 So welcome everyone to the Schubert seminar. Today, we're happy to have Rui Shung from University of Ottawa telling us about Chernsch-Warthkenfelsen classes of our flag varieties, theory rules, conjectures, and more. So please take it away, Rui. Yeah. Thank you for inviting me. Okay. Today, I'm going to discuss my recent paper joined with Neil Fenn and Peter Gore. And so let's start from the first part, the geometric background. And okay. Okay. I hope everyone loves counting. Okay. On the left hand side, we have enumerating problems. We have two principal, addition principal, and multiplication principal. It means to get the total number, we take a sum. To get the number, we take product. Right. So on the left hand side, it's very basic about counting. And okay. There is some geometric analog of the, of the two principal is, well, I want to claim that olacoloristics show the similar property as the left hand side. And first is let X be an algebraic variety over C and U be an open subset. And then we have the following, the following identity. We have the following identity, the olacoloristic of X equals to the sum of olacoloristic of U and its complement, the olacoloristic of its complement. It's followed from the long, the sequence of more homology. And secondly, if we have a fiber bundle, say X to be the total space, and B is the base space, F be the fiber. Then by the spatial sequence, we have the following identity, the characteristic of the entire space equals to the product of, or the characteristic of base space and the fiber space. Okay. So they are parallel in some sense. And the, okay, this is the first geometric background. And the second is consider the following two functions between varieties over C to the linear vector space over Q. And there are two functions. The first one is a space of constructable functions. It is defined by, it's just a, well, how to say, it's a function over X, which is spanned by, well, let me take this as a definition. It is spanned by the characteristic function of closed sub-variety. And it is a function, it means we assign each algebra of variety, not only, okay, we assign an algebra variety a linear space. We do not, we do not only assign for each algebra variety, we also define push forward for each proper morphism between two algebra varieties. The push forward is defined, well, you can check that the following definition is well defined. It defined such that the push forward or the characteristic of a closed sub-variety, the value at Y is a characteristic of its fiber. So it's defined by this. So in some sense, if you think it as an analog of the previous slides, this push forward is defined to be counting, counting along fibers but using all the characteristics. Okay, this is a constructable functions. And on the other hand, on the other hand, we have more homology. Well, roughly speaking, well, here I do not put more. So we can think that the algebra cycles lies in this more homology. And for proper morphisms, we define push forward, roughly speaking, its integral along fibers. Okay, they are also kind of parallel. They are both functions at least. So between these two functions, they are actually deeply related. So the most classical result in this, well, the most classic relation is the following. For the most projectile variety, X, we know that the characteristic, chi X equals to the push forward of the trend classes of the tangent bundle of X. And okay, this is a classic result. You went, well, found by Chen. And well, in terms of function, we have the following diagram. So characteristic function of X is pushed forward to chi X over the point. And on the other hand, the trend classes of tangent bundle is sent to, well, sent to the right-hand side. So they are equal. It means we have this diagram. And okay, I draw this diagram. It means, actually, I want to indicate that there would be some connection between this identity. So it was conjectured by Grassend and Lin and approved by McPherson that there is a natural transform. Well, here, a natural transform means it's commuting with push forward. There's a natural transformation from, well, the function of constructable functions to Burmese homology. And it's characterized by the following property. It maps the characteristic function of X to the trend class of the tangent bundle when X is smooth. So okay, this is a geometric background of this, well, for this CSM, now it's known as Schwarz-McPherson class. So CSM class means this. Okay, let me briefly explain an example. Okay, assume W is constructable in X. Let us denote CSMW to be the CSM of the characteristic function of W. It's actually inside the Burmese homology of X. Let's consider the project of line P1. And we know that topologically, it is a Sophia. And the right-hand side, okay, this big Sophia is a picture of P1. We can identify Burmese homology and the homology by Panhe duality. And we have the decomposition P1 is A1 union with this infinite. Okay, this point is infinite and the complement is just a copy of A1. Okay, and we know that the homology is, this ring is QX quotient by X squared. And X is a class of a point. Okay, and well, by our characterization of CSM class, P1, since P1 is most, this CSM class is just a C1, I'm sorry, it's not C1. In C is a total trend class. So this is a total trend class of tangent bundle, which is 1 plus 2X. And we can also compute for the point at the infinite. This