aa r X i v : . [ m a t h . R A ] J un ON FORMAL SCHUBERT POLYNOMIALS
KIRILL ZAINOULLINE
Abstract.
Present notes can be viewed as an attempt to extend the notion ofSchubert/Grothendieck polynomial of Lascoux-Sch¨utzenberger to the contextof an arbitrary formal group law and of an arbitrary oriented cohomologytheory.
Let F ∈ R [[ x, y ]] be a commutative one-dimensional formal group law over acommutative unital ring R and let h be an algebraic oriented cohomology theorywith the coefficient ring h ( pt ) = R . According to Levine-Morel [LM07] there is a1-1 correspondence between such F ’s and universal h ’s. Indeed, given an orientedtheory h the respective formal group law is determined by the Quillen formula forthe first charactersitic classes in the theory h of the tensor product of line bundles c h ( L ⊗ L ) = F ( c h ( L ) , c h ( L ))and given a formal group law F over R one obtains the respective universal theory h by tensoring with the algebraic cobordism Ω, i.e. h ( − ) := Ω( − ) ⊗ Ω( pt ) R, where Ω( pt ) → R is obtained by specializing coefficients of the universal formalgroup law.For example, additive formal group law F a ( x, y ) = x + y corresponds to theChow theory h = CH , multiplicative F m ( x, y ) = x + y − xy to the Grothendieck h = K and the universal one F u to the algebraic cobordism h = Ω. Observe thatin the first two cases h ( pt ) = Z and in the last case the coefficient ring is the Lazardring which is infinitely generated over Z .Let G be a split semisimple linear group over a field k containing a split maximaltorus T . Following [CPZ, §
2] consider the formal group algebra R [[ M ]] F := R [[ x ω ]] ω ∈ M / ( x , x ω + ω ′ − F ( x ω , x ω ′ )) , where M is the weight lattice, together with the augmentation map ǫ : R [[ M ]] F → R , x ω
0. From the geometric point of view R [[ M ]] F models the completion of theequivariant cohomology h T ( pt ) and the map ǫ is the forgetful map. Algebraically, R [[ M ]] F is non-canonically isomorphic to the ring of formal power series in rk ( M )variables.Consider the algebra of formal divided difference operators D ( M ) F on R [[ M ]] F and let ǫ D ( M ) ∗ F := Hom R ( ǫ ◦ D ( M ) F , R ) denote the dual of the algebra of aug-mented operators. The main result of [CPZ, §
13] says that if h is a (weakly bira-tionally invariant) oriented cohomology theory corresponding to F (e.g. h = CH , Key words and phrases.
Hecke algebra, elliptic formal group law, Kazhdan-Lusztig basis, Bott-Samelson resolution, Schubert polynomial, Grothendieck polynomial. K or Ω), then there is an R -algebra isomorhism(1) ǫ D ( M ) ∗ F ≃ h ( X ) , where X is the variety of Borel subgroups of G . Moreover, it was shown thatthe R -basis of h ( X ) consisting of classes of Bott-Samelson resolutions of Schubertvarieties corresponds to the basis of ǫ D ( M ) ∗ F constructed as follows:First, for each element of the Weyl group w ∈ W one chooses a reduced de-composition w = s i s i . . . s i r into a product of simple reflections and denotes by I w = ( i , . . . , i r ) the respective reduced word. Then one shows that the R -linearoperators ǫC FI w defined by composing ǫ with the composite of the formal divideddifference operators C FI w = C Fi ◦ . . . ◦ C Fi r form a basis of ǫ D ( M ) F [CPZ, Prop. 5.4].Finally, the elements A I w ( z ) give the desired basis, where z is the element of ǫ D ( M ) ∗ F dual to ǫC FI w ( u ) for some specially chosen u ∈ (ker ǫ ) dim X , w is theelement of maximal length and A i is the operator on ǫ D ( M ) ∗ F dual to the operatoron ǫ D ( M ) F given by composition on the right with C Fi [CPZ, Thm. 13.13].One of the major difficulties in extending the Schubert calculus to such gener-alized theories h (and, hence, the Schubert polynomials) is the fact that all men-tioned bases are non-canonical, i.e. depend on choices of reduced decompositions.Moreover, according to [BE90] they