A CHEVALLEY FORMULA IN THE QUANTUM K THEORY RING FOR THE G 2 FLAG MANIFOLD

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1 A CHEVALLEY FORMULA IN THE QUANTUM K THEORY RING FOR THE G 2 FLAG MANIFOLD MARK E. LEWERS AND LEONARDO C. MIHALCEA Abstract. Let X be the (generalized) flag manifold for the Lie group of type G 2 and let QK(X) be the quantum K theory ring of X. This ring has a basis given by Schubert classes, indexed by the Weyl group of type G 2 (the dihedral group with 12 elements). In this paper we give a conjectural formula for the 3 point K-theoretic Gromov-Witten invariants on X, where one of the Schubert classes classes involved corresponds to a simple reflection. This is used to obtain a Chevalley formula, i.e a multiplication of an arbitrary Schubert class by a class for a simple reflection. The products we obtain satisfy a certain commutativity/associativity property and their structure constants satisfy the K-theoretic version of positivity. Further, our rule coincides with an algebraic rule of Lenart and Postnikov, which was obtained using a different approach. 1. Introduction According to a famous classification theorem in algebra, the complex semisimple Lie algebras can be classified into 4 infinite series (types A n, B n, C n, D n ) and five exceptional finite series (types E 6, E 7, E 8, F 4, and G 2 ). To each algebra, one can associate a group and to each group a certain geometric object called a flag manifold. G 2 is considered the simplest of these finite series and we will denote by X to be the flag manifold for type G 2. One can associate several rings to the flag manifold X: the cohomology ring H (X), the K theory ring K(X), the quantum cohomology ring QH (X). Each of these rings has a basis given by Schubert classes, indexed by the elements w in the Weyl group W of type G 2. We recall that W is actually isomorphic to the dihedral group with 12 elements, although we will use a different realization of it, which is more suitable to our purposes. All of these rings are generated by Schubert classes for the simple reflections s 1, s 2 in W. Therefore, at least in principle, the full multiplication table in the rings is determined by a formula to multiply a Schubert class by a class for either s 1 or s 2. We call this a Chevalley formula. There has been substantial amount of work to find Chevalley formulas for each of the rings above, starting with Chevalley in 1950 s, see e.g. [4]. The similar formulas for the K-theory and quantum cohomology rings were found around 2000 s by Fulton and Lasoux, Pittie and Ram [6, 12] respectively Fulton and Woodward [7]. All these formulas can be expressed combinatorially in terms of the root system and the Weyl group for type G 2. For instance, in the previous year, in joint work with R. Elliott [5] we used the formulas in the literature to write down explicit Chevalley formulas for the rings H (X) and QH (X). The current paper is concerned with a further deformation of the rings above, called the quantum K theory ring QK(X). This means that if we take the multiplication in the quantum K ring, and we make certain specializations, we recover 1

2 2 MARK E. LEWERS AND LEONARDO C. MIHALCEA either K-theory, or quantum cohomology rings. We will explain this in section 3 below. The quantum K theory ring was defined by A. Givental [8] in 2000, and since then it has been connected to several areas of mathematics (algebraic geometry, combinatorics, representation theory) and to physics (quantum field theory, integrable systems). Unlike for H (X), K(X), and QH (X), there is no known general formula to compute multiplications in QK(X). Next, we will roughly explain our results. For u W, let O u be the Schubert class in the ring QK(X) which is determined by u. This can be defined using geometry but for this paper it can be thought of as a symbol associated to u. When u varies in W, the Schubert classes form an additive basis in QK(X) over the coefficient ring Z[[q 1, q 2 ]] consisting of formal power series in q 1, q 2 with integer coefficients. Let d = (d 1, d 2 ) Z 0 Z 0 be a (multi)degree. Then any element κ in QK(X) can be uniquely written as κ = d=(d 1,d 2),w W a w,d q d1 1 qd2 2 Ow where a w,d are integers and the sum is possibly over infinitely many degrees d. Since QK(X) is a ring, this means that the multiplication O u O v of any two Schubert classes can be again expanded as before, therefore we can define the structure constants Nu,v w,d by O u O v = d=(d 1,d 2),w W N w,d u,v q d1 1 qd2 2 Ow where denotes the multiplication in the ring QK(X). If we fix u, v, w W and d = (d 1, d 2 ), then there is a recursive formula to determine each Nu,v w,d which depends on 2 and 3-point K-theoretic Gromov-Witten invariants (KGW). These are certain integers which again can be defined geometrically, but for which there should be combinatorial formulas based on the root system of type G 2. In fact, Buch and Mihalcea [3] recently found explicit algorithms to determine the 2-point KGW invariants, which rely on calculation of the curve neighborhoods of classes O u. (We will define these in section 2.4 below.) But so far there are no explicit combinatorial formulas for the 3-point invariants. The purpose of this paper is to propose a conjectural formulas for any 3-point KGW invariant which involves a class O s1 or O s2. Together with the 2-point KGW invariants, these will be used to provide explicit conjectural formulas for the quantum K Chevalley formulas: O u O si = N w,d u,s i q d1 1 qd2 2 Ow ; i = 1, 2. In order to give evidence that this conjecture is related to the true (geometrically defined) quantum K ring, we check several properties which are expected to hold for the quantum K products: positivity, associativity/commutativity and finiteness. (While commutativity and associativity hold for geometric reasons, there is no known proof that the other properties hold in the geometric ring QK(X).) More precisely, we first checked that the products we determine specialize to the products in K-theory and quantum cohomology ring. In the K-theory ring K(X) it is known that when multiplying two Schubert classes, one is able to predict the sign of the coefficient for each of the Schubert classes in the product. A similar property should hold in quantum K-theory. In addition it is also desirable to check

