 Oh, I have to, yes, pull my screen back up. Yeah, so please go ahead, the second talk of the second half of Samir Saff's talk. OK, this is where we left off before the commercial break. So, right, how does this go? Oh, no, hang on. My, well, I guess it looks OK for you guys. So maybe I'll see if this sorts itself out. Oh, so sorry, the last next thing I'm going to tell you is just the insertion algorithm that proves the theorem. So we're going to first, I have to tell you how to label things, label a Conert diagram. And then I'll tell you what landing and bumping look like. And then we'll observe the row bump and mama, which is the crux of why this insertion works. So first, let's go back to Conert diagrams and that bijection. So if I start with a K-gross monion. So here's a K-gross monion. So what's K? The K is maybe 6. And I'll go let 0, 1, 3, 5, 6, 6 is my nu. That's my 6-gross monion remutation. So I can do this bijection by sort of looking in this second column maybe. And writing down, this is in row 6, this is in row 5, this is in row 3, this is in row 2. And that tells me what the entries are in the tableau here for the reverse tableau. 6, 5, 3, 2. That's the bijection. It's pretty easy. And it's really nice. One of the things is if you look at this first row, maybe I'll use the highlighter here. If you look at this first row, oh, OK. Well, now my things screwed up, so I can't quite see the first row. But if I look at the first row here, I can see these bubbles as the ones in row 6, row 6. And then I drop down to row 4, row 4, and row 3. And then I can look in maybe this one, the second row here. I can see 5, 5. And then I drop down to a 3, a 2, and a 1. And then I can guess what the neck going to be. It's just take the top bubble of each row that you have. And sure enough, yes, that's what this is. And then I have many colors I can use here. I'll do this one. And maybe yellow for this last one is here. OK, so I can kind of decompose my corner diagram into these threads. And what you'll notice is that what is this saying? It's saying that this is fully rectified. That rectification idea, that matching the bubbles between two consecutive columns, matching the bubbles from the left to the right, sorry, from the right to the left, they're all going to match up. And so nothing's going to move under rectification. So this is sort of the stable point of rectification. But what happens if I want to add a new bubble? So suppose I want to add a new bubble to this diagram. Well, then it's not going to be rectified anymore, especially if I add my bubble maybe out to the side like here. OK, so I'm going to keep this diagram the same. And I'm going to understand rectification by taking this bubble X and moving it. And saying, OK, well, what happens to X as I go? So X doesn't pair with anything. So again, X is a bubble. It's a new bubble I've introduced. It doesn't pair, so it's unrectified. So I'll have to move it left. That's why it's red. OK, so I move it left, and it's still not rectified. There's nothing in the column immediately to its left, weakly above it. Still nothing, still nothing. Oh, hey, now we've got some action. So the action we've got here is that this bubble pairs with this one. This bubble pairs with this one. This bubble pairs with this one. So who's left unpaired? It's not X anymore. It's now this bubble. So there's one bubble that's unpaired. So here's what I'm going to do. I'm going to bump X down, and I'm going to put helium in that red balloon and float it up to where X was. So they switch. And now X is unpaired, so X will move left. Now, let's check. X pairs, X pairs here. These guys pair, these guys pair. These guys pair. One unpaired bubble with this one. Same thing, X goes down, bubble goes up. And now we're going to rectify again. And now I'll check. And everybody's rectified. So we're done. We stop. We put a box around X. And what I have just done is literally RSK. So rectification, this idea of moving bubbles left, means something. It's not some contrivance. There's something really deep and magical going on here because RSK is deep and magical. So that's what's happening here. So I put the path here that X takes through the diagram in both paradigms. And you can see that, well, that's what X did. Yeah, that's exactly what the bumping path is in RSK. So again, I'm doing RSK on reverse tableau. So the five will have to bump the biggest thing, strictly smaller than it, which is the four, and so on. But this is a theorem with my grad student, Dan Joseph Quijada. We did something a little more general to try to multiply key polynomial. So this was month rule for key polynomials that we were proving. And we noticed this property of rectification. So that's kind of nice. That tells us that rectification is somehow important in this idea of insertion, because it's the move left. Connors rule is the move down. Rectification is the move left. So rectification is actually well-defined. This process is well-defined for any time it's Southwest. And in fact, if you use rectification, it can decompose a corner polynomial into damaged characters. But it's some kind of general thing. But in order to use it in this way to kind of understand insertion, we needed to take into account the fact that the Roth diagram is not rectified. It's not like a K-Gruss money case. It's not fully rectified. So when do we stop? Here we stopped when everything was fully rectified. But that's not the right answer in general. So we need to rethink, OK, how do we know when