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Quantum K-theory of Incidence Varieties

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n) and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov-Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov-Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov-Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive Littlewood-Richardson rule for the non-equivariant quantum K-theory ring of X. The Littlewood-Richardson rule in turn implies that non-empty Gromov-Witten varieties given by Schubert varieties in general position have arithmetic genus 0.


Introduction
Let X be a homogeneous space G/P defined by a complex semisimple linear algebraic group G and a parabolic subgroup P . Let d ∈ H 2 (X, Z) be an effective degree and M d := M 0,3 (X, d) be the Kontsevich moduli space parametrizing 3-pointed, genus 0, degree d stable maps to X. Let X u and X v be opposite Schubert varieties in X. Let M d (X u , X v ) := ev −1 1 (X u ) ∩ ev −1 2 (X v ) ⊆ M d be the Gromov-Witten variety of stable maps with one marking sent to X u and another to X v , and Γ d (X u , X v ) := ev 3 (M d (X u , X v )) be the closure of the union of degree d rational curves in X connecting X u and X v .
Buch, Chaput, Mihalcea, and Perrin [BCMP18] proved that when X is a cominuscule variety (certain homogeneous spaces with Picard rank one), is cohomologically trivial. This means the pushforward of the structure sheaf of M d (X u , X v ) is the structure sheaf of Γ d (X u , X v ) and the higher direct images of the first sheaf are zero. As a consequence, the (Tequivariant) K-theoretic (3-pointed, genus 0) Gromov-Witten invariants of X satisfy the identity , where χ T X denotes the T -equivariant pushforward along the structure morphism of X. Identity (2) says that these invariants defined on M d can be computed in the equivariant ordinary K-theory ring of X.
Conjecture 1.1. When X is of type A and X v is a divisor D, (1) is cohomologically trivial.
In this paper, we consider the incidence variety X = Fl(1, n − 1; n), which is a special two-step flag variety of type A and one of the simplest homogeneous spaces that are not cominuscule.
Our main geometric result is the following. is rationally connected.
Using a result of Kollár [Kol86] (see [BM11,BCMP13a]), Theorem 1.2 implies Conjecture 1.1 and the identity . The right hand side of (3) is easily computable using Lenart and Postnikov's Chevalley formula for the equivariant ordinary K-theory ring of G/P [LP07], because Γ d (X u , D) is either a Schubert variety or the intersection of a Schubert variety with D (see Section 2.2.2 for details).
Our main application is in studying the (T -equivariant) quantum K-theory ring of X. Quantum K-theory was introduced by Givental and Lee [Giv00,Lee04] as a K-theoretic analogue of quantum cohomology. The latter has been widely studied in theoretical physics and various branches of mathematics following the pioneering works of Witten [Wit90] and Kontsevich [Kon95]. Rationality properties of Gromov-Witten varieties often have implications in quantum K-theory, see for example [LP04,BM11,BCMP13b,CP11,BCMP13a].
Before stating our next main results, we shall introduce a bit more notations. (Opposite) Schubert varieties in X are indexed by W P := {[i, j] : i and j are unequal integers between 1 and n}.
Here [i, j] corresponds to the permutation w ∈ S n such that w(1) = i, w(n) = j and w(2) < · · · < w(n − 1). In order to state our formulas in a simple form, we define and for w = [i, j] ∈ W P we define where w := [i, j] ∈ W P is defined by i ≡ i and j ≡ j (mod n), d(w) := ( i − i n , j − j n ) ∈ H 2 (X, Z) = Z ⊕ Z, and QK T (X) q := QK T (X) ⊗ Z[q] Z[q, q −1 ] denotes the T -equivalent quantum K-theory ring of X with the deformation parameters q inverted. We let O [1] and O [2] be the classes of the structure sheaves of the two opposite Schubert divisors.
