 Okay, yeah. Welcome everybody to this last Schubert seminar this semester. Don't worry, we will continue in the new year. So again, today we have four speakers, all graduate students, and we will be on a tight schedule, so I'll just get started. Our first speaker is Assam Fagradin, who will tell us about permutation pattern avoidance and iterated fiber bundle structures on Schubert varieties. Please go ahead. So, thank you everybody and thank you the organizing committee to give me an opportunity to talk here. I'm really honored. In this time I divided my talk into two parts in the first part I'll talk about some bundle structures of Schubert varieties in the complete flag variety. And in the second part I'll talk about some properties of staircase diagrams. So, first, let's start with the field C. And by bracket and minus one, I mean the set of first and positive integers. And if we have a subset of bracket and minus one I say an ordered subset of bracket and minus one, then we can define a partial flag variety. So it is a couple of K plus one entries, where the entries are subspaces of CN, and they satisfy some satisfy a specific dimension conditions. So, there are two examples of partial flag by two steam examples. So, these are Tasmanian and complete flag variety. So for example, if the indexing subset of bracket and minus one is just a single tone. In that case, the partial flag variety is the Tasmanian of our dimensional subspaces of CN, whereas if it is the bracket and minus one. In that case, we get the complete flag variety. So, say we have two subsets of bracket and minus one, A and B, and B is contained in A. In that case, there is a projection map from the flatbread index by a to the flatbread index by B, and given by the projection map. And it's naturally a fiber bundle. And so this tells us that for the complete flag variety, we can have the iterated fiber bundle stacks. So if we have any nested collection of subsets of bracket and minus one, then it gives us an iterated fiber bundle stacks on the complete flag variety. So in this case, the projection maps, the fibers of each of these projection maps are isometric to some Tasmanians. So for example, if n equal to four, and say we have this collection of subsets of bracket three, then by this map, we get an iterated fiber bundle stacks on FL four. So we see whether this kind of properties hold for Schubert varieties. So let's define the Schubert varieties. So we fix a basis for CN and say EI is the subspace is spent by the first I basis vectors. The Schubert variety in the complete flag variety is a sub variety of the complete flag variety that satisfy some dimension conditions. And here is an example of Schubert variety index by 423. So we are using the one line notation of permutation. So if we do all these computations for this partition, for this permutation, then we will see that only this condition is needed to define this Schubert variety. Right. So you want to see whether Schubert varieties have fiber bundle stacks or not. So here we can see that a Schubert variety index by a permutation w has complete variable bundle stacks here. If we can find a nested collection of subsets for who is the projection maps induce fiber bundle structures on the Schubert variety. So here the this domains, these are the images of the Schubert variety under this projection maps. But it is not to that all Schubert varieties can have iterated fiber bundle structures for, for example. So this is the example we saw in the last slide, and this Schubert variety has complete parabolic bundle structures via this map. However, this one can show that this Schubert variety does not have complete iterated fiber bundle structure. In 1987, Ryan showed that the smooth Schubert varieties are iterated fiber bundles of gas monials. And later in 1990 lucky by and from that they showed that the, if the permutation about two patterns, 3, 412 and 4231, then the corresponding Schubert variety is a smooth and conversely. And using this to reserve him and gave generating cities to compute the smooth Schubert varieties. We also have gas money and Schubert varieties and gas money and Schubert varieties are indexed by partition. So suppose we have a partition, so that it's young diagram is contained in an R by N minus rectangle. In that case, we can define a gas money and Schubert variety. So it's a sub variety of the gas money and that satisfies some dimension conditions. We can also compute the dimension of the gas money and Schubert variety. So if the Schubert variety is XW then it's its dimension is the is the size of the partition, I mean is the number of boxes in the young diagram of the of the partition. And if there are any cells there, if any is big enough, I mean, at least two or more. In that case, is grass money and contains a core dimension one Schubert sub variety which is unique. So we call that core dimension one sub variety a Schubert divisor of the grass money and so for example if any seven and R is for then in dear 47. Grass money and Schubert variety indexed by the partition 443 has one core dimension, exactly one core dimension one Schubert sub variety. Okay. So we want to answer this question. So we want to see whether a Schubert variety in the complete flag variety are iterated fiber bundles of gas money and or gas money and Schubert dividers. So if we delete this part, then the question is already answered by Ryan in 1987. Sorry, 1990 by lucky by and and so on. So we are considering a bigger collection of Schubert varieties in the complete flag variety. And we found the answer. And the answer is, there is a correspondence. So we have a permutation avoiding these four patterns. So if a permutation about these four patterns, then the corresponding Schubert varieties have fiber bundles extracts where the fibers are either grass money and or grass money and Schubert dividers, and the convert is also true. And we also computed the number of sub sub Schubert varieties. So we got a generating function that here is the function. All right. So this is the first part of the talk. In the second part, I will talk about staircase diagrams and we will show how these are related to this fiber, these properties of Schubert varieties. So, to do that, I will recall some definitions. So we know that the, the symmetric group is a positive group with the set of generators from S one to SN minus one so these are the simple transpositions. And we also know that the coseter dimkin diagram of SN is a simple path with and minus one bar raises. And in order to define what a staircase diagrams. I'm