 Okay, welcome back to the second part of the Schubert seminar. Wei Hong will continue her talk on quantum k theory on incidence varieties. Okay, so yeah, let's, let's start talking about actually quantum k theory. So these k-theoretic Gromov-Wittern invariants can be packaged into the structure of a ring called the quantum k-theory ring. If you know quantum cohomology, this is the k-theoretic generalization of it. The quantum k-theory ring of X is an algebra over this formal power series ring, where we use the powers of the formal variables q1 and q2 to keep track of the components of the degree. Schubert classes form a basis for the quantum k-theory ring of X over this power series ring. Yeah, just like in quantum cohomology, Schubert classes form a basis. Knowing the all the k-theoretic Gromov-Wittern invariants is equivalent to knowing how to multiply in the quantum k-theory ring. Although some combinatorics needs to be worked out to go from one to the other. All of these work t-equivariantly. In order to state our formulas in a simple form, we're going to simplify the notation for a power of q times an opposite Schubert class by reversing the reduction mod n process to get an element in Wp-toda from the degree and the Schubert class. So this element lives in the equivalent quantum k-theory ring of X localized at the q variables because it is fine to allow negative components in the degree. Finally, for an integer i, we will write epsilon i for the character epsilon i bar. We have this equivalent Chevrolet formula, which tells us how to multiply a Schubert class and a Schubert divisor class. Again, of these indices live in Wp-toda, so the q variables are implicit in the formula. The non-equivariate case was worked out independently in Rosette's thesis. And a formula for two-step varieties, two-step flat varieties, is in the second version of Kuno, Lina, Naito, and Sogaki, which was posted on the archive a few days before my paper. In Schubert calculus, we are often in the situation where we have a natural basis given by Schubert varieties for some ring. When we multiply two basis elements, we can expand it as a linear combination of basis elements and the coefficients in the expansion are called structure constants. Because things are so nice, we often expect the signs of the structure constants to alternate with the co-dimensions of the Schubert varieties. This has been proved in various generalities and conjectured in other cases. In our case, some structure constants can be read out directly from our formula, and their signs are exactly as expected. Here, the L of u is the length of u considered as a value group element. It's also equal to the co-dimension of the opposite Schubert variety, xq. This integral is equal to the degree of q to the d in the quantum cohomology ring of x, where there is a grading. A result of cattle implies that the projection from x to the projective space induces a ring homomorphism from the equivalent quantum k theory ring of x to that of the projective space. Using this ring homomorphism and our equivalent Chevrolet formula, we can recover a known Chevrolet formula for the equivalent quantum k theory ring of the projective space. This was a result of my academic brother, DuVan Chang. In the non-equivarian case, I was able to obtain a little bit Richardson rule, which tells us how to multiply two arbitrarily Schubert classes. Here, chi of i greater than j is equal to one when i is greater than j and zero when i is smaller than j. This formula makes sense because when this inequality holds, x is never congruent to y mod n. And when the opposite inequality holds, x is never congruent to y minus one or y minus two. This guarantees that in each case, the indices on the right hand side actually live in WP Tota. The subtle point is that the converse here is not true. And again, the q parameters are implicit in this formula. The first two examples, when n is equal to five, the first two examples are of the first case where we get one term. And the last two examples are of the second case where we get three terms. Let's compute this example in detail. It's the product of a co-dimension for Schubert class with a point class. To compute our formula, we need to compute x equals two plus five minus one, which is six, and y equals one plus one minus five, which is minus three. Then we need to check which of the two inequalities hold. So x minus y equals nine, and because two is greater than one, five is greater than one. On the right hand side, we have five times one plus one, which is 10. Since nine is smaller than 10, we're in the first case. So we just get the single term 06 minus three. And the subtraction mod five gives us the Schubert class 012 and the degree 11, because we subtracted five from six to get one, and added five to minus three to get two. And again, the structure constants in our little with Richardson rule have expected signs. We have also studied the powers of Q that appear here with nonzero coefficients in the product of two Schubert classes. For example, it's known that for all GMAT P, there is always a unique minimal power of Q corresponding to the smallest degree of a stable curve connecting opposite Schubert varieties. In our case, we have that the powers of Q that appear always form an interval between a unique minimal degree and a unique maximal degree. Of course, not much can go wrong in our case, because the highest degree that we could possibly get is one one using cattle ring homomorphism. So with Richardson rule, we can recover a known little with Richardson rule for the quantum K theory ring of the projective space. This is a special case of book and me how she's theory rule for the grass mania. Now let me say a little bit about the proof of the geometric result. Recall that we want to show the general fiber of this restricted EV three is rationally connected. This general fiber is a three point promote written variety consisting of maps with the mark point sent to a Schubert variety and opposite Schubert divisor and some fixed point that is general in this two point curve neighborhood. Our first ingredient is a result of Graeber Harris star, which says that, given a dominant