 Yeah. Thank you, Leonardo. And thanks so much for inviting me here. So this is a, this is a new project that I haven't gotten a chance to talk about anywhere yet. And I'm really excited to talk about it. To this audience in particular. So this is a project is very much like a common at work in search of some algebra project and I think we've found the algebra now. So I mean, I'm very excited about this. It's a contextualizing some things that I've been wondering about for like almost a decade now. And I think we've found the correct algorithm to go with it, but we haven't really pushed very hard into exploring that story and so I hope to convince you that you should do that. Like, if you believe if you believe in combinatorics the way that I do and my co-authors Becky and Jessica, you believe in combinatorics that's telling you this is very nice interesting algebra to look at. And, you know, and this was a very diverse audience, maybe people don't believe in combinatorics as much as I do, but I hope to convince you that this is interesting algebra that you should think about and then you should tell me all about it. Oh, my slides won't go. Come on slides. There we go. So this is going to be a talk about spec modules, really. So, spec modules are the erupts for the symmetric group. They're labeled by partitions of and some ways of writing and as a sum of some non decrease non increasing sequence of positive integers. Okay, so it's easy to see that such objects correspond to conjugacy classes inside the symmetric group so the character theory is telling you that that's the right number of erupts. But there are like really nice ways to label, you know, canonically label your irreducible representations by these partitions and we'll sort of see some piece of that coming up. And so the like, why am I doing this a Schubert calculus seminar, like respect modules, but I want to think about things not just like, not just the like isomorphism class of this module but I want to like realize it very concretely. Like it's going to be made up of like explicit elements and the elements are just going to be like global sections of some line bundle on some partial flag variety and like depending on your lambda, you have like some choice and like which flag variety you're thinking about. And so we're going to do this all like very explicitly. And so the point is for me like, there's many ways to build up to build a spec module to build some irreducible representation. And when you do this, you don't just get, you know, just get like the module, like every time you're building this construction it, the construction gives you a basis or you end up with a distinguished basis. And maybe algebraically you don't really care about that so much but combinatorially you do very much care about the basis that comes with your object. And it's not canonical like different ways of building a spec module are going to give you different bases and combinatorics, of course we spend a lot of time looking at our distinguished bases thinking about how to change basis between things. And this gets you into some like deep material that got, you know, caused unlisted polynomials are essentially telling you like how to how to change basis for a spec module. We're going to like think about this as a object with with a basis stuck onto it. Okay, so I'm not going to think about all spec modules today I'm going to think about some non obvious two parameter family spec modules, coming from shapes that look like this so so there's some number K, we have two rows of size k, and then we have that stuck on a flagpole of height L. And these are the things we're looking at. Maybe not the family, you would first think to write down but I hope to convince you that this is a really nice interesting family of things to look at. And the first, the first place that these were sort of isolated these partitions at least, or like, look that as a thing. There's some work of Richard Stanley from the mid 90s. Based on observations of Kathy or horror and Andre Zilovinsky, looking at this particular set of partitions and it's all purely enumerative at that point. By ejecting some things with some other things, noticing some numerology, but I know it's been hints for a while that these are special shapes somehow. So, you know, we've been calling these flag shaped partitions for a while because it looks like a flag you have this thing and it's stuck on a flagpole. And now with all the flag varieties floating around that seems like a poor choice of names and so they're going to be pennants, right they're particularly long skinny flags on flagpoles. Here we go. This is the like only picture I could find online or something that actually had the flagpole attached. So it has to be preschool, apparently. Let's think about what's going on here. So, let's think about first the case l equals zero. So, l equals zero we have no flagpole. I'm just thinking about two by n rectangles. And in this case there's something really nice and like well understood, you get a basis of your spec module coming out of coming out of like temporarily leave ideas, and it fits into Cooper Berg story of Web's. Exactly like what you would get for the Web's associated to the Algebra SL2. So you're getting these, this diagrammatic basis is labeled by, you know, things that look like