 Okay, so welcome back. Let's start the second part of the Schubert seminar with Slava Naprienko telling us about so many sure functions. Let's go ahead. All right, so so far I talked about the sure functions and some of the ways how to represent them, some of the expressions that they have and then I talked about the factorial sure functions and I showed that they also naturally appear in geometry and in interpolation theory and they have the very same ways to be expressed. Now in a very quick run, I'm going to talk about two more families before I go to the new sure function family that I wanted to talk about. So in a very quick and sketchy way, I want to talk about the dual sure functions which is yet another family, never ending generalizations of the sure function, which also depends on the second set of parameters, but this time they are dual and this time they are not a polynomial when you restrict the variables. And the way how they appear naturally is when we are considering the when we are considering the hopf dual algebra of our equivalent caramology ring, we know that this one is represented, is a morphic to the ring of the factorial sure function. It turns out that there is some meaningful way how you can take the Poincaré dual and which corresponds to taking the hopf dual on the level of the caramology and roughly speaking it means that when you're considering something like a caramology hopf algebra this time is not a ring, it's going to correspond to the dual sure functions, sure factorial functions which were defined by Molyev. And so this is another very natural family of functions because it is just the dual to the equivalent caramology functions and so this is one way how they appear in nature and another way how they appear in nature is again in the interpolation questions and this time it is interpolation theory and this time the the theory is concerned not with the with the poles instead of zeros and it is a way you can think that there is a duality between zeros and poles and you can build analogous theory to the Newton interpolation of the symmetric functions but use the poles instead of the zeros so I again the dual situation and again in a very quick run I want to talk about the expressions and indeed as we as we hope there is the combinatorial expression where you can represent the this dual functions as the sum over over the block and this time the the weight is getting a little bit harder to write but one important feature of these new weights is that this time the weights appear in the denominator which corresponds to the fact that they are now sort of dual and the zeros you can think jump to be the poles again we indeed have the whale formula so you indeed can write the new functions as the determinant of this time it's going to be certain certain different kinds of monomials again this time is having the denominators rather than in the numerators and again divide by the again divide by the van der moen determinant and this gives us the determinant expression and again of course we have the gem belly where where we can write the our function the the dual function as the determinant of a completely homogeneous polynomials or duly in terms of the elementary symmetric functions so this is some properties that we see extend a lot to many of the generalizations of the sure functions when we consider in the dual sure functions we still have these properties so okay, so in the rest of the talk I think that I'm going to skip one of the familiar functions, which is super symmetric functions and I'm going to talk about the new familiar functions that I defined in my last paper and I want to talk about why I believe that this generalization is natural and why I hope to find the same relations to the Schubert calculus and interpolation theory as before and representation theory would be even better, but even for the factorial case it is not so clear for now so what is true about the functions that we looked before is going to be extended to the new family of functions and I guess the fourth part of the talk, the free fermionic sure functions and the name comes from the origin of the free fermionic six vertex model, which I'm not going to talk today about because it is interesting enough to talk about them being sure functions without how you actually can come up with them and these functions this time this functions this time depend on the two sets of parameters to the parameters a and b so you can think that it is the natural extension of the factorial function and the key feature that these new family of functions satisfies is the the following the following duality so these functions with the new two parameters they at the same time they at the same time generalize the factorial functions namely if you set parameter b to be zero oh there is a question in chat multiplication of the dual functions in the direct sum to the product of the determinant in the sorry is the multiplication of this sure function dual sure functions yeah, um I don't know I I can I can think about it at the end of the talk, but thank you for the question and let me continue so so this new family of functions at the same time generalizes the um the factorial functions namely if you put the parameter b to be zero you get exactly the previous simple factorial sure functions and at the same time if you set parameter a to be zero and just have b then you have the dual sure functions so in other words it is the generalization that at the same time captures the idea of the function and its dual and it has two of the parameters which generalize both of these directions at the same time and one the the reason I was talking about the expressions is because this new family of functions is having the all three expressions that we had before so it has the beautiful combinatorial formula where you can you can write the sure function as the the dual the three fermionic sure functions the double factorial that are depending on how you want to call it um again in terms of the tableau or which turns out to be even more natural in terms of the solvable lattice models which I'm not talking about and in any of these languages the combinatorics the new occurring combinatorics is capturing combinatorics at the same time of the factorial sure functions and the dual sure functions which is pretty surprising given that the combinatorics of the dual sure functions turned out to be way different in the paper of molif and the description in the tableau turned out to be quite unlike the description of the factorial sure function nevertheless the same property is true for these new functions moreover, of course, there is the well formula which again somewhat surprisingly with two parameters gives you these functions explicitly as the determinant and this time you have both numerator and the denominator with certain with certain powers and um and again you you can divide them by the well determinant and you you have the formula that at the same time captures the idea of the both of the functions of the uh dual and the factorial functions and of course there is also the gem valley formula which gives you the function as the determinant and you can describe it as the determinant in terms of the completely homogeneous polynomials which again are given by generating series which generalizes and unifies both the factorial and the dual sure functions and at the same time the elementary one so now I want to so now I just stated facts, but I want to bring some some motivation why I believe that these functions are natural in in the in the style of the sure functions so one of the perhaps most important identities that is true about the sure functions is the cachet identity cachet identity is perhaps the ultimate form the ultimate identity which captures the essence and the the combinatorics of the corresponding family of the sure functions and for the classical sure functions the cachet identity the the very very beautiful cachet identity is saying that if you are looking at the some of the products of the sure functions with two different set of variables they are going to be equal to the explicit and direct explicit