Identity (3) allows us to compute K-theoretic Gromov-Witten invariants and derive the following Chevalley formula for the equivariant quantum K-theory ring of X. By [BCMP16,Section 5.3], this formula uniquely determines the ring struture. Setting q = 0 recovers Lenart and Postnikov's Chevalley formula for equivariant ordinary K-theory [LP07] (see also [KZJ17] for a Littlewood-Richardson rule for the equivariant ordinary K-theory ring of two-step flag varieties). Theorem 1.3 ( . = Theorem 4.5). In QK T (X), for [i, j] ∈ W P , the quantum multiplication by divisor classes is given by A Chevalley formula for the equivariant quantum K-theory ring of the complete flag variety G/B has been proved by Lenart, Naito, and Sagaki [LNS20]. Kato has established a ring homomorphism from the equivariant quantum K-theory ring of G/B to that of G/P [Kat19]. Theorem 1.3 can also be proved by combining these results and working out the cancellations combinatorially. This approach is taken by Kouno, the ICERM Combinatorial Algebraic Geometry semester program. I thank ICERM and the organizers of this program for an inspiring semester.

Preliminaries
Varieties are not assumed to be irreducible, but rational, unirational, and rationally connected varieties are. Let f : X → Y be a morphism of varieties, then the general fibre is a fibre f −1 (u), where u is a closed point chosen from a suitable dense open subset U ⊆ Y .
2.1. Moduli Space of Stable Maps.
2.1.1. Overview. Given a smooth complex projective variety X and an effective degree let M g,m (X, d) be the moduli space of maps f from a projective nonsingular curve C of genus g with m distinct marked points to X such that f * [C] = d. There is a compactification whose points correspond to isomorphism classes of m-pointed stable maps f : C → X representing the class d, i.e., f * [C] = d, where C is allowed to be an m-pointed connected nodal curve of arithmetic genus g. Evaluating each f at the i-th marked point gives a morphism We write ev for ev 1 × · · · × ev m : M g,m (X, d) → X m . The contravariant functor M g,m (X, d) from the category of complex schemes to sets is defined as follows: given a complex scheme S, M g,m (X, d)(S) is the set of isomorphism classes of stable families over S of maps from m-pointed curves of genus g to X representing the class d. M g,m (X, d) is a projective coarse moduli space for M g,m (X, d).
2.1.2. Functoriality. The goal of this section is to specialize [BM96, Theorem 3.6] to the following.
Proposition 2.1. Given a morphism of smooth projective varieties ξ : Y → Z and d ∈ H 2 (Y ) + , there exists a natural transformation M g,m (Y, d) → M g,m (Z, ξ * d) given by composing stable families with ξ and collapsing components of source curves in families as necessary to make the compositions stable.
By [BM96, Theorem 3.6], there exists a pushforward (D, y, h) of (C, x, f ) of stable maps under (ξ, id, τ, id). Since (D, y, h) is a stable map over S from an m-pointed curve of genus g to Z of class ξ * d, this pushforward defines a map ρ S : M g,m (Y, d)(S) → M g,m (Z, ξ * d)(S). Moreover, this pushforward commutes with base change, making ρ a natural transformation. The proof of [BM96,Theorem 3.6] shows that the pushforward is constructed by composing stable families with ξ and collapsing components of the source curves in families as necessary to make the compositions stable.
Corollary 2.2. Given a morphism of smooth projective varieties ξ : Y → Z and d ∈ H 2 (Y ) + , there exists a morphism M g,m (Y, d) → M g,m (Z, ξ * d) given by composing stable maps with ξ and collapsing components of source curves as necessary to make the compositions stable.
2.1.3. Stable Maps to Flag Varieties. We are mainly interested in the case where g = 0 and X is a flag variety G/P defined by a complex connected semisimple linear algebraic group G and a parabolic subgroup P . Schubert varieties in X are orbit closures for the action of a Borel subgroup B ⊆ G.
Let X 1 , . . . , X l be Schubert varieties in X with l ≤ 3 (not necessarily in general position). We will be concerned with is either empty or unirational [BCMP13a, Proposition 3.2]. We will also work withM i.e., the closure of the union of degree d rational curves (images of degree d maps from P 1 ) in X that meet X 1 , and which is the closure of the union of degree d rational curves in X connecting X 1 and X 2 . Using automorphisms of P 1 , we can think of closed points in M 0,3 (X, d) as maps P 1 → X of degree d with marked points at 0, 1, ∞ ∈ P 1 .

