 Okay, welcome everyone. Welcome to the Schubert seminar. Today we're happy to have Professor Andru Weber from Warsaw telling us about the characteristic classes of b-orbit in square zero matrices. Please go ahead. Okay, so thank you. Thank you for the invitation and for letting me present my result. But before, maybe I have to say that this really, this period was not a good time for doing mathematics. You know, I walk up on first day morning and, you know, what happened and it was really the strange feeling like from some science fiction that a big European country attacked another, quite a big European country. And in fact, bombs were sent to the places that I know I visited and also I know people living there. And some bombs were really close to Polish borders so that was, and who knows what would be the next. So that's really not, I had problems with concentrating on mathematics, but slides were prepared so I can continue. And this is the joint work with my student, Piotr Rudnicki, about characteristic classes of orbit of Borel group in square zero matrices. So this is basically for GLM but can be generalized to other types, ADE basically. And there's some extension that is somehow related to my previous work with Richard Riemann, but this is only beginning for the stuff for elliptic classes. So I just basically tell you what was done with paper with Piotr Rudnicki and then we will see how much time is left. So this is really, we can start very elementary that we have square matrices and the condition is that for these matrices is x square equals zero. And I do not look like a variety given by equation because we have not reduced maybe scheme. I rather look at it as a B orbit, orbit of the Borel group and look at the closures of Borel orbits. So the varieties are reduced by definition. And I would concentrate on the orbits of this Borel group. Okay, the action is by conjugation. If you conjugate matrix with square equals zero then get again the matrix with square equals zero and I would concentrate on the matrices that are upper triangular, they are preserved by this action. But some, so this upper triangular case was better already started by several people. So I'll concentrate on that case and some possible extensions are around but it's easier to start with that case. So the first non-trivial is for the matrices of dimension three by three. So the upper triangular by three matrix, it's a matrix with A, B, C here. And the condition of being square equals zero is A, C equals zero. So this is that free variable. So it looks like two planes intersecting. And here you have zero and you have one orbit. The complement of this line. I'm sorry for my clock, it's late. So three orbits of rank one, two planes without a line, line without a point and the point itself is a zero, the rank zero matrix. Okay. So and why this space of square zero matrices is important? It contains a lot of information about the classical stuff like matrix Schubert varieties. If you have matrix and just K times L, not necessarily square matrix, you can embed it to a square matrices just as a block matrix. You can just make a block here and it automatically becomes a square zero matrix. And moreover, if you act on this by Borel groups from one side, you act by X upper triangular and from other side by inverse of Z. Then this action agrees with the action by conjugation. This is a simple calculus that conjugating this matrix by upper triangular is the same as left and right action by Borel groups. Okay. So it follows that all what I'm saying can be applied in the classical context of matrix Schubert varieties, but maybe I would prefer to say that this is an extension of the classical context. Okay. And the B orbits of square zero matrices were studied by Anna Melnikov quite a I think the 2000 something 2005 maybe the paper appeared. And it's really elementary that every orbit of Borel group among the square zero matrices contains a matrix that has only one somewhere, only one in one column, at most one in one column or at most one in one row. And in fact, this matrix is given by some combinatorial data. For example, you can call it pattern or linking diagram. So here you have row two and one in the row six. So two is connected with six and the matrix in fact says that the sixth vector is sent to the second vector and then the second vector says second basis vector to zero. And also here you have another one in the seventh column seven is sent to four and four is sent to zero. So the Melnikov shown that every orbit contains matrix of that shape and this matrix is unique. And from this combinatorial data, this is just a set of pairs set of these joint pairs and call it in the diagram. From that, you can easily read the dimension of the orbit. So this is the formula. So this is first R is the rank, which is number of pairs. So it looks like a dimension of grass mania, but from from that, you have to subtract the number of intersections. Here you have one intersection and also subtract them. You have some fixed points, those who are not not touched by arcs. So we have three, one, five, eight and over this each point you ask how many arcs you have above. So you have one and you have above two arcs passing over and here you have zero. So maybe another method would be just to draw lines from this and you could in that way you can count a total number of intersections. So in our case, the number the dimension would be the n is eight. So and the rank is two. So two times eight minus two and the number of intersections minus one and there are three. Here you have one arc passing over six points and here you have two. So this is two times six. That's 12 minus four. So that's eight. So it's easy to compute the dimension. Okay. So maybe another example is the, so let's take n equals four and the matrices of rank two correspond to the following patterns. Here you have one, two, three, four and there is one is connected with three and two is connected with four. The other one is, one is connected with four and two is connected with three and the last one one is connected with two and two is connected with three. Okay. So there are two rank two orbits and for the rank one orbit if you have one for example here, so you can show that generating other matrices using borough group you can obtain every matrix in this region. So you get of this kind provided this matrix is of rank one because the initial matrix was of rank one. So generating you still get rank one. Okay. So maybe I got it wrong. So you can almost every matrix of rank one obtain every almost every, but if you take a closure with the