Jennifer Morse, part1, `Nonsymmetric Macdonald polynomials with a small side of Schubert calculus'

Okay. >> Okay. Welcome everybody. Welcome everybody to our last uh Schubert seminar of the semester. We have uh Jennifer Morris from UVA who will talk about non-symmetric McDonald with a s small side of Schubert calculus. Thank you so much for inviting me. Um I always feel honored when I'm invited to anything Schubert Calculusy because I know so little. But anyway, we'll see. Um so I want to I' I've been working recently with my collaborators on something called non-ymmetric McDonald polomials. Um but what I wanted to focus on in this seminar is sort of a history of how I might have even been invited to this place which is because of work with McDonald polomials that inspired some things in Schubert calculus. And so that's kind of been my own history. And now with the new non-ymmetric stuff, I feel like there's a lot of interesting possibilities in terms of doing similar connections for that. So we'll see. It's a little bit too long. Um, but we'll do what we can and hopefully everybody will be able to get something out of it. Um, so everything is with my collaborators Jonah, Mark, Anna, and George. Um so I'll just before we talk about non-ymmetric McDonald polinomials I have to talk about McDonald polinomials and before that I'll just start sort of as a warm-up with sure functions I kind of always do that um so that I don't know it feels less scary for anybody who hasn't ever done it. So my my first page is about sure functions which everybody is familiar with hopefully. Um but I call it sure functions the awful because it's actually the least inspired definition I can think of for a sure function. Um I wanted to make sure to set the standards very low. Um so basically this is indeed a way that you can define the ch function basis for symmetric functions. Um it doesn't give much but it says okay it's it's the unique basis that's defined by two properties. one unit triangularity with respect to monomials and two orthogonality with respect to the whole inner product which is whatever some inner product on the symmetric function space that's defined in terms of the power basis none of the details are important really you should simply be taking away from this that okay there's a there's a basis that I can define with these two conditions but what what can we say about it what do we know do we have a feel for what this basis might look like what these functions are. So even to compute a simple example, you'd kind of have to do Graham Schmidt or some sort of yucky or fun depending on your point of view linear algebra. Um so that's sure functions the awful. And then the next thing of course is just to briefly review Sure Functions the great especially from a combinatorial position. Another way to think of them is that they're actually just sums of tableau. So write down all the tableau each monomial gives you each tableau gives you a monomial and you get the sure functions. So one of the things that this slide tells you is that that this definition though it isn't obvious constructs a family of symmetric functions that are actually positive sums of monomials. So anytime we see a positive sum of course as a comtorious we want to count. So the sure function is the sum of tableau. Um and so immediately you now know that the combinatorics of tableau is intertwined with sure functions. And it's not just the definition. The very second question you could ask second only to like what is a sure function would be okay if this is a basis for symmetric functions. What's the simplest symmetric functions I've ever encountered? Z1 plus z2 plus z3 plus z4 or whatever right? Like that's just like a very simple symmetric function. You can take powers of it. It's still symmetric. And you can ask what happens if you expand that in terms of sure functions. So you get some coefficients. And if you put that on the computer, it turns out that you'll find that those coefficients actually count standard tableau. So again, you see tableau coming up in questions concerning SH functions and symmetric functions. One of the things I love about this example, even though it's the simplest symmetric function you've ever seen, is that it is an algebraic identity that is equivalent to the Robinson Shenstead can correspondence, which I will controversially say is the most fundamental result in algebraic cominatorics in the 1900s. It basically associates words because this thing on the left is just a bunch of words and on the right you have pairs of tableau. And so this identity gives rise to this whole theory that that associates words to tableau that was developed over the period of you know decades and the sort of robust development that was primarily later shoots and burge but early on Robinson Shenstead and can um led to the possibility of answering more difficult questions. So the next and last sort of cominatorial example is that once you kind of know how to work with words and tableau and sure functions, you can go on to answer for example a harder symmetric function than the Z1 plus Z2 plus Z3 is the Sure function itself. And so you can ask what happens if I multiply sure functions and expand into the Sure basis. And it turns out amazingly those are positive and you can count them using Tableau. Okay. So from a combinatorial standpoint we love sure functions because of these beautiful