to land? How do we know when we stop and put a box around x? That's one thing. And then what can you bump? It's not true that you just bump anything, that you just make sure that you're rectified. Because, again, the Roth diagram is not fully rectified. But more than that, when I did this, I have a paper that was, I don't know, it's still in limbo in the referee purgatory, where I did this for key polynomials. And when I did it there, what mattered was not just that there was an x below in the rectification, but the label the x gets. There's some idea of labeling a diagram that's fundamental here. And when we label things, if we're trying to do this insertion for k, we can k land and k bump when the label isn't most k. So I need to explain to you about labels because it somehow gives you the answer to both of these questions. OK, so you'll have to pardon me for a digression into something that I've been told is very complicated and no one likes. But I was only told that by one person, so the jury's still out. But we need to talk about labels of Cronut diagrams. So I'll try to say this as simply as possible. So I started this with Dominic Searle. We defined the proper labeling for Cronut diagrams of composition states for the key polynomial case, the damage or care. And then I generalize this much more broadly to semi-proper, but for any diagram actually, not just Cronut diagrams or Roth diagrams. So what we do is we start with a super standard labeling. So this puts an R in row R. So this is the super standard thing. And now what you want to do is you want to think about doing Cronut's move to this label diagram and just following the label to just go. So if you push the six down, it's still a six. So here's what I mean. Take these three diagrams. Which of these, if any, is a Cronut diagram for this? That's sort of the question that I was interested in answering at the time when I first came up with labeling. So which ones are Cronut diagrams? Well, here's one way that you can try to answer this question. You can try to label it. Say, OK, well, this one had to have been the six. But I had to push that six in that column down because that's all that that could be. And I know this one's a three and this one's a two, and they didn't move because they're in the same position. So this one here has to be a three because there's only a three in that column. So where's the other three? Well, it could be here. Maybe I push the three together. And maybe this is a five and maybe this is a six. And then you just sort of check. It's like, well, could I do that? Sure, I push the three, I push the three, I push the six down to here, and then I push the five down to here. That's those are Cronut moves. So that guy's good. That guy's a Cronut diagram. I'll try to do the same thing over here with this one. I'll say, OK, well, this one's definitely a six. This one's definitely a three and a two. This one's 100% a three. Now, what's the next three? Here's the important point. It's not this. That's not a three. Because if you look over here, Cronut's rule says the only thing you're not allowed to do is push something that's not at the end of its row. You have to push this three first. You can't push this three first. So I can't get this three below that one. That's the crux of the idea, is that this three can't go below that one. OK, so where's the three? Well, it's here. OK, so what's this one? Well, maybe this is a six. Maybe this is a five. Can I get it? Like, just check. Is that something I can get? Well, I could push the six down to here. And then once I push the six down there, I just jump, jump, and then go, and then push that one down. I can get it. That's the Cronut diagram. OK, the last one. Let's analyze the last one. OK, same game. You're very good at this now. Three, two, this is a three. And now I am stuck. So this cannot be a three because you can't get that three below the other one. This cannot be a three. This one can't be a three either. Why not? Because Cronut pushes things down and not up. This is above row three. The three didn't just get helium. It goes down. This is not a three either. There is no way that I can label this diagram with respect to this guy that I started with. So this is not a Cronut diagram. And that's the point of labeling. It tells me when something is or is not a Cronut diagram. So this one's not. Yeah. Could you have made an example here where there would be more than one way to label one of the examples? So you want something like this where there are three ways to label it? Oh, OK. I guess I should just. It's a great question. It's a great question. And so this is this is maybe the big takeaway from the the transaction paper from 2022 that I have, which is that there could be lots of ways to label it there could be really good, pretty good and bad ways to label a diagram. So this is a Cronut diagram. The proper labeling is nice because there is an explicit algorithm that you do from right to left to label the bubbles. OK. And it's it's it's well-defined and it's prescriptive. There's no choice involved. You're always taking the minimum things. I put the five here, not here because I take the the smallest number I can at every step. And if I'm ever stuck and can't take something, then the labeling algorithm fail. It cannot be properly labeled, which means that it's not a Cronut diagram. So the proper labeling is nice because it gives you a test. Try to properly label it by this algorithm. And if you can do it, it's a Cronut diagram. And