 Okay, so welcome back to the to the Schubert seminar with Jakub Koncki on his second part of his talk. Please take it away. Okay, thank you. Now I want to talk about, I told you that equivalency theory is a really nice computational tool and I showed some way to define characterizing classes using resolution of singularities. Now I want to talk about some very specific characteristic class, which is stable envelope. It's defined in three types in kohomod G by Malik and Okunkov, k-tiori by Okunkov and Okunkov and Smirnov, and k-tiori by Agganj and Okunkov. And I want to focus only on the k-tioritical part of the story. Then, and I will swipe a lot of details under direct table envelopes are very broad subject and I will define them only for gmod B, only for some specific choice of parameters and I will give like, I want to give explicit formula for many things. Sorry. So what is a stable envelope. It is a class which depends on a parameter called slope. This parameter is a fractional character of the torus and the fixed point in gmod B. And to the choice of these two things, we assign a class in the equivalent k-tiori of gmod B with an additional parameter. So this class is defined axiomatically. So it's the only class which satisfies some sort of conditions, some set of conditions. Okay, we won't. I present you some general form of these conditions to define stable envelope this class. We all by localization theorem, we only need to say how the polynomials. Okay. One variable, how the following such polynomials look like for every fixed point W prime fixed point W prime and. Okay. First condition. Oh, and I probably should say the class that one of W. Some should somehow deform the fundamental class of the Schubert Variety X W. This is the idea. The first condition says that this class is zero outside of the Schubert Variety. If we take a fixed point outside then we obtain zero. Second one tells us that the middle of the cell. We have some explicit formula. This formula is actually some fundamental class of some variety in cotangent bubble, but let's just say that this is something explicit. And the remaining two axioms does have something what happened for the other point in the Schubert Variety. And we know that this polynomial is divisible by something explicit so it cannot be too small. And we have some restriction at monomials which may appear in this polynomial. So this polynomial can't be too big because only specific monomials can occur. And somehow these two things together with the fact that this class, this bunch of polynomials comes from the some global class in equivalence k theory. So you said that there's at most one class satisfying is for properties. Okay, what is, it's, it's easy to say that there's at most one class but it's harder to say that there exists class satisfying. Okay. And our aim is to find an element in the equivalent k theory of both some awesome. We probably some parameter such that if we push it forward. Then we obtained, we obtain exactly this class. Then we may study. Then we want this thing to be as explicit as possible. Then we may study a stable envelope using the geometry of what some else and varieties, and this but some lesson varieties are really well studied so we know a lot about that. Okay. So first, what is what was already known what was in homologies that problem was solved in free papers by Fahremani, Riemani, Varchenko, and Alufi Michal, Joshua Mansu, by something, some variation of terms for smart person class. So this is a theory for a special value of slope parameter, some, okay, small and the ample flow. This was solved by Alufi Michal, Joshua Mansu, Fahremani, and Weber and there is also my paper. And our goal with Andre was to generalize these results to the case of arbitrarily slow. There's also work in elliptic theory by Riemani Weber and Kuma Riemani. Okay. Just, I want a little reparameterization. I will use the notation mcw lambda for stable envelope. This is because I want this is stable envelope with some reparameterization. The reparameterization is harmless is on the level of this additional parameter, but I want this to have exactly the formulas. Okay, formulas are correct if I use this thing but it's easy to compute stable envelopes from this thing. Okay. So first let me review these results here. Now, they define something called motivic trend class of interior, which is element in the equivalent category of both some of them, which first. If we push it forward, then to the category of GMODB. Okay. Then we obtain the stable envelope. The thing is explicit in a sense that we may write the polynomials, which we have if we fix a fixed point. Great, great, great. Now, since I'm going to a party at 630 that's in the neighborhood. I'm going to sit here. I'm very highly appreciated. So if we have any fixed. I didn't hear sorry. We have fixed point here. Can you repeat the question maybe, maybe louder. All right. Not sure if there is a question. So let's continue and if there is a question maybe we can read it in chat. Okay. So, so we have a fixed point. Then this class, the polynomial defining this class can be explicitly given by rather easy formula. We decompose the fixed point the tangent variety. We multiply some as one dimensional tangent space as a sum of one dimensional representations. And this is just we multiply some factors for every one dimension. So here, when alpha of that cell is either one plus y divided by file or one plus y divided by file. I don't want to give explicit conditions here but they depend on how the boundary of what some of us look around the policy. Okay, so this is rather easy formula. We may using. Let's just remember off, we get some information about stable. Okay, and we want to generalize this to the case of arbitrage as well. So first let me give you an answer. Okay. I have some technical problems. So let me give you an answer. We want to know to obtain the stable envelope for any slow by pushing forward something from both some. So we take the answer for this specific value of slow. This is just the answer for lambda minus. And it turns out that it is enough to multiply it by some line. Okay. So this is again explicit. Okay, if we describe how this line bundle looks like this is explicit formula for some element in the equivalent category. Which when my kids gives us this table envelope for every choice. Okay, so how to obtain this line. If we have any character here. Then we may. There is classical that it induces some line bundle over GMOD B. So this is just we take trivial line bundle character lambda over G. And we may divide by B and it turns out that it's still line. Okay, this is, I think with minus lambda. We may. So we have some line bundle over GMOD B. We pull it back to what some else on variety. And it turns out that it can be uniquely written as a veil divisor