 Okay, so welcome back to the second part of the Schubert's seminar with Colleen Robichaux on the CM regularity in casualistic varieties. So now we can talk about some combinatorial formulas. And the first one and the main one that we'll be talking about today is for the case when we have a vexillary permutation. So these vexillary permutations are those permutations that avoid the pattern two, one, four, three. And one reason that we set out looking at the matrix Schubert varieties indexed by these vexillary permutations is they're particularly nice. So even noted in Fulton's paper in which he introduced these matrix Schubert varieties, they're very well behaved and there are many nice formulas, combinatorial formulas for the vexillary growth in the polynomials that can be helpful in computing these degrees or understanding them better. And I'm just going to explain our formula through this example here and it'll be clear what each of these terms is doing. And so for this permutation V, again, I have it in one line notation. And for our formula, we start out with, again, just drawing the ratha diagram. And inside of each ratha diagram box, I'm just going to store the corresponding entry of the rank matrix of that permutation. Or again, you can just think of it as storing in each box the number of bullets that are northeast of each of that box, okay. And so that's how we get numbers in boxes. And then what I want to do is I want to build a tableau out of this diagram. And so I do that by just shifting all of my these blue boxes northwest along these main diagonals to get this tableau here. And what I do from here is the degree of my growth in the polynomial is the coxeter length of our permutation, which I mentioned before is just the number of boxes in the ratha diagram, which is in this case 12, plus some extra stuff. And how we compute this stuff is by using an algorithm that's going to run over your integers in the interval from one up until, but luckily it's a very brief algorithm. So for the I equals one step, what we're going to do is focus on the subset of this tableau that has entries greater than or equal to one. So that's this subset of our tableau. And in this subset, I just want to find the longest anti-diagonal inside of this subset. And so that's these three boxes here. And so because there are three boxes in this longest anti-diagonal, that's going to contribute a three to my total computation. And that finishes up the I equals one step and I can move to I equals two. For I equals two, again, I look at the subset of my tableau that's greater than or equal to two. And I identify the longest anti-diagonal that's sitting inside of this sheet. And so that's just, again, this one box. So that's an anti-diagonal of length one, so that contributes this one. And if I do this for I equals three or beyond, you can see there's no entries in my tableau that's bigger than two. So any future applications of this algorithm aren't going to do anything. So in fact, we're finished. And this tells us that the degree of this growth need polynomial should be this coxsider length plus the sum of those longest anti-diagonals, which gives us a total of 16. So our algorithm just computes the coxsider length, then from this tableau computes these longest anti-diagonals and sums them up and that will give you the degree. Okay. Now I should also mention. We have a similar formula in our paper for permutations that avoid the pattern 1432. But shortly before we posted our paper, Pacenic, Spira, and Wygant actually posted a result that holds for general matrix sugar varieties, or general growth in the polynomial. So to compute the degrees of any growth in the polynomial that you could possibly want. Their result uses different techniques and is a very fun, fun formula. But so if you have any questions about it, I encourage you to ask Anna about it. But today we'll just be talking about these, these vexillary formulas, let's see. So for a bit of context into how we came up with these formulas, we were using just to understand better about what was going on. We were using the formula for the vexillary growth in deep polynomial that was given by Knudsen-Miller-Young in their paper, where they showed that these vexillary growth in deep polynomials can be computed using flagged set value tableau. And by sort of re, looking at these tableau, not as tableau, but by these other combinatorial objects called k-excited Young diagrams, that's how we were able to obtain our conjecture for what these formulas should be. But I won't go into such detail about these excited, these excited Young diagrams unless someone has a question at the end. So all in all, applying our formula for the degree of this vexillary growth in deep polynomial and connecting it back to our proposition that told us that the regularity of these matrix Schubert varieties is going to be that degree minus the coccister length. In our formula for these degrees, we already had the coccister length as a sum end. So we know, connecting back, that the regularity of these matrix Schubert varieties is exactly being computed by summing up these anti-diagnose in our tableau. And in our example that we were looking at before, this tells us that the regularity in this case is going to be four. So summing up that first diagonal of anti-diagonal of length three and the second one of one to give us a four without too much trouble. Now we've been talking about matrix Schubert varieties for some time now and now I want to sort of shift back into talking more generally about cost and leased dig varieties. And first we'll talk about the very nice setting of cost and leased dig varieties that are indexed by Grismani and permutations. So those are the permutations so that whenever you write them down in one line notation, there's a unique descent. So a unique point when you're reading across where there's a decrease in the entries. And one particularly nice fact about these Grismani and permutations is once you fix the symmetric group that they're living in, you can uniquely associate them with an integer partition. And so we'll be using that for our formulas. And in fact, for these Grismani and cost and leased dig varieties, these are actually isomorphic to vexillary matrix Schubert varieties. So this tells us that we're able to, again, use our formula from before for the vexillary matrix Schubert variety setting and then compute the regularities for these Grismani and cost and leased digs. Of course, these formulas aren't going to be very helpful if it's very difficult to map this pair of Grismani and permutations to our vexillary permutation. But as it happens, this association is pretty simple. And I'm just going to outline it here. So here we have our Grismani and permutations where I have just written down the integer partitions that they're in bijection with. And what we do is we just draw the young diagrams for each of these partitions where due to the condition on a Bruja order on this pair of permutations, we know that one young diagram is going to sit inside of the other. And what we do is in our diagram, we look at those shaded boxes here of the smaller partition. And I do all of these local moves. So I shift the shaded boxes south east along diagonals as much as I can. And what this will do is once we do this as much as possible with all of these shaded boxes still living inside of this game board defined by the larger permutation, this is going to allow us to precisely read off the Roth diagram for the vexillary permutation that will correspond with this pair. So this tells us that this Grismani and pair is going to act but precisely map to the vexillary permutation we had in our example previously. So the