Shiliang Gao, p1, `Tilted Richardson Varieties'

Okay. So, uh welcome everyone to the Schubert seminar. Uh before before we start the seminar, um so there's going to be one more uh one more talk uh for this year's edition uh which is going to be in two weeks. Jennifer Moors will be the last talk. But before that last talk today, we're very happy to have Shilang Gao from Cornell telling us about tilted riches and varieties. So please take it away Shirang. Right. Yeah. Uh thank you Leonardo for the introduction and invitation. Uh it's a pleasure uh to speak at the Schubert seminar. Uh so yeah today I'm going to talk about some joint work with uh Gian Ibok. Uh so the work really started back in like summer of 2022 when we were all at OPAC. Um and um uh I think it was back in 2000 uh 2023 like maybe in September we posted a version of our paper. Uh but that's um more or less like a third of the material that's uh in our uh newest version. Uh it's the archive uh 2602. Um yeah and um uh maybe some of you have seen uh you know one of us giving various version of this talk and um uh I have actually uh gave uh you know a abbreviated version of this talk at a sher at a previous Schubert seminar. Um but yeah so I hope u uh I hope uh we'll all uh uh hear about something new today. Um so yeah. Okay, wait. Um, yeah. Okay, there you go. Uh, yeah. So, uh, the talk will be basically three parts. Uh, first part I'll talk about quantum bha graphs. Uh, talk about, you know, some, uh, basic setting, uh, you know, looking at quantum call module of flag varieties and we'll define what uh, a tilted richerson variety is. Uh this will be a family of sub varieties uh in the flat variety that has u many uh amazing properties and we'll talk about uh uh this uh generalization of uh deodar decomposition that we came up with and along with uh some of uh its applications. Okay. So yeah, so we start uh with the uh complete flag variety uh uh that um this is the uh general linear group uh model on the right by subgroup that is the uh invertible upper triangular matrices. Uh so we can think of the uh think of a point on the complete flight variety as a chain of uh uh nested vector subspaces of c to the n. And we can uh think of it more concretely as the column span of the first I columns in invertable invertible matrix. Okay. So the uh comi ring is uh is generated by the sher classes uh that is the coji class of the sher varieties. uh it's a freez module and I know there are you know many conventions in terms of uh you know it's a sugar variety is opposite sugar variety to look at B or closures BM B minus or closures um that's not a big issue uh in today's talk uh uh and we won't you know get into uh uh that sort of details uh so if we take the product of uh two uh sher classes uh we and expanded uh back uh into uh linear combination uh or really uh non- negative uh integer combination of Schubert classes where the uh coefficients uh are known as the Schubert structure constant and the big open problem uh sher caucus problem is to find a com interpretation for uh all these coefficients. Okay. So uh moving on to the uh quantum coy ring. Uh so quantum coy ring is the tensor product of our coy ring with a uh polomial ring and it is uh now a free zq module uh again generated by our sugar classes. Now in the quantum call mod we uh have a product uh which will uh use the star uh which will denote by the star and uh it can again be uh uh the product can be expanded uh back into uh well now zq combination of um of sugar classes uh where the coefficient the uh c uvt uh will now uh be the uh chrome of weight right this is again going to be a non- negative integer and q to d is just going to be a monomial be the monomial q1 to d1 take the product all the way to q n minus one to dn minus one so d here will be a vector of uh dimension n minus one so for example uh if you take n equals 3 uh and let's say we take the product of uh sher class 213 with the sher class 231. Uh now in uh if we're uh dealing with uh the com ring you know this won't get us anything. Uh but in quantum coing uh we get uh this expansion q1 uh timeclass 231 plus q1 q2 * the tuper class 1 2 3. Now one of the uh uh one of the motivating question for us uh really for uh for our project uh is that you know what are the weights uh Q2D that appears uh in the quantum product and also you know what is the minimal of such uh Q2D and here by uh minimality uh we're thinking about uh you know Q to D1 Q to D less than equals to Q to D prime if Q to the D divides Q to the D prime or you know if we compare the two N minus one dimensional vectors uh um you know coordinate it's a partial order really and um um there are many uh uh uh answers uh to the to the uh to this minimal u degree uh question uh Hon Woodward uh expresses in terms of uh certain chains uh uh in the Bhawk graph uh pausing described this as um this minimal rate weight appearing in quantum quantum brohaga graph which we'll cover later and uh book Chong Le and Mihala uh in 2020 uh give a description uh in terms of u uh you know projecting onto various correspondence and we'll give a you know a cute little answer uh to this question for ourselves uh which actually