3 A QUANTUM K CHEVALLEY FORMULA 3 if multiplication by classes O si is both commutative and associative. To summarize our results: We predicted formulas for the 3-point KGW invariants of the form O u, O si, O w d d = (d 1, d 2 ), i = 1, 2; We used the former prediction to calculate all Chevalley multiplications O u O si for the quantum K theory ring. As evidence for the truth of the prediction, we checked that the structure constants Nu,s w,d i satisfy the sign predictability and commutativity/associativity conditions. We further compared our results to an algebraic conjecture given by Lenart and Postnikov [11] using λ chains in the Bruhat ordering on the Weyl group. We observed that these two approaches, which use different and independent techniques, give the same Chevalley formulas in QK(X) for type G Preliminaries 2.1. G 2 Root System. Our project studies certain rings associated to the flag manifold associated to G 2, primarily the quantum K theory ring. While we do not need to explicitly understand the group of type G 2, we will need to perform explicit calculations which involve the root system of this type. Note that most of the text in this section is taken from my Layman competition paper of last year [5]. Denote R the root system of type G 2. The roots belong to the hyperplane in E = R 3 given by the equation ξ 1 + ξ 2 + ξ 3 = 0. There are twelve roots, taken from [1]. The roots are displayed in Table 2, in terms of the natural coordinates in E. Each root α can be written uniquely as α = c 1 α 1 + c 2 α 2 where α 1, α 2 are simple roots and c 1 c 2 0. A root is positive (negative) if both c 1, c 2 are non-negative (resp. non-positive). The set of simple roots is denoted = {α 1, α 2 }, where α 1 = ɛ 1 ɛ 2 and α 2 = 2ɛ 1 + ɛ 2 + ɛ 3. For later purposes, we need to expand each root in terms of the simple roots. The full results for each root is shown in Table 2. The roots can also be understood geometrically. It is also important to define the roots in the following manner. Definition 2.1. Let R is the root system for G 2. defined as α = 2α (α, α) where (α, α) is the standard inner product in terms of (ɛ 1, ɛ 2, ɛ 3 ). The coroot α of root α is Note that ( α) = (α ) which means one needs to calculate only six coroots (preferably the positive ones). We denote the full set of coroots by R and define the set that holds the simple coroots: α 1 and α 2 for R. Table 2 shows the values for each of the coroots.

4 4 MARK E. LEWERS AND LEONARDO C. MIHALCEA 2.2. The Weyl Group. The Weyl group of G 2, denoted W, is the group generated by reflections s α, where α R. For simplicity reasons, we denote s αi = s i. Geometrically, s α is the reflection across the line perpendicular to the root α. For example, the reflection s 1 (corresponding to s α1 ) is the reflection across the line perpendicular to the α 1 axis (see Figure 3). For any root α, s α = s α. Since there are twelve elements in the root system, then only six unique reflections exist for the G 2 root system. It is known from J. Humphreys [10] that the Weyl group has the presentation W =< s 1, s 2 : s 2 1 = s 2 2 = 1, (s 1 s 2 ) 6 = 1 >. This means that every element in the Weyl group can be expressed as a product of simple reflections which correspond to the simple roots α 1 and α 2. Definition 2.2. Consider w W. A reduced expression for w is an expression involving products of s 1 and s 2 in as short a way as possible (via the relations in the presentation). If w W where w is a reduced expression, the length of w (l(w)) is the number of simple reflections (s 1 and s 2 ) that show up in the reduced expression. Therefore the order of the Weyl group must be finite and in fact has the following elements. W = {id, s 1, s 2, s 1 s 2, s 2 s 1, s 1 s 2 s 1, s 2 s 1 s 2, s 1 s 2 s 1 s 2,, s 1 s 2 s 1 s 2, s 2 s 1 s 2 s 1, s 1 s 2 s 1 s 2 s 1, s 2 s 1 s 2 s 1 s 2, s 1 s 2 s 1 s 2 s 1 s 2 } For simplicity, we denote s 1 s 2 s 1 s 2 s 1 s 2 = (s 1 s 2 ) 3 = w 0 1. From this it is simple to show that the Weyl group is isomorphic to the dihedral group D 12 (proof given in [5]). By this point it should be clear that it is possible to express the reflection for every root as a product of simple reflections. The table of results was determined for last year s Layman competition [5]). The table shows that all root reflections can be expressed as a product of simple reflections with odd length Moment Graph. With the Weyl group established, we are now ready to make the following definitions. Definition 2.3. The moment graph is an oriented graph that consists of a pair (V, E) where V is the set of vertices and E is the set of edges. Each vertex corresponds to an element in the Weyl Group (so if v V, then v W ). For x, y V, an edge exists from x to y, denoted by x α y if there exists a reflection s α such that y = xs α and l(y) > l(x). Definition 2.4. A degree d is a non-negative combination d 1 α 1 + d 2 α 2 of simple coroots. We will denote d = (d 1, d 2 ). What this is saying is that knowing the Weyl group for G 2 allows us to create a graph which encodes combinatorial properties of the reflections in W. The graph (constructed by myself and R. Elliott [5]) is an upward graph with the following properties. The vertices are the different Weyl group elements. This means there are twelve vertices. 1 For the identity element, I sometimes refer to it as 1 instead of id.

