Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties

Home , Flag (linear algebra), Identity matrix, Nonnegative matrix, Toeplitz matrix

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 2, Pages 363–392 S 0894-0347(02)00412-5 Article electronically published on November 29, 2002

TOTALLY POSITIVE TOEPLITZ MATRICES AND QUANTUM COHOMOLOGY OF PARTIAL FLAG VARIETIES

KONSTANZE RIETSCH

1. Introduction A matrix is called totally nonnegative if all of its minors are nonnegative. Totally nonnegative inﬁnite Toeplitz matrices were studied ﬁrst in the 1950’s. They are characterized in the following theorem conjectured by Schoenberg and proved by Edrei. Theorem 1.1 ([10]). The Toeplitz matrix ∞×∞ 1 a1 1   a2 a1 1  . ..   . a2 a1 .    A =  ......   ad . . .     .. ..  ad+1 . . a1 1     . . . .   . .. .. a a ..   2 1   . . . . .   ......    is totally nonnegative precisely if its generating function is of the form,

2 (1 + βit) 1+a1t + a2t + =exp(tα) , ··· (1 γit) i Y∈N − where α R 0 and β1 β2 0,γ1 γ2 0 with βi + γi < . ∈ ≥ ≥ ≥···≥ ≥ ≥···≥ ∞ This beautiful result has been reproved many times; see [32]P for anP overview. It may be thought of as giving a parameterization of the totally nonnegative Toeplitz matrices by

˜ N N ˜ (α;(βi)i, (˜γi)i) R 0 R 0 R 0 i(βi +˜γi) < , { ∈ ≥ × ≥ × ≥ | ∞} i X∈N where β˜i = βi βi+1 andγ ˜i = γi γi+1. − − Received by the editors December 10, 2001 and, in revised form, September 14, 2002. 2000 Mathematics Subject Classiﬁcation. Primary 20G20, 15A48, 14N35, 14N15. Key words and phrases. Flag varieties, quantum cohomology, total positivity. During a large part of this work the author was an EPSRC postdoctoral fellow (GR/M09506/01) and Fellow of Newnham College in Cambridge. The article was completed while supported by the Violette and Samuel Glasstone Foundation at Oxford.

c 2002 American Mathematical Society

363

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Now let U − denote the lower triangular unipotent n n matrices. One aim of this paper is to parameterize the set of totally nonnegative× matrices in 1 a1 1    a2 a1 1   X := x U − x =  ......   ∈ |  a3 . . .       . .. ..   . . . a1 1      an 1 ... a3 a2 a1 1   −     by n 1 nonnegative parameters. Let ∆n i(x) be the lower left-hand corner i i minor− of x X. Explicitly, we will prove the− following statement. × ∈ Proposition 1.2. Let X 0 denote the set of totally nonnegative matrices in X. Then the map ≥ n 1 ∆ 0 := (∆1,...,∆n 1): X 0 R −0 ≥ − ≥ −→ ≥ is a homeomorphism.

n 1 Note that ∆ : X C − is a ramified cover and the nonnegativity of the values → of the ∆i is by no means sufficient for an element u to be totally nonnegative. The statement is rather that for prescribed nonnegative values of ∆1,...,∆n 1, among all matrices with these fixed values there is precisely one which is totally− nonnegative. The proof of this result involves relating total positivity for these n n Toeplitz matrices to properties of quantum cohomology rings of partial flag varieties,× via Dale Peterson’s realization of these as coordinate rings of certain remarkable subvarieties of the flag variety. We show that the Schubert basis of the quantum cohomology ring plays a similar role for these matrices with regard to positivity as does the (classical limit of the) dual canonical basis for the whole of U − in the work of Lusztig [24]. This is the content of Theorem 7.2, which is the main result of this paper. The above parameterization of X 0 comes as a corollary. ≥ 1.1. Overview of the paper. The first part of the paper is taken up with intro- ducing the machinery we will need to prove our results. We set out by recalling background on the quantum cohomology rings of full and partial flag varieties, espe- cially work of Astashkevich, Sadov, Kim, and Ciocan-Fontanine, as well as Fomin, Gelfand and Postnikov. Their work is then used in Section 4 to explain Peterson’s result identifying these rings as coordinate rings of affine strata of a certain remarkable subvariety of the flag variety. The variety X of Toeplitz matrices enters the picture when theY + Peterson variety is viewed from the opposite angle (U −-orbits rather than U - orbits). We recallY the Bruhat decomposition of the variety of Toeplitz matrices. Each stratum XP has in its coordinate ring the quantum cohomology ring of a partial flag variety G/P with its Schubert basis and quantum parameters. In Section 5 we recall Kostant’s formula for the quantum parameters as functions on XB in terms of the minors ∆i and generalize it to the partial flag variety case. After some motivation from total positivity the main results are stated in Sec- tion 7. Theorem 7.2 has three parts. Firstly, the set of points in XP where all Schubert basis elements take positive values has a parameterization (q1,...,qk): k XP,>0 ∼ R>0 given by the quantum parameters. Secondly, this set lies in the −→

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smooth locus of XP and the inverse of the map giving the parameterization is analytic. Thirdly, this set of Schubert positive points agrees with the set of totally nonnegative matrices in XP . Proposition 1.2 stated in the introduction is proved immediately as a corollary. In Section 8 we make an excursion to recall what these results look like explicitly in the Grassmannian case, which is studied in detail in an earlier paper. We then use the Grassmannian components of the Peterson variety to prove that the top

P Schubert class σw0 is generically nonvanishing as function on XP . Conjecturally, the same should hold for the quantum Euler class, w W P σwσPD(w), which would ∈ imply that qH∗(G/P ) is reduced. The rest of the paper is devoted to the proof ofP Theorem 7.2. In the next two sections, parts (1) and (2) of the main theorem are proved. The main ingredient for constructing and parameterizing the Schubert positive points is the positivity of the structure constants (Gromov-Witten invariants). Computing the fiber in XP over a fixed positive value of the quantum parameters (q1,...,qk) is turned into an eigenvalue problem for an irreducible nonnegative matrix, and the unique Schubert positive solution we require is provided by a Perron-Frobenius eigenvector. The smoothness property turns out to be related to the positivity of the quantum Euler class, while bianalyticity comes as a consequence of the one-dimensionality of the Perron-Frobenius eigenspace. The final part of Theorem 7.2, that the notion of positivity coming from Schubert bases agrees with total positivity, is perhaps the most surprising. The problem is that, except in the case of Grassmannian permutations, we know no useful way to compute the Schubert classes as functions on the XP . In Section 11 we begin to simplify this problem by proving another remarkable component of Peterson’s theory. Namely, consider the functions on XB given by the Schubert classes of qH∗(G/B). Then when extended as rational functions to all of X, these restrict to give all the Schubert classes on the smaller strata XP . We prove this explicitly using quantum Schubert polynomials and Fomin, Gelfand and Postnikov’s quantum straightening identity. This last result enables us essentially to reduce the proof of the final part of Theorem 7.2 to the full flag variety case. The main problem there is to prove that an arbitrary Schubert class takes positive values on the totally positive part. This is done by topological arguments, using that the totally positive part of XB is a semigroup.

2. Preliminaries

Let G = GLn(C), and I = 1,...,n 1 an indexing set for the simple roots. Denote by Ad the adjoint representation{ − of G} on its Lie algebra g.WeﬁxtheBorel + subgroups B of upper-triangular matrices and B− of lower-triangular matrices in + G. Their Lie algebras are denoted by b and b−, respectively. We will also consider + + their unipotent radicals U and U − with their Lie algebras u and u− and the + maximal torus T = B B−.LetX∗(T ) be the character group of T and X (T ) ∩ ∗ the group of cocharacters with the usual perfect pairing <, >: X∗(T ) X (T ) Z × ∗ →+ between them. Let ∆+ X∗(T ) be the set of positive roots corresponding to b , and ∆ the set of negative⊂ roots. The fundamental weights and coweights are − denoted by ω1,...,ωn 1 X∗(T )andω1∨,...,ωn∨ 1 X (T ), respectively. We − ∈ − ∈ ∗ deﬁne Π X∗(T ) to be the set of positive simple roots. The elements of Π are ⊂

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denoted α1,...,αn 1 where the αm-root space gαm g is spanned by − ∈ 0 ..  .  m m+1 n 01 eα = δ δ =   . m i j i,j=1  0     .   ..     0   n 1   Let e := m−=1 eαm . A special role will be played by the principal nilpotent element f u− which is the transpose of e. ∈We identifyP the Weyl group W of G, the symmetric group, with the group of all permutation matrices. W is generated by the usual simple reﬂections (adjacent transpositions) s1,...,sn 1. The length function ` : W N gives the length of a reduced expression of w − W in the simple generators.→ There is a unique longest ∈ element which is denoted w0.

