Jennifer Morse, part2, `Nonsymmetric Macdonald polynomials with a small side of Schubert calculus'

Here we can finish. Okay. So, now we're going to talk about non-symmetric Macdonald polynomials, but sadly there's a lot of work to get to the point where I get to talk about anything new um because we're going to have to go back to the symmetric for a minute, but let me first define the non-symmetric world in the non-symmetric Macdonald polynomials. So, first of all, non-symmetric polynomials, what are they? They're just polynomials, Laurent polynomials. There's nothing particularly non-symmetric about them. They're just polynomials. Um and so now the bases are indexed by any integer vector, if you have n variables, an integer vector of length n. And just like I'm trying to kind of give you an analogy to what I started with with the sure functions, but now I'm in like this space of polynomials, bases indexed now by integer vectors, not partitions, and there's also an inner product on this space. Here it is. I don't think that you need to belabor it because it won't really be that relevant, but it exists. And then also what exists is an analog to sure functions. The difference here is that they're not self-dual. The The two The two sort of like sure bases for this space of non-symmetric Laurent polynomials um are called key polynomials and atom polynomials. And they have all of the beautiful features that sure functions have. They have combinatorics, representation theory, geometry. And you can You could also use these divided difference the isobaric divided difference operators to generate them. But most importantly for our talk is going to be that if you take the Weyl symmetrization, which if you want to think of it as this operator indexed by the longest permutation. That's not really usually how we think of it, but it is that. If you vial symmetrize an atom, it's either a sure function or zero. So, that means that if you have a function, a polynomial that's atom positive, and you vial symmetrize it, you get something that's sure positive. So, we think of it as a strengthening of sure positivity if you have something that's atom positive that's non-symmetric. So, that's kind of the key, but otherwise these are two bases very, you know, well-studied and that doesn't mean they're easy, but there's still unknown questions, but whatever. They do have all the nice features of sure functions. So, the non-symmetric McDonald's are well, actually Cheredniks, um because they're defined using his operators, but whatever. We're going to think of them in the very same way that we think of McDonald's and sure functions, which is that they are unitriangularly related to monomials now, and orthogonal with respect to this scary inner product with Q's, now Q's and T's just like before, um that reduces to the other one when Actually, here you have to sort of take a limit of Q goes to infinity and T goes to infinity. But, in any case, the idea is that like this does specialize to an atom, and you can compute an example, although we're lucky to have ChatGPT, these people had a lot stronger constitutions than us and probably did this by hand, but whatever. If you compute on, you're going to get something like you see. Again, it's not a positive sum. We probably wouldn't expect it to be because we already knew that the McDonald wasn't a positive sum. But, anyway, it looks like this. Um sorry, I forgot what I'm doing next. Oh, yeah. So, one of So, a couple things about them. Why Why did they arise? Well, this was as I said, Cherednik introduced some operators in the double affine Hecke algebra. These are polynomials which are eigen functions of his operators. So, that's why I say that there is a whole contingent of people who are working in the non-modified side. And now now so am I, I have to admit, but anyway, whatever. Um so, the other thing about these polynomials is that it you can hexametrize them to get a usual McDonald. And that's sort of pretty important, I think, in in the way that my work has gone in my life because you know, Cherednik had his work for the non-symmetric McDonald's, but one of the things that Knop and Sahi did was that they discovered that you could come up with these recurrences for the non-symmetric McDonald's that did not exist for symmetric McDonald's. So, they were able to prove nice, beautiful identities about non-symmetric McDonald's, and then hexametrize to get results about symmetric McDonald's. So, that's kind of a theme in my own work, which is like sometimes when you go to the like more complex setting, you're you get these tools that are not available to you in the restricted setting that help you even when you project back down. Anyway, that was some of the stuff that was going on. I've kind of forgotten. Oh, okay. But here here's like the big question in my you know, related to this talk in my life, which is we already hopefully are convinced now that the modified McDonald polynomials are these really great things that have the beautiful representation theory, the beautiful tableau combinatorics, and the K-sure functions. So, obviously I'm biased to the modified side. When the non-symmetric McDonald polynomials came out and it was known they hexametrize, so that's this arrow, to the McDonald's, the question naturally is like, well, is there some modified version of the non-symmetric McDonald that