 All right, so welcome back to the second part of all the worst talk. Okay, so like in the Tiro, in this three-year rectangular setting, we had our standard monomial basis that was sort of came out of things very nicely, very generally, but didn't have a lot of the combinatorial properties that we wanted. And then we were able to find instead of these special web bases that don't exist in general, but they do exist in these cases. And when they do exist, they're really nice. And I'm gonna say that in the pennant setting, there's also something really nice that's like a lot like a web basis. And here's what's going on. So I'm gonna think about completely arbitrary matrices of variables now. There's d by n matrices of distinct indeterminance. And now instead of looking at invariance of like SLD, I have to restrict to like a parabolic subgroup of SLD. There's like a really big parabolic subgroup. It just has like in the first two rows, I have these zeros that are forced. So I'm gonna think about invariance of this group. So this is acting on the left here, just multiplying. And I wanna know which things show up. And again, like invariant theory tells me what's going on here. I can write down generators. So one thing that's invariant under the action of P is all the d by d, the full rank, the biggest possible minors you could think about. But another thing that's invariant are some of the two by two minors, not all the two by two minors, but the two by two minors that are sitting at the top, the two by two minors that use the top two rows of my matrix of variables. And okay, so it's generated by these two things. So it has like two different kinds of plucary variables. And what this is corresponding to, like this is the homogeneous coordinate ring for a two-step flag variety. And it's almost like a completely general two-step flag variety, except my first, my first step has to be two planes. So I'm looking at, I have some fixed n-dimensional vector space, and I'm looking at two planes in my n-dimensional vector space contained in some d-dimensional planes sitting inside my n-dimensional vector space. d is sort of controlling how tall my flag pole gets. So I've gone from Grassmonyans to this special case of two-step flag varieties. And this is, I think, like the key thing that kept me confused for so many years was I was expecting to tell this story on a Grassmonyans always. And that's the wrong place to go. It lives here instead. Okay, so the thing is like, if I plucker embed this two-step flag variety, I think of it as living in a product of two projective spaces. So I have these, the d-by-d minors are giving me plucker variables on one of these projective spaces. My top justified two-by-two minors are giving me the plucker variables on the other projective space. So I have about two different, my grading is more complicated because I have two different kinds of variables. Okay, so now what I'm gonna do is I suppose to think about like all products of d-by-d minors and top justified two-by-two minors, but I'm not going to do that. I'm going to think about products where I use exactly one d-by-d minor and then a whole bunch of two-by-two minors, top justified two-by-two minors. And it's not like, you shouldn't like, see that this is the right thing to look at, but this is a very nice case. Just to think about these particular plucker variables, because I'm looking at, this is some line bundle here. This is like, this is O of one on one of my projective spaces. And this is like O of K minus one on the other one. And I'm tensuring them and stuff. And again, like I have this torus acting on my big matrix of variables that's scaling all the columns of this big matrix of variables. And let's make the numbers work out so that I can think about this all-ones weight space. Again, so now I'm going to think about products of minors where I take one huge minor that's as big as possible and a whole bunch of two-by-two minors with the property that among all these minors, every column of my matrix gets visited exactly once. So either it gets visited in the big minor or it gets visited in one of my two-by-two minors. And so this is some space of polynomials again. Okay, the definition is a little bit complicated but it is completely explicit. And these polynomials are the spec module. They're the spec module for this pennant shape. And you can write them down. And the action of the symmetric group is permuting the columns of your matrix still. So permuting the subscripts on your variables and all that. So standard monomial theory is still telling you a basis for this space that has a basis given by standard young tableau of the appropriate shape of your pennant shape. So you get some kind of object like this. You have your big flagpole, one, two, three, six, eight, 10. That tells me take the six-by-six minor using columns one, two, three, six, eight and 10. That's this thing. And then I have my flaggy parts of my pennant over here. And that's telling me to take a two-by-two minor and not just any two-by-two minor. Take columns four and five and rows one and two. I'm gonna take the top two rows, that's this. And then take this one, that's column seven, nine and the top two rows again. That's this thing over here. So this is some like very large polynomial. This is the polynomial that corresponds to this and their basis elements of the spec module. Okay, but again, like this is not a good basis. If I act by basically any permutation and permuting the numbers inside this tableau and it becomes kind of terrible and all sorts of things. And instead, there's a different basis that I'm gonna tell you about. This is like the main result of our project is that there's something like a web basis. So for the spec module, if I look at the set of non-crossing partitions with a certain number of blocks and with no singleton size blocks, all the blocks have size at least two, these are again a basis for this spec module. So I have pictures like this. And again, the long cycle is going to act in a way you can understand like up to up to some signs that I'm hiding. You just spin this thing around, okay? And the reflection is just doing W naught for you. So there's a lot of names here. Like I guess I sort of counted stuff. And so what Brandon Rhodes did was he built the spec module like not out of polynomials. He built it like with his bare hands. He said like, let's take the C linear span of these diagrams and then he like guessed rules. He said like, if I want to act on this diagram by this permutation, the answer is this thing that I guessed. And then many pages of