5 A QUANTUM K CHEVALLEY FORMULA 5 w 0 s 1s 2s 1s 2s 1 s 2s 1s 2s 1s 2 s 1s 2s 1s 2 s 2s 1s 2s 1 s 1s 2s 1 s 2s 1s 2 s 1s 2 s 2s 1 s 1 s 2 id Figure 1. The moment graph for G 2. The color code for the degrees is the following: Black-(1,0), Violet-(0,1), Red-(1,1), Green-(1,3), Blue-(2,3), and Orange-(1,2). The edges represent the root reflections associated to the G 2 root system. There are six different types of edges (different degree values) because there are only six reflections in the G 2 root system. Note that edges exist between Weyl group elements if the difference in the lengths is odd. The graph is oriented upwards which means that the bottom vertex corresponds to the element with the smallest length (id where l(id) = 0). The next row of vertices are supposed to have length 1 (s 1 and s 2 ). The length of these elements increases by one as you travel up the graph. The top vertex is where the element with the largest length is (w 0 where l(w 0 ) = 6. Note that for any vertex, there are only six edges connected to it and all six are unique. The moment graph for G 2 is displayed in Figure 1 on page 5. To help read the moment graph, a color code has been set up to represent the different types of edges Curve Neighborhoods. The importance of the moment graph can be realized with the following concept defined by A. Buch and L. Mihalcea [3]. Definition 2.5. Fix d = (d 1, d 2 ) a degree and u W an element of the Weyl group, The curve neighborhood, Γ d (u) is a subset of W which consists of the maximal elements in the moment graph which can be reached from u with a path of total degree d. In this paper we will often use the curve neighborhood Γ d (id), and we will denote it by z d.

6 6 MARK E. LEWERS AND LEONARDO C. MIHALCEA What this says is the following. Pick some w W on the moment graph and pick some degree d. Then for that particular w and degree d there is a path that can be taken to some v W where l(v) l(w) and the total degree of the path taken is less than or equal to the original degree d specified. From explicit computation, it is observed that the highest path for a given w W and degree d is unique [5]. Example 2.6. Consider w = id and d=(1,1). We want to determine the highest path from the identity where the total degrees traveled is no greater than (1,1). By inspecting the moment graph, there are three initial paths you can take from the identity. path d=(1,0). This will take you from id to s 1. At this point, you are not allowed to travel no more than d = (0, 1) upwards. Further inspection of the moment graph shows a path exists with degree (0, 1) that takes you from s 1 to s 1 s 2. We now have traveled a total degree of (1,1) and cannot move any more. path d=(0,1). This will take you from id to s 2. At this point, you are not allowed to travel no more than d = (1, 0) upwards. Further inspection of the moment graph shows a math exists with degree (1, 0) that takes you from s 2 to s 2 s 1. We now have traveled a total degree of (1,1) and cannot move any more. path d=(1,1). This path takes you from id to s 1 s 2 s 1. We have now traveled a total degree of (1,1) and cannot move any more. All possible paths have taken us to three different elements: s 1 s 2, s 2 s 1, and s 1 s 2 s 1. We are looking for the maximal element, i.e. the element with the longest length which here is s 1 s 2 s 1. So Γ (1,1) (id) = s 1 s 2 s 1. Table 3 in Appendix B shows the curve neighhorhoods at any degree d = (d 1, d 2 ) for any element w W (table taken from [5]). 3. Quantum K theory ring, QK(X) Recall that X denotes be flag manifold of type G 2. The quantum K theory ring, denoted QK(X) is an algebra over the formal power series ring Z[[q]] := Z[[q 1, q 2 ]], where q 1, q 2 are the quantum parameters. The ring QK(X) has a Z[[q]]- basis consisting of Schubert classes O w for w W. Let d = (d 1, d 2 ) Z 0 Z 0 be a (multi)degree. Recall from the introduction that every element can be written uniquely as a sum a w,d q d O w d=(d 1,d 2);w W where a w,d Z and q d = q d1 1 qd2 2. The degrees for the Schubert classes, integer coefficients, and quantum parameters are as follows. deg(a w,d ) = 0 deg(o w ) = l(w) deg(o u O v ) = deg(o u ) + deg(o v ) deg(q i ) = 2 and deg(q di i ) = 2d i deg(q d ) = deg(q d1 2 qd2 2 ) = 2(d 1 + d 2 ) Therefore we can state deg(a w,d q d O w ) = deg(q d ) + l(w).

7 A QUANTUM K CHEVALLEY FORMULA 7 Quantum K Theory QK(X) set q 1 = q 2 = 0 l(u) + l(v) = l(w) + deg(q d ) K Theory K(X) Quantum Cohomology QH (X) l(u) + l(v) = l(w) set q 1 = q 2 = 0 Cohomology H (X) Figure 2. This is a hierarchy diagram of rings associated with the flag manifold for type G 2. It shows QK(X) is a deformation of the rings K(X) and QH (X) which in turn are deformations of H (X). Furthermore, it shows the specialization needed in the product to go from one ring to the other. It is important to understand the addition and multiplication operators for this ring. The addition in this ring is componentwise: a w,d q d O w + b w,d q d O w = (a w,d + b w,d ) q d O w. w W w W w W The multiplication in the ring is much more complicated. The general rule for multiplying two Schubert classes is the following: (1) O u O v = q d Nu,v w,d O w w W,d=(d 1,d 2) where Nu,v w,d Z is a structure constant. An extra condition on the multiplication is that it is filtered, which means that l(w) + deg(q d ) l(u) + l(v). A significant feature of the QK(X) ring is that it is a deformation of the cohomology ring, H (X), the quantum cohomology ring, QH (X) and the K theory ring, K(X) in G 2. When analyzing the multiplication in QK(X), the following specializations can be made. if d = (0, 0), then Nu,v w,d = c w u,v where c w u,v is the structure constant in the K theory. In other words, if I take the product O u O v in QK(X) and set all the quantum parameters to zero, then we have the product O u O v in terms of the K theory ring. if l(w)+deg(q d ) = l(u)+l(v), then Nu,v w,d = c w,d u,v where c w,d u,v is the coefficient from quantum cohomology. What this means is that for a given product O u O v in QK(X) if you take the conditions with the lowest degree then you get back the equivalent product in the quantum cohomology ring.