motivates um the definition of Toby Richardson's and everything everything after that. Okay. So uh quantum bha graphs uh so the uh quantum bhagraph is a uh weighted directed graph uh whose vertices are permutations as and uh it has two uh types of edges. Uh so we have the uh strong blueha water edges uh double going to uh one of its strong blueha cover uh and we label uh these u edges with uh weight one and there's another uh special type of edge uh it's quantum edges uh where if we multiply on the right uh by tig and the lens go down exactly by the length of tig Okay. Then we uh give it the weight uh the product of qi all the way through q j minus one and it's a theorem by poss that uh there's a unique minimal q2d that appears in uh any uh quantum product and uh and such minimal q tod is the weight of any uh shortest path from u to v uh in the quantum prograph. And by shortest I just mean shortest. So the minimum weight pass is actually the same as the shortest path. Let me do an example. Right? This is a kind of complicated uh definition. Uh let's look at the quantum bhagraph uh on uh uh on S3. Uh so all six permutations and we have these uh black edges going up. These are the edges from our strong hot water. And we also have these blue edges going down with uh different Q weights uh that are our uh quantum edges. Now if we uh let's say if we look at uh u being uh 231 and v being 1 2 3 um then there are uh so the length of the shortest path from 2 3 1 2 3 is two and there are two ways to get there. uh we can either first go down to 213 and then go down to one two three or we can first go up to 321 and then go all the way down to one two3 and notice that uh the weight of uh these two path um first one's you know q2 * q1 second is 1 * q1 q2 are the same and that is in fact the uh minimum quantum degree that appears in the uh in the context. >> Okay. Uh >> that's just minimal number of steps. >> Yeah, minimal number of steps. Exactly. >> Nice. >> Okay. So, uh I promised a uh a formula uh you know yet another formula uh for uh for this uh minimal weight. Um now the the lot of words let me just uh do an example right. Uh so we know this uh minimum degree will be represented by some vector d1 through dn minus one and I'll tell you how to compute dk. Um so for example let's say u is the permutation 4 637 521 with permutation 5312 467 and let's say k equals 4. So we want to figure out d4. Now uh we'll construct a uh a lattice pass or uh kind of like thick pass uh where we'll move up uh well move up and to the right if the step is in uh the first four number uh in one line notation of view. uh three, four, six, and seven. And we'll move uh to the right and uh and down if uh it's in the first four uh number in one line notation of v. So in this case 1, two, three and five and notice that three is in both of them. So we'll just uh move towards the right and not do anything. Okay. So uh we can construct this path. Uh uh well we see one is in VK uh is in V4 uh but not in U4 so we'll go down. Two is in V not in U go down again. Three in both of them we move to the right. Four is in U4 um but not in V. We go up five we go down and then six seven we go up. uh and the depth of uh this path is just you know the uh basically the absolute value of the y-coordinate uh of the lowest point uh in this path and what we proved is that dk or in this case d4 uh is two okay and uh we can uh compute this for uh you know for k going from one through n minus And uh we can figure out uh all the decays uh all the minimum uh degrees uh uh all the components of the minimum degree. All right. Um yeah. So um so there's a uh there's a uh quantum analog of uh strong bhart uh of strong bhide intervals uh called the uh tilted intervals. uh this is uh a notion uh defined by Frankie for Posinkov back in uh 99 where uh the uh tilted height interval is the partial order u well the uh it's a partial order on the set um you know if we fix u and v and we look at all the possible permutations that's that uh that is on a shortest path from u to v okay so that's uh our ground set and we'll say w is less than or equals to w prime. So the partial order if uh w appear before w prime uh on the shortest path. Okay. So in particular you know our shortest path start with u. So u is really the smallest uh uh element in the partial order. Uh v is the last one uh that appear. So b is really the the largest one. And for example we earlier we looked at uh uh these uh shortest path from 2 31 to 1 2 3. Uh remember uh there's one going first to 23 and then 1 2 3 and another one going first to 321 and then 1 2 3. So uh the interval uh 2 3 1 2 1 2 3 is a rect two interval uh where uh that contain that also contain 213 and 321 >> that that name tilted bruhair interval is that your is that your name? >> Uh no this is uh introduced by branding passing. >> Okay. >> Yeah. >> Oh thanks. >> Yeah. And so you know a lot of the uh the naming uh in our paper is just uh throwing a tilted uh like uh like they did like they did. Okay. So um oh right and um so in the case where uh u is uh less than or equals to v in the strong bha order then the tilted bha uh interval is just the uh strong bha order uh strong bha interval