identities and we can play with tableau and etc. But of course the basis is notorious not just for the combinotaurics but because it comes into other areas. So I'll give you two examples. Every example is pertinent to the late to later in the talk. So you'll see them all again. Um so the first one is in representation theory and it has to do with an SN module happens to be the module we call the harmonics module. So it's just now we're not working with symmetric functions on this slide just multivariable polomials and I want to take the vector space that's made up of the span of all the partial derivatives of the Vandermon determinant. So that's what this is. You can compute them and you can get however many there are and then you take the span and you get a vector space. It's actually an SN module. If you take any harmonic and you permute the variables, you get a harmonic. So as an SN module, you can decompose it into its irreducible pieces. Each one indexed by partition because that's what indexes the irreducible representations of SN. So if you're interested in an SN module, you really just need to know like how many times does this thing appear in the module. So how many times does that partition or the representation or module associated to that partition occur in my SN module? Okay, so Ferdinius came up with a way to answer these or at least study these questions using symmetric functions. So this is just an SN module which happens to be multivariant polomials. But Forbinius said well irreducible representations of Sn are indexed by partitions. So if you map a partition to a sure function indexed by the same shape and take the sum then this result this positive sum of Sure functions is exactly equivalent to this decomposition by definition. just by definition it's kind of stupid because you have to know this decomposition right to map them down one one to one but if you then say like okay well what is this symmetric function here is it can I understand it and then maybe understand its sure decomposition then you can carry that back up to the module and so this example you may recognize from the f the slide that we discussed about the easiest symmetric function in the world it turns out that if you take the harmonics module module and you decompose it into its irreducibles and then map each irreducible to a sure function that symmetric function is exactly powers of z1 plus z2 plus z3 blah blah blah. So this we already said was the very manifestation of Robinson Shenstead can kuth and yet it tells you exactly how to decompose the harmonics module into its irreducible pieces. So I know that there's two of these because there's two tableau of that shape. Okay. So that's representation theory. And then now something I don't have to really spend much time on here is the other example or a third example. We have cominatorics. We have representation theory and then we have in Schubert calculus the grassmanian variety. And now you want to understand how to take intersections of Schubert subvarieties and each one is indexed by a partition and a rectangle. So it turns out that the coalology of the grassmanian is can be studied instead as a quotient in the symmetric function ring. So it's like symmetric functions modulo some ideal which I wrote there but whatever and the Schubert class in the coology can be identified with the Sure function and if you want to understand how to take the product of the Schubert classes then you have to understand how to multiply the Sure functions and mod out by that ideal and the sort of beautiful thing about how this works is we already discussed that we know how to do this because we know how to work with Tableau and words. And the nice thing about this ideal is that it just kills sure functions that are not indexed by partitions in a rectangle. So you can literally just work in the symmetric function ring and then any guy that's not indexed by a partition in the rectangle goes away and the others perfectly map this sure goes to that class. So the coefficients are the same. Okay. So my summary of sure functions is that you can introduce them with an uninspired definition but if you study them a little you'll find that they do warrant the name of this like prestigious and um wonderful basis combinatorally it's absolutely the heart of tableau cominatorics and in representation theory well the one example I gave there's many others is that they're associated to this harmonics module and you have the sum over words as tableau and then they also represent Schubert classes in the coology of the grass mona fine okay so McDonald polomials what are they all right well it turns out that you can define them in the identical way that we define sure functions in the uninspired way which is okay now you're still working in symmetric functions but your coefficients are in the field of rationals with two parameters which we call Q and T. Okay. Other than that what McDonald discovered is that there exists a basis that can be defined by two things exactly the same unit triangularity as sure functions unit triangular in terms of the monomial basis and orthogonality but now it's slightly different than the whole inner product. He's added this like rational sort of that's a scaler, right? But it's a rational in Q and T. Notice that this little addition here if Q is T, it goes away. So when Q is T, it's exactly the definition I gave for a