might have some of the interesting features that you have over here in the symmetric modified world. So, that's a that's a really natural question that over the years became even more compelling as more and more beautiful things arose around the McDonald modified McDonald polynomials. Um, well, here's the history of the problem. Um, very early on, Knop did find something a little bit like the funny positivity. Remember over here there was that funny basis and that funny positivity that very beautifully mapped to the sure positivity under the Garcia transformation. Knop studied these guys and he discovered that if you like do do some things like look at stable limits and find the right scalars that he could come up with some positivity conjectures there. And then they sat standing still and there was really not a lot of activity until 2019 and then Luke Lapointe started coming at it from the physics perspective um addressing similar questions. And then more recently there's been quite a lot of work by Bettsworth, Weising, Goodberry, and Orr not necessarily the question that I'm posing here, but more about particular stabilizations because that's kind of what you have to do. Early on, as I said, it was understood that you do have to look at sort of stable versions of this guy on the left. Whatever, I'll say a little bit more later. But and that's a lot of what their stuff has been done. They've been doing a lot of great stuff recently. Okay. So, that's kind of where it was and again, we're interested in this. So, now I have to go back all the way back to Catalan functions which are symmetric. So, this is another topic that is going to live in the symmetric world for a minute. And um so in the '80s, that was when the harmonics module was there and there was that was the harmonics module gave rise to this beautiful sum of sure functions with Q coefficients and then it led off to McDonald, all the stuff we talked about. Well, another direction was some of the work that was done by Lusztig. Basically, without the Q, this is a partition function and the coefficients of Z without this Q here, right? were given as the partition function generating function of Kostka. And Lusztig said, well, we can actually get Q weight So, you would get weight multiplicities by taking these You could write the weight multiplicities as a signed sum of those. Lusztig's went further and said, well, if you put a Q in here, then you can get more information and you can get this formula for Q weight multiplicities which is like the signed sum of these coefficients. So, this This is a This is just geometric series. If you expand it out, you get a positive sum Q positive sum of these like Laurent mine monomials. And the coefficients of them if you take a sign sum, give you the Q-weight multiplicities. Whatever. So, another way to to look at Lusztig's formula and this picture, which was a little bit of a different direction at the time, but more in the framework of what we've been talking about in terms of like sure expansions is that if you if you take this formula, right? And of course, Lusztig's interested in this thing. So are we, but anyway. By the way, he did this for like very very general types. But but I'm only sticking to the type A. So anyway, um if you take this identity and you Weyl symmetrize, essentially you just Weyl symmetrize the identity. So this is the Weyl symmetrization operator. But the thing about Weyl symmetrization is that it converts a monomial to a sure. But this is any vector. So it doesn't if it's a partition, it definitely goes to a sure function. If it's not a partition, it might introduce a negative. So anyway, if you Weyl symmetrize this, well times a monomial, then first you have to geometric expand it. Then you apply the Weyl symmetrization. So each of these monomials goes to well, some sure function, but you might get a negative. And sometimes you get zero. But the point is, the coefficients of those are this sign sum. Cuz like look at the Weyl symmetrization operator. Essentially, what it does is like converts it it gives you Lusztig's formula as a coefficient of sure well, sure series, right? Cuz these are kind of funny sure functions, right? They're allowed to have negative in their weights. They're just characters, that's fine. Now, hang on. So, this is kind of a cleaner picture of what I was saying. If you take the partition function, which would it's one over that, but I can put a monomial there, and you vial symmetrize it, you get the sure coefficient is this Lusztig formula for the Q8 multiplicity. But if you look at my example, so you get this thing, and those coefficients are what he wants. And it has negatives. But actually, if you kill these funny sure functions, the one with negatives, like what is that? I mean, of course it's something quite natural, but whatever. Kill the guys that don't have that have negatives, and look what you get in this example. This should look familiar to you. It was a the example that we did in the very beginning, the sum of sure function with tableau. So, this is actually a McDonald polynomial in a special case. Just as Q's. And so, the point is that this partition function this Q partition function, if you vial symmetrize it, leads somehow to a sure positive sum. Now, it kind of makes sense for people who've now studied this Q multiplicity formula, and they know it's