8 8 MARK E. LEWERS AND LEONARDO C. MIHALCEA From here you can do even further specialization (set l(u) + l(v) = l(w) in K theory and set d = (0, 0) in quantum cohomology) to reduce the multiplication products down to the cohomology ring. This is summarized in Figure 2 above. In addition, certain properties are expected to hold for multiplication in QK(X). Consider u, v, w W and degree d where l(u) + l(v) l(w) + deg(q d ). We expect to be able to predict the sign of the coefficient Nu,v w,d. Griffeth and Ram [9] showed that in K theory multiplication of X, one can predict the sign of the coefficient c w u,v by the relation ( 1) l(u)+l(v) l(w) c w u,v 0. (in K theory) In QK(X) we expect the following relation to hold (2) ( 1) l(u)+l(v) l(w) deg(qd) N w,d u,v 0 Since QK(X) is a ring, we expect that associativity for multiplication will hold. In addition, we expect O s1 O s2 = O s2 O s1 (Proof by inspection. See Table 1 on page 12). Therefore we expect the following relation to hold for any w W. (3) O s1 (O s2 O w ) = O s2 (O s1 O w ) Remember that multiplying two Schubert classes O u and O v has the filtered condition where if q d Nu,v w,d O w shows up for some in the product, then you have l(w) + deg(q d ) l(u) + l(v). If you fix u, v, only w and d vary. Since d Z 0 Z 0 then d is unbounded above. This means it is theoretically possible for the product O u O v to contain infinitely many terms. However in QK(X), products are expected to be finite which means after some degree d, Nu,v w,d = 0 for all d d. So we need to make sure our multiplication of two Schubert classes is finite. 4. Chevalley Formula and KGW Invariants 4.1. The Chevalley Formula. The quantum K theory ring is generated by Schubert classes for the simple reflections s 1, s 2 W, i.e. it is generated by O s1 and O s2. Theoretically it is therefore possible to develop the whole multiplication table for QK(X) by using a formula that multiplies a Schubert class O w where w W with either O s1 or O s2. This is called a Chevalley formula. Therefore the modification for the Chevalley product from Equation 1 is (4) O u O si = w W,d=(d 1,d 2) q d N w,d u,s i O w. In order write a formula for (4), one must first determine how to compute the structure constant Nu,s w,d i. According to A. Buch and L. Mihalcea [2], there is a recursive definition to determine N w,d (5) N w,d u,s i u,s i. = O u, O si, (O w ) d κ W ;d d >0 N κ,d d u,s i O κ, (O w ) d Here O u, O v, (O w ) d and O κ, (O w ) d are 3-point, respectively 2-point, K- theoretic Gromov-Witten invariants (KGW invariants). Note that the equation for Nu,s w,d i is recursive because for a given d, the equation calls structure constants with a strictly smaller degree. By recursion, this will continue until we hit d d = (0, 0)

9 A QUANTUM K CHEVALLEY FORMULA 9 and then we have Nu,s w,(0,0) i = c w u,s i. This definition calls on us knowing the structure constants in K theory. Using a table of K theory products by Griffeth and Ram [9], it is possible to determine these K theory structure constants point and 3-point KGW Invariants. Equation (5) depends on our knowledge of 3-point and 2-point KGW invariants. A. Buch and L. Mihalcea [2] calculated the 2-point KGW invariants but before we look at that, the following definition needs to be made. Definition 4.1. Let s i be a simple root reflection which corresponds to the simple root α i and consider u W. The anti-hecke product u s i is defined by { us i l(us i ) < l(u); u s i = u otherwise More generally, if v = s i1...s ik is reduced, then u v = ((...(u s i1 ) s i2...) s ik ). The idea of the anti-hecke product is that given some s i and w W, the product will return a Weyl group element (either us i or u) that has length no greater than u. This product guarantees that your output will not have a length greater than the element of highest length in the input. Now we are ready to look at the definition for the 2-point KGW invariant by Buch and Mihalcea [2]. (6) O v, (O w ) d = δ v zd,w where z d is the curve neighborhood for degree d evaluated from the identity. v z d is the anti-hecke product. { 1 v z d = w δ is the kronecker delta meaning δ v zd,w = 0 v z d w There is no outside information on how to determine the 3-point KGW invariant. Our original conjecture for determining this 3-point invariant (developed by L. Mihalcea) was incorrect because some of the structure constants it produced didn t satisfy all the multiplication properties. In particular, there were two problematic Chevalley products O s2s1s2s1s2 O s2 and O w0 O s2 where the associativity condition failed as well as finiteness. Note that for all the other Chevalley products tested, the multiplication properties held. By manipulating the good Chevalley products and using the multiplication properties, we determined the values of the 3-point invariant needed in order to make our structure constants satisfy the multiplication properties for all Chevalley products. The following conjecture is case by case. Conjecture Point KGW Invariant Below are listed four different cases for this conjecture. (1) Let u W {s 2 s 1 s 2 s 1 s 2, w 0 } and let s i W where i {1, 2}. Consider degree d = (d 1, d 2 ). Let w W. If d i = 0 then if d i > 0 then O u, O si, (O w ) d = c w u z d,s i ; O u, O si, (O w ) d = O u, (O w ) d.