Sure function. So all it is to get McDonald's polomial is to change the inner product so that you now have this little factor here. And that's McDonald polomial. But of course, we already decided that that definition of sure function was not exactly our favorite definition. But whatever we have to take what we can get. So McDonald computes some examples. Just like with the sure function, you remember we said, oh, when you compute examples, you see this beautiful sum of monomials and it's tableau and yay, life is great. So McDonald did that and sadly he did not see anything that looked even close to like something like a sum of tableau because you have like look at all these yucky negatives here. Not to mention you have rational functions, right? But he he's happy to clear the denominator. You can clear the denominator, but you still have negative yucky coefficients. So that many of us would look at that and decide probably not to proceed. But McDonald has a very high futility threshold and he was looked at it more closely and he said well there is a way first just get rid of the denominators and then there's a way to like combine the yiness into a basis. It's not the sure basis. It's not a power basis. It's not a basis you've probably worked with. I call it the funny basis because I don't even really know how to define it. It's like just not a basis you would ever really work that much with at least in this community. Um but some funny basis exists so that you can combine the terms and when you expand this McDonald polomial or a scalar multiple of it you get these beautiful sums of Q and T's and he conjectured that the basis defined in this weird uninspired way in fact has a QT positive sum of funny basis elements. Okay. So from a combinatorial viewpoint, you can say, "Okay, that's interesting. I wonder if I could prove his conjecture or figure out what those QT count blah blah blah." But it's pretty hard, right? Because first of all, we use this definition up here, which is already very inexplicit. Second of all, we know that when you take the monomial expansion, it's really ugly. So third of all, I don't even know what a funny basis element is. It's so hard. You know what I mean? Like, so how am I supposed to prove this? So, so Garcia had a proposal. He says, here you have this thing that's not monomial positive. It's yucky. And you want to prove funny basis positivity. If you instead take the McDonald polomial and just cross out the f and replace it with an s. Okay, then you have another a different polomial. It's obviously different because I've taken this yucky funny thing and replaced it with the sure function. Now whatever this new guy is which he calls the modified McDonald polomial whatever it is it should be sure positive if and only if McDonald's polomial was funny basis positive because it's a very stupid substitution right so you might think like how does that help because I already had to have the funny basis expansion to replace it with the sure function right so it does seem like it might not help But what do we know about sure functions? One thing we know is that each of these guys is a positive sum of tableau. There are positive sums of monomials, right? That was what we started with second. So that definitely means that this guy is a positive sum of monomials. A QT positive sum of monomials because these are these are not. So So that was one of the questions asked at the time. Could we in come up with a better definition than using this unit triangularity to just directly define these with combinotaurates because we know that there's the potential to define them as a QT positive something or other so it took I don't know like 15 years or whatever but Hayman Hagglin lure did do this and in fact there's a beautiful formula for McDonald polomials very much like sure functions McDonald polomials are a basis for symmetric functions. So they're indexed by partitions. But what you do is you take each row and you map it over to a tupil. So this is like a list of rows, one for each in this partition. And then you also allow not just this list of rows but any wiggling of those rows which we call ribbons but so like you take this guy and you're allowed to wiggle it into any skew shape that looks like a ribbon. So the McDonald polomial is and then you put tableau in them literally their usual tableau. So you sum over all the possible tableau you can get by filling tupils of these ribbon shapes. So that's what this is down here. It's a simple example and the these are all the possible tupils of two ribbons and a one box. And I fill them just with usual semi-standard tableau. Well, it's infinite but whatever. So each one just like Sure functions gives me a monomial just the weight. Of course, it's a QT sum. So each one of these guys contributes some QT power. The T is related to inversions. So you just look at the numbers on a diagonal and count how many are inverted. And then there's a weird off diagonal thing also, but whatever. It's basically like inversions. So that's what in is and the arm says, well, if I look at this tubole, how how much did we wiggle the shape? If you take this tupil the arm is zero but when you start bending your arm you get a contribution of Q. It's not important in the precise details just that you understand that McDonald polomials modified McDonald polomials are quite simple to write down. You just write down tableau