are canonical if and only if F has the form F ( u, v ) = u + v − βuv for some β ∈ R . In other words, the Bott-Samelson resolu-tions of Schubert varieties provide a canonical basis of h ( X ) only for Chow groups( β = 0), Grothendieck K ( β is invertible) and connective K -theory ( β = 0 isnon-invertible).In the present notes we try to overcome this difficulty and, hence, provide a canonical basis of h ( X ) by either(1) averaging over all reduced decompositions, i.e. over all classes of Bott-Samelson resolutions; or(2) exploiting the Kazhdan-Lusztig theory in the case of a special elliptic formalgroup law.Observe that approach (1) works only after inverting the Hurwitz numbers, e.g.over Q , but over Q all formal group laws become isomorphic. Therefore, one maysuspect that we simply reduce to the known cases of additive or multiplicativeformal group laws. But this isomorphism does not preserve the formal differenceoperators as well as many other structures, so this is not the case.Approach (2) seems to be even more interesting as it gives a canonical basisintegrally. However, we don’t know how to extend it to other examples of formalgroup laws. 1. Averaging over reduced expressions
Consider the evaluation map c F : R [[ M ]] F → ǫ D ( M ) ∗ F of [CPZ, § h ( X )) it coincides with thecharacteristic map induced by ω c h ( L ( ω )). In the case of the additive or multi-plicative formal group law it gives the characteristic map described by Demazure in[De74], [De73]. Moreover, according to [CPZ, Thm. 6.9] if the Grothendieck torsionindex τ of G is invertible (this always holds for Dynkin types A and C), then thekernel of c F is the ideal I WF generated by augmented W -invariant elements, and we N FORMAL SCHUBERT POLYNOMIALS 3 obtain an R -algebra isomorphism(2) R [[ M ]] F / I WF ≃ ǫ D ( M ) ∗ F , which in view of the results of [HMSZ] and [CZZ] relate the invariant theory of W with an F -version of the Hecke ring of Kostant-Kumar [KK86], [KK90], [Ku02]and Bressler-Evens [BE87]. In general, though the kernel of c F always containsthe invariant ideal I W , the induced map c F : R [[ M ]] F / I WF → ǫ D ( M ) ∗ F is neitherinjective nor surjective.Observe also that if τ is invertible, then we can identify the basis A I w ( z ) of ǫ D ( M ) ∗ F with the basis C I revw ( u ) of R [[ M ]] F / I WF , where u = [ pt ] corresponds tothe class of a point (see [CPZ, Thm. 6.7]). The latter suggest to define for each w ∈ W the following element in R Q [[ M ]] F / I WF :(3) P Fw := | red ( w ) | X I w ∈ red ( w ) C I revw ([ pt ]) , where the sum is taken over the set red ( w ) of all reduced words of w .The results of [HMSZ] and [CZZ] then imply that { P Fw } w ∈ W is the desired canon-ical basis over Q : Theorem 1.
The elements { P Fw } w ∈ W form a R Q = R ⊗ Z Q -basis of R [[ M ]] F / I WF and, hence, of h ( X ) ⊗ Z Q .Proof. By [HMSZ, Prop. 5.8] and [CZZ, Lem. 7.1] the difference ( C i C j ) ◦ m ij − ( C j C i ) ◦ m ij is a linear combination of terms of length strictly smaller than 2 m ij (here m ij is the exponent in the respective Coxeter relation). Therefore, each P Fw can be written as P Fw = ( C I revw + (products of smaller length))([ pt ]). So thematrix expressing P Fw in terms of the usual basis { C I revw } w ∈ W corresponding to afixed choice of reduced decompositions { I w } w ∈ W is upper-triangular with 1’s onthe main diagonal. (cid:3) Consider a root system of Dynkin type A n . Let { e , . . . , e n +1 } be the standardbasis with α i = e i − e i +1 the set of simple roots. Consider a ring homomorphism R [ t , t , . . . , t n +1 ] → R [[Λ]] F , given by t i x − e i . It is S n +1 -equivariant, therefore, it induces a map on quotients(4) R [ t , t , . . . , t n +1 ] /I → R [[Λ]] F / I WF , where I is the ideal generated by symmetric functions. By Hornbostel-Kiritchenko[HK, Thm. 2.6] this is an R -algebra isomorphism. Definition 2.