Incidence Varieties.
2.2.1. Definition and Basic Properties. From now on n ≥ 3.
has dimension 2n − 3. Let p 1 : X → Gr(1, n) ∼ = P(C n ) and p 2 : X → Gr(n − 1, n) ∼ = P(C n * ) be the projections. Schubert varieties in X are indexed by pairs of unequal integers [i, j] (we use square brackets to distinguish them from degrees), where i and j are each between 1 and n. We denote this indexing set by W P . X embedds in P(C n ) × P(C n * ) with image given by x 1 y 1 + · · · + x n y n = 0, where x 1 , . . . , x n are the standard coordinates on C n and y 1 , . . . , y n are the dual coordinates on C n * . Under this identification, the Schubert variety X [i,j] is the subvariety given by x i+1 = · · · = x n = y 1 = · · · = y j−1 = 0, and the opposite Schubert variety X [i,j] is the subvariety given by where The two Schubert divisors D Let G = SL(C n ) (of type A n−1 ), T ⊂ G be the maximal torus of diagonal matrices and B ⊂ G be the Borel subgroup of upper triangular matrices. For i = 1, . . . , n, let ε i : T → C * be the character that sends a diagonal matrix to its i-th diagonal entry. Then the root system of (G, B, T ) is and the subset of simple roots is ∆ = {α 1,2 , α 2,3 , . . . , α n−1,n } ⊂ Φ. The Weyl group W consists of permutations of {ε 1 , . . . , ε n }, and can be identified with S n . Let ∆ P = {α 2,3 , . . . , α n−2,n−1 } ⊂ ∆. ∆ P determines a parabolic subgroup P , which consists of block upper triangular matrices with 3 blocks sized 1, n − 2, 1. W P -the Weyl group of P -is the subgroup of W consisting of permutations of {ε 2 , . . . , ε n−1 }. We identify W P with the set of minimal coset representatives of W/W P by sending [i, j] to the unique permutation w such that w(1) = i, w(n) = j, and w(2) < · · · < w(n − 1). For u ∈ W , write ℓ(u) for the length of u. The Bruhat order on W gives W P a poset structure. X can be identified with G/P by the map where E i = Span{e 1 , . . . , e i } is the span of the first i standard basis vectors in C n . In P(C n ) × P(C n * ), gP has coordinates where C(g) is the cofactor matrix of g. Letφ : G → G be defined as Thenφ is an involution in the category of algebraic groups. Moreover,φ restricts to an involution of P and an involution of T which sends diag(t 1 , . . . , t n ) to diag(t n −1 , . . . , t 1 −1 ). In particular,φ induces an involution ϕ of G/P . Under our identifications, this is exactly the ϕ defined previously.
. When d 1 > 0 and d 2 = 0, since the image of every stable map of degree d is contained in a fibre of p 2 , and every two points in a fibre of p 2 is connected by a line and therefore the image of a stable map of degree d, Similarly, when d 1 = 0 and d 2 > 0, Given B = (B 1 , B 2 ) ∈ U , write B 1 = Span{b 1 }, and let f : P 1 → X be given by Then f has degree (1, 1), and f (0) = A, f (∞) = B. This shows Γ (1,1) (X 1,n ) ⊇ U . Since Γ (1,1) (X 1,n ) is closed, it must be the entire X. By the same reasoning, ) is the closure of the union of degree d rational curves in X meeting X [i,j] and D [k] .
When d k = 0, let C be a degree d rational curve in X, then C ∩ D [k] is either empty or C. In other words, C meets D [k] if and only if C ⊆ D [k] .
) is a Richardson variety, i.e., an intersection of two opposite Schubert varieties. Schubert varieties are known to have rational singularities [Ram85]. It then follows from [Bri01, Lemma 2] that Richardson varieties have rational singularities.
2.3. Lemmas from Algebraic Geometry. For completeness, we recall two likely well-known lemmas that will be used in our proofs. Lemma 2.6. Let φ : Y → Z be a dominant morphism between irreducible varieties over a field of characteristic 0. Then the general fibre is reduced.