Kostka-Foulkes, whatever. That's fine, but but still, by this definition that I showed you, all these negatives appearing really don't give you any reason a priori that you should have a pretty sum like this. Okay? But you do. And so, at the time, this was intriguing to people, and they said, "Okay, well, if that does give you a Q positive sum of sure functions, what happens if I change this product and don't take all of the roots? Right? Instead of taking all the positive roots, which was what the Q eight the the partition Q partition function was, take a subset of roots." And so, sort of the earliest case of people experimenting with this beyond the case of all positive roots, was Brouwer's work in '92. And this was like from a geometric perspective, he was looking at some special set of roots called parabolic roots, and the conjecture was that it was sure positive and that you could prove it by some higher homology vanishing. And then, lots of people sort of jumped in on that, and they looked at different cases, and like they were seeing all the sure positivity, and it was like really hard to prove. It was very, very hard to prove anything, even though people were attacking it from like lots of different ways, geometry, algebra, whatever. Okay, so on and on they're trying, and then around 2010, I think simultaneously, but separately, Chen Hamon and Pan Chiff came up with sort of the master conjecture was that anytime your roots lie above a Dick path, so these are like root e each one of these is a lattice point, and if you use those roots up in that thing there, then you get sure positivity. So, that was the conjecture, and we call these Catalan functions cuz well, you have a Catalan many of these Dick paths. So, if you take this expression up here, and this is roots underneath a Dick path, the conjecture was that that was K sure positive. It included the case that had been studied by Lustig, which were the Kostka folks polynomial or whatever. But it was a like, okay, kind of complicated. Anyway, why the heck do I care? Because you know me, I only care about K sure functions and McDonald polynomials. So, it's like, why do I care about this Catalan function stuff? Well, you can see the chart. You remembered how little progress we had with K sure functions with the T. I started working with Jonah. He's the one that made me aware of Chen Hamon's work. And um Anna Pun and Dan Summers, and we we came up with a new definition of K sure functions in terms of Catalan functions. And amazingly, that definition enabled us to prove that all the other definitions were the same except for the very first one. That's still open, but I wouldn't go near it with a 10-ft pole. But anyway, whatever. So, like this is why I cared. It's amazing that one new definition could do so much for you. So, this is just the quick summary of it. Catalan functions were you take any set of roots under underneath a above a Dick path. We found a really, really highly structured special Dick path associated to uh lambda and K and said that Catalan function is a K sure function. And the I mean, I'm not going to go into it, but the combinatorics of this path selection was just like so perfect that everything kind of fell out by like working with the root these roots and root expansions. So, that was that was pretty awesome, especially because it had been so long. Um we were really excited about that. It helped us prove a lot of that stuff. And then we did dabble a little bit in saying, well, of course when T is one, this is a new formula. And we know that when T is one, these are related to like the homology of the affine Grassmannian. So, we looked at like some other, you know, settings and we studied in particular a K-theoretic version of Catalan functions. Now you have just like some lowering ideals in there. Like another you have two root ideals and something that makes it inhomogeneous. And anyway, we were able to prove some positive branching property for that. But really, you know, Akita was he had a conjecture that motivated us and like they've really gone on and done much more than us. I mean, we we did a little bit and then we kind of walked away for now. Anyway, so can't remember my point. Oh, yeah, I'm getting there. I'm so sorry. You lost my train of thought. Okay, so whatever, this is not just like a brag session. It's actually going somewhere. Um, so we discovered that this Catalan function stuff was really useful and the Catalan function or at least for K-Schur functions. So, we could prove K-Schur functions this particular Catalan function was Schur positive. So, then we were like, well, what would we do next? What about this conjecture that these Catalan functions are positive for any root ideal. And that we could not do with the techniques we used for K-Schurs. The techniques for K-Schurs were highly reliant on the path having a lot of structure. Now here is a conjecture for any Dick path. And but what we ended up doing to prove it was something that really inspired the work with the non-symmetric Macdonald's. And so just briefly, what we did was we went to a bigger set, kind of like the Konopelchenko idea. The Catalan function is that you use the Weyl symmetrizer. But we started looking at not using the whole Weyl symmetrizer, but just partially symmetrizing. So, we took the same inside and then we said, "Well, just partially symmetrize." Of