2.1. Parabolics. Let P always denote a parabolic subgroup of G containing B−, and p the Lie algebra of P .LetIP be the subset of I associated to P consisting of P all the i I with si P and consider its complement I := I IP . We will denote ∈ P ∈ \ the elements of I by n1,...,nk with { } n0 := 0 =0foralli I and ∆+ is its complement. { ∈ | B ∈ } So, for example, ∆B,+ = and ∆ =∆+. ∅ + 3. The quantum cohomology ring of G/P 3.1. The usual cohomology of G/P and its Schubert basis. For our purposes it will suffice to take homology or cohomology with complex coefficients, so always H∗(G/P )=H∗(G/P, C). By the well-known result of C. Ehresmann, the singular homology of the partial flag variety G/P has a basis indexed by the elements w W P made up of the fundamental classes of the Schubert varieties, ∈ ΩP := (B+wP/P) G/P. w ⊆ P Here the bar stands for (Zariski) closure. Let σw H∗(G/P )bethePoincaré dual P P ∈ P class to [Ωw]. Note that Ωw has complex codimension `(w)inG/P and hence σw 2`(w) P P lies in H (G/P ). The set σ w W formsabasisofH∗(G/P ) called { w | ∈ }

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P the Schubert basis. The top degree cohomology of G/P is spanned by σ P and we w0 have the Poincar´e duality pairing

H∗(G/P ) H∗(G/P ) C, (σ, µ) σ µ × −→ 7→ h · i P which may be interpreted as taking (σ, µ) to the coeﬃcient of σ P in the basis w0 expansion of the product σ µ.Forw W P let PD(w) W P be the minimal · ∈ ∈ length coset representative in w0wWP . Then this pairing is characterized by

σw σv = δ . h · i w,PD(v)

3.2. Definition of the quantum cohomology ring qH∗(G/P ). The (small) quantum cohomology ring qH∗(G/P ) may be defined by enumerating curves into G/P with certain properties. This description is responsible for its positivity properties and is the one we will give here. For more general background there are already many books and survey articles on the subject of quantum cohomology; see e.g. [9, 13, 29, 30] and references therein. P Let I = n1,...,nk . Then as a vector space the quantum cohomology of the partial flag variety{ G/P}is given by P P qH∗(G/P )=C[q1 ,...,qk ] H∗(G/P ), ⊗C P P where q1 ,...,qk are called the quantum parameters. Consider the Schubert classes P P as elements of qH∗(G/P ) by identifying σw with 1 σw . We will sometimes drop the superscript P from the notation for the Schubert⊗ classes and the quantum parameters when there is no possible ambiguity. P P Now qH∗(G/P )isafreeC[q1 ,...,qk ]-module with basis given by the Schubert P P P P classes σw . It remains to give the structure constants σu ,σv ,σw d in P P P P P d P σv σw = σu ,σv ,σw d q σPD(u) u W P d∈Xk ∈N d to define the ring structure on qH∗(G/P ). Here q is multi-index notation for k di i=1 qi . 1 Consider the set d of holomorphic maps φ : CP G/P such that Q M → k 1 φ CP = di B−sni P/P . ∗ i=1 X h i d can be made into a quasi-projective variety of dimension equal to dim(G/P )+ M P P P di(ni+1 ni 1). To define σu ,σv ,σw first translate the Schubert varieties − − d P P P P P P ΩPu , Ωv and Ωw into general position, say to Ωu , Ωv and Ωw. Now consider the set d(u, v, w)ofallmapsφ d such that M ∈M f f f φ(0) ΩP ,φ(1) ΩP , and φ( ) ΩP . ∈ u ∈ v ∞ ∈ w Then d(u, v, w) is finite if dim(G/P )+ di(ni+1 ni 1)=`(u)+`(v)+`(w) f f − f and oneM may set − P # d(u, v, w)ifdim(G/P )+ di(ni+1 ni 1) M − − σP ,σP ,σP = = `(u)+`(v)+`(w), u v w d  P  0otherwise. 

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These quantities are 3-point, genus 0 Gromov-Witten invariants. By looking at d = (0,...,0) one recovers the classical structure constants obtained from intersecting Schubert varieties in general position. Therefore this multiplicative structure is a deformation of the classical cup product. We note that the structure constants by their definition are nonnegative integers. The quantum cohomology analogue of the Poincaré duality pairing may be de- P P fined as the symmetric C[q1 ,...,qk ]-bilinear pairing P P qH∗(G/P ) qH∗(G/P ) C[q1 ,...,qk ], (σ, µ) σ µ × −→ 7→ h · iq P which takes (σ, µ) to the coefficient of σ P in the Schubert basis expansion of the w0 product σ µ. In terms of the Schubert basis the quantum Poincaré duality pairing · on qH∗(G/P )isgivenby σP σP = δ , w · v q w,PD(v) where v, w W P ,andPD : W P W P is the involution defined in Section 3.1 (see e.g. [8],∈ Lemma 6.1). →

3.3. Borel’s presentation of the cohomology ring H∗(G/P ). Let G/P be realized as a variety of ﬂags of quotients as in Section 2.1, n n G/P = n1,...,nk (C )= C = Vk+1 Vk V1 0 dim Vj = nj . F { ··· → | } Then for 1 j k + 1, the successive quotients Qj =ker(Vj Vj 1) deﬁne rank ≤ ≤ → − (nj nj 1) vector bundles on G/P . Their Chern classes will be denoted − − (j) (j) ci(Qj)=:σi = σi,P .

By the splitting principle it is natural to introduce independent variables x1,...,xn

such that xnj 1+1,...,xnj are the Chern roots of Qj.So − (j) (3.1) σi = ei(xnj 1 +1,xnj 1 +2,...,xnj ), − − the i-th elementary symmetric polynomial in the variables xn +1,...,xn .Let { i i+1 } WP act on the polynomial ring C[x1,...,xn] in the natural way by permuting the variables. Then the ring of invariants is

(1) (2) (k+1) (k+1) [x ,...,x ]WP = σ ,...,σ(1),σ ,...... ,σ ,...,σ . C 1 n C 1 n1 1 1 n nk − h (j) i A. Borel [5] showed that the Chern classes σi generate H∗(G/P ) and the only relations between these generators come from the triviality of the bundle Q1 Q2 n ⊕ ⊕ Qk (which may be trivialized using a Hermitian inner product on C ). ···⊕ WP In other words, if J denotes the ideal in C[x1,...,xn] generated by the elementary symmetric polynomials ei(x1,...,xn), then

WP (3.2) H∗(G/P ) ∼= C[x1,...,xn] /J.

3.4. Schubert polynomials and elementary monomials for H∗(G/P ). For 1 i k + 1, deﬁne ≤ ≤ (j) (j) (3.3) ei = ei,P := ei(x1,...,xnj ), (k+1) the i-th elementary symmetric polynomial in nj variables. Then the ei ’s are the generators of the ideal J.Butfor1 j k, the element e(j) corresponds ≤ ≤ i

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under (3.2) to a nonzero element of H∗(G/P ), namely the special Schubert class P σsn i+1 sn . j − ··· j The polynomial deﬁned by

n1 nk nk 1 (1) (k) − − (3.4) cwP := e e 0 n1 ··· nk represents the top class σ P . w0 These are examples of the Schubert polynomials of Lascoux and Schützenberger, P WP [23]. The Schubert polynomials cw w W C[x1,...,xn] are, loosely speaking, obtained from the top one{ by| divided∈ difference}⊂ operators; see [23] or [27] for details. If P = B then Schubert polynomials cw are obtained for all w W , and the ones from above corresponding to G/P are just the subset consisting∈ of P all those for which w W . The key property of a Schubert polynomial cw is of ∈ course that it is a representative for the corresponding Schubert class σw. A different description, following [11], of the Schubert polynomials cw for w W P says precisely where these representatives must lie. They are those representa-∈ tives of the Schubert classes which may be written as linear combinations of certain WP “elementary monomials” in C[x1,...,xn] . (1) (k) Explicitly, let P be the set of sequences Λ = (λ ,...,λ ) of partitions, such (j) L (j) that λ hasatmost(nj nj 1) parts and λ1 nj.ToanyΛ P associate a polynomial, − − ≤ ∈L

(1) (1) (k) (k) eΛ = e (1) e (1) e (k) e (k) . λ λn λ λ 1 ··· 1 ··· 1 ··· nk nk 1 − − Let us call these polynomials P -standard monomials.TheseeΛ are linearly independent and span a complementary subspace to the ideal J.So

[x ,...,x ]WP = J e . C 1 n Λ Λ P ⊕h i ∈L Then the Schubert polynomial c is the (unique) representative in e for w Λ Λ P P h i ∈L the Schubert class σw .