10 10 MARK E. LEWERS AND LEONARDO C. MIHALCEA (2) Let u {s 2 s 1 s 2 s 1 s 2, w 0 } and consider s 1 W. Consider degree d = (d 1, d 2 ) and w W. The procedure to determine the KGW invariant is the same as in case 1. (3) Set u = s 2 s 1 s 2 s 1 s 2 and consider s 2 W. Consider w W. If d = (n, 2) where n N then 2 w = id O s2s1s2s1s2, O s2, (O w ) d = 1 w = s 2 ; 0 else for all other d = (d 1, d 2 ), use procedure in case 1. (4) Set u = w 0 and consider s 2 W. Let w W, m N \ {1, 2}, and k N \ {1}. For d = (1, 2), 2 w = s 1 O w0, O s2, (O w ) 1 w = s 1 s 2 s 1, s 2 s 1 s 2, s 1 s 2 s 1 s 2 s 1, s 2 s 1 s 2 s 1 s 2 d = ; 1 w = s 1 s 2, s 2 s 1, s 1 s 2 s 1 s 2, s 2 s 1 s 2 s 1, w 0 0 w = id, s 2 for d = (1, m), O w0, O s2, (O w ) d = δ s1,w; for d = (k, 2), use special procedure in case 3 (the first one listed) ; for d = (k, m), O w0, O s2, (O w ) d = δ id,w ; for all other d = (d 1, d 2 ), use procedure in case 1. With Conjecture 4.2 established, all the tools needed to solve for Nu,s w,d i present. Here is an example to show you how the computation works. Example 4.3. Consider the product O s1 O s1. Equations 4 and 5 respectively say that O s1 O s1 = q d Ns w,d O w and N w,d s = O s 1, O si, (O w ) d d=(d 1,d 2),w W κ W ;d d >0 N κ,d d s O κ, (O w ) d. Griffeth and Ram [9] tell us that there is only one K theory structure constant: when w = s 2 s 1, Ns s2s1,(0,0) = 1. We now begin looking at different degrees d. Consider d = (1, 0). = O s1, O s1, (O w ) (1,0) Ns κ,(1,0) d O κ, (O w ) d. N w,(1,0) s κ W ;(1,0) d >0 I will now break up the problem by first looking at the 3-pt invariant and then the summation term with κ and d. From Conjecture 4.2, we can reduce the 3-pt invariant (since d 1 = 1) O s 1, O si, (O w ) (1,0) = O s 1, (O w ) (1,0) = δ s1 z (1,0),w. From Table 3, z (1,0) = s 1 and the anti-hecke product s 1 s 1 is id. So O s 1, O si, (O w ) (1,0) = δ id,w. are

11 A QUANTUM K CHEVALLEY FORMULA 11 Our criteria for d is that d (0, 0) and (1, 0) d. This means the only possible degree value for d is d = (1, 0). Therefore we have Ns κ,(1,0) (1,0) O κ, (O w ) (1,0) = Ns κ,(0,0) O κ, (O w ) (1,0). κ W κ W Since there is only one nonzero K theory structure constant, let κ = s 2 s 1 to get κ W Therefore N w,(1,0) s N s2,(1,0) s Ns κ,(0,0) O κ, (O w ) (1,0) = Ns s2s1,(0,0) O s2s1, (O w ) (1,0) = O s2s1, (O w ) (1,0) = δ s2s 1 z (1,0),w = δ s2s 1 s 1,w = δ s2,w. = δ id,w δ s2,w. For w = id, Ns id,(1,0) = 1 and for w = s 2, = 1. I claim (by inspection) that for any other degree d, Ns w,d = 0. This means that in fact O s1 O s1 = q (0,0) N s2s1,(0,0) s O s2s1 + q (1,0) N id,(1,0) O id + q (1,0) N s2,(1,0) s = (1)(1)O s2s1 + (q 1 )(1)O id + (q 1 )( 1)O s2 = O s2s1 + q 1 ( O id O s2) O s1 O s1 = O s2s1 + q 1 ( O id O s2 ) Results. Table 1 on page 12 shows the complete Chevalley table for O u O si where u, s i W and i {1, 2}. It is now appropriate to check the table to see if all the multiplication properties hold. Sign Predictability: Upon inspection of Table 1, all the terms follow the criteria given by equation 2. Specialization: We compared the results given in Table 1 to those done for K theory multiplication by Griffeth and Ram [9] and for quantum cohomology multiplication by myself with R. Elliott [5]. Upon applying the specializations discussed in section 3 to our QK(X) product results, our results for K theory multiplication and quantum cohomology multiplication matched up. Associativity: Table 1 shows that indeed O s1 O s2 = O s2 O s1. With this condition met, we can now check the associativity condition given by Equation 3. For all 24 products listed in the table, they all meet this associativity condition. This was checked by using a small program in Mathematica that we wrote up. The code is available upon request. Finiteness: This condition appears to have been met but this will be discussed more in section Lenart Postnikov Conjecture. This goal of this paper is to predict what the 3-pt KGW invariants are in order to compute Chevalley multiplication in the ring. This is a geometric prediction in determining the multiplication table in QK(X). Recently, Lenart and Postnikov [11] conjectured an algebraic prediction to determine the multiplication table in QK(X). They predict directly the structure constants Nu,s w,d i, without using the recursive definition (5). Their method involves using λ chains in the Bruhat ordering on the Weyl group. In terms of type G 2, these λ chains correspond to ω 1 chains and ω 2 chains where ω 1, ω 2 are fundamental s O s2