Proof. Without loss of generality, assume Z = SpecA, Y = SpecB. Then the generic fibre is Spec Since the characteristic is 0, the generic fibre is geometrically reduced by [Sta21, Lemma 020I], and the conclusion follows from [Sta21, Lemma 0578].

Geometry of Gromov-Witten Varieties
By Corollary 2.2, there is a map ρ : M 0,3 (X, d) → M 0,3 (P n−1 , d 1 ) given by composing with p 1 and collapsing components of the source curve as necessary to make the composition stable. In particular, the diagram and )) be restrictions of ρ, where g ∈ G is not assumed to be general. Using the automorphism ϕ, we can often reduce to the case where (5) either d 1 < d 2 , or d 1 = d 2 and k = 1.
is surjective.
Proof. Let f 1 : P 1 → Gr(1, n) be a map of degree d 1 such that It suffices to show f 1 is in the image.
Being the preimage of the compliment of a linear subspace, U is open. Therefore, U is dense in P 1 . Since P 1 is nonsingular and X is projective, the mapf : U → X given bỹ can be extended to all of P 1 . The extended mapf has degree ( By attaching additional components to the domain as necessary using [FP97, Lemma 12], we get a stable map f : C → X of degree d with image f 1 . Note that the added components collapse after composing with p 1 . ) is non-empty if and only if i = n and in this case it is a point.
One checks that [n,1] ) and the conclusion follows.
Attach P 1 toC by identifying (1 : 0) with a and map this copy of P 1 to X by (s : Note that the added component, among others, collapses after composing with p 1 , so ρ(f ) = f 1 .
is rational by an argument similar to the proof of Lemma 3.4. It follows that M e (Y 1 , Y 2 , Y 3 ) is rational.
On the other hand, since ρ g,k 3 is closed and surjective by Corollary 3.3 and M d1 ( ) is irreducible by Lemma 3.5, at least one component maps surjectively to it. Therefore, exactly one component Y maps surjectively to ) over which fibres of ρ 2 are irreducible and of expected dimension is non-empty, ρ 2 (Z) ∩ U is also non-empty and therefore dense in ρ 2 (Z).
By generic flatness, the general fibre of Z → ρ 2 (Z) has dimension When k = 1, (7) is equivalent to Since Z doesn't map surjectively, (7) must be an equality, which is equivalent to m ′ = m. This implies Z contains an irreducible fibre of ρ 2 , but then Z meets Y , which a contradiction.
Theorem 3.7. The general fibre of is rationally connected.
Proof. When d = 0, the domain and target are both isomorphic to X [i,j] ∩ D [k] and ev 3 is an isomorphism. When d > 0, without loss of generality, we assume (5). Case 1: d 1 = 0. Assume M d (X [i,j] , D [k] ) is non-empty. By Lemma 2.5 and Lemma 2.6, there is a dense open subset U ⊆ Γ d (X [i,j] , D [k] ) such that for all u ∈ U , ev −1 3 (u) is reduced and every irreducible component meetsM d (X [i,j] , D [k] ). On the other hand, sinceM 0 (p 1 (X , g.X [n,1] ) for general g. By Corollary 3.3, Lemma 3.5 and Proposition 3.6, the restriction is a surjective morphism between complex irreducible varieties and the target is rational. By Lemma 2.5 and Lemma 2.6, the general fibre is reduced and every irreducible component meetsM d ( Example 3.8. When d 1 = d 2 = 1, for a general point C = (C 1 , C 2 ) ∈ X, ev 1 : Proof. Without loss of generality, assume k = 1. Let Since C is general, we may assume We shall construct a unique preimage µ : where v 0 ∈ A 1 , v ∞ ∈ C 1 are unique and nonzero. Let µ : P 1 → X be given by Then µ is the unique degree (1, 1) morphism µ : P 1 → X such that Proof. This follows from Theorem 3.7 and [BM11, Theorem 3.1] (see also [BCMP13a, Proposition 5.2]).