course, then what the heck does this mean? Because that said, "Take a Schur function that's ugly and throw it away." I don't have any Schur functions anymore because I've partially symmetrized. It's not even symmetric. But we have the key polynomials, which are like Schur polynomials for the non-symmetric world. And so this was important here and it's important to our newer work, which is that it turns out that this truncation you expand a non-symmetric polynomial into keys, and then you might have negatives cuz they're they're indexed by any integer vector, and you throw away the guys that have it negatives. So, that's all it is. If you have the key expansion, it's an easy operator, but unfortunately getting some of these guys into keys is a little bit yucky, right? So, I'm not saying it's it's easy. But anyway, that was what worked. And so that was kind of what enabled us to prove um that the non-symmetric versions in particular cases were atom positive, and remember that implies Schur positivity when you symmetrize. So, the key tool here was that in the non-symmetric setting, there's a rotation theorem that allows for inductive techniques that you couldn't get when you restricted to the symmetric case. So, anyway, we did that. Kato went on to do the geometric version of our sort of algebraic conjecture. And we have combinatorial formulas. So here I I say two positive outcomes. I actually don't mean Cado's work, even though that is a highly positive outcome, but not of me. Um the other positive outcome is that we discovered like I was trying to allude at in the previous slide, that this polynomial truncation of keys is like an amazing operation in its own right. So we define an operator that says, take any polynomial, divide by these powers that we've been seeing all over the place, and then take this polynomial truncation. It's a very similar to what we've been doing already. But now just think of it as an operator. And so when you go back to this long-standing mystery about non-symmetric McDonald polynomials, the missing thing is that well, since we don't have a really concrete funny positivity to just replace with nice positivity, you have to rely on knowing what that operation is. What is the operation to take you from the not modified sort of yucky side to the pretty side that has a potential for representation theory and K-sure functions and super calculus. And this operation that I just defined, well, you might think it's the right one and you apply it to the non-symmetric McDonald and then like you get something yucky and then you cry and go home and give up. No. Of course you don't, because you wouldn't want to do McDonald polynomials or super calculus if you were that personality type. So instead, you know that all these people have been doing amazing things with different stable versions of that. And so it turns out, now I know I'm like going over time, so I'll just like move a little bit more quickly, that if you take, we call it an R non-symmetric McDonald, so instead of like non-symmetric, just the first R variables are non-symmetric and then it's symmetric beyond that. But you can pick any R. But so it's like a refined kind of a refined approach where you work in an R R non-symmetric space. And when you take those Macdonald polynomials and you apply our operator well, I computed an example at the bottom and you start to see something really pretty, positive sums of monomials with Q's and T's. So, that's highly suggestive that this guy could be the right operator. And so we define the image of one of these R symmetric Macdonald polynomials to be a modified R non-symmetric Macdonald. So, we call it modern just because it's like an acronym. So, this is like a modern Macdonald, modified R non-symmetric. And you see that it's positive here. And not only that, but we can complete this commutative diagram because when we looked at examples and applied the vial symmetrization, we got an actual modified. Which is sure positive. So, this suggested that our modern Macdonalds are atom positive. And that they vial symmetrize to the modified Macdonalds. So, of course that brings a lot of questions forth and then I'll just say like we do have a beautiful tableau formula very much like the Macdonald polynomial formula. You again take tuples of tableau. But the only thing is now they're flagged. So, each row has a maximum bound on the letter that you can get. So, you get a finite number of them and you get a non-symmetric Macdonald polynomial with the same Q and T. It's still the arms wiggling and the inversions. And so I guess I'll stop. Um But anyway, so I hope like kind of the main thing for me is that like of course I have been obsessed and I have like a one-track mind about McDonald's and especially K-Schurs, but I'm I'm trying to say, I don't know if successfully, that I really love that the ties that have come out of it with Schubert calculus and now that we have this modified non-symmetric framework, I just don't see why there won't be amazing things related to Schubert calculus coming out of these non-symmetric modern McDonald's or whatever. So, anyway, there's a lot of work to be done and some people are already doing some great stuff. They're so fast, it's incredible. It's so humbling. Anyway, but thank you so much for listening and I'm sorry to go over. >> That was really great. Thank you so much.