We define an F -Schubert polynomial π Fw to be the image of P Fw inthe quotient R [ t , . . . , t n +1 ] /I via the isomorphism (4).If F has the form F ( u, v ) = u + v − βuv for some β ∈ R , i.e. exactly theformal group law for which the respective composites C I w do not depend on choicesof reduced words of w , then for β = 0 (resp. for β = 1) π Fw coincide with therespective Schubert (resp. Grothendieck) polynomials of Lascoux-Sch¨utzenberger(e.g. see [Fo94], [FK]) and for arbitrary β we obtain polynomials studied in [FK]and [Hu12].Observe that under this isomorphism the class of the point [ pt ] corresponds tothe class of the polynomial t n t n − . . . t n and the formal divided difference operator KIRILL ZAINOULLINE C i ( u ) = ux − i + s i ( u ) x i = (1 + s i )( ux − i ) (here x − i = x − α i = x e i +1 − e i ) corresponds tothe operator(5) C i ( f ) = (1 + σ i )( fρ i ) , where σ i swaps t i and t i +1 and ρ i is the formal power series given by t i − F t i +1 = F ( t i , ı ( t i +1 )), where ı ( x ) is the formal inverse series of x .By definition, operators C i are R [[ M ]] W i F -linear (here W i = h s i i ) and satisfy[CPZ, Prop. 3.13]: C i ( uv ) = C i ( u ) v + s i ( u ) C i ( v ) − κ i s i ( u ) v and C i (1) = κ i , where κ i = ρ i + ı ( ρ i ) ∈ R [[ M ]] F . We also have C i ( t j f ) = t j C i ( f ) for j = i, i + 1and C i ( t i ) = t i t i − F t i +1 + t i +1 t i +1 − F t i = t i − F t i +1 + F t i +1 t i − F t i +1 + t i +1 t i +1 − F t i = F ( ρ i ,t i +1 ) − t i +1 ρ i + t i +1 κ i . Using these formulas one can compute the polynomials π Fw .Our goal now is (using these formulas) to express each polynomial π Fw as a linearcombination of sub-monomials of t n t n − . . . t n with coefficients from R . Example 3.
We can write an arbitrary formal group law F as F ( x, y ) = x + y − xyg ( x, y ) . This implies that z + ı ( z ) = g ( z, ı ( z )) . If F ( x, y ) = x + y + a xy + a xy ( x + y ) + O (4), then g ( x, y ) = − a − a ( x + y ) − a ( x + y ) − a xy + O (3)and ı ( x ) = − x + a x − a x + O (4) . So, after substituting, we obtain x − F y = x + ( − y + a y − a y ) + a x ( − y + a y ) + a x ( − y )( x − y ) + O (4) == ( x − y ) − a y ( x − y ) + a y ( x − y ) − a xy ( x − y ) + O (4) , and, hence,( x − F y ) + ( y − F x ) = a ( x − y ) − a ( x + y )( x − y ) + O (4) , ( x − F y ) + ( y − F x ) = 2( x − y ) − a ( x + y )( x − y ) + O (4) , ( x − F y )( y − F x ) = − ( x − y ) + a ( x + y )( x − y ) + O (4) . Combining these together we get g ( x − F y, y − F x ) = − a − a a ( x − y ) − a ( x − y ) + a ( x − y ) + O (3) = − a − ( a a + 2 a − a )( x − y ) + O (3) . We then obtain r ( x, y ) = F ( x,y ) − yx = 1 + a y + a y ( x + y ) + a y ( x + y ) + a xy + O (4) . Hence, r ( x − F y, y ) = 1+ a y + a y ( x − a y ( x − y ))+ a y (( x − y ) + y )+ a ( x − y ) y + O (4) == 1 + a y + a xy + ( a − a a )( x − y ) y + a y (( x − y ) + y ) + O (4) , N FORMAL SCHUBERT POLYNOMIALS 5 and, therefore, r ( x − F y, y )+ yg ( x − F y, y − F x ) = 1+ a