3.5. Astashkevich, Sadov and Kim’s presentation of qH∗(G/P ). The presentation for the quantum cohomology ring qH∗(G/P ) analogous to Borel’s presentation of H∗(G/P ) was first discovered by Astashkevich and Sadov [2] and Kim [18]. A complete proof may be found in Ciocan-Fontanine [8]. In special cases these presentations were known earlier, e.g. for Grassmannians [4, 36], and in the full flag variety case [16, 7]. The generators of qH∗(G/P ) will be the generators of the usual cohomology ring σ(j) (embedded as 1 σ(j)) along with the quantum parameters qP ,...,qP . i ⊗ i 1 k Here i, j runs through 1 j k +1 and1 i nj nj 1.Letusfor (j) ≤ ≤ ≤ ≤ − − now treat the σi and qj as independent variables generating a polynomial ring [σ(1),...,σ(k+1) ,q,...,q ]. C 1 n nk 1 k − Definition 3.1 ((q,P)-elementary symmetric polynomials). Let i Z and l (l) (l) (1) (k+1)∈ ∈ 1, 0,...,k +1 . Define elements Ei,P = Ei C[σ1 ,...,σn nk ,q1,...,qk] recursively{− as follows.} The initial values are ∈ −

( 1) (0) (l) E − = E =0 foralli, and E = 0 unless 0 i nl, i i i ≤ ≤

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(l) (l) and we set σi =0ifi>nl nl 1 and σ0 =1.For1 l k +1and0 i nl (l) − − ≤ ≤ ≤ ≤ the polynomial Ei is deﬁned by

(l) (l 1) (l 1) (l) (l 1) (l) (l) Ei = Ei − + Ei −1 σ1 + + E1 − σi 1 + σi − ··· − nl nl 1+1 (l 2) +( 1) − q E − . − l 1 i nl+nl 2 − − − − (l) If the ql aresetto0andtheσi are as in (3.1), then this recursion deﬁnes the (l) elementary symmetric polynomials ei . Remark 3.2. This is a basic recursive deﬁnition of the quantum elementary symmetric polynomials. See [8] for a host of other descriptions. And here is also one other curious one to add to this list. (j) (j) (j0 ) Order the variables σi lexicographically, so that σi <σk whenever j

(j+1) (j) (j) (j+1) nj+1 nj +1 σnj+1 nj σnj nj 1 = σnj nj 1 σnj+1 nj +( 1) − qj. − − − − − − − The qi commute with everything. Now add a central variable x and deﬁne poly- nj nj 1 (j) nj nj 1 1 (j) nomials pj(x)=x − − + σ1 x − − − + + σnj nj 1 . Then expand the product ··· − −

pm(x) pm 1(x) p1(x) · − · ··· · (j) and write the resulting coefficients in terms of increasing monomials in the σi (monomials with factors ordered in increasing fashion) by using the commutation nm d relations. Then for d nm the coefficient of x − gives the (q,P)-elementary ≤ (m) symmetric polynomial Ed . (3) (3) For example, in type A2 for the full flag variety case, the polynomials E1 , E2 (3) and E3 turn up as coefficients in

3 2 (x + x3)(x + x2)(x + x1)=x +(x1 + x2 + x3)x

+(x1x2 + x2x3 + x1x3 + q1 + q2)x +(x1x2x3 + x1q2 + x3q1).

(i) (i) Theorem 3.3 ([8, 2, 18]). The assignment σj 1 σj and qi qi 1 gives rise to an isomorphism 7→ ⊗ 7→ ⊗

(1) (k) (3.5) [ σ ,...,σ ,q,...,q ]/J ∼ qH (G/P ), C 1 n nk 1 k ∗ − −→ (k+1) (k+1) (l) where J is the ideal (E1 ,...,En ). This isomorphism takes the element Ei P for 1 l k to the special Schubert class σsn i+1 sn . ≤ ≤ l− ··· l An immediate question raised by this theorem is how to describe Schubert classes in the picture on the left-hand side of this isomorphism. This is answered by a quantum analogue of the Schubert polynomials.

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3.6. Quantum Schubert polynomials. In the case of G/B a full theory of quantum Schubert polynomials was given by Fomin, Gelfand and Postnikov [11]. This was later generalized to partial flag varieties by Ciocan-Fontanine [8]. Note that the quantum Schubert polynomials for partial flag varieties are not special cases of the full flag variety ones, due to lack of functoriality of quantum cohomology (but see Proposition 11.1). There is also a different construction of (double) quantum Schubert polynomials due to Kirillov and Maeno [20] which has been shown to give the same answer. We give the definitions following [11] and [8].

Definition 3.4 ((q,P)-standard monomials). As in Section 3.4, let P be the set (1) (k) (j) L of sequences Λ = (λ ,...,λ ) of partitions such that λ hasatmost(nj nj 1) (j) − − parts and λ nj.ToeachΛ P associate an element 1 ≤ ∈L E = E(1) E(1) E(k) E(k) [σ(1),...,σ(k) ,q ,...,q ]. Λ (1) (1) (k) (k) C 1 n nk 1 k λ λn λ λ 1 ··· 1 ··· 1 ··· nk nk 1 ∈ − − − These elements are called the (q,P)-standard monomials. Remark 3.5 (Example). The (q,B)-standard polynomials were introduced in [11]. They are the monomials of the form (1) (2) (n 1) E E E − B,j1 B,j2 B,jn 1 ··· − where 0 jl l for all l =1,...,n 1. ≤ ≤ − Let V denote the C[q1,...,qk]-module spanned by the (q,P)-standard monomials, V = [q ,...,q ] E . C 1 k C Λ Λ P ⊗ h i ∈L Then [σ(1),...,σ(k) ,q ,...,q ]=J V. C 1 n nk 1 k − ⊕ Definition 3.6 ((q,P)-Schubert polynomials). The quantum Schubert polynomial CP [σ(1),...,σ(k+1),q ,...,q ] is defined to be the unique element of V whose w C 1 n nk 1 k ∈ − P coset modulo J maps to the Schubert class σw under the isomorphism (1) (k+1) [σ ,...,σ ,q ,...,q ]/J ∼ qH (G/P ). C 1 n nk 1 k ∗ − −→ Remark 3.7 (Example). From (3.4) it follows immediately that the (q,P)-Schubert polynomial representing the top class σwP qH∗(G/P )isgivenby 0 ∈ n1 nk nk 1 P (1) (k) − − (3.6) CwP = EP,n EP,n . 0 1 ··· k 3.7. Grassmannian permutations. A Grassmannian permutation of descent m P P is an element w W d for the maximal parabolic Pd with I d = d .Aspermu- tations on 1,...,n∈ these may be characterized by { } { } w W Pd w(1) <

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The classical Schubert polynomial for wλ,d is just the Schur polynomial cwλ,d = sλ(x1,...,xd); see e.g. [27]. Therefore by the Jacobi-Trudi identity

(d) (d) (d) eλ eλ +1 eλ +c 1 10 10 ··· 10 − e(d) e(d) e(d)  λ0 1 λ0 λ0 +c 2 2− 2 ··· 2 − cwλ,d =det ,  ..   .   (d) (d)  eλ +c 1 eλ   c0 ··· ··· c0   −  where λ0 is the conjugate partition to λ and c = n d (see [28]). Repeatedly applying the identity − (m) (m+1) (m) ej = ej xm+1ej 1 − − of elementary symmetric polynomials to one column in the determinant at a time, one obtains (d) (d+1) (n 1) e e e − λ10 λ10 +1 λ10 +c 1 (d) (d+1) ··· (n 1)− e e e −  λ0 1 λ0 λ0 +c 2 2− 2 ··· 2 − cwλ,d =det .  ..   .   (d) (n 1)  eλ +c 1 eλ −   c0 ··· ··· c0   −  Expanding out this determinant gives an expression for cwλ,d as linear combination of B-standard monomials. Therefore the quantization is simply given by

(d) (d+1) (n 1) E E E − λ10 λ10 +1 λ10 +c 1 (d) (d+1) ··· (n 1)−  Eλ 1 Eλ Eλ +−c 2 C =det 20 − 20 ··· 20 − . wλ,d .  ..     (d) (n 1)  Eλ +c 1 Eλ −   c0 − ··· ··· c0    3.8. Quantum Chevalley formula. The Pieri formula for H∗(G/P )ofLascoux and Sch¨utzenberger was generalized to the qH∗(G/P ) setting by Ciocan-Fontanine in [8]. We will only need the following simpler case.

Theorem 3.8. For h l 1,...,k set τh,l = snh snl+1 1snl 1 ≤ ∈{ } P · ··· · P − − · ··· · snh 1+1 and qh,l = qh qh+1 ql.Letnj I and w W .Then − · · ··· · ∈ ∈ P P P P σ σ = <α,ω∨ >σ + qh,lσ . snj w nj wsα wτh,l α ∆+ h,l 1,...,k ∈{ } wsX∈ W P 1 hXj l k α ≤ ≤ ≤ ≤ `(ws )=∈`(w)+1 `(wτh,l)=`(w) `(τh,l) α − This is a reformulation of a special case of Theorem 3.1 in [8].