morphism of complete complex irreducible varieties. If the target and the general fiber are both rationally connected, then the source is rationally connected. On the other hand, given a stable map to x, we can compose it with the projection to get a map to the projective space. If the original map has degree D equals D one D two, then the composition will have degree D one. Now that we may need to collapse some components of the source curve to make the composition stable. A result of Baron and Manon says that doing this actually gives us a morphism from the modular space of degree D. Maps with target x to the modular space of degree D one maps with target the projective space. We can restrict this morphism to the three point promote written variety, which we want to show is rationally connected. A point in this form of written variety is a map with the mark points sent to the Schubert variety, the Schubert divisor and the fixed point. Now that we are composing with the projection will give us a map to the projective space with the mark points sent to the projection of the Schubert variety, the projection of the Schubert divisor and the projection of the point. In other words, the target of the restriction is this three point group of written variety for the projective space. We can also show that both the target and the general fiber of this restriction are rational. In fact, they are both parameterized by sections of line bundles. Now, if we can also show that this map is surjective, and the source is irreducible, then we can apply Weber Harris star to draw our conclusion. We need to be a little careful, because this map is not surjective in general. In fact, it can happen that the target has higher dimension than the source. What we can prove instead is that if D one is the smaller of the two components of the degree, then this restriction is indeed surjective. And ultimately, we can always reduce to this case, because there is an evolution of acts, given by swapping the two projective spaces and relabeling the coordinates. This evolution swaps the two divisors and the two components of the degree. Similarly, with some work, we can also show the source is irreducible. So we are done by Weber Harris star. What about other type A flag varieties? What can we say about their quantum K theory ring? Well, for the full flag variety, FLN, where the sequence of As is the full sequence from one to M minus one. We have this known Chevrolet formula, which expresses the product of a Schubert class with a Schubert divisor class as an alternating sum over non-empty paths in the quantum A-brew hat graph. This formula was first proved by Lienard and Mano for multiplying quantum growth and deep polynomials. And later, Lienard, Naito and Silgacchi proved that quantum growth and deep polynomials represent Schubert classes. So combining these two results give us this theorem. Now, what are quantum brew hat graphs or quantum A brew hat graphs? Well, a quantum brew hat graph is a directed graph on the symmetric group with edges labeled by pairs ij such that i is smaller than j. Each such pair corresponds to a transposition tij, which swaps i and j. Each edge in the quantum brew hat graph carries a weight. There is an edge between two permutations if they differ by a transposition tij and either their lengths differ by one or their lengths differ by the length of the transposition. In the first case, the edge points towards the longer permutation and carries weight one. In the second case, the edge points towards the shorter permutation and carries weight this product of Q variables. As an example, let's draw the quantum brew hat graph for S3. We have vertices one, two, three, two, one, three, one, three, two, two, three, one, three, one, two, and three, two, one. The first type of edges point upwards and we have these. The second type of edges point downwards and we have these. The edge from three, two, one to two, three, one carries weight Q1, for example, because we're swapping the first two numbers. Similarly, the edge from two, three, one to two, one, three carries weight Q2. And the edge in the middle from three, two, one, two, one, two, three carries weight Q1 times Q2 because we're swapping the first and the third numbers. We work with paths in the quantum brew hat graph. For example, here we have a path following this edge, this edge, and this edge. The weight of a path is simply the product of the weights of its edges. So this particular path carries weight Q1 times Q2 times one. This example is for S3, but for us, the symmetric group will be the infinite symmetric group. Anything else is the same. We'll use a subgraph of the quantum brew hat graph called the quantum A brew hat graph, whose edges consisting of pairs such that the I is at most A and the J is greater than A. And finally, we will give the set of edges in the quantum brew hat graph a total order using this definition. It's just a variant of the lexicographic order. So let's revisit this theorem. So we see that in each salmon, each salmon corresponds to a non-empty path in the quantum A brew hat graph. And the permutation at the end of that path gives us a sugar class. The length of that path gives us a product of Qs, and the length of that path determines the sign. Using cattle's ring homomorphism, we can get a formula for the quantum k-fury ring of a partial flat variety, but it will have cancellations. So we expect that in general, we don't need to sum over all of these paths, but rather we just need to sum over paths satisfying some additional conditions depending on the A's. And when we do that, we can actually get a cancellation free formula. And this is exactly my conjecture. And as that, we need to impose these three additional conditions on the paths. Note that these conditions depend on the A's, where we set A0 to be 0, and Am plus 1 to be n. In the case of the full flat variety, these conditions become vacuous. So we recover the theorem on the previous slide. And this conjecture has been verified by my computer program for m up to 4 and n up to 8. Yeah, this is everything I wanted to say. Thank you very much for your attention and happy Pi Day. Thank you very much.