and then you compute the inversions. You compute the arm wiggle and you're done. It's a QT sum. That's great because at least that seems a lot easier to approach sure positivity than this funny basis stuff. Turns out it actually wasn't that easy, but at least it gave the illusion of it. So anyway, so that that's good because now we don't have to talk about any scalar products or anything. We can literally just write down a McDonald polomial. Okay, but what about the fact that the real conjecture was this was supposed to be sure positive and then going back to McDonald's it's funny basis positive. So what you really want not necessarily just this beautiful combinatorial formula but you want to know what the sure coefficients are. Well, okay, that that is harder. But it was actually Garcia's idea to switch the funny basis to sure was really a lot deeper than just oh well then I'll have a monomial positive thing. He was thinking about the harmonics module. So remember the harmonics module was this linear span of the partial derivatives of the Vandermon. And we saw that if you mapped an irreducible to a sure function, well, we saw it without this Q. It was just the sum of sure functions with tableau and it related to words. You can put in extra structure here. You have an isomorphic copy of this module. But of course, the actual harmonics in here don't have the same degree as the harmonics in there. These are degree two and those are degree one. So if you want to know more than just oh there's two isomeorphic copies of this. If you want to know like I have one with polomials of degree 1 and one of polomials of degree 2 you can actually get that information by simply mapping still the sure function to its shape the shape the irreducible to the shape but then the degree two gives you the power of q. Okay. So now you get a Q positive sum of sure functions. So anytime you take a graded SN module and you take an irreducible and map it to Q to the graded piece times that Sure function of course you'll get a Q positive sum because you're summing the irreducibles. So this picture with the Q was completely worked out when McDonald polomials came along. It was already known and in fact the Robinson Shenstead can worked not just to prove that words went to the tableau but that the inversions which was what you get when you do this can be mapped to charge. So it this this was sort of like a very very complete and beautiful beautifully well understood picture. The other thing that was understood when the McDonald polomials were introduced is that this was a special case. McDonald polomials have a Q and T and the modified McDonald's right they they look like this QT positive sum of sure functions and in a special case you actually got exactly this. So the idea in Garcia's head was if we want to prove sure positivity of the McDonald, the modified McDonald's, we think they're QT sure positive. So is there a module that now has a bgrading? So before we just had the linear spans of the Vandermon, these were polomials in X. If we have something with polomials in x and y then if you record the q power could record the degree in x as before and the t power could record the degree in y. So the same map from the irreducible to a sure function but now attached to a qt monomial recording the by degree of the sn module and that's a question that that garcia and hmon worked on. Is there such a module? We know that in a special case it's this harmonics module and they proposed one and eventually Hmon was able to prove that in fact the modified McDonald was its verbinous image and what they did was they they sort of beefed up the Vandermon determinant to have X's and Y's and there was one for each partition. So using the linear span of beefed up Vandermons in two sets of variables, they got a an SN module, the symmetric group acts on the X variables and the Y's simultaneously. And Haymon proved that the Fbinous image is this modified McDonald, which by definition means that you have sure positivity because the Fbinius image is a sum of Sure functions reflecting the decomposition. So that's amazing and that was done around the same time as the cominatorial formula actual actually actually um but what's still open is okay you know that there's one of each of these guys for each tableau and you know it's true positive now so you know that there's just a QT monomial for each tableau the the Hmon Hegel lore formula says the modified McDonald is the sum over the tupils of tableau with arms and inversions. Is there a combinatorial way to prove what these coefficients are by associating some T power and Q power to each tableau? It's like you know it has to exist. But anyway, it's still open. So that's a hard problem. Obviously one that is near and dear to my heart so near and dear to my heart that I have been working on it since I became a mathematician as a graduate student obviously that doesn't speak very well to my success as it is still open and that was in the '9s anyway so what was I doing in the '9s I'll tell you about that I was like obsessed with this problem it's a beautiful problem of course it wasn't even a theorem yet that that you could do it but we do know that for each tableau you should be able to find a Q and T in the modified McDonald polomial. So, we know that and we know that in a special case like this one, oops, I just hit the eraser. In this special case, you only