 All right, so welcome everyone to the Schubert seminar. Today we're happy to have Slava Naprienko from Stanford University talking about so many Schur functions, please go ahead. Hello, thank you so much for the invitation. And indeed I'm gonna talk today about so many Schur functions and how they appear everywhere in math. And I want to talk about how to generalize some of the Schur functions and why some generalizations are important. And I guess the main topic and the main theme of my entire talk is going to be trying to address and answer this question. And the question is, what is Schur? Many of the functions we feel that they deserve the name Schur and many of the functions they satisfy similar properties. But because they have different origin, they have different ways how you can look at them. So since the first part of the talk is introductory, I'm gonna talk about the classical family of the Schur functions, how they appear and what property they satisfy. And then I'm gonna go forward towards more general Schur functions. So the story begins with the classical Schur functions. Oops, sorry. The story begins with the classical Schur functions and the classical Schur functions which are denoted by a slender and parameterized by partitions is a symmetric function which happens in many, many different contexts and probably the most important context for this seminar is the context of the Schubert calculus. So if we look at the Grassmanian and you can think of the infinite Grassmanian, you can think of the finite Grassmanian and you look at the thermology ring of the Grassmanian. It turns out that the thermology ring is naturally even morphic to the ring of the symmetric functions and moreover, the Schubert class in the thermology ring exactly represented by the Schur function. And so this alone is already a very good reason to try to understand what Schur functions are because if we want to understand the combinatorics of the Grassmanian and being able to answer all of the questions from numerative combinatorics, we just want to understand what these Schur functions are because just the Schubert classes are represented by the Schur functions in the form of the linear basis of the ring of the combinatoric ring. Another source of the Schur functions is the representation theory. If you look at the representation theory of JLN or SN, the symmetric group, the general linear group or the unitary group, you will find that the characters of irreducible representations are also being equal to the Schur functions. So in other words, if we look at the character of the irreducible representations, it turns out that irreducible representations, polynomial reducible representations are often parameterized by partitions in these cases and the characters are again equal to Schur functions. So this is already two independent contexts where Schur functions appear naturally and it gives a very good reason to try to understand them in detail. All right, probably some other reasons which I'm not going to address in detail in this talk is the fermion-Bason correspondence which happens in the mathematical physics and integrability and some of the others are in numerative combinatorics and so on. So there are many, many different reasons to look at the Schur functions and try to understand the structure. So in particular, in the Schur calculus, if you want to understand the multiplication on the ring of the carmology and then expand the product, again, in terms of the Schur functions, you will look at the constants which are called the little with the richer concoctions and they play the important role. So this is some different context where we can find the Schur functions in the wild. Some natural questions like Schur calculus, the carmology ring of the Grasmanian or some natural questions like the character of the reducible representations. So now I want to talk about how to actually study and how to actually write down Schur functions, how to actually express them and where do these expressions come from? I want to talk about the expressions. And so the first way how you can write down the Schur functions is combinatorial and this is perhaps the easiest way how to do it. In fact, it is so easy that this is exactly what I always give to my students in research experience for undergraduate students in the programs because this is something you can give to anyone who doesn't know anything at all and nevertheless they can start feeling that there is some deep in the mental structure. So if you want to describe the Schur function, it turns out that it is just enough to look at the tableau which is just diagrams parameterized by partitions and you fill them with numbers which is called the semi-standard tableau. I'm not gonna go into detail because I want to talk a little bit of the big view and talk about the different generalizations but roughly you look at the combinatorial data, you assign weight to each combinatorial data, oops. And then you sum over all of these things. As a result, you have very, very explicit expression for the Schur function. It is so easy that you can give it to anyone who can write numbers in the table and you can write them down explicitly and it is very down-to-earth definition but of course it has some drawbacks and one of the obvious drawbacks is from this definition it's not even clear that the function is symmetric. If I give you some sum over all of these partitions and you look at all of these terms and sum them together, why would the sum be even symmetric? And how do you understand any of the other properties from the combinatorial sum? So it has some positive and negative sides and another definition which is probably some of the people learned first, I'm gonna call it a whale formula. It is the definition of the Schur function as the determinant. You can just give it explicitly as some determinant or rather as the ratio of two determinant and in the denominator we have the wonder month. So I'm gonna just write explicitly the expression for the wonder month and this definition is great for other purposes. This definition is allowing you to see the explicit expression for the Schur functions in terms of the variables but it has another drawback. If you expand the determinant and look at the terms in the expansion of the determinant you have lots of sine consolations. The first formula inherently just positive you just write Schur function as its monomials at every term that happens in the Schur expansion and the formula because of that becomes extraordinary explicit and you just write Schur function as one term plus another term and so on and it is just a positive sum. On the other hand, this one is more obscure because now we don't understand the terms so well but now we have the closed expression in terms of the determinant. The third one, which probably going to be the last one that I'm going to talk about is the definition in terms of determinant or I'm going to call it a Jambelli. So Jambelli definition is always the definition of the Schur function in terms of the very same Schur functions but of smaller order. So in particular the