CSM class is just the class of a point, so it's X. And since, okay, we can compute then the CSM class of the complement, this part, A1, it just is a difference. Since here, this is 1, P, and this is 1, infinite. And so the difference is 1, A1. So as a result, the difference of them is just the CSM class of A1. So it's 1 plus X. This is an example. Okay, this is a geometric background of CSM classes. And okay, let's turn to flag varieties. Okay, actually P1 is an example of flag varieties. So we will concentrate on the classical flag variety. It is a variety parametrized following datum. It's a chain of vector spaces in CN such that each space has dimension, V i has dimension i inside CN. And for each element, it's called a flag. That's the reason it's called flag variety. And the classical result is we can decompose flag variety into disjoint union or sugar cells. So it's parametrized by a symmetric group for 1 to n. And okay, here I use the opposite sugar cell. And so that each sugar cell is equal to an isomorphic to a fine space of core dimension equal to the length of the number of inversion. And okay, now the question is how to compute the CSM class of a sugar cell. Actually, the CSM class of supercells over flag varieties are computed by Alofi and Mihashi. Mihashi is using the Bussamel solution. And actually they show that CSM classes can be computed by the majority operators. Okay, and this is the background. And now let's turn to our first result. The following is our first mean result. Recall that the chain classes of the dual of case tautological bundle say, okay, say r with k hat, it means the chain classes, the r's chain class of the dual of case tautological bundle. It can be represented by the elementary symmetric polynomial in x1 to xk. So that may be built by er, x1 up to xk. And our first theorem is for any permutation u in SN. And okay, we have CSM class of sugar cell corresponding to u. And if we take product with this class, it equals to the following combinatorial result. And we have the following combinatorial formula. The CSM expansion of the left hand side is equal to, well, it's also equal to some CSM class, the sum is firstly, multiply three. And it sum over the so-called decreasing path gamma or less r start from u to, well, to its end. So here r is a degree of this class. And okay, decreasing path is we are tracing the path in the following diagram. The diagram is given as follows. So the vertex is just a symmetric group. And we put an arrow between two permutations with a label tau. If w equals to u times TAB, TAB is a transposition of a and b for some a and b. And a is no greater than k. And b is strictly greater than k. And the length of w is greater than the length of u. So this actually means the orientation of the arrow. And moreover, the label tau is defined to be ua. So a is the smaller number exchanged. And tau is ua. u applied to the smaller index changed. Okay, and this is our first theorem. Okay, and for this diagram, I have some beautiful picture to show to us. Okay, this is a graph when n equals to 4 and k equals to 2. I'm sorry, here k is a k-thotological bundle here. It means a number of variables. So this example is when n equals to 4 and k equals to 2. And for example, if we want to find a decreasing path, decreasing path means the label is decreasing. Let me find one decreasing path. Okay, for example, this 2, 1, 3, 4 goes to 4, 1, 3, 2. Then this is 2. Then we find the label in smaller than 1. We can go to here, go to here. And perhaps it also has some other choices. Also this and this. So the result will be, I mean, the set some class of 1, 2, 3, 4 times e r x1, x2 equals to the set some class of these three classes. If I compute it correctly, if I did not miss anything. This is the first diagram. We say that the diagram is very symmetric. And secondly, another example is this. This is an example when n equals to 4 and k equals to 1. It's also very symmetric. Okay, and I want to mention a phenomenon that actually we can prove that CSM class of Schubert cell in, I mean, in module flagrarity, the lowest term, geometrically it is a high, it's a cycle or high, highest dimension equal to the Schubert class. So it basically means if we take the lowest component of both sides, we will get a period rule for Schubert classes, which is found by Sotili. And in that case, if we take the lowest term both sides, we will get the pass such that each step, it increased dimension by 1. So because of the dimension, because of degree reason. So I want to show you what, show you if I remove. So just one naive question. What is the role of tau again? I don't see any mention of tau in the in the phone. Here tau, here I mean decreasing means the label is decreasing. Oh, the tauts have to be decreased. Okay, okay, thanks. Yes, exactly. Thank you for your question. Yes, okay, and if we take the lowest term, we will get the period formula for Schubert classes, which is classical and well, not that classical. And I want to show you a phenomenon that, okay, the diagrams here looks very symmetric, right? Here we have four components and they reflect together. We have reflection, yes, symmetric, symmetric. And for this, well, it's more symmetric. It's very beautiful, right? We have this symmetric, symmetric. But if we focus on Schubert class, it means each step we can only go, each step we can increase the length only by one. So let us remove the