if you can't, then it's not. Semi-proper labeling turns out to be really convenient when you're messing with the labels, when you're doing things like, oh, let's do a crystal move and see if it preserved the fact that I'm in the set of Cronut diagrams. It did. How? Why? Because it's semi-proper. The semi-proper labeling are nice in that regard. They give us other ways than just this canonical labeling. And then there's just garbage labeling like this. And again, it's garbage. Why? Because of the threes. Those threes are the problem. The idea is that you kind of descend. But here you notice that the sixes don't descend. That's OK, because this six is really trying to be a three. And there's some sense in which we can make that precise. I'm not going to make it precise in this talk. But this six really wants to be a three because hanging out with the threes. So it doesn't mind that this six is below it. But these two threes are stuck together. They can't be out of order. The left one has to be above. OK, so there's some notion of labeling. You can label a diagram in lots of different ways. I'm interested in all diagrams from here on out will be properly labeled. And again, I'm not telling you what the algorithm is. In this talk, we only have so much time. And you only have so much brain capacity. I'd rather tell you what you're here for, which is the Shuber thing. So I can now define the algorithm. Let me tell you how to land. And I'll tell you how to bump. So I'm going to look at cover relations. Because I kind of know when to land. I land when I'm at a cover relation. So this is my diagram D of U written three times. I have given you the super standard labeling of it. I put a R in row R. I have highlighted the dots for cover relations. These are all the cover relations that I have. I'm sorry, in this example, I'm doing k equals 4. And the dots are all in column 7. The dots are all in column 7, k is 4. And I want to kind of understand these. And I'm going to understand them from the rule. So what does the rule say? Well, when I go to do the cover relation, these are the diagrams that I get labeled now. So the box is in the right spot, according to my rule of the lattice permutation tableau. But now I've given you the super standard labeling of these guys. So you can ask yourself, well, what happened? Well, what happened here? What happened was that 6 became a 2 down here. What happened was these 6s became 3s. Oh, sorry, these 6s became 3s. These 6s became 4s. OK, so let's do that. We've just done a conor move. That's important. And again, that's like how I came up with this rule, because I was playing with conor's rule. We did conor move, and we got those down there. So now our diagrams are matching up a little bit. So these are conor diagrams. Great. Now what do we do? Well, now we need to do this box, but I don't like where that box is. That box is not in the column that I record with the decoration. So Bear's Run and Citeal told me to decorate it with this column. So I want to do that. I'm going to add that column there. I'm going to add the box there. Then these diagrams kind of look a lot like these diagrams, but they're not quite right, because I haven't rectified yet. Fine. There is a bijection. It's not a hard bijection, and you can probably figure it out just by looking at these pictures quite honestly. It gives me a bijection. So these sets here are in bijection. This set here maps to this one, but I have to subtract all the ones. So if I wanted to be a bijection, I subtract the ones that I get from 7, 2. And then I get a bijection. So I subtract all the ones that are over here that live in here, because you'll notice that this diagram here is actually a conor diagram of this. OK. So that means everything that's a conor diagram for this middle one is actually also a conor diagram for this one. So I have to subtract that overlap, and then I get a bijection down here. So this union is not disjoint. That's OK, though, because we have a bijection. And we know how to describe it. You take the smallest r that you could. I'm thinking of the 2 and the 3 and the 4 as r. You take the smallest one you could, and that's what you map to. So these are the diagrams at which I'm going to land. This is my notion of landing. I land basically if I'm the box. If I'm in any one of these diagrams and I am the box, that's what this label of x equals r mean, right? Because here, the 6 and the box become 2. And then over here, the 6's and the box become 3, and so on. So I want the label of x to be whatever the box should have been, and that's how I know to land. Let's do some examples. So here, here's an x. It's labeled, and I put in the dot. I guess I didn't do all of them, but just so you could see what's going on. I'm trying to land here. This is a cover relation. The 4 or 5 is a cover relation, OK? So that's fair game. And I want to ask myself, can x land? Well, if x lands, I have to be able to call x a 4 once I do the switch. But really when I'm up here in one of these diagrams. Well, I can, because these would be 4s. I push the 5 down. That's semi-proper. The 5 got pushed over top of the 4s. That's fine. And the 4 got pushed down here. This is a semi-proper labeling. The box is labeled 4. This is a landing spot. If I got my x to here, stop. You're done. Put a box on it. You're done. And another example, what if x was here? So notice that we get the same diagram here, but I've labeled it differently. So to Anders' point, you can label it differently. And it's important