 Right, so welcome to the second half of Professor Scott or stock on Catalan. Varieties. Please take it away. Yeah, yeah, yeah, this is with the difficult situation Katalan, but we change the title because Katalan variety was already defined so. Anyway, so let me try to say something. So let me try to write the dinking diagram. So, in Taipei, N minus one, but anyway, we have something like this, and we have fundamental ways. This is supposed to be epsilon one, the standard choice. This is supposed to be epsilon one plus epsilon minus one. But if we consider everything as GL and we will have pi n that is supposed to be this one. This one is basically the determinant characters. And if we affinize the situation and so imagine that we have some extra vertex and something like this, then we get the affine. Here it's actual things and we would like to have lambda zero, lambda one, lambda two, and lambda n, and so far, we have some lift of the full situation from here to here. So this defines some nonstandard, because the reason I call nonstandard is that so this determinant character usually does nothing on this simple algebra theory, so it should not correspond to something like this. But this time, we prefer this kind of things. So, in other words, our nonstandard lift since lambda is supposed to be mi of pi i, i equal one to n to lambda, that is supposed to be i equal one to n of mi of lambda pi. So this is a nonstandard lift and the theorem. So, sorry, is it a typo that you only have n minus one on the left hand side and you have. Yes, yes, yes, yes, yes. Sorry, this doesn't give my section. Thank you. And what they call rotation theorem is that so particular case of this one. So this is the Mac Dick pass. So maximum one, which I explained to learn a few minutes before this written like this. Many, many brackets are supposed to be close. And where the idea are the module operators. And so, this is so, you know, you have lambda. Some character. So anyway, so the. So, I need to explain but somehow the effect of the module fun, the module operator of F is supposed to be somehow F minus s i of F life or alpha i is some news. In that indexed by I and in general, sorry, for general. So we remove some of this wrong. Some region. Like here. Here. Here. Here. And add. Extra. In. Another thing here. And that is supposed to be here. It's not so easy to tell what happens, but let me try to figure out what happens in example. So any is supposed to be five. We would like to take some zero zero zero zero zero. And we would like to have some extra things. So the things. Like this. In this situation, if you write down the four things. It's still on the way. So, maybe you notice five or endless frequently. Anyway, so we are supposed to see the boxes here. And this is the effect of this is supposed to erase some of them like this. And try to put some. Some extra. So this is what happens in a lot of rotation so and the reason they call the rotation so is that so this one, and this one, and this one, and this one, somehow change a shifting some index by one by one. And this can be rewriting in terms of the affine thinking diagram auto in that to rotate the circle, but somehow we have getting get the leader leader with this thing in order to analyze the situation. So far. Yes. And file. Yes. This coefficient and file because you have lifting and one to M three and four, but the voice and file. And five is different. So, so this one. So, so this is this one determinant character. We make a non standard lift. So the determinant character of this one has a low in affine thinking data. We originally will have M one and two and four right. So five n equal five. No, I mean the coefficient of fundamental weight. Yes. And one, one, and it's five will war fundamental weight right. Yeah, for fundamental weight but we have five fundamental affine weight. But how you have this lifting as the note. Lifting. I'm asking you started from this one. I started to some effective is the determinant of the determines. We can artificially add the lambda to determinate ideal. And this has no effects in the practical no effect when considering things in X, but it can have some meaning. If we try to compactify or something like that. In this capital lambda. The lambda is now fine right and that to leave to this is the official lambda zero zero lambda zero is the coefficient of this makes sense. This makes sense for you. No, not so much. But sorry, for type eight we have a minus one what is this right. Yes, yes. And what is this one. We see that this determinant character as the ends but the final but it's of course this is not true. But the neighbors if you consider the weights weight consideration, it is reasonable. Okay, okay, okay. So this is only for type eight. Unfortunately, yes. So let me try to do geometric realization. This is in fact, this is supposed to be module operator. This effect of something called the module function called the eye. And so, you know, lambda I, this is the character of modules. So it means that so we can replace some di to be I, and the lambda I to 10 sub product of the corresponding character use some module that's called n psi of lambda, such that such that H of psi of lambda, this supposed to be the greatest character of M of psi of lambda. And in addition, so if we examine the construction, we can say, so direct sum of lambda in P plus of M of psi of lambda is a link, and such that if we consider the approach of this construction. So this approach direct sum of this one. So this means that this torus is something that the characters is supposed to be. And this is the action torus action display. And the generalization of the usual construction of projected spaces in algebraic geometry, and we can consider this one, and what we can say formally is that. So if you write like this X of psi. And for formally formal consideration case that X psi this projective and normal algebraic variety. And so on such that plus this line one do or X psi of lambda, such that H zero, a good grade the character H zero of X psi of all X psi of lambda. So this is supposed to be H of psi of lambda. So, and this says that this X psi and its line bundles that natural candidate this X psi. So for X psi lambda may have something to do has may have something to do with X psi of X, but it's relation is not so clear because just having a normal algebraic variety not sufficient to say some something. Okay, and the point is that so in the rotation theorem expression. It is not too easy to resize information about X, but it is just, it just says that we have a successive P1 vibration that the map to X psi. We do not know which two points are supposed to collide and so on, and that to use some that might produce some singularities and so on. But if we try to tweak this expression. We can rewrite rotation theorem, actually, into the following form. So, some, some, something is some, something coming from here, and but so the next terms are very specific. So, lambda one, and the zero D4 D3 D2, that is the same as original. But now, M2, lambda two, the zero D3 of M3, lambda three, D1 D0 D4. E of M4 D2 zero E of M5 lambda zero. So what is it, what is it, so instead of doing this, so in the in the rotation theorem expression, so this these two are, so these two are not common. But we can, we can try to modify this, this notation theorem by using some, some blade relations and so on, to make up some other expression, such that these ways are supposed to be paired together. And so this kind of somehow rewriting makes the situation a little bit better. Because somehow, some, some of the indices like to somehow zero one, zero one, and some zero four, zero four, and common on each other. So this particular reason implies the following description, that is our main theorem. So, if we do, if we carry out this, another expression, we can show that x psi this successive projective space bundle. In particular, it is smooth. So if we know that if you know some specific variety is smooth, then we can try to do, carry out some tangent space console comparison and so on to say something. And if we do this, we get the following. So this one is actually a sub right and open subset of this one. This sister attracting point set of x setting inside of x psi, and this is, this is a particular connected component. Maybe I will put this one is supposed to be a particular connected component. And this sister is kind of this sister itself, maybe it is better to recall, and this is a great idea. And this consideration make us possible to say that if we consider if we consider the co homology of this x and the pie star of psi of x of lambda, that is something we are supposed to consider. And this is actually the union of co homology of x of psi of all the x psi of lambda. This is not a coincidence, but we can compare the character to show show that these are related by restriction and this. This is not this one. Sorry, plus M of pi n. So we can conclude by C of M of pi n, in order to, in order to remove the effect of determinants twist on this, this particular in jail for I is not on the lambda in people. And we can, so we can conclude that the substance. So we can include that to some part of the, the vanishing part of something, something predicted by she was on all. And this part says that. And this device. The company defined company and maybe this is what I wanted to explain today. And it is better to stop here. And I'm happy to discuss a little bit more after, after finishing this session. All right. Thank you very much.