 Right, so welcome back to the second part of the Schubert seminar with a talk by Rui Xiang, please take it away Rui. Thank you. So, in the rest of my talk, I would give some applications of our results. Well, the first application is hoop formulas. So, let me briefly review the classical hoop formulas. So, basically, the number of standard, say, wow, in terms of representation series, the dimension of irreducible representation of a symmetric group is given by hoop formula, hoop length formula. So, as an example, I take a 3-2, this partition, we say that it is represented by three boxes and two boxes, three boxes and two boxes. And the standard long dot blue means we fulfill the table by 1 to n and the number of boxes. And we assume that in each row and each column, it's in each row and each column, it's increased and fulfilled by 1 to n. Okay, for example, well, just to have a, just to try yourself, you will find that for this partition 3-2, there are five standard yam w. Say, it's given on the left hand side. And so the number of standard yam w is five. And on the right hand side, there is one way to compute it without enumerating. And it's given by this expression. Okay, the numerator is five factorial, so five is the number of boxes. And the denominator is the product of hoop lengths. And, okay, what is hoop lengths? So, to each box of this yam diagram, we can compute in the so-called hoop lengths. For example, at this box, for example, as presented here, we can draw a hoop and the number of boxes touched by this hoop, touched by this hoop, is the hoop length. So this is four boxes attached, so it's four. And for this, this box, there are three boxes touched, so it's three. The similar reason for this one is two, this one. So the product of length is just like this, four times three times one times two times one. So above is five times four times three times two times one. Cancer, cancer, cancer, so rest five. So it's true, and actually it's true in general. So for arbitrary large partition, we have the same formula, right? This n factorial divides the product by hoop lengths. And okay, it's a very classical result. And for this, actually I want to, there's a very beautiful proof using super calculus. It's a Naroos method. So Ikita and Naroos observed that the classical hoop formula is a shadow of a equivalence of a formula of a grass mania. Okay, let me first describe this formula. D is over grass manian. D is a divisor, C1 or 1. It's a divisor. If you know geometry, then you go to the first-gen class of a tautological bundle and take minus. Okay, D is this, and we can expand it in terms of superclasses, equivalent superclasses. The first term is this class again, but the coefficient is the localization of this D at lambda. And the rest term is the coefficient is the same as the non-equivalent coefficient is by adding a box on this lambda. So all possible way to add a box on it. And wow, if we use this formula and using localization, using some geometry, then we can prove the classical hoop formula. So, okay, this method is very beautiful. And actually it can be generalized to, for example, classical class types, type B and type D. And also it can be generalized to K theory. It can also be generalized to SSM classes just recently by Naro, Mihashi, and Su. Okay, and here I draw a picture. So here is a classical hoop formula. We add one box each time. It means this time, the non-equivalent term, we have each box adding each time. So we have this classical hoop shape. And for K theory, it's by obtained by adding horizontal strip. Actually, I don't know very well about this. But anyway, look at this paper. And for SSM class, so for SSM class, it's closely relates to CSM class. And each time it's obtained by adding a boundary hook. And to, okay. So it has a lot of generalization. And our generalization goes to another direction, another dimension. We replace a divisor here by a higher degree classes. So for this generalizations, they all use divisor. They use degree one classes. And here we use higher dimension, higher degree classes. And so that each time we add not only one box, we add more. Okay, let me introduce another interesting formula, which is not that well known. There is also a hoop formula for domino double by forming and rule of which, okay, the formulation is can be illustrated by two examples. So the first example is a partition for four. Okay, so. Okay, it is obtained by divides the ship by dominoes, as we see here. And we also requires, we also put number inside each domino, such that it increase each row and each column. So we can define it properly. Okay. And another example is four, three, one. And we see that since they have the same number of boxes. But we see that the number of different right the number is six. This number is two are true is true. And we have a very interesting formula is very similar to the classic book for a hoop formula is. Well, the numerator is eight double factorial. So eight is a number of boxes and double factorial means is eight times six times four times two. Each time we minus two. And the denominator is a product or even hook lenses, or even hook lenses. So for example, for this for full partition. Well, the following double records, records the hook lens at for each