that you can. I have to allow for this. This is actually the proper labeling over here, but here x can be labeled by 4 so I can land. And a non-example, this x does not land. Why does it not land? Because if I try to move to the semi-proper diagram, I can't do it because this cannot be a 4. Can't get the 4s up high enough for this to be a 4 because it's in row 5. I have a flagged condition, right? Ponert moves things down. So that can't be a 4. So I can't do the same trick. This is not there. I can try to properly, I can do the proper labeling algorithm and I'm not even in this set. And that's why I can't do it. And so therein lies the beauty of the labels. They tell you that you're in the right set to land and they tell you based on your label whether or not you can land. So now we know about landing. We're looking at these diagrams, not these, but we just map with a simple bijection. So you do your insertion, you land when you get here and then you map by a simple bijection to go here before you move on. So that's how we stop. That's our stop condition, land. Now I have to tell you how to proceed. You proceed with bump. So how do you bump? Well, there are two things you can bump. You can bump anything small. Definition of small is your labels at most K. And also, by the way, you're gonna switch X and Y. So if X is bumping Y, you wanna switch them. X gets the label that Y had. So that means that if you're switching X with maybe a 12, then X cannot be above row 12 because X is gonna become a 12. So it can't be above row 12. So you can bump anything with small label, but not when you're too high up. Be careful of that. You can also bump. This is really the crux of it. There's this weird bumping condition. You can also bump something that would have a small label if you were in one of these sort of trying to land diagrams. Let me give you an example and I'll really focus on this one because that's the more interesting case. So here K is three and here's my diagram. And here's my X and I wanna know, can X bump four? Well, four is bigger than K. So not by this rule. But I wanna ask myself, if I were somewhere out to the right of this, right? Maybe I was trying to do, let's see, a landing with four and three. Okay, so the four dots like maybe here and the three dot is here. So that's a cover relation. So I could try to land with four, three. And if I did that, this four would become a three. So because of that, because it would become a three, that's X is free to bump the four because the four really wants to be a three, okay? And you don't have to believe my it really wants to be argument. This is exactly what you need to make this thing objective. It's the right answer. It's the only right answer. Okay, so we can bump that four. Here's a non-example. Can I bump this four? No, this four still wants to be a three. But once you put the four up here, it can't be a three anymore, okay? If you put it up here, it's not a three because it won't have that property anymore that it forms a thread down with a three. So it won't be a three anymore so you are not allowed to bump that one. This is a non-example, can't bump that one. So what's the algorithm? Well, you start by inserting X way to the right of your diagram and land as soon as you can. If you can land, land. That's the first thing you wanna check. Can I land? If so, yes, you're done, great, good job. If you cannot land, then look for something you can bump. I say here to bump the lowest because otherwise you'll just be doing more bumps but you don't have to bump the lowest, I guess. It'll end up that way. So save yourself a little time and take the lowest thing but bump whatever you like. Bump in your column. If you can bump, bump. If you cannot land and you cannot bump, then go left. And that's that rectification idea. So if you cannot land and you cannot bump, you go left. Let's turn these into actual examples. Okay, so here's the diagram that I begin with and here's my X way out to the right of my diagram. And X comes in, there's nothing to do. It can't land, it can't bump until it gets above the four and then it has the potential. And in fact, it does bump here. And so we'll call that a bump. That's a case two. And we bumped it and now what? Well, now if you look at this, the dot up here is a six and the dot down here is a two and a six two is a cover and it can land. So we're done. That's the insertion in that case. What about this bottom guy? I started with a similar diagram. I started with the X out here, the X gets above the four and it cannot bump the four. So it just keeps fumbling along and it ends up getting here above the three. And the X can happily bump a three and we're only doing it in row three. We can't bump the two and we couldn't bump the two when we were here because the two is not allowed to go above row two. But the three can happily go to row three to bump the three. Now you can bump the two. Now bump the two. Now you can land, you can do a one five landing. So that's where the bullets are. You can do one five landing to land and you're done. You get a buck. That's the algorithm. Now we'll say, so, let's see. It took mental and me 32 pages currently, although we're still adding figures and examples. It's currently 32 pages. You state this precisely and to prove that it is well-defined, just well-defined. So you might wonder, why don't we just use bears on Billy insertion on Pipedream? It takes them a paragraph to state their rule and it takes them about another paragraph to prove everything