regularity of the Cos-dom-Lucig that's indexed by this Grismani and pair is again going to be 4. And now we can get into a couple of applications to this fact that we can just use our vexillary formula to compute the regularities for these Grismani and Cos-dom-Lucig. And the first application that we can find is looking at a conjecture that was posed by community luxury by Sastry Shashadri in a 2015 paper of theirs. And so what they were doing is they are taking these varieties YW that they construct by intersecting a Grismani and Schubert variety with the opposite big cell in the Grismani. And they look at the fiber of the restriction of this natural projection along this space of matrices here. And in their 2015 paper they were studying free resolutions of these varieties in terms of certainly comological computations. And at the end of the paper they conjecture the regularities of these varieties. And in their conjecture they conjecture for very particular conditions on the Grismani and permutation that's indexing these permutations which I can get into detail about if anyone is interested. There are just restrictions on the partition that's associated you can think. And they conjecture that the regularity of these coordinate rings should be a weighted sum of the differences between the parts of the partition that's indexing the Grismani and permutation that's indexing this variety. But on its face these varieties that they were working with are actually just custom-lustig varieties. So by taking this to be your other permutation that this bigger permutation in your pair the varieties that they were looking at are actually isomorphic to the custom-lustig indexed by this special really big permutation and the Grismani and permutation that they were originally working with. But we actually know something more. In fact these Grismani and custom-lustig varieties where the other permutation the larger permutation that you're indexing by is so big this is actually just isomorphic to a Grismani and matrix Schubert variety. So again we're able to apply our formula to see compare it with this conjecture. And so in the case which we have this Grismani and permutation I can draw out it's Rotha diagram and fill each box with the rank and compute anti-diagnose. So for the I equals one step that would give me this anti-diagonal here of length three and then I have a anti-diagonal of length one for the I equals two step and another anti-diagonal of length one for the I equals three step to give me a regularity of five according to our previous theorem that we've discussed before in the case of vexillary permutations which the set of vexillary permutations naturally contains the set of Grismani and permutations. So we can compare this computation to their conjecture so for theirs the partition that's associated to this Grismani and permutation is three two two zero and so we can compute this weighted sum. And in this case it would give us seven so this tells us that in general this this conjecture might not hold but in comparing the structure of the conjecture it will actually often give us an upper bound for the regularity of these Grismani and matrix sugar varieties. There are cases in which it will not be an upper bound but often it will be. Another application is to compute the regularities of one sided mixed letter determinants and so for those we start with a matrix of indeterminants where the indeterminants are all in a sort of young diagram shape here that's I showed it here with the L region and what we do to give us our generators that to build our ideal that will cut out this variety is on the sort of southwest border of these indeterminants I can mark points these RIs where these RIs are going to be some non-negative integers and those marked points dictate that I want to take the RI sized Northwest miners from that corner. So in this setting if I set this R I equal to three that would contribute all of the three miners of this shaded region into my generators for this defining ideal and if you're very familiar with one sided mixed letter determinant ideals it should be no surprise that they're isomorphic to vexillary matrix sugar varieties and actually as they were introduced by Avyan Karan the 80s to in fact study singularities of grass money and nature for varieties that should again not be a surprise that these are isomorphic to grass money and cost on lucid varieties. So we can again apply our theorem for the vexillary setting to compute the regularities of all one sided mixed letter determinants varieties in the same way in ongoing work I'm looking at computing regularities for two sided mixed letter determinant ideals and so what these how these varieties are defined is instead of having a young diagram a straight shape of indeterminants xijs when if I have a skew shape of indeterminants so crushing in this northwest corner and define my miners in the same way to give me my generators of this defining ideal in this setting these are again going to be caused on lucid varieties where now your permutations that are indexing this variety are going to be three two one avoiding and these varieties are still going to be homogeneous which is great news and what's interesting is in the previous examples that I've talked about we found that they were secretly isomorphic to certain matrix super varieties but there are many cases of these uh these two sided mixed letter determinant varieties that are not isomorphic to matrix super varieties so these are settings in which one cannot appeal for example to the patin expire why get formula for example to compute the regularities and so we have to use some different different tools so presently I'm working at giving a construction to compute the regularities in this setting so for these three two one three two one avoiding caused on lucid varieties I should mention that cotton holler and go potty have a formula that gives constructions for the regularities or a related invariant to regularities this a invariant for particular two-sided mixed ladder determinant varieties but they there are certain conditions on where you can place the marked points in their setting and to do this I'm going to use when looking at the k polynomials that have been identified for these caused on lucid varieties and these are the unspecialized growth in big polynomials introduced by wooing young and they're defined in a very similar way to ordinary growth in big polynomials except is this additional indexing permutation v is going to restrict uh which of the pipe dreams uh that will actually contribute monomials to the ending polynomial that we're building and so what they say is that we start with again the ratha diagram for w and v as we have before and for the starting board I'll start by looking at v and shade all of the ratha diagram boxes for v and left a line those and that is going to stay static throughout all of the combinatorial diagrams that will build on top of this sort of game board that's defining some shaded region and then we do just the normal the normal pipe dreams all of the normal pipe dreams of w on top of this game board that's been defined by v and what their formula says is the only pipe dreams that are going to contribute terms to this unspecialized growth in big polynomial are going to be those pipe dreams in which all of the pluses are living inside of the shaded region that's defined by by v so in this case uh because this plus is occurring outside of the region and this plus is occurring outside of the region they don't contribute which would tell us that in this case the unspecialized growth in polynomial would just be an x1 so much smaller but I'll just I'll just stop there all right thank you very much