arrows, which has a degree, the length increase more than one. Then what will we get? We see that they are all symmetric. If we remove everything, well, it looks not very symmetric, right? So it basically means, okay, let's say another diagram. For this, if we remove the arrows, which increase more than one, then it is not symmetric, not that symmetric. So I want to say that considering CSM classes would make the question more symmetric and perhaps sometimes it would become easier. This is what I want to say. So compare this graph. So take Lewis term, break the symmetry. So we consider CSM class by it's more symmetric, I want to say this. Okay, now let's turn to our next theorem. Our next theorem is equivalent PR rules. So actually we proved the rigidity theorem, which states that the equivalent coefficients are controlled by non-equivalent coefficients. So, okay, as an implication, we achieve the following equivalent PR rule. So now what we replace? We replace the left hand side by equivalent CSM classes. And the formula, we want to find the CSM expansion of the left hand side. And all the settings are the same, except this is equivalent. The sum is also over decreasing passes, but we do not have any restriction on the length. So we just let the path go decreasing, the label is decreasing. And the sum, wow, this is the CSM class and this is endpoint of the path. And this is coefficient. It's an elementary symmetric polynomial in T of this delta K, U, and gamma. And this is defined to be, I mean, this set is defined to be Ui, from i is from one to K, which is such that Ui is not equal to wi. So it's actually, it's an image of element from one to i, which is not moved to the same point by U and w. So it's a little bit hard to describe it in English. So anyway, we can, and the index is r minus length gamma. It means the sum actually over, the sum is actually over passes of length, no greater than r. And in particular, when length is equal to r, we get one. So it recovers our previous result. Okay, this is aqua-variant period rule. And it also recovers some, if we take the lowest term, we also recover some aqua-variant period rule for superclasses. Okay, and the next one is aqua-variant Banag and Nakayama rules. So actually, we can also obtain an aqua-variant MN rule for CSM classes. So in this case, we take P, r, x, k. It's just the Newton's power sum of k variables, x1 of power k add to xk of power, or x1 of power r up to xk of power r. And well, geometrically speaking, it appears in the, it appears in as a sum of a trend character of the topological bundle. So anyway, let's state our theorem about MN rule. So CSM classes, this is our friend CSM class corresponding to you. Corresponding to you. And times this prxk, it's equal to the following expression. Well, first, the first term is localization of this term at you. So it's tuk. And the rest term is sum over all the w. w can be written as u times eta for an r prime plus one cycle. eta is r prime plus one cycle. And moreover, u, we need that there exists a path from u to u eta. Okay, this is a description of u. And actually we prove that in the sum, how to say that there will be no reputation for this kind of w. So if there exists, then well, there exists some kind of unique path. But we need to put restriction on the path anyway. And the coefficient is, there's a homogeneous symmetric polynomial. And the index is u times mu w. And mu w is indexes that is mapped to different index by u and w. So it is. And we do not forget to apply u. So this is our formula. And of course, it's a little bit messy, but it's a result of our theorem. That's equivalent coefficients can be controlled by non-equivalent coefficients. So when we prove this, we first prove the non-equivalent version. And then apply our formula to get the equivalent version. Okay, this is our main theorem. And then let's discuss generalization. So actually we proved more general rule for for polynomials or hook shapes. Which including circle class or tautological bundle. So the following picture maybe shows what happened. So as I explained for e, the sum is taking over the decreasing path. It means the label is decreasing. And on the other hand, if we consider the circle class of tautological bundle, say the for h, the pair rule for h is sum over the increasing path over the graph I described. And for hook shape, for short polynomials or hook shapes, it's sum over passes which increasing firstly and decreasing. And how many steps it increase and how many steps it decreases depends on the shape of the hook. So we can see actually this is a hook shape somehow. Okay, this is generalization. And actually our formula we obtained is a generalization of the following theorems. A lot of formulas. So firstly is the Charlet formulas for CSN classes, which is proved by Alofi, Ehashiyan, Sherman, and Xu. And secondly is the Schubert-Pierre rule by Soteli. And third is the equivalent Schubert-Pierre rule by Robinson. And also Li, Ravi Kumar, Soteli and Yang. Actually they found a geometric proof of the equivalent Schubert-Pierre formula. And actually in our case up to I cannot figure out, up to now I cannot figure out a geometric proof of it. Anyway, and also the Schubert-Emmel rule and also our result also generalized Schubert-Emmel rule due to a horizon and Soteli. So this is our main results and generalizations. So I think it's a good time to take a break. Very good.