about it, not just that it's well-defined. Okay, and that's fair. The reason is theirs doesn't work. It just doesn't work on Pipedream. You're not doing the bumping in the right way and the Pipedreams don't work. And there may be a million ways to do Monk's rule on Pipedream. And I feel like Nantel and I found them all quite frankly, as we were trying to translate this algorithm that I had into Pipedream and we could not do it. And the crux of it comes down to the fact that the labels don't translate across nicely to the pipe. Even through that bijection that I gave you, we couldn't get the label to work with the pipe. It just became a contrived mech. So Pipedreams don't satisfy the row bumping lemma, Konert diagrams do. Well, let me do an example. I'm gonna take, let's see, we'll take X. Let's do X and row two. So here's my X, okay? I'm gonna start with X and row two. And here I've written the Roth diagram just so you can kind of look for those potential landing spots that we have. And you can move over and you can ask yourself, okay, so X is in row two. What's it gonna do? So X is gonna move over. So X moves over and gets on top of it. So it can't land here because it would be landing on a three and the threes are too high. So it moves over to here and can it bump the six? Well, it can't bump the six because the sixes could maybe become threes, but then there's another three that's too low. So it can't just switch with that. So it's gonna keep trundling. This is not a landing column because the dots below K. So it's not a place we could land. So it gets to here. It gets to here and it says, oh, finally I can bump the three. So it's gonna bump the three and then it's gonna land. It's gonna bump the three and then it's doing a landing right here with these guys. And so that's what we get. Oh, wait, hang on. Sorry, it takes a higher one. It lands with a four. I have the five, right? It lands with four. So it lands here and then I can properly label the diagram here. Things shifted over because of this shift that we get when we do this and we relabeled things in here. And so then let me insert another two. I insert a two again. I'm in this case. So my landing column will be to the right. Of course it'll be to the right because I'm not gonna get past this thread of threes. Now when I get to the six, I can bump it and I do bump it and I land. Now let me insert, say in row one. And again, in row one, I'm never getting past those threes. So there, I have another three. And let me insert higher. Let me insert in row four, okay? When I insert in row four, it's really interesting because I'm way too high to bump onto anything. And then I end up coming past the six and getting to this column and then I can bump the four, the three, the two and I land at a one. And you'll see that that's exactly this case of the bumping lemma. And then I keep going and I can keep inserting things. And then I have this nice example. And what have I really done? Well, if you look at the numbers that I inserted, what do I insert? I inserted a two, a two, a one, a four, a three, a two and a four. Let's do RSK real quick with that. So if I do a two and I record it, oh, sorry. Yeah, we'll record it this way. I'll record it with a four and then a two, I record it with a four and then a one, I record it with a four. And now I have a descent or an asset so I'm gonna bump to the four, bump the two and I record it with a three. The three, bumps the two and I record it with a three. Let's see, I'm at the two, the two bumps the one and I record it with a three. The four, another descent is gonna bump three, bumps the two. So there's gonna be two, that's a three, that's a four and I record it here with a two. What I've really done is this is a tableau and I've just inserted this tableau into that diagram because this is a tableau that has when I do reverse RSK with a super standard guy then this is the word that I get. So that I've just multiplied by a tableau. The recording tableau will always be a lattice permutation tableau cause it inherits the latticeness from this and I'm running short on time so I won't go through this one too much but here I have this as my recording tableau. Same insertion diagram but let's switch the recording tableau. This two is the last thing to be inserted. It's the first thing to be removed. I'm gonna be uninserting from this column. The, I pop out a four, I can uninsert from that three and I'm gonna pop out a two. I'm gonna uninsert from this three and I pop out a four and then a four and then a one and then a two and then a two which gives me another example of the insertion. Here I got a different recording tableau with a different insertion and a different tableau that I multiply here. So that's it, that's our theorem and thank you very much for your time and attention. I'll leave you with a few references. If you actually wanna work with this formula at all, Franken and Tellpaper is beautiful and it tells you everything about like the K-Bruhaw order that you wanna know in terms of like if you just take U and W, you can tell right away they have a static test for whether they're related in K-Bruhaw order and how to get the change efficiently. My conjecture is on the archive and yeah, I had even in Tella thought we'd have the paper done by now but we just keep adding examples because it's complicated combinator, but I think it's nice. Anyway, thank you very much for your attention. Thanks very much for a very nice, very colorful talk. Let's thank the speaker. Thank you.