they'll be the same thing and um every shortest path from you to me in that case will all be uh these uh strong bruha order edges Okay. So, uh in the case of strong bruha order uh we have the uh criterion which uh can tell us you know uh can tell about well characterize the strong bro. Uh now we have a uh well a generalization of strong brha order and we want to give the same thing. Uh so uh so let a be an integer from 1 through n. Uh we define this total ordering of one through n where we set a is the smallest and then uh a minus one is the largest and we just uh go around we make one uh larger than n is necessary. And if uh we have two uh k element sets of 1 through n uh i and j, we can uh first order uh each set uh using this total order of n. And then we can uh impose a partial ordering on uh these k sets where we say i is less than equals to j under this uh uh shifted order. If uh well I am is uh less than or equals to jm under the uh shifted order a for all m from uh one to k. So if a is one this is just the uh uh just the uh usual uh partial ordering of ket that we're familiar with. Okay. Um uh yeah and in other words uh really for any given a uh uh this is just a um a permitted copy of uh of young uh so yeah so uh proposition um so if you have me any two permutation u and b I can always find a sequence a uh that is a sequence of n minus one integers um from one to n uh So that uk so the first k numbers in one line notation is less than or equals to vk under this shifted order ak and moreover uh given any such a so any a such that u is less than or equals to b. um for uh and we can we can tell if a permutation w is in the uh two tip or hot water by comparing uh uh comparing it under this uh new order with uh U and B. If it's in between then it's in the typical interval. If it's not it's not. And one may ask you know how do we uh choose this uh ak uh well uh these are precisely the indices uh of these uh lowest points uh in the uh in this latice pass that we constructed. Uh so here uh in the case uh us uv are these uh two permutations in S7 we can have a four to be either three or four or six. to any one of them and we can uh yeah and for each k we can pick a k uh using uh using that and um we have uh a lot of um u a's that we can uh we can choose from uh so just another example uh if you take u to be the permutation 2 31 v123 as we've seen earlier uh we can set a to be the sequence 2 And we can check that uh you know two is less than or equals to oh shoot uh uh that should be a one. Um yeah so uh uh 2 is less than or equal to one under this shifted order two and 23 is less than or equals to one two uh under this uh shifted order three. Uh and we can also check that 213 lies in between uh uh using this. Okay. Um yeah. So uh we've got a u uh a version of uh um criterion uh for our uh tip for hot water. Uh so the way uh one can uh one way uh that one can think of um criterion is that it gives rise to these uh rank conditions that we can use to define sugar and opposite sugar varieties. And so uh now that we have uh uh this uh tilted analog uh we can define what we called uh tilted riches some varieties and tilted riches themselves. I I'll just uh uh do an example to explain the definition. Uh so let's say we take U to be the permutation 421. Uh this is represented by the stars. V is a permutation 3142. Uh so we're using column spans. So um that's the three one four and two. And in this case uh we uh uh choose our sequence a uh to be 4 to2. Uh now u is less than or equals to b uh let uh under this uh order a. And we represent the a using these uh sort of uh red um lines uh to uh we think of uh these red lines as representing the ceiling um uh in a matrix. Um so uh now we want to define uh a bunch of uh rank conditions. I'll explain uh how that how that goes. Uh so in the case um so for each k we'll define some rank condition on the first uh k columns and here we'll just do the example where k equals two. Um so we start from row two which is uh so uh if we look at the ceiling uh in column two uh this is right above uh uh row two. So we start from row two. Um so we'll require that uh the rank of this green region will be less than equals to the number of stars in the green. Uh so in this case zero and uh we'll require the rank of this 2x two matrix uh to be at most one uh and the rank of this uh uh 3x2 matrix to be at most two and similarly uh so these are the equivalent of uh uh conditions defining uh opposite sugar uh opposite sugar properties or really the uh uh the northwest uh rank conditions and we have the southwest rank conditions. Uh so we have uh the um this 1x two matrix the rank uh here to be less than or equals to the number of uh blue dots which is one. Uh the rank of uh this specific 2x2 uh is most one again and um the rank here is most two. So there are already uh six rank conditions uh for uh each of uh k equals to one two and three. And if we replace all the rank uh at most uh condition with rank uh exactly equals uh that give us the uh definition for the uh tilted riches self to the rich varieties. Um and in particular if uh all the a's uh if u is less than or equals to v uh in strong order and we can take a to be just