xy +( a − a a ) xy ( x − y )+ a xy (2 y − x )+ O (4) . Observe that there is a relation 2( a − a a ) = 3 a in the Lazard ring (the onlyrelation in degree 4) which gives2[( a − a a ) xy ( x − y ) + a xy (2 y − x )] = 2 a xy ( x + y ) . In particular, if 2 is invertible and x = t i , y = t i +1 for the type A , then t i t i +1 ( t i + t i +1 ) is in the ideal I generated by symmetric functions in t , t , t , meaning thatfor i = 1 , C i ( t i ) = t i t i − F t i +1 + t i +1 t i +1 − F t i = r ( t i − F t i +1 , t i +1 ) + t i +1 κ i = 1 + a t i t i +1 . Example 4.
Using the formulas above we obtain the following expressions for π Fw in the A -case for an arbitrary F (this agrees with computations at the end of [HK]and [CPZ]) π F = C ([ pt ]) = t t , π F = C ([ pt ]) = t , Indeed, C ( t t ) = t t t − F t + t t t − F t = t t C ( t ) = t t and C ( t t ) = t t t − F t + t t t − F t = t ( t t − F t + t t − F t ) = t (1 + a t t ) = t .π F = C ([ pt ]) = t + t + a π F , π F = C ([ pt ]) = C ( t t ) = t , Indeed, C ( t ) = C ( t ) t + t C ( t ) − κ t t = (1+ a t t )( t + t )+ a t t = t + t + a t t and C ( t t ) = t C ( t ) = t (1 + a t t ) = t And for the element of maximal length w = (121) = (212) we obtain C ([ pt ]) = 1 + a π F , C ([ pt ]) = C ( t ) = 1 + a t t = 1 + a π F . Indeed, C ( t + t + a t t ) = t C (1) + C ( t ) + a t C ( t ) == t ( − a − ( a a + 2 a − a )( t − t ) ) + 1 + a t t + a t (1 + a t t ) =1 + a t t − a t ( t − t ) = 1 + a t t = 1 + a t as t ≡ t t and t ( t − t ) is in the ideal ( t ( t − t ) ≡ t ( t + t ) ≡ t ≡ π Fw = 1 + a ( t + t t ) . Observe that the twisted braid relation (which leads to the dependence onchoices) of [HMSZ, Prop. 5.8] then coincides with C − a C = C − a C . KIRILL ZAINOULLINE A special elliptic formal group law and the Kazhdan-Lustig basis
Consider an elliptic curve given in Tate coordinates by(1 − µ x − µ x ) y = x . The corresponding formal group law over the coefficient ring R = Z [ µ , µ ] is givenby (e.g. [BB10, Example 63]), F ( x, y ) := x + y − µ xy µ xy and will be called a special elliptic formal group law. Observe that by definition,we have F ( x, y ) = x + y − xy ( µ + µ F ( x, y )) , so a = − µ and a = − µ . By [HMSZ, Theorem 5.14] for the type A n the algebra D ( M ) F is generated byoperators C i , i ∈ ..n , and multiplications by elements u ∈ R [[ M ]] F subject to thefolowing relations:(a) C i = µ C i (b) C ij = C ji for | i − j | > C iji − C jij = µ ( C j − C i ) for | i − j | = 1 and(d) C i u = s i ( u ) C i + µ u − C i ( u ),Recall that the Iwahori-Hecke algebra H of the symmetric group S n +1 is (af-ter the respective normalization) an Z [ t, t − ]-algebra with generators T i , i = 1 ..n ,subject to the following relations:(A) ( T i − t − )( T i + t ) = 0 or, equivalently, T i = ( t − − t ) T i + 1,(B) T ij = T ji for | i − j | > T iji = T jij for | i − j | = 1.Observe that T i ’s appearing in the classical definition of the Iwahori-Hecke alge-bra in [CG10, Def. 7.1.1] correspond to tT i in our