this you sort of non crossing matching diagrams I have and points around my circle and I'm pairing them up in some way, such that all of the lines don't they pair wise don't intersect. There's something really nice going on there. And we're going to let our partition have a flagpole it's going to become L be bigger than zero, and you're going to get a corresponding basis of objects that look like this. So we've taken what was a partition of our n things into pairs. And now we're partitioning the blocks that are potentially bigger than size to. So we're looking at non crossing set partitions that have no singleton blocks inside them. So it's, you're looking at, you know, you're taking n things and you're breaking them into pieces, such that the convex hulls are paralyzed non intersecting. And this might look like some sort of funny combinatorial accident but I claim it there's more to it than that. Like there's an SL three version of the story where I won't get too deeply into this but you're looking at, you know, these sorts of things. There's some kind of planar bipartite. Trivalent graph here, giving you a basis element. And, and I claim that there's a corresponding story that sort of a marriage of the SL three web story and the pennant story you get you get pictures like this. It's some kind of planar hypergraph that is bipartite in some sense and trivalent in some sense and elliptic in some sense and they're, they're giving you something very analogous this is a sort of ongoing combinatorial work in progress. I don't understand the story but I claim it's very nice and so this is not like some one off accident. This is a pennant case here or there really is something interesting going on. The history is sort of how we got into this as we were, you know, we're thinking we're motivated by some combinatorics problems are motivated by these dynamic algebraic combinatorics issues that we're thinking of some sort of action on tableau trying to understand dynamics of, of actions on tableau. And these should be the tools that you use to do that. And somehow like we understand kind of the main line of the argument there but we're getting stuck in some places and so we thought we should go back to this pennant case where we understand the combinatorics and we should figure out the algebra and get some more tools so that we can return to these cases and with some extra power. Okay, so I'm not going to actually tell you like what the tableau dynamics are, because most people here probably don't particularly care about these applications. But I think like webs or something that people here should care about. And maybe you do care about them and if you don't care about them you probably should be caring about them. They do all sorts of great things. So they're not just you know telling you about spec modules but there's like the there's a quantum group floating around here. And from that you're getting quantum Lincoln variants so from in this SL2 case, this is giving rise to the Jones polynomial. And the point is that, you know, if you look at a link diagram, and you zoom in on your link diagram, a little neighborhood of it is going to look basically like a matching. So you can use the sort of diagramatics to expand that and obtain an invariance. So if you look at an analogous over here in SL3 case I'm not really sure what to call it but you get something from kind of Cooper Berg invariant. Here is some kind of SL3 quantum Lincoln variant. And so you could sort of hope to do something. I'm telling you I have these things that are basically webs and so you should try to do analogous things in these other places. So, you know, webs are doing nice things on the geometry of springer fibers, springer fibers in general like they're associated to some Jordan type, and they're hard to understand really what they look like. But in these like special Jordan types where you have corresponding webs, you can make things very explicit, you can understand really how the various irreducible components interact with each other. And there's also like really nice cluster algebra stories associated to all these webs. So you should care about webs now I'm giving you some things that are very much like webs. Let's, let's talk about this story. Some more as a, as a sort of warm up to tell you what's going on there. And then we will take our break after we do that and then we will see how to how to fly people on the things and get this tenant story over here. So let's think about that case and see where this is all coming from. I want to think about a polynomial ring and two n variables, and I want to look at the polynomials that are invariant under SL two. So the way I'm thinking about this is that I'm arranging my variables as a two by n matrix or those two by n matrix of distinct and determinants. And I'm just letting SL to act by left multiplication here. So it does some things to my entries of this matrix, and I'm interested in polynomials that are like unaffected by this multiplication. And if you think about this a little bit it's not so hard to see some invariant polynomials. For example, if I look at this two by two minor. That's going to be the same as this two by two minor. That's just something you can compute explicitly. And that's not just like that one like all the two by two minors of this matrix are