Jambelli sum of the Jambelli as identity are given us the Schur function in terms of the complete symmetric polynomials which could be given for example by generating series or just computed explicitly or doily we have the expression in terms of the elementary symmetric functions. So this is three different ways how you can just write down the Schur function give it to someone and so that they could look at this but before I go further, I want to take a moment and appreciate these three definitions and perhaps to think how long trivial it is for someone who doesn't know anything about the Schur functions and their applications why these three definitions would give you the same answer. If you look at these definitions, this is an explicit combinatorial sum from which it is not even clear that the resultant sum is going to be symmetric. Why would you write such a sum as the determinant with appropriate cancellations and why would you be able to write the very same sum as the determinant in terms of the smaller Schur functions and is it, which one is preferable? Which one is more natural and others just some of the formulas that you can derive from the first one. So this is a non-trivial question and the different authors take different approach. For example, Stanley in enumerative combinatorics in the second volume just starts by defining the Schur function as the combinatorial sum. And he writes that the following definition might be not motivated. Look at the Schur function, it is just sum over sum tableau and you just write some weights in a tableau and just enjoy your function, but you have no idea why would you do this? On the other hand, the whale formula, this is the approach taken by McDonald. You can immediately see that it is the ratio of the anti-symmetric function by the symmetric function and by some simple analysis, you immediately understand that it means that the function should be symmetric. And moreover, it is immediately clear that such functions form the linear pages of all of the symmetric functions. So this is a really good definition, but it has a drawback that it is not as explicit as the expansion in terms of the determinant. And it turns out that this definition is very hard to generalize. It is very nice that for the Schur functions, we have such a simple formula in terms of the determinant, but it turns out that for more complicated functions, there is no such simple expressions. But meanwhile, the combinatorial expression or the expression in terms of the Jumbellis formula is gonna generalize better. Okay, give me a second. So what we have at the moment is a family of Schur functions which appear in very natural contexts like in Schubert calculus and representation theory and there are different ways how you can see this Schur function. So I want to say that these properties are in some sense very fundamental and crucial. And the further we're gonna go, I'm gonna talk about generalizations. We'll see that these properties extend to every other Schur family. So let me go to the second family. And again, the second family is having a very good relation to the topic of the seminar. The second family is our first generalization is the factorial Schur functions which this time I'm gonna write the parameters here which this time depends on a new set of parameters on new parameters A. And how can you find these functions in the wild? What are they living their own life in various parts of math? Well, so one of the most important findings is that these functions describe them again the Schubert calculus. So again, a very, very natural thing to do is when we consider our Grassmanian and we want to consider the carmology ring, we find that the torus of the JLN is acting on the Grassmanian in a very natural way because it could just act on the planes by the torus. And it turns out that you can extend the carmology theory and instead of computing the regular carmology, you compute the something called equivariant carmology of Grassmanian where T stands for the action of the torus. And it turns out that this more refined version of the carmology, so again, very natural thing, we just look at the Grassmanian with the action of the torus. It turns out that it is isomorphic to the factorial symmetric functions which I will give you the explicit expressions for these functions in a moment. But of course, what is most important is that again the Schubert class is represented by the factorial Schur function. In other words, here is the very natural origin for the factorial Schur functions. And again, it gives the motivation to study the properties explicitly, in particular, the Richardson-Litterwood coefficients which were studied in many of works of the Sagan and Molyev, which was later used by Knudsen and Tao to give the combinatorial realization for the multiplication in the equivariant carmogenings of the Grassmanians. So there's another very natural family of the functions, but now I want to have a question. Is there representation theory origin for such functions? Before we, sorry, my pen seems to be more functional. So before we saw that the regular Schur functions, they appear both in the Schubert calculus and in representation theory. But now when we extend our definition to include the factorial Schur functions, there is a very natural extension to the Schubert calculus. Just look at the Grassmanian, but only with the action of the Taurus and look at the carmology. But at least I don't know what is the corresponding extension of the theory of the representation theory, what kind of representations we want to consider such that their characters would be naturally represented by the factorial Schur functions. So we lost one of the motivations to study these functions, but it turned out that we found another one and somewhat irrelevant. And another motivation to look at these functions is the interpolation theory. Interpolation theory of symmetric functions was developed by Sahi and Akhenkov and studied by Alshansky and many other people. And in short, the question is, if I have the symmetric function, I want to interpolate it, I want to expand it in terms of some coefficients and some bases. And it turns out that the factorial Schur functions form an extremely and extraordinary grade basis for expansion of the symmetric function any symmetric function in this basis. And the reason is because of the factorial parameters, the factorial Schur functions, they satisfy some vanishing property that makes the theory looks a lot like the standard, the classical Newton interpolation. I'm not going to go into detail about this, but what I want to point out is that now we have a new motivation, a new source where such functions appear naturally in particular in Akhenkov work. I don't remember the year. Akhenkov classified all possible schemas for interpolations of the symmetric functions. If we want to interpolate symmetric functions and have some good properties for the interpolation nodes, what could they be? And I don't doubt that there's only three families that satisfy these good properties and factorial Schur functions is just a natural family, one