notation, where t = q − / .Following [HMSZ, Def. 5.3] let D F denote the R -subalgebra of D ( M ) F generatedby the elements C i , i = 1 ..n , only. In [HMSZ, Prop. 6.1] it was shown that for F = F a (resp. F = F m ) D F is isomorphic to the nil-Hecke algebra (resp. the0-Hecke algebra) of Kostant-Kumar.Comparing the relations for D F and H we see that for R = Z [ t, t − ][ t + t − ], µ = 1 and µ = − t + t − ) there is an isomorphism of R -algebras (see [CZZ1] and[LNZ] for the case of an arbitrary root system)(6) H [ t + t − ] ≃ D F given on generators by T i ( t + t − ) C i − t, i = 1 ..n. By definition of (6) the involution on H (sending t t − and T i T − i )corresponds to the involution on D F obtained by extending the involution t t − on the coefficient ring. Observe that each push-pull element C i = t + t − ( T i + t ) isinvariant under this involution.Consider the Kazhdan-Lusztig basis { C ′ w } w ∈ W on H (e.g. see [CG10]). Recallthat it is unique and does not depend on choices of reduced decompositions. Afterthe respective normalization we have C ′ w = T w + X v We define the special elliptic polynomial π sew to be the image in Z [ t, t − ][ t , . . . , t n +1 ] /I of the element t + t − ) l ( w ) C w − ([ pt ]) via (4).We expect polynomials π sew to play the same role (in the special elliptic case) asthe Schubert (resp. Grothendieck) polynomials for Chow groups (resp. K ). Example 6. For the type A we obtain π sei = C i ([ pt ]) , π seij = C j C i ([ pt ])and for the element of maximal length we obtain exactly the twisted braid relation π sew = ( C + µ C )([ pt ]) = ( C + µ C )([ pt ]) = 1 . Remark 7. It would be interesting to see(1) that π sew = 1, for the element of maximal length w .(2) whether π sew corresponds to the class of an actual resolution of the respectiveSchubert variety X w . References [BE92] Bressler, P.; Evens, S. Schubert calculus in complex cobordisms. Trans. Amer. Math.Soc. 331 (1992), no.2, 799–813.[BE90] P. Bressler, S. Evens, The Schubert calculus, braid relations and generalized cohomology. Trans. Amer. Math. Soc. 317 (1990), no.2, 799–811.[BE87] Bressler, P.; Evens, S. On certain Hecke rings. Proc. Nat. Acad. Sci. USA 84 (1987),624–625.[BB10] V. Buchstaber, E Bunkova, Elliptic formal group laws, integral Hirzebruch genera andKrichever genera , Preprint arXiv.org 1010.0944v1, 2010.[CZZ] B. Calm`es, K. Zainoulline and C. Zhong, A coproduct structure on the formal affineDemazure algebra , Preprint arXiv:1209.1676, 2013.[CZZ1] B. Calm`es, K. Zainoulline and C. Zhong, Push-pull operators on the formal affine De-mazure algebra and its dual , Preprint arXiv.org 1312.0019.[CPZ] B. Calm`es, V. Petrov and K. Zainoulline, Invariants, torsion indices and oriented coho-mology of complete flags, Ann. Sci. ´Ecole Norm. Sup. (4) 46, no.3, 2013, 405–448.[CG10] N. Chriss, V. Ginzburg. Representation theory and complex geometry. ModernBirkhauser Classics. Birkhauser Boston Inc., Boston, MA, 2010. 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