product the the cachet the cachet product and the origin of this identity is the how duality which could be thought as the peter well duality and so the is related to the peter well duality and the origin of this identity is in representation theory in representation theory you can look at the action of the what did I change it you can look at the action of the glm cross glm and it turns out that the the symmetric power of the of the representations which could be which you could write as matrices like this it should be on the right side is expanding directly as the the representations of the corresponding corresponding module so this is just taken characters of this of this identity which is more interesting than a combinatorial identity for the two functions themselves so so this is a very very natural very very beautiful very great identity which just captures really a lot of properties about the sure functions and the rsk algorithm is giving you the direct projection here Oh, yeah, you're right glm. Thank you so now the question is can we extend this cachet identity for the factorial functions in other words is it true that if you are having the factorial Function and you sum them with the factorial sure function of the second set of variables you are getting the explicit product It turns out to be not the case. It turns out to be that the cache identity does not naturally there is no a natural cache identity that would Connect the factorial sure functions and for the longest time people couldn't find it until Molyev did and it turns out that the the natural way to think about this Identity on the in terms of the sure functions is aspiring In in fact, you are thinking that you are taking one function and the dual functions and match them together And then when you sum over all of the lambdas, you are getting the explicit product And then there is some explicit explicit product of the of the similar kind So in other words, what What this identity Is really telling us is that the sure function and the dual sure functions. We need to pair them together And it just happened that in the simple situation there when we're considering non factorial sure functions They are just self-dual. So you can just pair them with themselves and get the cache identity So I I didn't talk about the supersymmetric sure functions, but I really cannot Fight the resistance and the temptation to write the cache identity for the supersymmetric sure functions because of how beautiful it is The cache identity for the supersymmetric sure functions is the extension of the classical Identity and it goes like this. We have the sure functions which now depend on two sets of variables on the variables and the super variables and the cache identity in this ultimate form Is this uh, it was proven by Pirelli and uh, Rager And it follows from the uh, supersymmetric auduality This is the the ultimate beautiful, um Cache identity for the supersymmetric sure functions You can see how many symmetries They are and it turns out that uh, it specializes of course to the regular cache identity and to the dual cache identity And so this is a very satisfying piece of Of mouth. It just looks so symmetric and so right And uh, perhaps the biggest reason why I believe that the new family of functions that are introduced and Advertised is because they satisfy the following cache identity, which is very very striking It turns out that um, I guess it's theorem if you're considering the supersymmetric versions of the function that described to you Now they depend on two parameters a and b And then you have the second set of variables and they depend on parameters a and b It turns out that we have the cache identity And perhaps the most mysterious part is that the right side of the cache identity Is independent on both sets of the parameters in other words despite our cache identity is Despite both of our functions depend on the infinitely many parameters Which at the same time unite the parameters a and b Nevertheless, the cache identity gives the perfect pairing which is independent of the nature of the deformations in other words This uh, this new functions and their natural dual they form the perfect pair pairing which takes Get rid of them all of them deformations And and perhaps the most important part in here is that this dual sure functions Which originally for example in work of molif they were just defined as some functions which satisfy the cache identity And which you know give give you the dual hopf algebra these functions Their dual Is expressed in terms of the functions themselves And it was not possible to do without the second set of parameters because you need to change the parameters a and b in other words This functions as the space of these functions it is self-dual in other words if you connect the The factorial functions and the dual functions into one bigger space then this bigger space contains both Contains the dual functions are expressed in terms of the functions themselves in other words this This space is really contains all of their functions that I described above our special cases in particular um, I guess that uh, I can just write it real quick that if you If you uh, no, I was writing before they they generalize both the Factorio and the dual sure function and of course if you set the parameters a and b to b zero They generalize the supersymmetric sure functions as well In other words this family generalizes all of the sure family That were before and they generalize it in such a way that Only when you have two parameters b and a you can see the duality between these two functions Which was not possible to see in uh in the case of the dual sure functions and the factorial sure functions because for the factorial sure functions you define a new A new family and it's absolutely not clear How they're related to each other It turns out that only if you have the second set of parameters and you understand that there is a hidden symmetry You you you can understand that uh, this is actually a zero And this is actually zero a And of course once you write something like this you realize that the The true identity needs to be a b and the dual function needs to be ba And uh, so it is just swap into parameters a and b which turns the function into its dual And so this is very similar to what happens in the mcdonald polynomials where mcdonald introduced two polynomials mcdonald introduced the polynomials which depend on the parameters q and t And they satisfy the similar Symmetry when you are considering the cache identity you are considering the cache identity with the With the second set of polynomials where you exchange the parameters q and t In other words, the second parameter is showing what is the natural Pairing and dual it is exactly the second set of polynomials with the exchange parameters and uh, I guess I have a couple of minutes more and uh in this Last couple of minutes. I just want to speak late and something that I'm trying to understand We know that the factorial sure functions they describe the Co invariant Caramology of the grass manian We know that the dual functions describe the dual hopf algebra now i'm really curious what What kind of theory Do I need to have? perhaps with the action of the two With the two parametrized families of groups on the grass manian such that I would get the new family of the double factorial sure functions or pre-chermonic sure functions And the feature of this parameterization and characterization would be such that the transition from the Caramology to the Caramology Would correspond to the switch of the parameters from a and b to b and a In other words, we would just exchange the action of the parametrized groups in order to understand how to jump from one to another and Then this family would Connect the properties in a way the properties of both the Caramology and Caramology But this is speculation on which I'm working now and try to learn more about the Schubert calculus and I was coming originally from combinatorics and so I I hope that I will learn something new on this seminar and that would be The end of my talk. Thank you Thank you very much