space of singular matrices. Okay. And for the orbits of rank two, maybe if you are interested in equations. So this these two configurations are given by equations just the rank conditions. So you can find yourself what is the rank condition if two rectangles are overlapped and then you can have rank two and otherwise rank one and this this this configuration says that everything is contained in the rectangular which is above above the diagonal and this is other case. And here there is extra equally extra extra condition that this matrix if you put here generic things this matrix might be not of square zero for example that one you have a b c d e and if you take a square you get equation of this type. So this is the condition. So to make it short it's in a paper of Anna Melnikov that the equations are the equations of the closure of the orbit are the ideal is generated by rank conditions and square zero with this non trivial for some configurations and for some follow automatically and there is a minimal for the fixed rank in fixed dimension there is a minimal orbit and it looks like this it's minimal because it has maximal number of crossings and also maximal number of can say bridges passing over the over this region every every artist passing over that and it corresponds to orbit that here you have to have to have zero and there you um the closure of the orbit is something that's a linear space and the orbit itself is some is is a consist of a matrices that here on this diagonal are non-zero entries okay and the the pattern is I mean the matrix the representative of the orbit it looks like that and then you generate okay and also there is an algorithm how to obtain a minimal orbit of the fixed rank by applying the simple reflections so the algorithm algorithm goes like this that's okay so here you have some configuration so there is one arc and another arc and the rule goes like this you have to so there are several ways you can do it but there is one at least one canonical then first move left legs to the left so this is a left leg and this is a left leg and you want to arrive to a position when the left leg is a position one and the second left leg is in the position two so just move this first first first left leg left leg is in the good position the second one you have to move so you move and then you have to move the right legs so this this other one should be transported here so you transport it just applying simple transpositions and you arrive here and somewhere on the way you have to jump I mean you have to switch this but that's it has to happen somewhere if they were in a not good order okay so you the right leg is on the second right leg is in the good position and then first like right left you have to still move it a little bit so you land up here so this is kind of algorithm you can do a similar algorithm switching the role of left and right first to move right legs and then left and still there are a lot of freedom the more possible choices in fact if you have this kind of pattern I checked out exactly three ways to three different permutation that's conjugate this one because you can consider this as the first arc this is the second and this third and then you have to skip there or other way around you consider this as the first this as the second and this are the first so you have to skip with this on the left side or there's another way that's something in the middle so the algorithms combination of them between the finding the canonical presentation of the permutation but also some some some positions you can simplify okay so as the singularities of these varieties are for us not I mean we work with singularities but for matrix Schubert varieties we know that for this I'm not sure at least in the type C there's a counter example but at least the singularities are rational and there's this resolution so I will tell about the resolution in the moment but first I have to recall the standard thing that for the g-orbit or important g-orbit you have a standard resolution at least in type E so if you have a matrix with square zero so from this matrix you can extract its image so if this matrix of rank r then the image is the subspace of of rank of dimension r so it defines an element of the grass mania so from a matrix a you you you obtain a element of grass mania and to determine this matrix completely you have to have a map from cm to the image from cm to the image but the square equal zero implies that the map vanishes on the image that's why you have map from the quotient cm by the image to the image so image is the tautological bundle and exactly this guy is a cotangent bundle so that's why you have a smooth resolution even a symplectic resolution of the g-orbit in the potent g-orbit and we are interested not in g-orbit but in the Borel orbit so still the Borel group it's Borel equivarian so Borel group acts on that on that resolution and this orbit is sent to some Schubert Schubert cell here downstairs so it's easy to to guess that you need to resolve this Schubert cell and finally you arrive to a problem of resolving some bundle over Schubert cell and it recently appeared then a paper of Bender Eperon just with a full description of this kind of resolution for the square zero matrices and for the type A D E everything works fine but I would just concentrate on the type A so the what is the what is the algorithm so the statement is that you take the minimal orbit so this is a minimal orbit orbit of Frank R and I already told you that this is a matrix set of matrices of that form so this is vector space so it's smooth maybe I should say for the type B C and G F so this might be non smooth so you would have to resolve it it's another another step to do but for the type E it's a vector space and then resolve like you resolve the Schubert varieties so this is you obtain construction of Botz Samuelson resolution so then you get bundle over Botz Samuelson resolution written that way that is iterated twisted product of minimal parabolic so this is standard maybe I don't have to recall it here so anyway this resolution is of the form of the bundle over Botz Samuelson variety and when this pi is something that's pi times minimal orbit is them that we are interested in okay so maybe I can stop now because now I would move to more homological part there the first was just a description of the situation geometric situation then I would move to the co-homological invariance so let's make five minutes break sounds good thank you very much any questions before the break all right if there are no questions let's take a break for five minutes okay