Chevalley Formula for T -equivariant Quantum K-theory
See [CG97] for an introduction to equivariant K-theory and [BCMP16, Section 2] for a summary of the (equivariant) (quantum) K-theory of flag varieties. We will use the same set-up and notations as in [BCMP16]. 4.1. Pullbacks along Morphisms of Algebraic Groups. Let G, H be algebraic groups and f : G → H a morphism. Let Z be an H-variety. Then Z is also a G-variety with the action given by g.z := f (g).z for all g ∈ G, z ∈ Z, and an H- is a module over the Grothendieck ring K H (Z) of H-equivariant vector bundles over Z, and f * is compatible with the module structures in the sense that the diagram commutes.
Lemma 4.1. Let G, H be algebraic groups and f : G → H a morphism.
(1) Let λ : H → GL(V ) be a representation, and [V λ ] the corresponding class in K H (point). Then (2) Let Z be an H-variety and Ω ⊆ Z an H-stable closed subvariety, then (3) Given φ : Y → Z a flat H-equivariant morphism of H-varieties, the diagram Proof. This is clear.

Chevalley Formula for T -equivariant Quantum K-theory.
Notation 4.4. From now on, when we write "≡" or " ≡", it shall be understood that the equivelance is taken modulo n. Let and the subscript q stands for localization with respect to {q d : d ∈ Z 2 }. We let Recall that K T (X) is a free module over K T (point). The classes O w , w ∈ W P form a basis. For w ∈ W P , let O ∨ w ∈ K T (X) be the basis element dual to O w in the sense that . We extend the definition to allow for w ∈ W P by letting For i = 1, 2, we will write O [i] for the class of O D [i] in K T (X).
The goal of this subsection is to prove the following T -equivariant Chevalley formula and derive some immediate consequences.
Theorem 4.5. In QK T (X) q , for [i, j] ∈ W P , the quantum multiplication by divisor classes is given by The Chevalley formula for the equivariant ordinary K-theory of X can be recovered from Theorem 4.5 by interpreting O w as 0 unless w ∈ W P . We rely on [LP07] for this special case. The formula is derived from [LP07, Corollary 7.1 and Corollary 8.2] using that (α 1,2 , . . . , α 1,n ) is a reduced ε 1 -chain in the sense of [LP07, Definition 5.4].
Note that Corollary 3.9 gives us the following "quantum-equals-classical" type result, which allows us to compute some equivariant K-theoretic Gromov-Witten invariants using the Chevalley formula for equivariant ordinary K-theory.
For the second equality, see [BM11, Section 4.1] and the references therein. The third equality follows from the projection formula. The last equality follows from Corollary 3.9. The rest follows from Lemma 2.3.
Notation 4.7. For σ 1 , σ 2 ∈ K T (X) and w ∈ W P , we write I T (σ 1 , σ 2 ; O w ) for I T d(w) (σ 1 , σ 2 , O w ) and , with the convention that they are 0 when d(w) ≥ 0. We will only allow the third argument to be outside of K T (X), and the semicolon is there to remind the reader of it. We do the same in the non-equivariant case.
When specialized to the non-equivariant case, the following proposition says that for general g and k = 1, 2, In other words, when M d (g.X u , D  Proof. Assume i < k and j ≥ l. If k ≤ n, then k = k > i, and by Corollary 4.6 (14) By Section 2.2.2, (15) n ≥ a = k > i and b ≤ j.
By the Chevalley formula for the equivariant ordinary K-theory of X, The conditions (15) implies a = n > b when i + 1 = j = n and either a > i + 1 or b < j when i + 1 = j < n. If k > n, then by Corollary 4.6 (17) If l < 1, then X [a,b] = X and the right hand side of (17) equals 1. Otherwise By Section 2.2.2, a ≥ i and b = l ≤ j. By (16), the right hand side of (17) equals 1.
The other two cases of (12) are similar and the proof is omitted. (13) follows from (12) by applying Lemma 4.3 and Lemma 4.1.