box. So this is a five, four, three, two, and four, three, two, one. And the even hook lens, as I put to both frame on them. So it's four, two, four, two. So it's not meter is four, two, four, two. Let's check. Two to cancer for cancer cancer. So it's six. Yes, six equal to six. Good. And for this similar formula can be found. So this is six. This is four, three, one, four, two, one, one. It's fine. And this box contains this four boxes are even. So it's eight double factor divided by six times four times four times two. So there are only two. So this formula. Wow. It's also a hook lens for. And okay. This is no result. And okay. By our emerald. We can deal with the general case of our Rimi hook double. Using our. Equivalent emerald. So our formulation is a follow. So for a skillship. So for a skillship. Of size. We have the following. Lauren. Expansion. The left hand side is a. Okay. Okay. For this and this. They are both super class. Equivalent. Equivalent. Equivalent. Equivalent. And this, they are both super class Aquamarine super class and localized at the point. Lambda. And we know that localization. Is a function of. The function of Aquamarine parameters. And we also need to specialize. The current parameters. So we said to him to be the. To I is a current parameter. So why are you sorry to interrupt? There is a question out there. If, if this form in Lula formula has a Q analog, if you happen to know that? Oh, I don't know. Well, maybe we need the correct formulation of equivalent version. Okay. Oh, actually, I don't know maybe. Q analog, not the key variant analog. Oh, KK analog. Yeah. Q analog. Q analog. So maybe, maybe it's related. Ah, Q analog. Like segre motivi classes, I would imagine. Ah, yes. Something along this extra parameter Y. Probably, yeah. Yeah. I think I'm not an expert, but sometimes the sum over, instead of taking just counting the compute sum over Q of some span of something. And span means, well, we take height into consideration. I think, yes. There would be some, yes, yes, yes. So I guess there would be some Q analog. Yes. Oh, it's an interesting question. Yes. Okay. Now, let us describe this. So the left-hand side is geometric. It's localization and spatialization through some Z. And Z, why I use Z since actually, I want to think Z as a complex variable. And we can expand it at the Rousseau unity. The primitive R-Rousseau unity. It has the following Lorentz expansion. Okay. Okay. This is the lowest term. And the coefficient is a number of ribbon. So the R-Rimi hook double, the number of R-Rimi double device sum. Well, this is actually, if you like, you can think it as dr factorial r. So for example, when r x2 to two is double factorial, when equal to three, then you deduct r each time. You minus r each time. So anyway, and plus minus. So we can control this sign anyway. We can control this sign. And while there are some remainder, well, the rest terms we do not know how to control. Okay. We can have some expression, but maybe it's not that regular. So maybe for this term, it's also interesting. So at least as a lowest term, the boron expansion is closely relates to the number of ribbon double. And for this, this is skew shape actually. When lambda equal to the empty partition, then we can use Galois theory and we can recover this formula, this forming and rule. Rule of. Okay. This is our formulation of hook formula. So as the central theorem, the main theorem we use is equivalent M and rule. Okay. And actually, I did not describe it in my talk since actually we consider it over grass mesh. So before my theorem, I stated over flakorite. You can easily transform it to a grass mesh. Okay. This is our hook formula. And then let's discuss positivity conjectures. Okay. For positivity, there are a lot of positivity conjectures which were calculus and CSM class maybe have a lot. Since we can formulate different kind of positivity. So first is the structure constant of CSM class. So the following conjecture was formulated by several authors by Mihashi, Knusen and the broadcasting and Kuma. And it states that the structure constant for CSM classes is positive. It means the product goes to CSM classes is non-negative expansion, non-negative combination of CSM classes. And this conjecture was recently proved by Sherman Simpson and Wall. And actually they probe it using the theory of power shifts. So it's a, well, as I remarked below, so precisely the conjectures, the conjecture should be stated in terms of SSM classes. And actually they approve the program or more general geometric result. And while in our case, we overflag variety, non-equivalent CSM class and SSM class differ by certain sign modification. So okay, the conjecture stated here is equivalent to the reference here. So sorry for the confusion since I do not want to introduce SSM class in this talk. Okay, the first, of course, the first positivity conjecture should be made on structure constant. And another positivity conjecture, well, no, it's not a conjecture, okay. So it's a relation between CSM class and Schubert classes. So from the formula, recursion formula, we have, we can expand CSM classes in terms of Schubert classes. So okay, as I said, the lowest term is just Schubert class and plus some higher terms. So it's, we can prove that