Диссертация (1136188), страница 29

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. . ; xn for any root a. By relabeling x1 ; . . . ; xn we can assume thata ¼ e1 e2 . Then, for any monomial m ¼ x1k1 x2k2 . . . xnkn , we haveA^a ðmÞ ¼ x3k3 . . . xnkn A^a ðx1k1 x2k2 Þ:Then exactly the same argument as the one above for A shows that A^a ðx1k1 x2k2 Þ is a powerseries in x1 and x2 .Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism795.1.

Algorithm for computing the products of Bott–Samelson resolutions. We nowproduce an explicit algorithm for computing the product of the Bott–Samelson classesZI in terms of other Bott–Samelson classes, where I ¼ ða1 ; . . . ; al Þ. The key ingredient isour algebraic Chevalley–Pieri formula (Proposition 4.3), which can be reformulated asfollows:l Pc1 LðlÞ A1 . . . Al Ze ¼A1 .

. . Aj1 Aj c1 Lðlj Þ Ajþ1 . . . Al Ze ;j¼1 where lj ¼ sj1 s1 l (in other words, c1 Lðlj Þ ¼ ½sj1 . . . s1 c1 LðlÞ ) and the operatorAj is defined as follows:Aj ¼ Aaj ¼1 ð1 saj Þ:c1 Lðaj ÞWe can compute Aj on any polynomial in the first Chern classes by the same methods as Aj(see the end of Section 3). Note that for the Chow ring Aj ¼Aj (this follows from Lemma3.4 and the fact that sj aj ¼ aj and c1 Lðaj Þ ¼ c1 Lðaj Þ for the additive formal grouplaw), but for the algebraic cobordism ring this is no longer true.

More generally, for any polynomial f ¼ f c1 Lðm1 Þ ; . . . ; c1 Lðmk Þ in the firstChern classes of some line bundles on X , we can compute its product with A1 . . . Al Ze byexactly the same argument as in the proof of Proposition 4.3:ð5:0Þf A1 . . . Al Ze ¼lPj¼1A1 . . . Aj1 ½Aj sj1 . . . s1 ð f ÞAjþ1 . . .

Al Zeþ A1 . . . Al ½sl . . . s1 ð f ÞZe :Note that the last term on the right-hand side is equal to the constant term of the polynomial ½sl . . . s1 ð f Þ (which is of course the same as the constant term of f ) multiplied byA1 . .

. Al Ze . In particular, for f ¼ c1 LðlÞ this term vanishes modulo S. Here and below,by the ‘‘constant term’’ of a polynomial in L½x1 ; . . . ; xn we mean the term of polynomialdegree zero (the total degree of such a constant term might be negative since the Lazardring L contains elements of negative degree). Note that all elements of L H L½x1 ; . . .

; xn are invariant under the operators si , and hence commute with the operators Ai . For anarbitrary reductive group, the constant term of an element f A W ðX Þ is defined as theproduct of f with the class of a point.It is now easy to show by induction on l thatfA1 . . . Al Ze ¼PQaJ ð f ÞAi Ze ;JHIi A I nJwhere aJ ð f Þ for the k-subtuple J ¼ ðaj1 ; . . . ; ajk Þ of I is the constant term in the expansionfor ½sl . . . sjk þ1 Ajk sjk 1 . . .

sj1 þ1 Aj1 sj1 1 . . . s1 f , which is invariant under si (for all i) andhence equal to ½Ajk sjk 1 . . . sj1 þ1 Aj1 sj1 1 . . . s1 f . Indeed, we first use formula (5.0) aboveBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4680Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismand then apply the induction hypothesis to all terms in the right-hand side except for theQlast term, which already has the form aJ ð f ÞAi Ze for J ¼ j. We geti A I nJA1 . . . Aj1 ½Aj sj1 .

. . s1 ð f ÞAjþ1 . . . Al Ze¼ A1 . . . Aj1P aJ ½Aj sj1 . . . s1 ð f ÞJHI nf1;...; jg¼PQaJ 0 ð f ÞAi Ze ;J 0 HIi A InJ 0QAi Zei A InðJWf1;...; jgÞwhere the last summation goes over all subsets J 0 of I that do contain j but not1; . . . ; j 1. Plugging this back into formula (5.0) we get the desired formula. Combiningthis with Theorem 3.2, we get the following formula inW ðX Þ for the product of the Bott–Samelson class ZI with the first Chern class c1 LðlÞ in terms of other Bott–Samelsonclasses:ð5:1ÞPbJ ðlÞZI nJ ;c1 LðlÞ ZI ¼JHIwhere bJ ðlÞ is the constant term in the expansion for ½Aj1 sj1 þ1 .

. . sjk 1 Ajk sjk þ1 . . . sl c1 LðlÞ :We changed the order of the si when passing from aJ to bJ since ZI ¼ Al . . . A1 Ze .Note that for J ¼ j we have bJ ¼ 0, and for J ¼ ðaj Þ we have bJ ¼ ðl; bj Þ since thec...slÞcLðsLðsj sjþ1 . . . sl lÞ1jþ1l1is equalconstant term in Aj c1 Lðsjþ1 .