4. Quantum cohomology rings as coordinate rings 4.1. ASK-matrices. We define with some minor changes an n n matrix A[k+1] (1) (k) × with entries in C[σ1 ,...,σn nk ,q1,...,qk] introduced by Astashkevich, Sadov − and Kim in [2] and [18]. Setting nk+1 = n and n0 = 0 define first (nj nj 1) − − ×

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(j) (nj nj 1) matrices D by − − (1) (1) (1) σ σ ... σ (j) 1 2 n1 0 0 σ − − − nj nj 1 0 0 . ··· . − .− −  ··· ···  . . .  (1) . . (j) . . . D = . . and D = . .  . .  . . (j)   . .  . . σ2   . .   − (j)    0 ... 0 σ   0 0   − 1   ··· ···     [l]  for 2 j k +1,andlet D be the nl nl block matrix made up of diagonal ≤ (1)≤ (l) × blocks D ,...,D . Furthermore define nl nl matrices × 0 . .. nl [l] 1  [l] nm+1 nm nm 1+1 nm+1 f = and Q := ( 1) − qmδi − δj . . . − i,j=1  .. ..     10   Then set  A[l] := f [l] + D[l] + Q[l]. The coefficients of the characteristic polynomials of the A[l] satisfy precisely the (l) same recursion as the (P, q)-standard symmetric polynomials EP,i. Explicitly, we have [l] nl (l) nl 1 (l) det(λ Id A )=λ + E λ − + + E . − P,1 ··· P,nl (k+1) (k+1) In particular the relations E1 = = En = 0 of the quantum cohomology ring are equivalent to ··· (4.1) det(λ Id A[k+1])=λn. − [k+1] Let us call the matrices in gln of the same form as A (with the same pattern of 0and1entries)ASK-matrices. They form an affine subspace P in gln.Let P be the (nonreduced) intersection, A N

P = P , N A ∩N of P with the nilpotent cone in gl . Its coordinate ring is denoted ( P ). A N n O N Then (4.1) implies that the map ( ) [ σ(1),...,σ(k) ,q ,...,q ]de- P C 1 n nk 1 k O A −→ − ﬁned by A[k+1] induces an isomorphism (1) (k) (4.2) ( ) ∼ [ σ ,...,σ ,q ,...,q ]/J. P C 1 n nk 1 k O N −→ − The statement of Theorem 3.3 may therefore be interpreted as

(4.3) ( P ) ∼ qH∗(G/P ). O N −→ 4.2. Peterson’s theorem. All the affine varieties Spec(qH∗(G/P )) turn out to be most naturally viewed as embedded in the flag variety (or in general in the Langlands dual flag variety) where they patch together as strata of one remarkable projective variety called the Peterson variety. This is the content of Dale Peterson’s theorem which we will deduce here explicitly for the type A case. Definition 4.1. Let b++ := g and π++ : g b++ be the projection α ∆+ Π α ∈ \ → along weight spaces. Let f gl be the principal nilpotent f [k+1] from above. P n Then the equations ∈ ++ 1 π (Ad(g− ) f)=0 ·

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deﬁne a closed subvariety of G invariant under right multiplication by B−.Thus they deﬁne a closed subvariety of G/B−. This subvariety is the Peterson variety for type A.Loosely, can be described by Y Y

1 = gB− G/B− Ad(g− ) f b− Ceαi . Y ( ∈ · ∈ ⊕ ) i I X∈ n Let v1,...,vn be the standard basis of V = C .Then vi1 vij 0 < { ∧···∧ | i1

v ωm = vm+1 vn. − ∧···∧ m Deﬁne rational functions Gi = Gsm i+1 sm on G/B− in terms of matrix coeﬃ- cients of the fundamental representations− ··· by

m (g v ωm sm i+1 sm v ωm ) (4.4) Gi (gB−)=Gsm i+1 sm (gB−):= · − | − ··· · − . − ··· (g v ωm v ωm ) · − | − m Note that sm i+1 sm v ωm = vm i+1 vm+2 vn and Gi (gB−)maybe written down− simply··· as a· quotient− of− two∧ (n m)∧···∧(n m)minorsofg. − × − Theorem 4.2 (D. Peterson). (1) For any parabolic subgroup WP W with ⊂ longest element wP deﬁne P as (nonreduced) intersection by Y + P := B wP B−/B− . Y Y∩ Then on points these give a decomposition

(C)= P (C). Y Y P G (2) For each parabolic P there is a unique isomorphism

(4.5) ( P ) ∼ qH∗(G/P ), O Y −→ which sends G to σP . snj i+1 snj sn i+1 sn − ··· j − ··· j m Remark 4.3. If P is the parabolic subgroup, then Gj is a well-deﬁned (regu- + P lar) function on the Bruhat cell B wP B−/B− precisely in the case m I = ∈ n1,...,nk .Infactwehave { } k + ( ni) B wP B−/B− ∼ C i=1 , −→ P n1 n1 n2 nk gB− (G (gB−),...,G (gB−),G (gB−),...,G (gB−)), 7→ 1 n1 1 nk or in other words,

+ n1 nk (B wP B−/B−)=C[G1 ,...,Gn ]. O k n1 nk Let P C[G1 ,...,Gn ] denote the ideal deﬁning P . J ⊂ k Y Proof. (1) is proved in [33]. See also Lemma 2.3 in [35]. We will deduce (2) very explicitly from the ASK presentation. Begin by deﬁning a particular section

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+ + u : B wP B−/B− U of the map x xwP B− in the other direction. For → 7→ (l) l =0,...,k we have n (nl+1 nl) matrices U deﬁned by × − Gnl 1 Gn1 Gn1 Gn1 nl 1 2 n1 1 . . ···. . − . .. n1 .. .   01G1 .  ...... n1  .  . . G n  2  G l Gnl   .   2 nl   . n1   . .   . G   nl . .   1  G . .   1   1  (0)   (l)  . . .  U =  0  ,U =  1 .. .. .       .   .   .   0 .. Gnl Gnl     1 2     .     .1. Gnl     1     1  . .    . .   .     .  0 0       0 0   ··· ···     ··· ···  where 1 l k. Then the matrix ≤ ≤

 

u =  U (0) U (1) U (k)   ···                deﬁnes a map u : B+w B /B U +. It follows using Remark 4.3 that this map P − − is indeed a section. That is, → + gB− = u(gB−)wP B−, for gB− B wP B−/B−. ∈ Consider the matrix

˜ 1 n1 nk A = u− fu gln(C[G1 ,...,Gn ]). ∈ k A direct computation shows that modulo the ideal P defining P the matrix A˜ J n1 nkY is an ASK matrix (i.e. it is an ASK matrix over C[G1 ,...,Gn ]/ P ). Also it is k J clear that the characteristic polynomial of A˜ satisfies det(λ Id A˜)=λn,sinceA˜ + − is conjugate to f. Therefore the morphism B wP B−/B− gln defined by A˜ restricts to a morphism −→ ˜ (4.6) A P : P P |Y Y →N from P to the variety of nilpotent ASK-matrices. Y n (1) (k+1) For the inverse define a map ψ : [G 1 ,...,Gnk ] [σ ,...,σ ,q,...,q ] C 1 nk C 1 n nk 1 k nj (j) → − by ψ(Gi )=Ei . Applying ψ to the entries of u we obtain a matrix uE with entries in [σ(1),...,σ(k+1) ,q,...,q ]. Then the recursive definition of the E(j) C 1 n nk 1 k i translates into the identity−

1 [k+1] (4.7) uE− fuE = A + M

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(1) (k+1) [k+1] of matrices over C σ1 ,...,σn nk ,q1,...,qk ,whereA is the matrix deﬁned − in Section 4.1, andhM the n n matrix giveni by × (k+1) 0 0 En ··· ··· . . (k+1) . . En 1  . . −. M = . . .  . . . .  . .  . . (k+1) . . E2   (k+1) 0 0 E   ··· ··· 1    This identity implies that ψ induces a map of quotient rings

˜ (1) (k+1) (k+1) (k+1) ψ : ( P ) C σ1 ,...,σn nk ,q1,...,qk /(E1 ,...,En ). O Y −→ − h i This map together with (4.2) deﬁnes an inverse P P to (4.6). Thus ψ˜ is an N →Y isomorphism and everything follows from Theorem 3.3. Remark 4.4. Actually, Peterson’s results are more generally stated over the integers. And here in particular the analogous theorem over Z holds, with the exact same proof. We have stayed over C in our presentation since that is all we will require.

4.3. Toeplitz matrices. The stabilizer of f under conjugation by U − is precisely the (n 1)-dimensional abelian subgroup of lower-triangular unipotent Toeplitz matrices,− 1 a1 1     .. X := (U −)f = x = a2 a1 . a1,...,an 1 C  .    − ∈    . .     . .. 1      an 1 a2 a1 1    − ···       Let us take the matrix entries a1,...,an 1 as coordinates on X, thereby identifying  −  (X)=C[a1,...,an 1]. For 1 m n 1let∆m (X) be deﬁned by O − ≤ ≤ − ∈O am am 1 − ··· .. .. . n m am+1 . . . (4.8) ∆m =det(aj i+m)i,j−=1 = , − ...... am 1 − an 1 am+1 am − ··· where a =1anda =0ifl<0. Let 0 l + + XP = X B wP w0B . ∩ We recall the following explicit description of the XP X. ⊂ Lemma 4.5 ([35] Lemma 2.5). As a subset (not subvariety) of X, XP is described by

XP = u X ∆i(u) =0 i n1,...,nk . { ∈ | 6 ⇐⇒ ∈{ }}

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Note that the map n 1 ∆=(∆1,...,∆n 1):X C − − → 1 n 1 has the property ∆− (0) = 0 (in fact over any field). And ∆∗ : (C − )= { } O C[z1,...,zn 1] C[a1,...,an 1] is homogeneous, if the generators are taken with suitable degrees.− → Therefore ∆− is a finite morphism; see e.g. [17]. Theorem 4.6 (D. Peterson). (1) Define P := B−w0B−/B− and P := . X Y∩ X X∩Y Then the isomorphism U − B−w0B−/B− defined by u uw0B− iden- → 7→ tifies X with and also XP with P for each parabolic P . X X 1 (2) The map (4.5) induces an isomorphism of ( P ) with qH∗(G/P )[q1− , 1 O X ...,qk− ] giving 1 1 (4.9) (XP ) ∼ ( P ) ∼ qH∗(G/P )[q− ,...,q− ]. O ←−OX −→ 1 k In particular, each P is open dense in P . X Y For a proof of this when P = B see Theorems 8 and 9 in [21]. The general case is analogous, and for (2) see also the proof of Lemma 5.1 below.