it's actually has Z coefficients. And actually, actually the coefficients are non-negative. This is theorem by Alofi Mihashi, Schurman-Suhl. So basically it means this coefficient are non-negative. So geometrically speaking, it means the CSM class is effective. CSM class is effective. Okay, and the proof use the theory of D modulus to relate CSM classes with one by D modulus. Perhaps I should also put a remark here. So actually more general, so actually in the paper, more general theorems are proved. So it's proved that the S-S-M class is, well, after modification, sign modification is probably effective. Anyway, so there are two theorems on CSM classes. And from this, geometrically speaking, CSM class is effective. So it's nature to ask, it's conjectured by Kuhlma. The CSM class or any Richardson variety is effective. So, well, we need to be careful here. Richardson variety is the intersection of Schubert variety and opposite Schubert variety. So when we mean for Richardson cell, it means here XU circle means it's a Schubert cell and Y we use in our talk is opposite Schubert cell. And we take the intersection, take the intersection, well, let us call it by Richardson cell. Okay, then we consider the CSM class of it and it was conjectured by Kuhlma that it's effective. That is, well, it can be explained as a non-negative expansion of Schubert classes. And actually we proved a weaker form of it in type A that this class CSM class of Richardson cell is monomial positive. Okay, so for this we actually, okay, on the right hand side it's sum over all type A sum over all permutations. And this Schubert class, we know that it can be represented by Schubert polynomial and Schubert polynomial corresponding to W in SN is a polynomial whose monomial is well, sorry. So we can expand. Okay, let me just say to them, our zero. So the CSM class of Richardson cell is the sum over the non-negative combination of the four monomials X of power delta. Well, this is actually delta one, delta n is X1 of power delta one up to Xn of power delta n. And delta is less than n minus one up to one zero. So it's, I mean, first we know that Schubert classes since it can be expanded, can be represented by Schubert polynomial. And Schubert polynomial is known to be monomial positive. And so it would be a shadow of this conjecture. Okay, so it's a weaker form. And probably, well, for this conjecture it's stronger, maybe much stronger than our zero, but our zero, maybe one step. Okay, and actually we proved that the Kuhn-Mers conjecture is equivalent to the following result. For any permutation U and V, the CSM class, product with Schubert class is non-negative combination of CSM classes. So, okay, actually here is this constant, maybe it is not good to call it structure constant. So since it's mixed, so I want to say the CSM expansion of Schubert times CSM is non-negative. This is our, well, we proved that this statement is equivalent to Kuhn-Mers conjecture. So this is maybe more commentary show from some point of view. This statement is equivalent to this statement. Okay, and actually we also formulate the equivalent version of this conjecture. Since for this it can be viewed as a generalization of the conjecture, generalization of classical positivity of Schubert product. The classical geometric proof of non-negativity of structure constant of Schubert class. If we take the lowest term of both sides, here becomes Schubert class and here becomes Schubert class of suitable, of compatible degree. And then it becomes a classical transversal intersection product. Okay, and okay, this is our conjecture. And also, since for the Schubert side, there exists a equivalent analog. And so we also formulate the equivalent version of this conjecture. It says these days that we can put equivalent, we can change every class by equivalent analog. So equivalent CSM class and equivalent Schubert class and equivalent CSM class. And the coefficient we, I mean the structure constant is non-negative linear combination of positive roots, the monomial positive roots. So we formulate this conjecture. And actually, these two conjectures with our peer rule is a spatial case of it. So for spatial V, I mean, okay, for spatial V, it can be E r x1 up to xk. For spatial V, it can be represented by E r xk. So our, okay, recall our formula is sum over sum, sum path. And the sum is over, I mean, the right-hand side, it's just sum over sum path and it's a CSM class of the end of the path. So it's automatically positive, right? It's automatically non-negative. And similar for this, for this maybe since actually in my talk, I did not state our equivalent peer rule for equivalent CSM class times equivalent Schubert class. But we, well, it's a little bit long, the expression, but we can prove that the coefficients, when V is spatial, spatial means it's correspond to spatial Schubert class which corresponding to, okay. So it comparatorially is double E r. So it's double E r, so it's x1 up to xk and T1, Tn. And, okay, our formula shows that the coefficient is actually by some here. So, okay, our peer rule is, okay, in total just our peer rule can be used to check for spatial, which this is our serum and sand application. So that's all, so thank you. Thank you very much. Thank you for a beautiful talk.