. . sl lÞ ¼c1 Lðaj Þto ðsjþ1 . . . sl l; aj Þ (see the proof of Proposition 4.3, and Proposition 4.1 for the definitionof b i ), which is equal to ðl; b j Þ. So the lowest-order terms (with respect to the polynomialgrading) of this formula give an analogous formula for the Chow ring as expected.We now have assembled all necessary tools for actually performing the desiredSchubert calculus.

Namely, to compute the product ZI ZJ we apply the following procedure(which is formally similar to the one for the Chow ring). We replace ZJ with the respectivepolynomial RJ in the first Chern classes (using Theorem 3.2 together with the formula forZe ) and then compute the product of ZI with each monomial in RJ using repeatedly formula (5.1). Note that formula (5.1) allows us to make this algorithm more explicit than theone given in [4] (see an example below).The naive approach to represent both ZI and ZJ as fractions of polynomials in firstChern classes and then computing their product is less useful.

In particular translating theproduct of the fractions back into a linear combination of Bott–Samelson classes will bevery hard, if possible at all.5.2. Examples. We now compute the Bott–Samelson classes ZI in terms of theChern classes xi for the example X ¼ SL3 =B where B is the subgroup of upper-triangularmatrices. We then compute certain products of Bott–Samelson classes in two ways, byBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism81hand and then using the algorithm above together with formula (5.1). Note that only thesecond approach generalizes to higher dimensions.In SL3 , there are two simple roots g1 and g2 . In X , there are six Schubert cyclesXe ¼ pt, X1 , X2 , X12 , X21 and X121 ¼ X (here 12 is a short-hand notation for ðg1 ; g2 Þ,etc.).

Each XI except for X121 coincides with its Bott–Samelson resolution RI . Note thatin general RI and XI do not coincide even when XI is smooth. (By the way, for G ¼ GLnthe first non-smooth Schubert cycles show up for n ¼ 4.)Computing ZI as a polynomial in the first Chern classes. We want to express ZI as apolynomial in x1 , x2 , x3 using the formulasZsi1 ...sil ¼ Ail . . . Ai1 Re ; 1 Re ¼ c1 Lðg1 Þ c1 Lðg2 Þ c1 Lðg1 þ g2 Þ :6Note that in computations involving the operators Aa it is more convenient not to replacec1 LðaÞ with its expression in terms of xi until the very end.Let us for instance compute R1 as a polynomial in x1 , x2 , x3 modulo the ideal S generated by the symmetric polynomials of positive degree: 1R1 ¼ A1 Re ¼ ð1 þ s1 Þc1 Lðg2 Þ c1 Lðg1 þ g2 Þ6 1 1 ¼ c1 Lðg2 Þ c1 Lðg1 þ g2 Þ ¼ F wðx2 Þ; x3 F wðx1 Þ; x3 :332 3u and F ðu; vÞ ¼ u þ v þ a11 uv þ a12 u 2 v þ a21 uv 2 , whereWe have wðuÞ ¼ u þ a11 u 2 a112a11 ¼ ½P 1 and a12 ¼ a21 ¼ ½P 1 ½P 2 ([14], 2.5).

Thus 11 F wðx2 Þ; x3 F wðx1 Þ; x3 ¼ F ðx2 þ a11 x22 a11 x23 ; x3 ÞF ðx1 þ a11 x12 a11 x13 ; x3 Þ331¼ ðx2 þ x3 þ a11 x22 a11 x2 x3 Þðx1 þ x3 þ a11 x12 a11 x1 x3 Þ3¼ x32 ;since ðx3 x2 Þðx3 x1 Þ ¼ 3x32 mod S, and ðx2 þ x1 Þðx2 x3 Þðx1 x3 Þ ¼ 3x33 ¼ 0 mod S.So the answer agrees with the one we got in Remark 2.7.Here are the polynomials for the other Bott–Samelson resolutions:R212 ¼ 1 þ a12 x12 ;R12 ¼ x1 ½P 1 x12 ;R121 ¼ 1 þ a12 x1 x2 ;R21 ¼ x3 ¼ x1 x2 ;R1 ¼ x32 ¼ x1 x2 ;R2 ¼ x12 ;Re ¼ x12 x2 :Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4682Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismNote that the Bott–Samelson resolutions RI in this list coincide with the Schubert cyclesthey resolve if I has length e 2.

The corresponding polynomials RI are the classical Schubert polynomials (see e.g. [16] and keep in mind that his xi is equal to our xi ) except forthe polynomial R12 .In general, polynomials RI can be computed by induction on the length of I . E.g. tocompute R212 we can use that R212 ¼ A2 R21 and R21 ¼ x3 . Hence,R212 ¼ A2 ðx3 Þ ¼ 1 þ a12 x2 x3 ¼ 1 þ a12 x12 :The middle equation is obtained using the formula Aðy1 Þ ¼ 1 þ a12 y1 y2 þ from Section2.1 and the observation that all symmetric polynomials in x2 and x3 of degree greater than2 vanish modulo S.Computing products of the Bott–Samelson resolutions. Let us for instance computeZ12 Z21 .

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