5. The quantum parameters as functions on XP P After applying (4.9), the quantum parameters qj may be expressed (up to taking some roots) in terms of the functions ∆i from (4.8). In the full ﬂag variety case, that is on XB where all ∆i are nonvanishing, Kostant [21] has given the following formula: B ∆j 1∆j+1 − (5.1) qj = 2 . (∆j ) This generalizes as follows to the partial ﬂag variety case. P Lemma 5.1. Let the quantum parameters qj be regarded as functions on XP via 1 1 the isomorphism (XP ) = qH∗(G/P )[q− ,...,q− ] from (4.9).Then O ∼ 1 k nj+1 nj nj nj 1 P (nj nj 1)(nj+1 nj ) (∆nj 1 ) − (∆nj+1 ) − − q − − − = − . j nj+1 nj 1 (∆nj ) − − The proof of this lemma which we give below is an adaptation of Kostant’s proof of the formula (5.1).

Proof. Let y XP .Thenyw0B− P and we have ∈ ∈X + yw0B− = uwP B− for some u U . ∈ 1 Without loss of generality u may be chosen such that Ad(u− ) f is an ASK-matrix + · A P .SinceuwP B− = yw0B−, we can ﬁndu ¯ U and t T such that ∈N ∈ ∈ 1 1 1 y = uwP− w0− tu¯− . We have 1 1 1 1 (5.2) Ad(¯u− ) f =Ad(tw0w− u− y) f =Ad(tw0w− ) A. · P · P · The right-hand side may be expanded to give n 1 1 − Ad(t− w0wP ) A = mifi + higher weight space terms, · i=1 X

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where explicitly

nj+1 nj P ( 1) − αi(t)qj (y)ifi = n nj, mi = − − αi(t)ifn i/ n1,...,nk . ( − ∈{ } This follows from the isomorphisms in Section 4. On the other hand sinceu ¯ U +, ∈ the left-hand side of (5.2) implies that all the mi must be equal to 1. Therefore we have the identities

nj+1 nj P 1 αn nj (t)=( 1) − qj (y)− j =1,...,k, − − αi(t)=1 n i/ n1,...,nk . − ∈{ } Thus t is determined up to a scalar factor λ by the qi(y)’s. Let Tk+1 be the (n nk) (n nk) identity matrix and Tj the (nj nj 1) (nj nj 1)matrix − × − − − × − − 1

n nj . Tj =( 1) − qj(y)qj+1(y) qk(y) .. , − ···   1   for 1 j k.Thent may explicitly be described by  ≤ ≤

Tk+1  

 Tk  t = λ   .    ..   .       T1    Now  

∆nj (y)=(y v1 vn nj +1 vnj +1 vn) · ∧···∧ − | ∧···∧ 1 1 1 =(uwP− w0− tu¯− v1 vn nj vnj +1 vn) · ∧···∧ − | ∧···∧ 1 1 = ωn nj (t)(uwP− w0− v1 vn nj vnj +1 vn) − · ∧···∧ − | ∧···∧ n nj nk nj nk 1 nj nj+1 nj = λ − qk(y) − qk 1(y) − − qj+1(y) − − ··· and the identity follows. 6. Total positivity

AmatrixA in GLn(R) is called totally positive (or totally nonnegative)ifall the minors of A are positive (respectively nonnegative). In other words A acts k by positive or nonnegative matrices in all the fundamental representations Rn (with respect to their standard bases). These matrices clearly form a semigroup. The concept of totally positive matrices is in this sense more fundamentalV than the naive concept simply of matrices with positive entries, which overemphasizes the standard representation. Total positivity for GLn was mainly studied in and around the 1950’s by Schoenberg, Gantmacher-Krein, Karlin and others, and has relationships with diverse applications such as oscillating mechanical systems and planar Markov processes. More recently, G. Lusztig [24] extended the theory of total positivity to all reductive algebraic groups. His extension rests around a beautiful connection with

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canonical bases (for ADE type). This point of view on total positivity was part of the motivation for the main result of this paper, stated in the next section. It goes as follows. Let us consider the lower uni-triangular matrices U − (staying with G = GLn, to avoid making further definitions). Let U −0 be the set of totally nonnegative ≥ matrices in U −. And define the “totally positive” part of U − by + + U>−0 := U −0 B w0B . ≥ ∩ Now the canonical basis of the quantized universal enveloping algebra q− defined by Lusztig and Kashiwara gives rise, after dualizing and taking the classicalU limit, toabasis of the coordinate ring (U −). Lusztig proved that the canonical basis has positiveB structure constants forO multiplication and comultiplication, using his geometric construction of −. This is the main ingredient for the following theorem. Uq Theorem 6.1 ([25]; see also Section 3.13 in [26]). Suppose u U −(R).Then ∈ u U − b(u) > 0 for all b . ∈ >0 ⇐⇒ ∈B The functions b are matrix coefficients of U − in irreducible representations ∈B of GLn with respect to canonical bases of these representations (obtained from the canonical basis of q−). This theorem is a reformulation of a result from [25], which holds for any simplyU laced reductive algebraic group. Philosophically, U − is a variety with a special basis on its coordinate ring (even a Z-basis if we were to define U − over the integers) which moreover has nonnegative integer structure constants. And by Lusztig’s theorem, the question regarding which u U − have the property that all b are positive on u has a very nice ∈ ∈B answer, namely the totally positive part of U −. We ask the same question for the components P of the Peterson variety (or Y equivalently for the XP U −), whose coordinate rings are naturally endowed with Schubert bases also with⊂ positive structure constants (as enumerative Gromov- Witten invariants). And remarkably we discover total positivity again in the answer. The corollary, the parameterization result for totally nonnegative finite Toeplitz matrices stated in the introduction, also illustrates a common feature in total posi-

tivity. For example there are natural parameterizations of U>−0 (introduced in [24]) which are related to combinatorics of the canonical basis and have been studied extensively. See e.g. [12] for a survey.

7. Statement of the main theorem

The varieties we are studying lie either inside GLn or G/B−. By their real points we mean coming from their split real form, GLn(R) and the real flag variety. We consider the real points always to be endowed with the usual Hausdorff topology coming from R. The positive parts will be semi-algebraic subsets of the real points. Following [24], the totally positive part (G/B−)>0 of G/B− is defined as the + image of U under the quotient map G G/B−. By a result in [24] this agrees >0 → with the image of U>−0w0B−.So + (G/B−)>0 = U>0B−/B− = U>−0w0B−/B−.

The totally nonnegative part (G/B−) 0 is the closure of (G/B−)>0 inside the real ﬂag variety. ≥

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Using Peterson’s isomorphisms (4.5) and (4.9) we may evaluate elements of qH∗(G/P ) as functions on the points of P and P ,orXP . Y X Definition 7.1. Let the totally positive part of P be defined as Y P,>0 := P (R) (G/B−) 0 B−w0B−/B−. Y Y ∩ ≥ ∩ This automatically lies in P ,sowealsoset P,>0 := P,>0. Finally, compatibly X X Y with this definition, set XP,>0 := XP (R) U −0. ∩ ≥ We define the Schubert-positive parts of P , P and XP , also compatibly with the various morphisms between them, by Y X Schub P P P,>0 := x P (R) σw (x) > 0allw W , Y { ∈Y | ∈ } Schub P P P,>0 := x P (R) σw (x) > 0allw W , X { ∈X | ∈ } Schub P P XP,>0 := x XP (R) σw (x) > 0allw W . { ∈ | ∈ } These are all semi-algebraic subsets of the real points of P , P and XP , respectively. Y X

P P P k Theorem 7.2. (1) The ramified cover π = π =(q1 ,...,qk ): P (C) C restricts to a bijection Y → P Schub k π>0 : P,>0 R>0. Y → Schub P (2) P,>0 lies in the smooth locus of P , and the inverse of the map π>0 : YSchub k Y R>0 is analytic. YP,>0 → (3) The two notions of positivity agree. That is, Schub = P,>0, YP,>0 Y Schub Schub Schub and also = = P,>0 and X = XP,>0. YP,>0 XP,>0 X P,>0 Remark 7.3. We conjecture that all the analogous results to those stated in Theo- rem 7.2 should hold true in general type. The stabilizer of the principal nilpotent f in that case should have a totally nonnegative part with a cell decomposition coming from the Bruhat decomposition. And there should be an analogous relationship with Schubert bases for quantum cohomology rings qH∗(G∨/P ∨) of partial flag varieties of the Langlands dual group, via the Peterson variety for general type. B The bianalyticity of π>0 is equivalent to the nonvanishing on XB,>0 of the quantum Vandermonde function introduced by Kostant [21], Section 9, and [22]. This function on XB is expressed as the determinant of a matrix whose entries are alter- nating sums of minors. There is no obvious reason why it should be nonvanishing on XB,>0. But Theorem 7.2(2), which is proved by completely different means, implies that the quantum Vandermonde must take either solely positive or negative values on XB,>0. We expect that the values will always be positive, but have checked this so far only in very low rank cases. We now deduce the corollary stated as Proposition 1.2 in the introduction.