uh all ones then uh this is exactly giving us uh uh the riches and variety and uh uh riches and cells. Okay. Uh so uh the uh uh uh theorem uh we proved back in 23 is that uh the definition of totes and varieties and uh to riches cells does not depend on the choice of a. So for whatever a such that u is less than or equals to b we get the same exact uh variety. uh so we'll just lose the a in our definition uh in our notation and denote them as uh tuv and tuv and uh yeah so here are some special cases that maybe I've mentioned um yeah so if u is the identity uh then we know v is lar greater than equals to u in stronger hot order and um uh in this case the t riches variety is the super variety And similarly if V is w not this is the opposite for ID and like I mentioned earlier if uh they're comparable in strong order we just get uh riches riches. Okay. Um yeah so uh uh back in 23 we proved uh some uh geometric properties uh of these uh tilted richersons um so we've showed that uh terson variet is close sub variety uh and the uh t richer cells are open uh in uh uh in tuv and a coordinate plaque or a tix Right. Ew lies on the tilted riches variety if and only if uh it is uh adamant in the tilted bhau order. Uh so the so like uh the uh uh strong bhau order case where a permutation is in a strong bhau uh interval if and only if it's on the richerson variety. This is a generalization of that. Uh so we have a formula for a dimension. Uh dimension is exactly the length of the shortest path from u to v uh on the quantum bhack graph. uh like the richerson varieties. Uh we have a uh a decomposition of uh tilted riches varieties into t riches cells uh where the disjoint unions over all uh tilted brha intervals uh that lies inside uh the interval UV. Uh we have uh the closure condition. So uh TV is the closure of uh TV circ and that uh uh it is irreducible. So uh basically uh everything that uh uh you know all these uh geometric properties of uh riches varieties uh we can just directly generalize to uh tilted riches and just a oops uh just a spoiler that uh uh the proof idea is generalizing the dar de composition uh which I'll talk about later in the Okay. Um yeah. Um so uh so in the case of Richard variety the uh the co-omicass of riches and variety uh can be uh uh is actually the product of uh two shar classes. Um so uh we can think of uh you know understanding the decomposition of a richerson class into shubber cotus uh well we can understand the sher cautus problem as um you know understanding the uh expansion of Richardson class into sher classes and now we have uh some sort of you know quantum analog of um or quantum generalization of riches uh can we say something about called logic class. Uh well the answer is yes. Uh but first let's u look at something called curve neighborhoods. Uh this is defined by uh book chapu mihan parent 2013. So uh if we fix two permutations and fix a degree uh the two pointed curve neighborhood is the union of all degree d rational curves passing through super variety x2 and opposition uh opposition variety x2 and super variety xv. Um so if d is zero then uh what we're getting is well uh this will be the union of all points uh uh that's on uh the sher and opposite super variety. So that will just give us uh the richest variety and uh when uh uh a degree you know Q2D appears in uh the quantum product uh the uh coicas of these uh curved neighborhoods uh actually tells us something about uh the uh the product. So if we look at the product and we look at uh the terms that has uh well in the expansion um that's q2d uh uh we look at those terms and we multiply by 1 / c for some um non uh for some positive integer c that is the co class of our uh two pointed curve neighborhood and um so uh uh so the curve neighborhoods You know I defined it as you know union of these curves sounds very mysterious but in a lot of cases we know what these are. uh in the Gmanian the curved neighborhoods are poss uh in commun uh well they are uh you know more generally in commenced retentive varieties and uh in the complete black variety case as I mentioned earlier if the are all zero uh this is just the uh richerson variety and If uh one of the di is one, all the others are zero. Uh this is again a rich variety with you know uh with uh different uh label by two different permutations. Um this is work of lean hot and if uh use the identity then uh we're getting a uh sugar variety and um we know this uh uh the the curve neighborhood is going to be empty unless uh d is at least the uh uh de mean the minimal quantum degree coordin but I mean in general we don't quite know uh which flag lives in a uh two-pointed curve neighborhood. And so uh our theorem will be proved is that if we look at the two-pointed curve neighborhood in the minimal quantum degree this is exactly uh the tilt riches and variety pub and that we prove we've proved that uh it's called class is precisely the uh QD mean um uh part of uh the quantum product uh of the two sugar classes. So there are no um one over C emo. So we can the C is actually just one. All right. I think I'll stop here for uh the break and um All right. Yeah. So uh let's take maybe a a 4 minute break until 507.