12 12 MARK E. LEWERS AND LEONARDO C. MIHALCEA O w O s i w W,d=(d 1,d 2 ) qd N w,d u,s i O w O id O s 1 O s 1 O s 1 O s 1 O s 2s 1 + q 1 (O 1 O s 2) O s 2 O s 1 O s 1s 2 + O s 2s 1 O s 1s 2 s 1 O s 2s 1 s 2 + O s 1s 2 s 1 s 2 + O s 2s 1 s 2 s 1 O s 1s 2 s 1 s 2 s 1 O s 2s 1 s 2 s 1 s 2 + O wo O s 1s 2 O s 1 O s 1s 2 s 1 +O s 2s 1 s 2 O s 1s 2 s 1 s 2 O s 2s 1 s 2 s 1 +O s 1s 2 s 1 s 2 s 1 +O s 2s 1 s 2 s 1 s 2 O wo O s 2s 1 O s 1 2O s 1s 2 s 1 O s 2s 1 s 2 s 1 + q 1 (O s 2 2O s 1s 2 + O s 2s 1 s 2 ) + q 1 q 2 ( 2O 1 + 2O s 1 + O s 2 O s 2s 1 ) O s 1s 2 s 1 O s 1 O s 2s 1 s 2 s 1 + q 1 (O s 1s 2 O s 2s 1 s 2 ) + q 1 q 2 (O 1 O s 1 O s 2 + O s 2s 1 ) O s 2s 1 s 2 O s 1 2O s 1s 2 s 1 s 2 + O s 2s 1 s 2 s 1 2O s 1s 2 s 1 s 2 s 1 2O s 2s 1 s 2 s 1 s 2 + 2O wo O s 1s 2 s 1 s 2 O s 1 O s 1s 2 s 1 s 2 s 1 + O s 2s 1 s 2 s 1 s 2 O wo O s 2s 1 s 2 s 1 O s 1 O s 1s 2 s 1 s 2 s 1 +q 1 (O s 2s 1 s 2 O s 1s 2 s 1 s 2 )+q 1 q 2 (O s 2 O s 1s 2 O s 2s 1 +O s 1s 2 s 1 ) O s 1s 2 s 1 s 2 s 1 O s 1 q 1 O s 1s 2 s 1 s 2 + q 1 q 2 (O s 1s 2 O s 1s 2 s 1 ) O s 2s 1 s 2 s 1 s 2 O s 1 O wo + q 1 q 2 2 (O 1 O s 1) O w 0 O s 1 q 1 O s 2s 1 s 2 s 1 s 2 + q 1 q 2 (O s 2s 1 s 2 O s 2s 1 s 2 s 1 ) + q 1 q 2 2 (O s 1 O s 2s 1 ) + q 2 1 q 2 2 ( O 1 + O s 2) O id O s 2 O s 2 O s 1 O s 2 O s 1s 2 + O s 2s 1 O s 1s 2 s 1 O s 2s 1 s 2 + O s 1s 2 s 1 s 2 + O s 2s 1 s 2 s 1 O s 1s 2 s 1 s 2 s 1 O s 2s 1 s 2 s 1 s 2 + O wo O s 2 O s 2 3O s 1s 2 3O s 2s 1 s 2 + 2O s 1s 2 s 1 s 2 O s 2s 1 s 2 s 1 s 2 + q 2 (O 1 3O s 1 + 3O s 2s 1 2O s 1s 2 s 1 + O s 2s 1 s 2 s 1 ) + q 1 q 2 2 (O 1 O s 2) O s 1s 2 O s 2 2O s 2s 1 s 2 2O s 1s 2 s 1 s 2 + O s 2s 1 s 2 s 1 s 2 + q 2 (O s 1 2O s 2s 1 + 2O s 1s 2 s 1 O s 2s 1 s 2 s 1 ) + q 1 q 2 2 ( O 1 + O s 2) O s 2s 1 O s 2 3O s 1s 2 s 1 +O s 2s 1 s 2 3O s 1s 2 s 1 s 2 4O s 2s 1 s 2 s 1 +5O s 1s 2 s 1 s 2 s 1 +4O s 2s 1 s 2 s 1 s 2 5O wo +q 1 q 2 ( 2O 1 +3O s 2 2O s 1s 2 +O s 2s 1 s 2 )+q 1 q 2 2 ( O 1 +2O s 1 O s 2s 1 ) O s 1s 2 s 1 O s 2 O s 1s 2 s 1 s 2 + 2O s 2s 1 s 2 s 1 3O s 1s 2 s 1 s 2 s 1 2O s 2s 1 s 2 s 1 s 2 + 3O wo + q 1 q 2 (O 1 2O s 2 + 2O s 1s 2 O s 2s 1 s 2 ) + q 1 q 2 2 (O 1 2O s 1 + O s 2s 1 ) O s 2s 1 s 2 O s 2 3O s 1s 2 s 1 s 2 2O s 2s 1 s 2 s 1 s 2 +q 2 (O s 2s 1 3O s 1s 2 s 1 +2O s 2s 1 s 2 s 1 )+q 1 q 2 2 (2O 1 2O s 2) O s 1s 2 s 1 s 2 O s 2 O s 2s 1 s 2 s 1 s 2 + q 2 (O s 1s 2 s 1 O s 2s 1 s 2 s 1 ) + q 1 q 2 2 ( O 1 + O s 2) O s 2s 1 s 2 s 1 O s 2 3O s 1s 2 s 1 s 2 s 1 + O s 2s 1 s 2 s 1 s 2 3O wo + q 1 q 2 (O s 2 3O s 1s 2 + 2O s 2s 1 s 2 ) + q 1 q 2 2 ( O 1 + 3O s 1 2O s 2s 1 ) O s 1s 2 s 1 s 2 s 1 O s 2 O wo + q 1 q 2 (O s 1s 2 O s 2s 1 s 2 ) + q 1 q 2 2 ( O s 1 + O s 2s 1 ) O s 2s 1 s 2 s 1 s 2 O s 2 q 2 O s 2s 1 s 2 s 1 + q 1 q 2 2 (2O 1 2O s 2) O w 0 O s 2 q 2 O s 1s 2 s 1 s 2 s 1 +q 1 q 2 (O s 2s 1 s 2 O s 1s 2 s 1 s 2 )+q 1 q 2 2 (2O s 1 2O s 1s 2 2O s 2s 1 + 2O s 1s 2 s 1 + O s 2s 1 s 2 O s 1s 2 s 1 s 2 O s 2s 1 s 2 s 1 + O s 1s 2 s 1 s 2 s 1 + O s 2s 1 s 2 s 1 s 2 O wo ) Table 1. Chevalley table for quantum K theory ring. weights in the G 2 root system. From Lenart and Postnikov s paper [11] on page 33, they define the chains as the following: ω 1 chain: (γ 1, γ 2, γ 3, γ 4, γ 5, γ 3 ) ω 2 chain: (γ 6, γ 5, γ 4, γ 3, γ 2, γ 5, γ 3, γ 4, γ 5, γ 3 ) where γ 1 = α 1, γ 2 = 3α 1 + α 2, γ 3 = 2α 1 + α 2, γ 4 = 3α 1 + 2α 2, γ 5 = α 1 + α 2, and γ 6 = α 2 (the γ s are just the positive roots for type G 2 ). Note that the conjecture calls for ( ω i ) chains, not ω i chains where we have the following: ( ω 1 ) chain: ( γ 3, γ 5, γ 4, γ 3, γ 2, γ 1 ) ( ω 2 ) chain: ( γ 3, γ 5, γ 4, γ 3, γ 5, γ 2, γ 3, γ 4, γ 5, γ 6 ) Before we look at the conjecture, we need to define the following operator first.