unaffected by acting by SL to this thing. Okay, so here's a very old theorem not really sure who to attribute anything to. But in fact this is essentially everything. What this ring of invariance is generated algebraically by the two by two minors there's nothing else sitting inside there. They're not, you know, linearly independent or anything they have relations, but they are a generating set. So if I proge this invariant ring, I'm just getting my grass money and this is giving me my grass money and have two planes in n space, which is a thing I like very much. And so these minors these minors are turning into my poker variables. This is not usually how people tell, you know the story of the poker embedding but normally you're thinking of what's going on here is you have your homogenous coordinate ring is going to be some quotient. I'm taking a polynomial ring in my poker variables, modular some relations module of the poker relations. And what I've done here is I set everything up as a subring of something else. The point is that there are relations among these two by two minors, and those relations are just exactly the poker relations that that you would have anyway. And the point of doing things this way is that now I'm not dealing with equivalence classes, like I really have a literal polynomial associated to everything. Okay, so we said that we're spanned by products of two by two minors products of looker so poker manomials. So I want to think about how can I write down such a poker manomial. How do I write down a product of two by two minors. Well, I just need to tell you which minors you're using to tell you a two by two minor of a two row matrix I just need to tell you which columns you're thinking about. And so I could just write down a two row array, where in every column of my array I'm going to tell you which two columns of the matrix to think about. The columns here are not the columns before. So like if I have this this to five column here that's telling me, you know, you should think about this, this minor, a minor that uses the second column, and the fifth column of your matrix of variables. And so you, you know, you take these seven columns they tell you to take seven minors and multiply them together. And this is some seven fold product of looker variables. And we can like, you know, organize this, this array a little bit, right, we might as well arrange so that all of the columns are are increasing. Why not. We can certainly sort of write them however we want. These are, these are greater than signs, not wedges. It doesn't matter like, you know, multiplication is commuted it so I can reorder my columns wherever I want. So I might as well like arrange for my first row here to be weekly increase I can just like sort the columns in some way. But the second row I have basically no control over the second row is just, you know whatever it is it has no nice conditions on it. But every one of these poker monomials I can encode by like some kind of array like this. I sort of try to make it look nice, but the second row looks like kind of a mess. But it's a way of like writing down things and I know that if I write on all these things and that spans my invariant space. There's an alternative way I could try to, you know, tell you which poker monomial I'm thinking about, which is, if I'm telling you a two by two minor I'm telling you a pair of columns. Why don't I just take, if I have eight columns, I'll just put eight dots around a circle. And whenever I'm thinking about the IJ, look record and that let me just draw an edge from I to J. And so this picture here is encoding the same thing as this array up here. Here is my two five edge. It's just telling me to think about the column two column five minor. So I have these two different sort of, I have this sort of, I don't know, this one's easier to type and this one's sort of easier to think about visually. So maybe the like, the key thing of standard monomial theory is to say well I have these products, these poker monomials that span my space, but they're not linearly independent I would like to actually have a basis. And so if I look at M fold products of poker variables. This is spanning some finite dimensional space like what is the dimension of that space, which, which one should I look at to have a basis. And the answer turns out to be to look at all these arrays that we wrote down before. And in general the second row was a complete mess, but let's just restrict the tension to the ones where the second row happens to also be weekly increasing just like the first row was. And it turns out that these are linearly independent of each other. And they, they're also spanning. So they're actually a basis for this thing. And then you know the dimension of everything because there's some kind of combinatorial formula for counting these tableau these are called semi standard tableau. And, and you have some formula for counting them and you know the dimension of this guy. So that's not quite what I want to do I want to. Well, first of all I need an action somewhere and then I want to like find my spec module inside this story. And so what we want to notice is that there is also an action here of a rank and porous so I haven't fordible diagonal matrices, right multiplying my matrix, and they're just scaling the columns of