defining an L function so this is. So by L function what I mean, the L often interval in the, in the, in the path is the, so if it is the left most index of the verdicts. So for example, if I'm considering an interval from S two to S five, or from as to say a stand that L of that interval is just to. Okay. And from now on, I will call an interval a block. By block we mean a connected sub graph of the, of the path down. So now, I'm going to define what a staircase diagram is. So here I'm only considering the staircase diagrams of type A, there is a general version of the status diagram in other type. So it's a partially ordered set of blocks. So that is five four conditions, and the conditions are one block cannot be contained in other. If two blocks are, if a block is covered by other, then their union is also blocked. So I mean their union is connected. If there are two blocks whose union is connected, then the blocks are comparable. So if we have a chain of blocks, and if we take the left most indices of the blocks, then they will follow either ascending or descending chain. So I will give an example to explain this for. So we have a mixture of staircase diagram. So I'm considering this staircase diagram so with five blocks, one from S one past three, the other one plus two, and so on. So here for simplicity, by one I means as one by two I mean as two. So we see there, one blog is no blog is contained in other. In the first two blocks, we see that the block 234 is covered by 134. And if we take that union then we get 124 pieces connected in the in the gamma. And if we if we choose a chain from five, six and six, six, seven, eight, they follow the chain, the block five, six and six. And if we take the left most index five and six, they follow an ascending order, and so on. So, so this is a perfect example of an staircase diagrams. And there is also a notion of the state notion of labor staircase diagrams so by labeling, we mean a function from the staircase diagrams to the group. Here I'm not going to define what the function is because it's technical, but what the importance of this function is that if we have a labeling. And it, it, it gives a super variety corresponding to the diagram so is diagram corresponds to any specific super variety. So if we have a leveling lambda of a diagram D, then it gives an element in the group in the in the in the group of our mutations. And we have a permission then we have a super variety in the complete Friday. And this element is determined by a linear extension but it is independent of the choice of the extension. And the blocks in the diagram, determine the fibers of the fiber bundle structure. And the partial order of the loss determine the sequence of the fibers. So I will give an example of labeling. It's called maximal labeling. So what it is. So, so we have a say we have a function from the diagram to the group. And it sends its block to the maximal element so by maximal element I mean the maximal element in the subgroup generated by by the. So if we have a subset of the set of generators and say uj is the unique maximal element then in maximum element, and if we send is blog to the corresponding maximal element of the group then it will satisfy all conditions of labeling and this labeling is called maximal labeling. In 2017, they showed that there is a bisection between the smooth sugar varieties and the set of staircase diagrams. And, and, and the correspondence is given by by a labeling and and the labeling is the maximum labeling and we notice that the maximum labeling is unique for any, any staircase diagrams. So here I'm considering another labeling I call is divisor labeling. So by divisor labeling, I mean the labeling is either maximal, or I say almost maximal. So if the, if it is maximal then the length is the maximal. Otherwise, I say it's lens can be one less than one less than the maximum. So I've been it's any question. Sorry. Thank you. So, it is also important that it in a staircase diagram, we can always have the maximum labeling, but some blocks only can only have the maximum labeling not the divisor, not the non maximal labeling. So, it turns out that if we have a single block of length say m, then the number of divisor labeling is n minus one. So one is maximal, its length is three, and there are two divisor levelings, and one is the maximal. One is the maximal, and the other one is one less than maximal and is given by if you multiply the maximal element in the left hand side bias to. And, and in the first case, the, the Schubert variety we, we get is the Schubert variety indexed by 4321. And in the second case, the Schubert variety we get is indexed by 423 m. So, these one is smooth, because it is maximal, and this one is not small, but it does have iterated fiber bundle structures, where the fibers are either gas monials, or gas monials. And we found another result is, so these are the four patterns that we considered before, we said that if the permutation avoid these four patterns, then the corresponding Schubert variety has complete parabolic bundle structures with fibers, either grass monial or grass monial Schubert devices. And here's another result we get is that the set of permutations avoiding these four patterns are in by section with the set of divisor level staircase diagrams of type A. So, to summarize what I so far discussed here is this. So we are considering these sets. The first set is the set of permutations avoiding these four patterns. The second set is the set of Schubert varieties in the complete flat variety such that the Schubert varieties have iterated fiber bundle of grass monials or grass monials were divisors. The third set is the set of divisor level staircase diagrams, whose support is contained in in the set S12SN. So these three sets are by vector. And we can compute the size of the sets is by this generating function. And that's it for today. And thank you everyone. If you have any questions. Yeah, thank you very much for a very nice talk. Maybe I have a tiny little question. I think we have one minute. Is there any motivation for having fibers being a divisor. So the, so the motivation I got us because So these are already answered. What would happen if I considered a bigger collection of Schubert varieties, whether in that case we also get the fiber bundle of stats or not. So here we, we only consider the Schubert varieties, we have fiber bundle structures, where the fibers are either that's monium or or core dimension one in the Tasmanian. So. Okay. Okay, thanks very much. And thank the speaker again. Thank you.