Corollary 7.4. Let X 0 denote the semi-algebraic subset of X(R) of totally nonnegative unipotent lower-triangular≥ Toeplitz matrices. Then the restriction n 1 ∆ 0 : X 0 R −0 ≥ ≥ −→ ≥ of ∆:=(∆1,...,∆n 1) is a homeomorphism. −

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Proof of the corollary. By Theorem 7.2 we have homeomorphisms P P k (q1 ,...,qk ):XP,>0 R>0, → P one for each parabolic P . By Lemma 5.1 the qj ’s are related to the ∆nj ’s by a k transformation which is continuously invertible over R>0. By this observation, and since X 0 = XP,>0,wehavethat ≥ n 1 F ∆ 0 : X 0 R −0 ≥ ≥ −→ ≥ is bijective. So ∆ 0 is continuous, bijective, and a homeomorphism onto its image ≥ 1 when restricted to any XP,>0. Since ∆ is ﬁnite it follows that ∆−0 is also continuous. ≥

8. Grassmannians etc. The quantum cohomology rings of Grassmannians have been studied much more extensively than those of partial ﬂag varieties; see for example [4, 15, 36, 37]. The Grassmannian case can be regarded as a kind of toy model for this paper. The main results stated in the previous section generalize properties from the Grass- mannian case, which were studied by elementary means, basically playing with Schur polynomials, in [35]. We will brieﬂy recall what happens in that case.

(1) Let d,n be the transpose of X¯P (upper-triangular rather than lower- V d triangular Toeplitz matrices), notation as in [35]. And let red( d,n)bethe O V reduced coordinate ring of d,n. Then an incarnation of Peterson’s theorem V says red( d,n) = qH∗(G/Pd). O V ∼ (2) The points of d,n are those V 1 a1 ad 0 ··· ..  1 a1 .  . .  .. .. a  u =  d  . .   .. a .   1   1 a   1  1      for which d 2πi d d 1 (mj ) p(x)=x + a1x − + + ad = x + ze n ··· j=1 Y for some z C and integers 0 m1 <

where sλ is the Schur polynomial associated to λ.

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2πi d 1 d 1 d 1 − − +1 − (4) Let ζ = e n ,andsetu 0(t)=u(tζ− 2 ,tζ− 2 ,...,tζ 2 ). Then ≥ ∼ u 0 : R 0 ( d,n) 0 : t u 0(t) ≥ ≥ −→ V ≥ 7→ ≥ is a homeomorphism, where ( d,n) 0 denotes the totally nonnegative ma- V ≥ trices in d,n. (5) The valuesV of the Schubert classes on the u(t) are given by a closed (hook- length) formula, which explicitly shows them to be positive for t>0. d+1 n (6) The quantum parameter q is given by q(u(x1,...,xn)) = ( 1) x1 .In n − particular, q(u 0(t)) = t . ≥ From the proof of Theorem 4.2, in particular from inspection of the matrix u introduced in (4.2), we see directly that = via u uw B . Notice also d,n ∼ Pd Pd − that (4) and (6) give the parameterizationV by quantumY parameters7→ of Theorem 7.2 in this special case. Peterson has announced in [33] that all the quantum cohomology rings qH∗(G/P ) are reduced. To prove this amounts to showing that the element w W P σwσPD(w) ∈ is a nonzerodivisor in qH∗(G/P ) (see also [1]). This is because, for example, if σ P ∈ qH∗(G/P ) is nilpotent, then so are all µσ for µ qH∗(G/P ), and the corresponding ∈ multiplication operators Mµσ on qH∗(G/P ) have vanishing trace. But computing

these traces by Poincaré duality gives tr(Mµσ)= µσ w W P σw σPD(w) q =0 ∈ and therefore σ w W P σwσPD(w) = 0. It is in fact sufficient to show that · ∈ P w W P σwσPD(w) is generically nonvanishing on P (see Lemma 10.1). Apart∈ from theP Grassmannian case where everythingY is very explicit, and the P full flag variety case treated in [21], where the Peterson variety B is irreducible, I Y do not know a proof that w W P σwσPD(w) is generically nonvanishing. But with the help of the explicit results∈ above we can prove at least the following lemma which will come in handyP later. P Lemma 8.1. The element of ( P ) defined by σwP takes nonzero values on an O Y 0 open dense subset of P (C). Y P Proof. Since P is open dense in P , it suffices to show that σwP is nonzero on an X Y 0 open dense subset of P (C). By Theorem 4.6 we may furthermore replace P by X P ¯X XP . So let us identify the Schubert classes σw with rational functions on XP and prove that the top one is generically nonvanishing. Pm Let Pm denote the maximal parabolic with I = m and let C be an irre- ¯ {P } ducible component of the closure XP = P P XP 0 .IfI = i1,...,ik ,then 0⊇ { } ¯ k (∆n1 ,...,∆nk ):FXP (C) C −→ n 1 is finite, as pullback of the finite map (∆1,...,∆n 1):X(C) C − .There- − −→ fore the restriction of (∆n1 ,...,∆nk )toC is surjective and C intersects all of the subvarieties X¯ of X¯ . Pni P Now in qH∗(G/P )wehave P P P P P σw = σs1 sn σs1 sn σs1 sn . ··· 1 · ··· 2 · ··· · ··· k Let x XP . Then tracing through Peterson’s isomorphisms gives ∈ P nj σs1 sn (x)=Gnj (xw0B−), ··· j nj where Gn is as in (4.4). This function extends to XP X¯P and is seen to be j nj ⊂ nonvanishing there using the explicit description of X (see (2) above). Since Pnj

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¯ P any irreducible component of XP meets XPn ,wehavethatσs1 sn is generically j ··· j nonzero on X . The same holds therefore for σP as the product of the σP . P wP s1 sn ··· j 9. Proof of Theorem 7.2(1) P k We must ﬁrst check that π>0 actually takes values in R>0. This follows from the following observation.

Pn Lemma 9.1. Let Pn be the maximal parabolic deﬁned by I j = nj ,andset j { } Pnj P v = w W nj to be the longest element. Then v W P and we have the 0 ∈ ∈ following relation in qH∗(G/P ): P P P P σs σv = qj σvτ , nj · j,j

where τj,j = snj snj+1 1snj 1 snj 1 +1. ··· − − ··· − Proof. Let α be a positive root such that <α,ω∨ >=0.Soα = αh + + αl for nj 6 ··· some h nj l. By the Chevalley formula, σvs appears in the expansion of the ≤ ≤ α product only if `(vsα)=`(v)+1. If h

sends both αnh and αnl = snh snl+1 1(αnl 1) to negative roots we must have h = l = j. So by quantum Chevalley’s··· rule− the− only possible quantum contribution P P P P to the product σs σv is qj σvτ . It follows by a direct check that this term nj · j,j does indeed appear (as of course it must, since the product cannot be zero by the same arguments as in Lemma 8.1). P Now we would like to show that π>0 is actually surjective. For this fix a point k P Q (R>0) and consider its fiber under π = π . We may regard ∈ P P RQ := qH∗(G/P )/(q Q1,...,q Qk) 1 − k − 1 as the (possibly nonreduced) coordinate ring of π− (Q). Note that RQ is a finite- dimensional algebra with basis given by the (image of the) Schubert basis. We will P use the same notation σw for the restriction of a Schubert basis element to RQ.

Lemma 9.2. Suppose µ RQ is a nonzero simultaneous eigenvector for all linear ∈ operators RQ RQ which are deﬁned by multiplication by elements in RQ.Then → 1 there exists a point p π− (Q) such that (up to a scalar factor) ∈ P P µ = σw (p) σPD(w). w W P X∈ Proof. Consider the algebra homomorphism

RQ C −→ which takes σ RQ to its eigenvalue on the eigenvector µ. This deﬁnes the C-valued 1∈ point p in π− (Q). Now let us write µ in the Schubert basis, PD(w) µ = mwσ ,mw C. ∈ w W P X∈

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P For σ RQ,let σ Q C denote the coefficient of σwP in the Schubert basis ∈ h i ∈ 0 expansion of σ. Then by quantum Poincaré duality we have w w w w mw = σ µ = σ (p) µ = σ (p) µ = σ (p) m1. h · iQ h iQ h iQ Here m1 must be a nonzero scalar factor (since µ = 0), and the lemma is proved. 6 We continue the proof of Theorem 7.2(1) our immediate aim being to find a 1 Schubert positive point p0 in the fiber π− (Q). Set P σ := σ RQ. w ∈ w W P X∈ Suppose the multiplication operator on RQ defined by multiplication by σ is given by the matrix Mσ =(mv,w)v,w W P with respect to the Schubert basis. That is, ∈ P P σ σ = mv,wσ . · v w w W P X∈ k Then since Q R>0 and by positivity of the structure constants it follows that ∈ Mσ is a nonnegative matrix. Furthermore let us assume the following lemma (to be proved later).