my matrix. So I can look at, you know, a weight space let me look at the all ones weight space. So this means, I'm going to look at products of two by two minors, where my two by two minors involved every column of this matrix exactly once. So every column is going to show up and one of my minors and only one minor. Well, if I look at that piece. That piece, as a set, it's preserved by permuting the columns, I have this right action of the symmetric group, my symmetric group acting by multiplication on the right, it's permuting the columns here, and it preserves this space of invariance. And it's not just like, it's an SN module, it's not just any SN module that happens to be irreducible and it happens to be the spec module that you maybe would have hoped that it would be, it is the one for the two rectangular shape that M M. And this is not I think how people usually teach about spec modules, but I think it is like the best way to think about what they are, like, and how it's very concrete the elements of this are like some very explicit polynomials that you can literally write down. So equivalent classes of anything going on here. So, there's a basis here. So if I'm looking at products of two by two minors where I touch every column exactly once. That means that in my corresponding semi standard tableau, I should be writing down every number between one and exactly once. But semi standard tableau shape M M, where I have bijectively filled the cells with the numbers one, two, and n equals to M. So these are called standard tableau. And of course we also know how to count these and very famously these are the special case there's Catalan many of them so I'm looking at some Catalan dimensional spec module in this setting. And so and they have a very explicit basis for it right. I look at this tableau here, and it says, first take the column columns one and four and take that two by two minor, then take the columns two and five and take that two by two minor. Take the three and seven columns look at that minor take the six and eight columns right down that minor nine and 10 columns. And just take these five two by two minor isn't multiply them together. So here's one of my basis elements, you can literally just write it down or you can your linear algebra one on one students can write down this polynomial. Nothing hard about it. So, I would like to understand the action of the symmetric group on this thing. And in particular for whatever reason, I'm interested in this long cycle. Well, it's one of the sort of the simplest permutations you could think about. It's a coxer element and sort of maybe the easiest coxer element to write down, except for the inverse of it. And let's think about how this acts on my module and in particular let's think about how it acts on my module with respect to the basis that appeared out of standard monomial theory. Well, what it's doing what I'm doing is I'm permuting my columns. So in this case I'm cyclically shuffling my columns around. It's coming from some like cyclic shift map on a grass mania. And so what does that do. If to the basis element my basis element looks like one of these tableau, and now I'm re indexing the columns. So all I need to do is puff all the numbers up by one, except the number 10 is going to turn into a one. I should get this thing, but that's not really the right kind of thing because I like I didn't use do my conventions properly. I should take this column and I should, you know, reorder it to have the one on top should go 110 instead of 10 one. And then I should sort my columns then so that 110 column goes to the front. So I'm going to end up with this. But this is not standard. My second row is no longer increasing. So I took my basis element and I hit it with this cyclic shift map and I ended up with some like other product of bookers that was not a basis element. So we use the quicker relations to rewrite this in terms of things that actually standard like just keep doing these quicker relations think of uses straightening laws for writing this is some sum of standard tableau. And you're going to get a whole bunch of stuff you're going to get all sorts of things and it's sort of not obvious what they're going to be. And it's kind of terrible. We took like the easiest of permutations to think about, and we let it act on a basis element, and we ended up with something kind of horrifying. And it's sort of telling us that maybe we should be thinking about a different basis for this world so I would like to find a better. Ask a question. Yeah, the first step is supposed to be easy. The from the first tableau to the second tableau. This one. Yeah. Yeah, I just take my tableau. So I'm okay I'm cheating a little bit there's some science that I'm glossing over. It's like a minus one to the something that I'm ignoring. But basically all I want to do is I want to take these numbers and I want to commute these numbers according to my permutation. And I'm going to get something that's like a tableau except it satisfies no increasing those conditions anywhere. And then recognition just doesn't tell me how to get from one to the other, even if I think about a cycle. I say, based my pattern recognition doesn't tell me how to get from the left one to the middle one, even if I think about a cycle for mutation. So all I do is I have this