have one parameter because that was the harmonics module case. And you know, it's you know what it is. It's called charge. And I can take this tableau and I can compute this by very easily thanks to lasco. That's so much to know and it seems so like you could definitely answer this question. Okay. So of course I sat in the corner tried tried tried and so in the late 90s I was with uh Alen Laskco and loop lup point trying this problem and while we didn't solve it we did study these tables a lot and we made an observation so the observation is like this guy only involves T and he's pretty easy immediately all the others have T's and Q's they all or at least all the others have Q's here you have only T's But what we noticed was if you you can look at not just the first one but but the first three all these guys which are indexed by partitions with no more than two columns. So restrict yourself just ignore this stuff. What we discovered was that remember we felt like t sums of sure functions are easy. It's just a singly degraded something. It's charge. And we we saw that if you looked at that set, there were these sums of sure functions involving just t's. And then the coefficients were still positive. Okay? And so one for each of these shapes, one of these bubbles for each of those shapes still had positive coefficients in Q and T. So we were like, oh wow, maybe we can figure out what this guy is and then this is easier. But of course it's a little bit disappointing that it only works for two bounded shapes. So that's not that's like special case city. But what we observed is you and it doesn't right you see that there's this Q here. That guy really ruins it for us because this is definitely not here. So you definitely can't just take the three shape. So what we discovered though was that if you want to consider all of the guys with at least at most three columns you have the same phenomena but it breaks apart the two shape case. So here you have this bubble for just two shapes. When you go and add another McDonald you now get smaller bubbles that are making up the case for the higher level. So what we saw experimentally can be summarized here which is that for each K and that'll be the width how wide you want to take your McDonald. So you're going to take the the span of McDonald's where you only have shapes no bigger than K columns. If you do that they decompose into some other elements positive sums of T's. So these have only T's in them. That's not a very good color, but that's a T, right? So, it's like there's a basis for the space of the span of the this restricted space of McDonald's that has that involves t positive sums assurers and the coefficients are still positive and they are embedded in each other. So, that was like the big conjecture when I was a graduate student. And again, we thought this is awesome because we'll just figure out what these guys are, no problem. And then we'll just get figure out what the coefficient this these smaller coefficients are. So then after two decades, we couldn't even find a definition where we could prove almost any of this. So we said, what are these things? Like what is the definition of this thing? It's a supposedly a basis. It's just a t positive sum of shurers like how hard could it be right and in our original paper we proposed a definition this one we the branching is the phenomena that the k version can be decomposed positively in terms of the k plus one okay by by our original definition we could show it was a t positive sum of sure functions we couldn't show it was a basis and we couldn't show anything else so then we kept we said what about a different definition. What about a different and like various people tried and introduced different definitions. You can see that like at some point it was even so sad that we picked up several collaborators and proved even less. So like over time, right, the situation didn't really improve at all. It just like was stagnant. Lots of different ways to look at what we thought they were, but none of these could be proven to even be the same as each other. So, it's like really really a problem. So, you're thinking you spent 20 years on that and like you really got nowhere. And so, since I'm still in math, obviously I was doing something else. And so, what happened was Luke and I decided, okay, well, like what if we let T be one? Can we give maybe it's easier to work with these guys when T is one. So again they're they're the guys are these things coming up in McDonald polomials. When t is one they're just sums of sure functions. Well it turns out actually t is one's not that much easier. However something amazing happens which is that when t is one the linear span of McDonald polomials. So now or the linear span of these kure functions simply becomes this ring. it's a ring otherwise it's not a ring and it's the ring generated by H1 up to HK E1 up to EK however you want to think of it and so we found a definition where we could finally prove like at T is one you have a basis for this sub ring but what was really kind of exciting and probably the reason that we stuck with it was that when T was one since it was a ring you could multiply them together and then we saw that the coefficients in terms of that basis were positive. So that was like okay now we're really starting to see a lot of the phenomena that the sure functions have and we kind of have a responsibility