Lemma 9.3. Mσ is an indecomposable matrix.

Given the indecomposable nonnegative matrix Mσ, by Perron-Frobenius theory (see e.g. [31], Section 1.4) we know the following. The matrix Mσ has a positive eigenvector µ whichisuniqueuptoascalar (positive meaning it has positive coeﬃcients with respect to the standard basis). Its eigenvalue, called the Perron-Frobenius eigenvalue, is positive, has maximum absolute value among all eigenvalues of Mσ, and has algebraic multiplicity 1. The eigenvector µ is unique even in the stronger sense that any nonnegative eigenvector of Mσ is a multiple of µ. Suppose µ is this eigenvector chosen normalized such that µ Q =1.Thensince the eigenspace containing µ is 1-dimensional, it follows that µhisi a joint eigenvector for all multiplication operators of RQ. Therefore by Lemma 9.2 there exists a 1 p0 π− (Q) such that ∈ P P µ = σw (p0) σPD(w). w W P X∈ P P Schub Positivity of µ implies that σw (p0) R>0 for all w W . Hence p0 . ∈ ∈ ∈YP,>0 Also the point p0 in the ﬁber with this property is unique. Therefore we have shown modulo Lemma 9.3 that Schub k (9.1) P,>0 R>0 Y −→ is a bijection. Finally we complete the proof of Theorem 7.2(1) by proving the lemma.

Proof of Lemma 9.3. Recall that σ = w W P σw. Suppose indirectly that the ∈ P matrix Mσ is reducible. Then there exists a nonempty, proper subset V W P ⊂ such that the span of σv v V in RQ is invariant under Mσ. We will derive a contradiction to this statement.{ | ∈ } First take any element v V . Then the top class σwP occurs in σ σv with ∈ 0 · coeﬃcient 1 by quantum Poincar´e duality. Therefore we have wP V . 0 ∈

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Next we deduce that 1 V . Suppose not. Then the coeﬃcient of σ1 in σw σwP ∈ · 0 must be zero for all w W P ,orequivalently ∈

σw σwP σwP =0 · 0 · 0 Q D E P for all w W . But this also implies σw σwP σwP = 0, since the latter ∈ · 0 · 0 q P D E k is a nonnegative polynomial in the qi ’s which evaluated at Q R>0 equals 0. ∈ Therefore σwP σwP =0inqH∗(G/P ) by quantum Poincaré duality. This leads to 0 · 0 a contradiction with Lemma 8.1, that the element σ P is generically nonzero as a w0 1 function on P . So V mustY contain 1. Since V isapropersubsetofW P , we can find some w/V . In particular, w = 1. It is a straightforward exercise that given 1 = w W P there∈ exist α ∆P and6 v W P such that 6 ∈ ∈ + ∈ w = vsα, and `(w)=`(v)+1. P P Now α ∆ means there exists nj I such that <α,ω∨ >= 0. And hence by ∈ + ∈ nj 6 the (classical) Chevalley formula we have that σs σv has σw as a summand. But nj · if w/V this implies that also v/V ,sinceσ σv would have summand σs σv ∈ ∈ · nj · which has summand σw. Note that there are no cancellations with other terms by positivity of the structure constants. By this process we can find ever smaller elements of W P which do not lie in V until we end up with the identity element, so a contradiction.

10. Proof of Theorem 7.2(2)

We need to show that P,>0 lies in the smooth locus of P . Consider the map Y Y (k+1) (k+1) n n E = (E ,...,En ) : C C[q1,...,qk] , 1 −→ h i k and its evaluation at Q =(Q1,...,Qk) C , ∈ n n EQ =evQ E : C C . ◦ −→ n (m) Here the coordinates 1,...,n of the source C are the σi =: nm 1+i.Let − ∂E(k+1) J := det i [σ(1),...,σ(k+1) ,q,...,q ], E C 1 n nk 1 k ∂j ∈ − !i,j

which at q = Q evaluates to J =det ∂(EQ)i [σ(1),...,σ(k+1)], the EQ ∂ C 1 n nk j i,j ∈ −

Jacobian of EQ. Let us also denote by JE and JEQ the classes these functions deﬁne via (3.5) in qH∗(G/P )andinRQ = qH∗(G/P )/(q1 Q1,...,qk Qk), respectively. − − P 1 P 1 Note that the zero-ﬁber of EQ equals (π )− (Q), and a point p (π )− (Q)isa ∈ smooth point of P if the JEQ (p) = 0. The smoothness assertion of Theorem 7.2(2) follows from theY following lemma.6

1W. Fulton and C. Woodward [14] have in fact recently proved that no two Schubert classes in qH∗(G/P ) ever multiply to zero. This result can also be recovered as a corollary of Theorem 7.2, since the product of two Schubert classes must take positive values on P,>0 and hence cannot Y be zero.

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Lemma 10.1. The element JE qH∗(G/P ) is expressed in terms of the Schubert basis by ∈

(10.1) JE = σwσPD(w). w W P X∈ Proof. The main ingredient for this lemma is a result from [6] or [36]. But we begin P by checking the normalization. Following [18] we have JE = JE = W . h iq Q Q | | In fact, in terms of the Chern roots J is expressed explicitly by E0 (1) (k+1) J = (x x ) [x ,...,x ]WP [σ ,...,σ ], E0 i j C 1 n = C 1 n nk − ∈ ∼ − (i,j),i nm

and hence represents the Euler class in H∗(G/P ). Therefore, JE0 0 = G/P χG/P = P h i W .Butbyitsdegree JE = JE is a constant, independent of Q. | | h iq Q Q R Now given the normalization as above, [36, Proposition 4.1] says that ResEQ (˜η)= n n η ,where˜η (C )andη RQ = (C )/((EQ)1,...,(EQ)n)istheclass h iQ ∈O ∈ ∼ O represented byη ˜. Putting this identity together with [36, Lemma 4.3] we obtain the identity

tr(Mκ)= κJE ,κRQ, Q Q ∈ where Mκ is the multiplication operator by κ on RQ. On the other hand this trace may be computed from Poincar´e duality by

tr(Mκ)= κ σwσPD(w) . * w W P + X∈ Q Comparing the two expressions for all Q and all κ it follows that

(10.2) JE = σwσPD(w) w W P X∈ as required.

P It remains to prove that the inverse to π>0 is analytic. This follows from the following lemma.

Lemma 10.2. Choose local coordinates y1,...,yk in a neighborhood of p0 XP,>0. P ∈ ∂qi The Jacobian =det is nonzero at the point p0. J ∂yj P Proof. Let Q = π (p0). Let R = qH∗(G/P )andletI R be the ideal (q1 Q1, ⊂ − ...,qk Qk). The Artinian ring RQ = R/I is isomorphic to the sum of local rings − RQ = x (πP ) 1(Q) Rx/IRx.Andforx = p0 the local ring Rp0 /IRp0 corresponds ∼ ∈ − in RQ to the Perron-Frobenius eigenspace of the multiplication operator Mσ from the aboveL proof. Since this is a one-dimensional eigenspace (with algebraic mul-

tiplicity one), we have that dim(Rp0 /IRp0 ) = 1. Therefore any nonzero element r Rp /IRp has the property r(p0) = 0. But the Jacobian gives a nontrivial ∈ 0 0 6 J element in Rp0 /IRp0 , since its residue at p0 with respect to I is nonzero (see e.g. Chapter 5 in [17]).

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11. The Schubert classes as rational functions on Y To compare Schubert-positivity with total positivity we need to make a closer study of the functions deﬁned by the Schubert classes. The following proposition is one of the most striking features of the Peterson variety picture of quantum cohomology. As far as I understand, it can be extracted from Peterson’s statements in [33] or [34] on the connection between each of the qH∗(G/P )’s and the homology of the loop group ΩK of the compact real form of G. We will give a direct proof here for type A. Proposition 11.1 (D. Peterson). Let w W and let σB be the corresponding ∈ w Schubert class regarded as a function on B .Letσw be the rational function on the Y B Peterson variety = B deﬁned by σw B = σw .Thenσw is regular on P if w W P . AndY in thatY case we have |Y Y ⊂Y ∈ e e P e σw P = σw ( P ). |Y ∈OY Our proof of this proposition uses the following lemma. e Lemma 11.2. Suppose that j IP and j +1 / IP . Then the rational function B j j 1 j 1 j ∈ ∈ qj Gi 1Gl −1 Gi−2 Gl vanishes on P . − − − − Y P Proof. Let gB− P .Then(g v ωm v ωm ) = 0 precisely if m I , and in this case ∈Y · − | − 6 ∈

m (g v ωm sm i+1 sm v ωm ) Gi (gB−)= · − | − ··· · − (g v ωm v ωm ) · − | − B P is well deﬁned. Also (5.1) implies that qm is well deﬁned on P whenever m I , and is given by Y ∈

B (g v ωm 1 v ωm 1 )(g v ωm+1 v ωm+1 ) − − − − − − qm(gB−)= · | · 2 | . (g v ωm v ωm ) · − | − Therefore we have