Definition 4.9. Following [BCMP16], for σ 1 , σ 2 ∈ K T (X), we define the formal sum and we define the automorphism Ψ : QK T (X) q → QK T (X) q of K T (point)((q 1 , q 2 ))-modules by We will write Γ d (X u ) for Γ d−d(u) (X u ) when d ≥ d(u) and set Γ d (X u ) = ∅ when d ≥ d(u). Then we have By [BCMP16, Proposition 2.3], for σ 1 , σ 2 ∈ K T (X), the quantum product is given by Recall from Section 2.2.1 that we identify W P with the set of minimal coset representatives of W/W P .
Definition 4.10. For v ∈ W P , we define I(v) to be the set of u ∈ W P such that the interval [u, v] in W between u and v is contained in W P , i.e., The next lemma is a special case of [Deo77, Theorem 1.2 and Section 3].
Lemma 4.11. Let f be any function on W P with values in an abelian group and define function g on W P by Then for all v ∈ W P , .
we have the following in particular.

Now we give explicit computation of I(v)
. Some examples are illustrated in Figure 2.
Proof. This is clear.
Proof. By reasoning as follows, the conclusion can be read from the poset W P . When n = 5, the poset structure of W P is illustrated in Figure 1. If a− b = 1 or 2, then neither [a− 3, b] nor [a, b + 3] is in I(v), and that at least one of them is in W P ∩[u, v] except when we are in either of the following cases: (1) a − b = 1, u = [b − 1, a + 1].
(2) a − b = 2, u = [b, a]; By Lemma 4.13, u ∈ I(v) unless we are in one of those two cases. The Bruhat interval [u, v] in W contains 8 elements in case (1) and 6 elements in case (2). W P ∩ [u, v] contains 7 elements in case (1) and 6 elements in case (2). We conclude that u ∈ I(v) in case (1) and u ∈ I(v) in case (2).
where T X is the tangent bundle of X, and d(u) c 1 (T X ) is the degree of q d(u) in the (graded) quantum cohomology ring of X.    Proof of Theorem 4.5. Without loss of generality, we may assume [i, j] ∈ W P . We will prove (19) By [BCMP16, Proposition 2.3], this is equivalent to (9). We will prove (19) by comparing the coefficient of In order to compute the coefficient of O w on the right hand side of (19), we list curve neighborhoods in Table 1 through Table 5 (see Section 2.2.2). Table 1. Assume i < n and i + 1 = j.
When i + 1 ≡ j, using the definition of Ψ (equation (18)), Table 1, and Table 2, we see that the coefficient of O w on the right hand side of (19) equals (a) 1 − [C εi−ε1 ] when we are in one of the following situations: Table 2. Assume i = n and j > 1.
, which we will now compute and compare with (a) through (i). By Corollary 4.12, By Proposition 4.8, S k is easily computable using Lemma 4.14. For example, when 1 < a − 1 = b + 1 = j < n, For another example, when a ≡ 1 and b = 0, ([a, b])). We shall consider the three cases in Lemma 4.14 separately. Table 6, Table 7, and Table 8 list values of S k in each case, respectively. When the conditions in the tables are not satisfied, S k = 0. Table 6. Values of S k when a − b = 1, 2. either b = j or b = n − d 2 n for some d 2 > 0 k = a 1 k = a − 1 and a ≡ 1 −1 Case 1: Assume a − b = 1, 2. Table 6 says that out of all S k , only S a and S a−1 can be nonzero when a ≡ 1, and only S a can be nonzero when a ≡ 1. Therefore, we can reduce (20) to a finite sum. When a ≡ 1, equation (20) becomes By Table 6, S a and S a−1 are nonzero only when either b = j or b = n − d 2 n for some d 2 > 0, and in either case they are 1 and −1, respectively. Therefore, (2) 1 − [C εi−ε1 ] when a = i < n and b = n − d 2 n for some d 2 > 0; (3) [C εi−ε1 ] when a = i + 1 ≤ n and b = j; (4) [C εi−ε1 ] when a = i + 1 < n and b = n − d 2 n for some d 2 > 0; (5) 0 otherwise. When a − b = 1, 2 and a ≡ 1, the situations (a1), (a2), (b1), (b2), (d), (e1), (e2), (f3), and (i) are realizable while the other ones are not. In all realizable situations, the value of b] ) agrees with the coefficient of O w on the right hand side of (19).