13 A QUANTUM K CHEVALLEY FORMULA 13 Definition 4.4. Let α be a positive root in the G 2 root system. Define c 1, c 2 Z such that α = c 1 α1 + c 2 α2. Consider any w W. The quantum Bruhat operator Q α is defined by ws α if l(ws α ) = l(w) + 1, Q α (w) = q α ws α if l(ws α ) = l(w) 2(c 1 + c 2 ) otherwise, For the purposes of this paper, all you know about this operator is that it is a deformation of the Bruhat operator they define for K theory. At this point we have all the tools needed to understand the Lenart Postnikov Conjecture (page 35) 2. Conjecture 4.5. Lenart Postnikov Conjecture Fix a simple reflection s i. Let (β 1,..., β l ) be a ( ω i ) chain of roots. Then O si O w = (1 (1 Q β1 ) (1 Q βl ))O w. We then computed all 24 products. What our results showed is that their Chevalley multiplication table matches ours perfectly. If our geometric approach yielded the same result as their algebraic approach, then this says we either have found the actual QK(X) Chevalley multiplication table or we are on the right track. 5. Future Work There is still future work that needs to be done in this QK(X) ring. Back in Section 4.3 we labeled out the different properties of multiplication that were met in the ring. One property we didn t check off though was finiteness. As stated, an issue that comes up with these KGW invariant calculations is that they depend on your (multi)degree d = (d 1, d 2 ) Z 0 Z 0. Since d is not bounded above, it is logical to wonder whether the number of nonzero structure coefficients Nu,v w,d is finite i.e. does your product have finite amount of nonzero terms or not? We expect that the quantum K-theory products to be finite so this is an important issue that needs to be resolved. The proof is still being worked out but that is due to the large number of cases to check rather than the difficulty of the problem (simple induction arguments have been working so far). In addition, further work needs to be done on determining the presentation and relations for this ring. We believe that it is possible to determine it if one knows how the multiplication works in QK(X) (which we are on the right track in determining) and if you know the presentation and relations of the various specialization rings i.e. that of K(X) and QH (X). Knowing the presentation and relations for these allows us to check whether the presentation and relations for QK(X) will specialize down to K(X) and QH (X). Appendix A. G 2 Root System Data This appendix contains data on the G 2 root system. Table 2 contains information on all the roots in terms of natural coordinates, simple roots, positivity, and the corresponding coroot. Also contained in here is information on the fundamental weights for G 2. 2 I am calling it the Lenart Postnikov Conjecture as a means to draw attention to its usefulness in this paper.

14 14 MARK E. LEWERS AND LEONARDO C. MIHALCEA 3α 1 + 2α 2 α 1 + α 2 2α 1 + α 2 α 2 3α 1 + α 2 α 1 α 1 (3α 1 + α 2 ) α 2 (2α 1 + α 2 ) (α 1 + α 2 ) (3α 1 + 2α 2 ) Figure 3. The root system for G 2 where each node is a root. This is represented in the basis where the blue lines represent the coordinate system. Fundamental Weights ω 1 = 2α 1 + α 2 ω 2 = 3α 1 + 2α 2 Natural Coordinates E Basis ± Simple Roots Basis Coroot α (ɛ 1, ɛ 2, ɛ 3) (α 1, α 2) α = µα 1 + να 2 ɛ 1 ɛ 2 + α 1 α1 = α 1 ɛ 3 ɛ 1 + α 1 + α 2 (α 1 + α 2) = α 1 + α 2 ɛ 3 ɛ 2 + 2α 1 + α 2 (2α 1 + α 2) = 2α 1 + α 2 ɛ 2 + ɛ 3 2ɛ 1 + α 2 ɛ 1 + ɛ 3 2ɛ 2 + 3α 1 + α 2 α2 = 1 3 α2 (3α 1 + α 2) = α α2 ɛ 1 ɛ 2 2ɛ 3 + 3α 1 + 2α 2 (3α 1 + 2α 2) = α α2 (ɛ 1 ɛ 2) α 1 ( α 1) = α 1 (ɛ 3 ɛ 1) (α 1 + α 2) ( α 1 α 2) = α 1 α 2 (ɛ 3 ɛ 2) (2α 1 + α 2) ( 2α 1 α 2) = 2α 1 α 2 (ɛ 2 + ɛ 3 2ɛ 1) α 2 ( α 2) = 1 3 α2 (ɛ 1 + ɛ 3 2ɛ 2) (3α 1 + α 2) ( 3α 1 α 2) = α α2 ( ɛ 1 ɛ 2 2ɛ 3) (3α 1 + 2α 2) ( 3α 1 2α 2) = α α2 Table 2. The table with information on the Root System for Type G 2. Appendix B. Curve Neighborhoods This appendix contains the curve neighborhoods for all the Weyl group elements. In order to write in a concise method, we need to define the following: l, m = 0, 1, 2, 3,... N, M = 1, 2, 3,... N, M = 2, 3, 4,... If w W then Γ (0,0) (w) = w so we won t include that condition in the tables (except for w o ). These were calculated by myself and R. Elliott [5].