permutation w. And so I'm just gonna say this is to and this is going to become w of two. All I'm doing is I take these numbers here and I feed them into the permutation and I get some, you know, some array of numbers here that is like, not not the sort of thing you like to call a tableau. And then you like do some stuff to make it look great and there's some signs that get introduced, and then you apply the straightening laws over and over again. And something falls out. And you get this big linear combination of things that actually look correct. But like you don't want to do this right you're trying to understand the action of this like very simple element and you had to do a lot of work. What I'm going to do is find a basis where, you know, the long cycle is going to act in a way we can understand with our with our own brands. Okay, and this is one place that webs come in. So, if I'm thinking about this spec module. What we learned before is that it was going to be spanned by products of pluckers corresponding to matchings. So I take the numbers one to end, and I put them around a disk. And I matched them up in some way. So every edge of that matching that tells me some looker and I have this ij edge that tells me think about the two by two minor that is columns I column J. So, most of these are going to have you know things like this they're going to be like lots of edges all crossing each other everywhere. Just like before most of our tableau were a disaster in the second row. And it turns out that if you restrict to those particular matchings were none of the lines cross each other. You look at these non crossing matchings. Those are linear independent and there's a right number of them and their basis of the space. And it's a better basis in some sense. One one way in which it's better is if I want to understand what does my long cycle due to a basis element it. Again I'm glossing over some signs, but it's going to send it to another basis element. So some big linear combination just takes a basis element to a basis element, and it's even sort of obvious which one it goes to. You just take your diagram and you rotate it. So the long cycle is just being by like spinning spinning these pictures around up to some signs and you know also w not is acting really nicely w not just reflecting your diagram pick it up and you flip it over. So, this basis like it's not a permutation representation in general but it. You know, some of your favorite permutations are acting very nicely right in ways that you can understand through pictures and you don't have to do a lot of arithmetic to understand what's going on. And let's see that this is like genuinely a different basis I mean I guess we know that because things are acting nicely but like if I. So the looker looker relations. They're telling you how to uncross your diagrams, so I'm going to think about the case this is 1234. So, this picture here is telling me to take the 13 minor and the two for minor so it looks like this tableau. 121324. This one is telling me to take the one for minor and the two three minor. And this one over here telling me to take the 12 minor and then the 34 minor. Okay, so you want to notice that these are not the same basis because the basis given by non crossing math chains, that's these two things. So these are the two pictures that are not crossing. Whereas, if I look at the corresponding tableau. This tableau that's now I wish I had more colors. This tableau is bad. The first row is nice but the second row goes for three. So the tableau, the standard monomial basis would be to take this one, and this one. And the polynomials here, each of these polynomials is a product of two quicker variables, but they satisfy this three term relation. And so any two of them are going to be a basis and the web basis is to take these two to be the basis elements and the standard monomial basis to take these two on the outside to be your two basis elements. And then you're either rewriting this one in terms of the other two, using like the straightening law on your tableau, or you're rewriting this one in terms of those two, using your uncrossing rules. So that's what's going on here. Well, let's briefly. Just a quick question. Move on. I feel like the plate when you're trying to understand as an action the thing that you should start with our simple translations, simple reflections, first versus these other actions and I was curious if the, which basis is it easier to see the action of a simple reflection. Basically, everything is bad and the standard monomial basis, you can't really see the action of anything. It's going to be possible to understand that. So I think the right way to like find the thing that you're finding good basis is to think about these guys as a coxer element and the nw not. Once you've done that. Nice things happen for the simple reflections also. So what your simple reflections are doing is you're taking a diagram like this, and you're taking two adjacent things like seven and eight, and you're just like introducing a crossing. Right. And then, then you know how to undo it. Yeah, okay, yeah. Okay, and this is like a temporary leave thing to do. So, like the diagrammatic ends up being really nice, sort of all the way around you can understand the action. So it's not going to be a permutation representation