B j j 1 j 1 j (g v ωj+1 v ωj+1 ) − − − − qj Gi 1Gl 1 Gi 2 Gl = · | 3 − − − − (g v ωj v ωj ) · − | − (g v ωj sj i+2 sj v ωj )(g v ωj 1 ,sj l+1 sj 1 v ωj 1 ) · · − | − ··· · − · − − − ··· − · − − (g v ωj sj i+2 sj 1 v ωj 1 )(g v ωj 1 sj l+1 sj v ωj ) . − · − | − ··· − · − − · − − | − ··· · − P P Now (j +1) / I and j I imply that (g v ωj+1 v ωj+1 ) = 0 while ∈ ∈ · − | − (g v ωj v ωj ) =0on P . Hence the above expression vanishes on P . · − | − 6 Y Y

Proof of Proposition 11.1. If w = sh i+1 sh 1sh,thenwehave − ··· − h σsh i+1 sh = Gsh i+1 sh = Gi − ··· − ··· and the Proposition holds in this case by Theorem 4.2. Let w W P and consider e P ∈ the quantum Schubert polynomial Cw written as a linear combination of (q,P)- standard monomials as in Section 3.6. So

CP,w = mΛEP,Λ,mΛ C. ∈ Λ P X∈L

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(j) nj In EP,Λ replace each factor EP,i with the corresponding rational function Gi to deﬁne GΛ. Then as a function on P , Y P σw = mΛGΛ P . |Y Λ P X∈L We now use the “quantum straightening identity”, [11], Lemma 3.5,

(j) (j) (j+1) (j) (j) (j+1) (j) (j) (j) (j 1) (j 1) (j) Ei El = Ei El Ei 1El+1 + Ei 1El+1 + qj Ei 1El −1 Ei −2 El , − − − − − − − P (j) (j) to rewrite σw . Note that a factor Ei El may occur in a (q,P)-standard monomial P P (j) nj EΛ only if j I and j +1 / I . If we replace the Ei ’s by Gi in the above identity and apply∈ Lemma 11.2,∈ then we get

nj nj nj+1 nj nj nj+1 nj nj Gi Gl P = Gi Gl Gi 1Gl+1 + Gi 1Gl+1 P . |Y − − − |Y B But the function σw on B (or equivalently the rational functionσ ˜w ( )) may be obtained from theY expression m G we had for σP by repeated∈KY Λ P Λ Λ w substitutions of the kind ∈L P nj nj nj+1 nj nj nj+1 nj nj (11.1) Gi Gl Gi Gl Gi 1Gl+1 + Gi 1Gl+1, −→ − − − nj nj until the resulting expression has no more summands with factors of type Gi Gl . (These transformations correspond to the classical straightening identities which areusedtoturntheP -standard monomial expansion of cw into the B-standard monomial one.) But the substitutions (11.1) do not aﬀect the restriction to P .So Y we are done.

Proposition 11.3. For the Grassmannian permutation w W Pm deﬁne the ra- ∈ tional function Gw on G/B− by

(g v ωm w v ωm ) Gw(gB−):= · − | · − . (g v ωm v ωm ) · − | − Then

Gw = σw ( ). |Y ∈KY B Proof. By Proposition 11.1 it suﬃces to show that Gw B coincides with σw .But e Y this follows from A. N. Kirillov’s explicit formula for the| corresponding quantum Schubert polynomials (see Section 3.7) together with Peterson’s Theorem 4.2, and inspection of the matrix u from (4.2) in the case where P = B.

Pi Corollary 11.4. (1) If y B,>0, then for any i I and w W we have B ∈Y ∈ ∈ σw (y) > 0. P (2) If y P,>0 then q (y) > 0 for all i =1...,k. ∈X i Proof. (1) is an immediate corollary of Proposition 11.3, since for any g U +, ∈ Gw(gB−) is a quotient of nonzero minors of g. Part (2) follows from Proposi- tion 11.3 along with Theorem 4.6 and Lemma 9.1.

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12. Proof of Theorem 7.2(3) We begin with a partial converse to Corollary 11.4(1) in the full flag variety case. B Lemma 12.1. Let y B be such that σw (y) > 0 for all w W .Theny B,>0. Schub∈Y Schub ∈ Schub ∈X In other words, = B,>0. And therefore also X XB,>0. YB,>0 XB,>0 ⊂X B,>0 ⊂ B Proof. ByLemma9.1wehavethatqi (y) > 0 for all i =1,...,k. Therefore y B.Nowwemaywritey = xw0B− for some x XB. It remains to prove that ∈X ∈ B x U − . The positivity of all the quantum parameters q implies by (5.1) that ∈ >0 i ∆j(x) > 0 for all j =1,...,n 1. Now by Proposition 11.3 the positivity of the B − σw for the Grassmannian permutations w of descent d in W implies the positivity of all the d d minors with column set 1,...,d and arbitrary row sets. But this × { } suffices to determine that x is totally positive; see e.g. [3]. Schub Proposition 12.2. XB,>0 = XB,>0. Proof. By Lemma 12.1 we have the following commutative diagram: Schub X , XB,>0 B,>0 → &.n 1 R>−0 where the top row is clearly an open inclusion and the maps going down are restrictions of πB. By (1) of Theorem 7.2, which is already proved, the left-hand map to n 1 R>−0 is a homeomorphism. It follows from this and elementary point set topology Schub that XB,>0 must be closed inside XB,>0. So it suffices to show that XB,>0 is connected. For an arbitrary element u X and t R,let ∈ ∈ 1 ta1 1  .  (12.1) u := t2a ta .. . t  2 1   . .   . .. 1   n 1 2  t − an 1 t a2 ta1 1  − ···    So u0 =Idandu1 = u,andifu XB,>0,thensoisut for all positive t. ∈ Let u, u0 XB,>0 be two arbitrary points. Consider the paths ∈ γ :[0, 1] XB,>0 ,γ(t)=uu0 → t γ0 :[0, 1] XB,>0 ,γ0(t)=utu0. → Note that these paths lie entirely in XB,>0 since XB,>0 is a semigroup (as the intersection of the group X with the semigroup U>−0). Since γ and γ0 connect u and u0, respectively, to uu0, it follows that u and u0 lie in the same connected component of XB,>0, and we are done. Schub Schub Corollary 12.3. P,>0 = P,>0 = P,>0 = P,>0 and in particular also Schub Y X X Y XP,>0 = XP,>0 . Schub Schub Proof. The identity P,>0 = P,>0 follows from Lemma 9.1. It remains only to XSchub Y show that XP,>0 = XP,>0 . We begin with the inclusion .LetX 0 = X U −0. Then clearly ⊆ ≥ ∩ ≥

(12.2) XB,>0 X 0 ⊆ ≥

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is an inclusion of closed subsemigroups of U −. We show that this is actually an equality. Suppose x X 0; then for any u XB,>0 and ut defined as in (12.1), the ∈ ≥ ∈ curve t x(t)=xut starts at x(0) = x and lies in XB,>0 for all t>0. Therefore 7→ x XB,>0 as desired. As a consequence, using Proposition 12.2, we have ∈ Schub (12.3) XP,>0 = XP XB,>0 = XP X . ∩ ∩ B,>0 P Now consider the Schubert classes σw qH∗(G/P ) as functions on XP .ByPropo- P ∈ Schub sition 11.1, σw =˜σw XP ,andσw takes positive values on XB,>0 .Letuschoose | P P x XP,>0. Then by (12.3) we have also σw (x) 0 for all w W . On the other ∈ P P P k ≥ ∈ hand Q := π (x)=(q1 (x),...,qe k (x)) R>0 by Corollary 11.4. But we have seen in Section 9 that there is only one∈ Schubert nonnegative point in the fiber P 1 P (π )− (Q), and that that one is strictly positive. Thus in fact σw (x) > 0 for all P Schub w W and XP,>0 XP,>0 . ∈ ⊂ Schub It remains to show that XP,>0 , XP,>0 is surjective. Consider again the proper map → n 1 ∆=(∆1,...,∆n 1):X C − − → n 1 defined in Section 4.3. Its restriction ∆ 0 =(∆1,...,∆n 1):X 0 (R 0) − is ≥ − ≥ → n≥ 1 surjective, since the image must be closed and contain ∆ 0(XB,>0)=R>−0 . From Theorem 7.2(1) along with Lemma 5.1 we know≥ that the further “restriction” of ∆, P Schub k ∆>0 =(∆n1 ,...,∆nk ):XP,>0 (R>0) , → is bijective. So we have the following diagram: Schub XP,>0 , X → P,>0 &.∼k R>0 P where the downward arrows are given by ∆>0 and its restriction. By the surjectivity of ∆ 0 we also have that the left-hand map is surjective. This implies the desired ≥ Schub equality, XP,>0 = XP,>0 . Acknowledgements Dale Peterson’s beautiful results presented by him in a series of lectures at MIT in 1997 were a major source of inspiration and are the foundation for much of this paper. It is a pleasure to thank him here. Some of this work was done during a very enjoyable and fruitful stay at the Erwin Schrödinger Institute in Vienna, and sincere thanks go to Peter Michor for his hospitality. Finally, I would like to thank Bill Fulton for his kind invitation to Michigan, where some of the final writing was done.

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