like calculations to make sure that this is actually an SN action. And then many more pages of calculations to make sure this is the spec module that you were hoping it would be. And it's all kind of amazing and magical. And somehow like, I mean, that's great, but it seems to come out of nowhere. And so like what I want to do is make it very explicit like to find this basis sort of sitting in nature, this representation in nature. And so what we're going to do is we're going to actually write down for every one of these we're going to find an appropriate invariant polynomial such that the action is something you can understand. There's a very recent work now with Brandon Rhodes and Jesse Kim where they again find this spec module sort of in nature. They find it living inside a fermionic co-invariant ring, sorry, a diagonal co-invariant ring. So the diagonal co-invariant story like this is like Garcia, Heyman, Parchesi story, all sorts of amazing things coming out of it. And they think about this fermionic version where you have a skew commuting variables and inside they find this spec module and the basis that comes out of that story is somehow like this non-crossing partition basis. I don't understand how their story is related to our story but next week Jessica and Brendan and I will all be at a band fork shop together. And so maybe we'll understand how these two stories play into each other. Okay, so I'm running out of time but let me quickly tell you like what polynomials you write down. So I'm going to get a polynomial for every set partition doesn't have to be non-crossing. The non-crossing ones are going to be a basis but I have a polynomial in general. So I take some kind of picture like this I have all these blocks, I have these things blocks no singletons and my polynomial is going to be a sum over some kind of tableau. We call these Reiner Schumazono tableau because that's where we found them. I'm not sure if they existed earlier. The tableau that look like this, they like have not so many conditions on them. They look kind of like Swiss cheese, it's okay. What we're going to do is for each column of this funny switch to use tableau, we're going to, which is each block of this picture over here, we're going to write down a minor but something really weird is going on. So here I have a block of size four, a block of size four, a block of size two. I'm going to take a minor of size four, a minor of size four and a minor of size two. That doesn't look like it lives in the correct space. I was supposed to be thinking about things of pennant shape that are supposed to be one really big minor times a bunch of two by two minors. So it looks like it's in the wrong place. In fact, it is in the wrong place. And even worse, like this four by four minor is not using the top four rows of my matrix. So it's not even the right kind of thing. It's like not a plucker variable. And this one is also not a plucker, little one is. Okay, so like I'm taking the wrong kinds of things and the wrong sizes and I'm multiplying them together. This seems very bad. But the point is I don't just take one of these. I'm going to take, there's a bunch of tableau associated to my set partition like this, the sort of ways of shuffling these boxes around. And for each one of them, I write down one of these polynomials. And the polynomials don't independently live in my spec module. But when I add them all up, then they do and they do. And so this sum here I claim is an element of my spec module. And so how do you see this? Well, there's some kind of strange relation that these polynomials satisfy. It's some sort of way of moving your block sizes around. So it doesn't change the number of blocks, but it does change how big they are. And so you can take your thing where you have blocks of sort of the wrong sizes and use this recurrence, which is some sort of amazing fact about determinants to like gradually rewrite things in terms of diagrams where you really do have one big block and then a bunch of pairs. So you can take one of these things and write it as a linear combination of things that do obviously live in the appropriate spec module. And if you take this big relation, you can specialize it to a rule for uncrossing things. And so you can then take one of your crossing diagrams and use this rule to gradually rewrite it as a linear combination of things that don't cross. And that shows you that the non-crossing diagrams are spanning. The polynomials for non-crossing things are spanning your set and there's the right number of them, as we already know. So they're a basis of this thing. Okay, so let me just sort of try to like sum up where the story is. So like for every set partition, you have one of these polynomials and variant under this parabolic. And if you look at the non-crossing ones, like those are linear basis. And now you can understand like really what the SN action is. It's just like take the things, take the points around your circle and literally permute them. Like take the variables of your polynomials and just permute them. And so you can like actually compute all sorts of things now. You don't have to like undo these crossing rules one of the time. You just write down your polynomial and you wanna expand it in your basis of polynomials, which you just like do by a triangularity. So it becomes like much more computable. And you can understand things like these enumerative questions we carried about to begin with like symmetry classes of set partitions. So homework for everyone before you go away. Like somehow this is a story just about spec modules so far but there should be quantum groups in here somewhere. So someone please tell me where to put them. And once you have that, like maybe this is a pipe dream but like it's supposed to tell you about quantum Lincoln variants. And I think the thing that if you got anything in this way like what you would get would be for a spatial hypergraph. So you have some kind of hypergraph which you think of topologically and you embed it in like the three sphere and think about going on there. And there should be some kind of cluster algebra structure going on here as well because there isn't the other cases and maybe weak evidence for this like existing is that very old work. There were some like tropical freeze diagrams and freeze diagrams are like very special cluster algebras, like the easiest possible cluster algebras think about in the tropicalization also bad. Okay, so I'd better stop because I've run really long and here's a freeze diagram for you. Well, thank you very much for a beautiful talk.