } + y 10 [s 1 s 2 s 1 ] + y 10 [s 2 s 1 s 2 ] α 10 [s 2 s 1 s 2 s 1 ] [s 1 s 2 s 1 s 2 s 1 ] 2 = { y 32 [s 1 ] + (y 22 + y 32 )[s 2 s 1 ] (y 11 + y 21 + y 22 )[s 1 s 2 s 1 ] } + (y 01 + y 11 + y 21 )[s 2 s 1 s 2 s 1 ] α 01 [s 1 s 2 s 1 s 2 s 1 ] [s 1 s 2 s 1 s 2 s 1 ][s 2 s 1 s 2 s 1 s 2 ] = { [1] [s 1 ] [s 2 ] + [s 1 s 2 ] + [s 2 s 1 ] [s 1 s 2 s 1 ] [s 2 s 1 s 2 ] } + [s 1 s 2 s 1 s 2 ] + [s 2 s 1 s 2 s 1 ] [s 2 s 1 s 2 s 1 s 2 ] 2 = y 10 [s 1 s 2 s 1 s 2 ] α 10 [s 2 s 1 s 2 s 1 s 2 ] 5. References [BGG] I.N. Bernstein I.M. Gel fand and S.I. Gel fand Schubert cell and cohomology of the spaces G/P Russ. Math. Surv. 28 (3) (1973) [Br] M. Brion Positivity in the Grothendieck group of complex flag varieties J. Alg. 258 no. 1 (2002) [Bou] N. Bourbaki Groupes et algebres de Lie Chapt. IV-VI Masson Paris [Ch] C. Chevalley Sur les decompositions cellulaires des espaces G/B in Algebraic Groups and their Generalizations: Classical Methods W. Haboush and B. Parshall eds. Proc. Symp. Pure Math. Vol. 56 Pt. 1 Amer. Math. Soc. (1994) [CG] N. Chriss and V. Ginzburg Representation theory and complex geometry Birkhäuser Boston [D] M. Demazure Désingularisation des variétés de Schubert généralisées Ann. Sci. École Norm. Sup. 7 (1974) [FL] W. Fulton and A. Lascoux A Pieri formula in the Grothendieck ring of a flag bundle Duke Math. J. 76 (1994)

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