connected in the boundary. Okay, so if we only said that these things should be should be. Okay, connected in the boundary of what some else on the exceptional locals. Then there is only one by divisor that we have a quality here. And this is really classic this AI are computed by the Chevrolet formula. Okay, and to obtain their rational numbers. And to obtain our desired bundle, we just round up all the eyes. Okay, so this is all fairly explicit. And this was our solution. This allows to compute stable envelope at all fixed points using what some else on, but this is painful, because there are a lot. If you remember in the last three months rock. If you remember in the last three months of formula to compute the push forward there is some over all fixed points, but there are a lot of fixed points in the box and resolution, but there is better way. There is something. Some inductive procedure. One, this is can define operator using this geometry of Boston also acting on the equivalent K theory of dream of B, such that the operator applied to the KQ of a smaller fixed point. Okay. If we fix a simple reflection. We define operator to S lambda, such that if we apply TS lambda the stable envelope of W and so lambda, then we obtain the stable and develop of W s and love them and some different slow. Okay, this is nice because you can compute stable envelope of any point, applying this procedure many. We're starting with the motive stable envelope of identity, which is really simple thing. And using this operators. And this is a procedure which Mathematica can do for you like you feel and code this operator. Okay, and I only want to say that similar things were done for many other characteristic classes. Which are connected to Schubert varieties and using put some lesson I don't suddenly have time to prove this. So my only give you an answer. That what we obtain is some twisted version of the major lipstick operator. Okay, if we only assume that this. Okay, there's big formula where alpha S is some bundle over GMO B and LS is a root corresponding to the reflection as, and we've only set in this formula that this exponent here is zero. We obtain standard, the major lipstick operator. And what we proved is that if we use exactly this TS lambda, then this computes the stable envelope of longer root from the stable envelope of smaller of shorter room. If we put one, which it a little and put lambda equals zero, then this thing here becomes standard them as your lipstick. And this your theorem here is just the recursive formula of me how to show man so, and so that was all. Okay. This all follows from some inductive construction of both summers and varieties. Okay. At what hour I should stop like I have two minutes or 12 minutes. You really have 12 minutes I think the seminar is one hour. Okay, we have 12 minutes so it's great okay so sorry I panicked because I think I have two minutes. Okay, we'll be able to say a little how to obtain such formula. Okay, so it turns out that if we have a simple route. Okay. Element of variable and a simple route. And then I have a few unbound then the. Okay, and WS is greater than W longer than W, then the big sugar variety, big, big both summers and variety is a few unbundled over both summers and variety of smaller. Okay. The composition of a fixed point set to a big. Okay, both summers and variety as two copies of fixed points set to the small both summers and variety. And if I want to compute. If I have some a in the category of big both summers on. If I want to compute its push forward to the, okay, to the category of black variety, then by left to remember off. I obtain some some over some fixed points of the big. But some of something I won't write everything as this whole correction. But using this decomposition here, I can decompose this sum into two sums over fixed points of this small both summers and resolution. And again, using left to three man row. It somehow happens that this two sums a compute push forwards. Okay, this is before by WS, and this is before from the smaller but some of some of some to others classes. So this is the idea how to connect things which comes from the big but some of some resolution with things which comes from the small but some of some resolution. Like, you have this become kind of the composition of the fixed point set. And then because the push forward is given by something over fixed points. And pushing forward something from here is pushing forward of some two things from here. Okay. And in this formula, this. Okay, these are exactly this two factors for response. And to guess this operator. If you do the calculation very carefully then you obtain. Okay. Okay, the other thing is that using slightly different construction of but some of some, you may obtain some other operators called left. We did the material stick. And again, if this is long formula. But if we put this exponent here equal to zero. Then, we obtain standard the material left the material stick of me how to generalize and so. And it turns we proved that if we apply this operator. To the stable envelope of point w. Then we obtain stable envelope of a longer route. Now, now we obtain from w as w in the preview. Okay. Now from W obtained as W in the previous slide from W and tail we obtained W s. Okay. And if we put lambda equal zero. Then. This formula is a formula of halogen and so. So, maybe to conclude all these things. This twisted them as your standard them as your operators compute stable envelopes for for this canonical value of slow parameter. If you want to compute them for different values of slow parameters, then you need to change a little. This, the major logic operators. And it turns out that only thing we need to do is to change exponent in one part. And before our work we defined the twisted operators and the formulas. And maybe the last thing. There is something in the theory of stable envelopes called wall crossings. And this describe how stable envelope depend on parameters like slow or some implicit parameters. And it turns out that our inclusions connect the stable envelopes for different choices of parameters. So they are a we are able you to compute the same matrices using this inclusion. For example, this inclusion here. So you compute the stable envelope from so s lambda from stable envelope from so lambda. And if you work a little you can give comp using this thing you can tell how the fully tell how the stable envelope depends on slow parameter. Well, our matrix and this recovers some results of so down so. On the other hand, this inclusion here. It's not so obvious from this form, but can fully tell you how this table envelope depends on something called weight chamber. But I think I don't want to go to the details here. And I think this is all high. Yes, yes, sorry. All right. Well, thank you very much. Any questions.