There is a conjecture (see [BCMP16] for example) about the positivity of structure constants in QK T (G/P ). In the case of incidence varieties, [BCMP16, Conjecture 2.2] specializes to the following statement.
−εr ] − 1 : r = 1, . . . , n − 1]. It follows from Theorem 4.5 that the conjecture holds when O v is (a power of q times) the class of a Schubert divisor. Equivalently: Proof. Without loss of generality, we may assume u = [i, j] ∈ W P . One checks that when i + 1 ≡ j, Using [Kat19, Theorem 2.18] with the projection p 1 : X → P n−1 , we recover the following Chevalley formula for the equivariant quantum K-theory of projective spaces, which is a special case of [BCMP16,   Corollary 4.21. In QK T (P n−1 ), for k = 1, . . . , n,

Quantum K-theoretic Littlewood-Richardson Rule
The goal of this section is to prove the following Quantum K-theoretic Littlewood-Richardson Rule and some corollaries.
Lemma 5.3. Let R be an associative ring with unit 1. Let S ⊂ R be a subset that generates R as a Z-algebra. Let M be a left R-module. Let µ : R × M → M be any Z-bilinear map. Assume that for all r ∈ R, s ∈ S, and m ∈ M we have: (1) µ(1, m) = m; (2) µ(rs, m) = µ(r, sm). Then µ(r, m) = rm for all r ∈ R and m ∈ M .
We will also use the following lemma.
Corollary 5.5. For d ∈ H 2 (X) + , a general g ∈ G, and u, v, w ∈ W P , M d (g.X u , X v , X w ) has arithmetic genus 0 whenever it is non-empty.
We shall need the following simple lemma.
Therefore, nχ(a > b) ≤ a − b, which is a contradiction.
When i > j and k > l, we have n < x − y = x − y + n(d 1 + d 2 ) < 2n and therefore, either x > y and d 1 + d 2 = 1 or x < y and d 1 + d 2 = 2. When i > j and k < l, or, i < j and k > l, we have 0 < x − y = x − y + n(d 1 + d 2 ) < n and therefore, either x > y and d 1 + d 2 = 0 or x < y and d 1 + d 2 = 1. When i < j and k < l, we have −n < x − y = x − y + n(d 1 + d 2 ) < 0 and therefore x < y and d 1 + d 2 = 0. In all cases, (41) holds.
The proof of this is completely analogous and is therefore omitted.
For the quantum cohomology and quantum K-theory of all homogeneous spaces G/P , it is known that there is a unique minimal power of the quantum parameters q that appear with nonzero coefficient in the product of any two given Schubert classes. This minimal power is the same in quantum cohomology and quantum K-theory, and corresponds to the minimal degree of a stable curve connecting opposite Schubert varieties [BCLM19]. For the quantum cohomology of cominuscule flag varieties, it is known that the powers of q that appear with nonzero coefficients in the product of any two given Schubert classes form an interval between two degrees [BCMP21,Pos02]. However, this is not true for the quantum cohomology of complete flag varieties. A counterexample is the product of [Y 164532 ] with itself in the quantum cohomology ring of Y = Fl(6), where the maximal powers of q with nonzero coefficients are not unique and the powers of q with nonzero coefficients do not form an interval [BCMP21]. For the quantum K-theory of type A complete flag varieties, it is known that there is a unique maximal power of q that appear with nonzero coefficient in the product of a Schubert divisor class and a Schubert class; however, the powers of q that appear with nonzero coefficients don't always form an interval [LNS20]. It remains an open problem whether there is always a unique maximal power of q that appear with nonzero coefficient in the product of two given Schubert classes in the quantum K-theory ring of a homogeneous space. In the case of incidence varieties, it is immediate from Theorem 5.1 that: Corollary 5.8. In QK(X), for u, v ∈ W P , the powers of q that appear with nonzero coefficients in O u ⋆ O v form an interval between two degrees.
Using [Kat19, Theorem 2.18] with the projection p 1 : X → P n−1 , we get the following quantum Ktheoretic Littlewood-Richardson rule for projective spaces, stated in Notation 4.20. This is a special case of [BM11, Theorem 5.4].