of them. In other words, it is another great source and inspiration and motivation to look at these functions. And now I'm going to talk about the corresponding expressions and whether they generalize nicely to the factorial Schur functions because we want to look at them because they appear in all of these areas of math. All right. So if we're looking at the expressions as before, we have the combinatorial expression and the combinatorial expression is if you want to give the factorial Schur function again, you can do it very easily. You, you know, you can do it very easily. You again consider the semi-standard tableau. Again, fill it with numbers the very same way how we did before. But this time we just assign weights in a little bit more refined way. Instead of just writing X1, we remember where the entry one occurred. So if it occurred here, I'm not going to explain why we have certain coefficients, but we can write the corresponding weight where each term remembers where in the combinatorial data it occurred. And because of this, it is perhaps something like this. Because, yeah, and we do the summation. Of course, because of this, now we have the refinement and this definition gives us immediately understanding that these Schur functions very naturally generalize the regular ones. If you set all of the a's to be zero, then you're just gonna collect the very same weight that you were doing before. You don't have any of the equivalent parameters and you go back to the regular Schur function. So it corresponds exactly to the fact that if your torus acts trivially, then you are just considering the regular carmology ring. Okay, so as you see, we still have the combinatorial expression for the factorial Schur function. And the combinatorial expression became a little bit more complicated, but still pretty easy. If we look at the whale formula, it turns out that it also generalizes very well. If you want to write the factorial Schur function as the determinant, you can still write it as the ratio of the determinants. But this time, instead of just having the variable x, we have the factorial power of x, which is just some shifted version of the very same power. And then the formula is pretty much identical. We again divide by the wonder month and our formula, our determinant formula remains the same. Okay, and of course the very same way we extend the Giambelli definition. And indeed the factorial Schur functions could be given as the determinant in terms of the complete factorial Schur functions, which could be defined by some simple generating series or even explicitly, or doily we can expand it in terms of the elementary factorial Schur functions. So you see the very same properties, they remained intact, but I want to point out that there is something very interesting happening and probably this is how I'm gonna make a break after I explain this. When we look at the Schur functions, at the original Schur functions, these three definitions, the combinatorial, the whale formula and the Giambelli, they come from different sources. So you can imagine that the combinatorial definition is really coming from the representation theory of GLN. Because you can think of the tableau as parametrization of the weights of an irreducible representation. So then it is very natural to write the Schur function as the sum over all of the weights. And it makes the combinatorial formula very natural object. You just sum over all of the weights in the irreducible representation, the character. And that's how you get the combinatorial formula. On the other hand, Giambelli formula, for example, is naturally coming from places like sugar calculus because the way how you can find these formulas is by considering the churn classes of the tautological bundle and it is very natural language to describe the entire ring in terms of the complete homogenous functions or the elementary symmetric functions. In other words, and the whale formula is coming from even another source which probably could be called crystals and some other structures, which I'm not talking about. But what I want to point out is that the way why these definitions create different definitions are natural, it's very different reasons. For example, the combinatorial definition in terms of the tableau in a way it doesn't make any sense in terms of in sugar calculus because what each monomial even means, we cannot expand if we look back here, we see that in our identification, the Schur functions represent the sugar classes in the caramology ring. But if we want to expand the Schur function in terms of the monomials, what could it even mean on the side of the caramology ring? You probably could embed it into something bigger but it is not immediately a not immediately natural object because we don't have any space into which monomials would live naturally. But in JLN it is a very natural object. So what I want to say is that these definitions are coming from different places. And when we go to the factorial Schur functions, Jambelly formula again could be thought to be coming from the Schuber calculus, but now we don't really understand what is the origin of this definition? What is the origin of this definition? Why this definition as the combinatorial sum of tableau is natural because as I pointed out before, it's not very clear what is the corresponding representation theory such that we would have some reducible representations and then we take their characters, we would get these factorial Schur functions. We would love to have some theory where we would have things like this. But as I understand, and there is no such theory, at least we couldn't come up yet. And but nevertheless a very similar formula that does a very similar job is there. And well formula is having something very similar, a connection to here if you describe the combinatorics of the representation theory. So in other words, we see that the Schur functions when you generalize them and when you can see there are more general versions coming from different places, they still have these three properties even if we don't know why. Even if we don't know why we have some of these properties, nevertheless they do. And this is going to be the feature for the further generalization I'm going to talk about. And at the end I'm going to talk about the ultimate generalization. Oh yeah, I see there is in the chart. Yeah, yeah, you can embed it to the matrices and then there is monomial expansion. I guess that you need to embed your symmetric situation into something non-symmetric to see this monomial. But I don't know if there is a correspondence between the representation theory because there is this result that connects the Schubert calculus and representation theory of JLN. You can show that these things, they have some sort of factorial connection. Meanwhile, in the case of the factorial Schur functions, it is very natural to consider Schubert calculus but it's not very clear what is the corresponding representation theory. Anyway, I think that it's going to be a stop for me and we can take the break. Very good, okay, so we'll take a five minute break.