in general like when I apply this thing I'm going to get two terms popping out. But it's not so hard to see which ones they are. Thanks. That's a good question. Okay, so let me briefly say like what the three row version of this is and then we will like take our, take our break. Three row version I'm going to take a three by n matrix of distinct indeterminance. And I'm thinking about SL three invariance in the same way. And again it's going to be spanned by products of maximal minors and other three by three minors instead of two by two minors. And so I have that space. And again, I have this tourists have this for us here and it's acting, scaling the columns, multiplying on the right. And I'm going to look at this all ones weight space again. So I'm going to look at products of three by three minors, where every column of this matrix is going to show up in exactly one of my three by three minors. So in particular, and had better be multiple of three for this to make any sense. So when I do this. I got some space of invariant polynomials, some particular white space here, and again it carries a symmetric group action is a symmetric group also, you know, multiplying on the right permuting these columns around, and that preserves the set of polynomials where I touch every column exactly once, because if I permute the columns I'm still touching them all exactly once. And I then have some SN module. And again this is just my spec module is like another very concrete realization of the spec module. And standard monomial theory again is going to tell me how to write down a basis for the space, and I have my spec module as a bunch of polynomials. It's going to have a basis given by standard young tableau that are going to be three row rectangles. So M is the, you know, the number of quicker variables I'm thinking about. And this is telling me what to do so here I'm going to take the columns 13 and five of my matrix. I'll take my columns to six and 10 there's another three by three minor by 4811 columns my seven nine 12 columns, I have these four three by three minors. They are disjoint in terms of which columns they think about every column shows up in one of them I mean apparently n equals 12. And that's some explicit polynomial and it lives in my spec module. And the action of SN is easy to understand in the same way I'm just taking these numbers and feeding the numbers into the permutation and getting a new tableau. Right, or on the polynomial, I'm just taking my variables, these variables, and I'm just re indexing them. I'm sending X I to X sub w I, and why I go to Y sub w I, and all that. So I'm just getting this very concrete action of the symmetric group on some like very concrete polynomials. You have a question in there. Oh, okay. You've made some noises. Okay, so there's. So again, again, like I have this basis. If I hit it with the long cycle, they hit it with the long cycle like it's a disaster right like, or hit it with basically any permutation as a disaster like I permute the numbers and I get something that's not a standard tableau. And like I try to unwind it like lots of terms appear. Like you can do it and you can compute it by hand but it takes a while. There's a nice pictorial story in the special case that let's you see what's going on a bit better. There's like lots of names on this but I'm not really sure what to say. This is the basis of SL three web so there's these kind of diagrams that look like this, and I'm not going to tell you like exactly how one does it. The every such diagram there's some like invariant polynomial that you're supposed to write down. You look at this graph and this graph is some sort of a recipe for feeding determinants into each other. And in this basis, one way to see this is a nice basis is if I act by the long cycle, then instead of getting some like big linear combination of basis elements, I just get one thing, like maybe some assign attached to it, and it's just like what you get by spinning this thing. You're just rotating your diagram, relabeling things and w not is doing a really nice thing is just picking it up and flipping it over. This one's symmetric so you don't always do that. And so like you can, you can do this and you get like nice combinatorial stuff going on a nice algebra attached to all sorts of things. And it ends up being very useful. And like for a long time people have been looking for like a good definition of webs and higher ranks. So Cooper bird Cooper works like in general tape but he's only thinking up to rank to think what SL three can think about some other things but like what are you doing with SLK. And people have made definitions and like they're probably even the right definitions but they, they don't quite do the things that we want them to do for like our combinatorial purposes. And instead, I'm going to, I'm going to tell you like we'll take our break here break break time. We're going to go back to this penance story, and I'll tell you how to see this penance story as like some explicit set of polynomials. And there's going to be something like a web basis there like I'm going to have these, these planar diagrams and every planar diagram is going to like give me some corresponding polynomial that I can work with very explicitly. The same things will happen. Okay, so let's stop there for now.