to like work this out. So I I do have a natural stopping point in a couple minutes. Is that okay? Okay. So briefly before we rest it turned out that we could relate when t is one these we call them quir functions to quantum coomology so as far as I understand quantum komology is a Q deformation of coology so as a linear space all you're doing is thinking about coology and taking your sche classes and putting Q coefficients as a sum But the product is where it becomes much trickier. One way to think about the quantum coology structure is that the ring structure is defined by the structure constant. So we say let's define a product by saying if I take these two Schubert classes and multiply them their expansion coefficients are be going to be given by this thing called a Gromov Whiten invariant. So some like complicated numbers whatever you guys know way more about them than I do um that especially at this time were very very difficult even to compute were used to define well it was an associative ring and they called it quantumology. So in the case when you take the grass manian this is actually isomorphic again the symmetric functions modulo and ideal I always feel a lot better when we talk about algebra so anyway um so for me like I'm I perk up immediately but sadly this ideal is pretty yucky right like it's not like the ideal that you use to get the coology the grassmanian you now have like this stuff in there. So you could take a sche class and say okay it's going I'm going to map it to a sure function and I can get the product by taking two sure functions multiplying and mod out by the ideal. Yeah, you can do that but a sure function modulo this ideal is not nice. It's not easy to do that. It's hard. The ideal doesn't have a good description in terms of regular SH functions. So what we discovered though is that if you use the kir functions it works exactly the way it works with the shure functions for coalology the grassmanian where when you take a quir function product and you expand in terms of quesir functions that if you map it down modulo this ideal here anybody who's not indexed by the nice shape is zero and otherwise you get exactly one positive q power of a sche class. So that means that these coefficients are exactly these coefficients which we said were these growit invariance. So this is like mind you I I didn't know very much at all at this point. I know only slightly more now but anyway um so I was like that's that's awesome. Obviously, I'm not going to lose my job. But also, I was really curious because it was like, but there's all these sure functions, right? For every partition that doesn't have a width bigger than K, I have a Kure function. And so many of them go to zero when I work here. And I was talking to Mark Shazono, who is awesome because he he's like knows a lot about McDonald's and about super calculus. So, he's like my perfect um resource. But anyway, so I was like, why are there so many of these? What what could the rest of the coefficients be? And so that's when he told me about all this stuff with Peterson and he talked about the aphine grassmanian um and that apparently it was known that if first of all that you have a homology you have a ring structure on the homology of the aphine grassmanian and that this is isomorphic to the quantum coology of flags. Remember what Luke and I did was only the grassmanian and so so you have this setup where you can study quantumology of the flags which by the way means you can also ask the question about how to multiply Schubert polomials which is even even when Q is zero we don't get it very well cominatorilially but anyway so you have this isomorphism from the homology of the aphine grassmanian of SLN but it was known that that was iso orphic to none other than exactly this ring that we had been studying because it was the Kir function were lying a basis for that ring. So then you know Mark and I decided that it had to be that the quesure functions were representing these Schubert classes in the homology of this aphine grassman and and Thomas Lamb who was like a whippers snapper at that point like immediately proved it um that when t is one this definition that Luke and I gave for quirer functions just when t is one could be representatives for this aphine grass money. So this is my summary and then we'll stop. So I mean unfortunately there's still a lot more but we'll at least stop for a minute. So the the summary of modified McDonald polomials is that you have this basis defined in what I think is an uninspired way right very difficult to get a handle on it but when you modify it using this Garcia modification you get this beautiful basis that's combinatorally a sum of tableau ribbon tupal tableau involving inversions and stuff you you get the Garcia Haymon modules. So it has this beautiful representation theoretic interpretation and of course the proof required geometry way beyond my comprehension that Haymon had to use Hilbert schemes and stuff like this to prove that. And then finally the last piece is that we found these quir functions by studying McDonald's and preserving this positivity and getting this like branching phenomena. And when t is one you see that they are these representatives for the aphine homology of aphine grassmonia. Okay. So would you like to take a stop now? >> Yeah. Yeah. This seems like a really good time. Maybe we could stop the recording and