15 A QUANTUM K CHEVALLEY FORMULA 15 id s 1 s 2 Γ (N,0) (id) = s 1 Γ (N,0) (s 1 ) = s 1 Γ (N,0) (s 2 ) = s 2 s 1 Γ (0,N) (id) = s 2 Γ (0,N) (s 1 ) = s 1 s 2 Γ (0,N) (s 2 ) = s 2 Γ (N,1) (id) = s 1 s 2 s 1 Γ (N,1) (s 1 ) = s 1 s 2 s 1 Γ (N,1) (s 2 ) = s 2 s 1 s 2 s 1 Γ (1,N )(id) = s 2 s 1 s 2 s 1 s 2 Γ (N,N )(s 1 ) = w o Γ (1,N )(s 2 ) = s 2 s 1 s 2 s 1 s 2 Γ (N,M )(id) = w o Γ (N,M )(s 2 ) = w o s 1 s 2 s 2 s 1 s 1 s 2 s 1 Γ (N,0) (s 1 s 2 ) = s 1 s 2 s 1 Γ (N,0) (s 2 s 1 ) = s 2 s 1 Γ (N,0) (s 1 s 2 s 1 ) = s 1 s 2 s 1 Γ (0,N) (s 1 s 2 ) = s 1 s 2 Γ (0,N) (s 2 s 1 ) = s 2 s 1 s 2 Γ (0,N) (s 1 s 2 s 1 ) = s 1 s 2 s 1 s 2 Γ (N,1) (s 1 s 2 ) = s 1 s 2 s 1 s 2 s 1 Γ (N,1) (s 2 s 1 ) = s 2 s 1 s 2 s 1 Γ (N,1) (s 1 s 2 s 1 ) = s 1 s 2 s 1 s 2 s 1 Γ (N,N )(s 1 s 2 ) = w o Γ (N,N )(s 2 s 1 ) = w o Γ (N,N )(s 1 s 2 s 1 ) = w o s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 2 s 1 s 2 s 1 Γ (N,0) (s 2 s 1 s 2 ) = s 2 s 1 s 2 s 1 Γ (N,0) (s 1 s 2 s 1 s 2 ) = s 1 s 2 s 1 s 2 s 1 Γ (N,0) (s 2 s 1 s 2 s 1 ) = s 2 s 1 s 2 s 1 Γ (0,N) (s 2 s 1 s 2 ) = s 2 s 1 s 2 Γ (0,N) (s 1 s 2 s 1 s 2 ) = s 1 s 2 s 1 s 2 Γ (0,N) (s 2 s 1 s 2 s 1 ) = s 2 s 1 s 2 s 1 s 2 Γ (N,M) (s 2 s 1 s 2 ) = w o Γ (N,M) (s 1 s 2 s 1 s 2 ) = w o Γ (N,M) (s 2 s 1 s 2 s 1 ) = w o s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 w o Γ (N,0) (s 1 s 2 s 1 s 2 s 1 ) = s 1 s 2 s 1 s 2 s 1 Γ (0,N) (s 2 s 1 s 2 s 1 s 2 ) = s 2 s 1 s 2 s 1 s 2 Γ (l,m) (w o) = w o Γ (l,n) (s 1 s 2 s 1 s 2 s 1 ) = w o Γ (N,l) (s 2 s 1 s 2 s 1 s 2 ) = w o Table 3. Curve Neighborhoods for every degree at w W for G 2. References [1] N. Bourbaki, Lie Groups and Lie Algebras: Chapter 4-6 (Elements of Mathematics), Springer, 2008 [2] A. Buch and L. Mihalcea, Quantum K Theory of Grassmannians, Duke Mathematical Journal. 156 (2011), no. 3, pag [3] A. Buch and Leonardo C. Mihalcea, Curve neighborhoods of Schubert varieties, to appear in Journal of Differential Geometry, available on arχiv: [4] C. Chevalley, Sur les décompositions cellulaires des espaces G/B, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, With a foreword by Armand Borel, pag [5] R. Elliott and M. Lewers, Quantum Schubert polynomials for G 2 flag manifolds, in preparation. [6] W. Fulton and A. Lascoux, A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J. 76 (1994), no. 3, pag [7] W. Fulton and C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), no. 4, [8] A. Givental, On the WDVV equation in quantum K-theory. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J. 48 (2000), [9] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 (2004), no. 8, pag [10] J. Humphreys, Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics) (v.9), Springer, [11] C. Lenart and A. Postnikov, Affine Weyl groups in K-theory and representation theory, Int. Math. Res. Not. IMRN, 12 (2007), Art. ID rnm038, 65. [12] H. Pittie and A. Ram, A Pieri-Chevalley formula in the K-theory of a G/B-bundle, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), pag