Christian Gaetz, p2 Combinatorial invariance for the coefficient of q in Kazhdan-Lusztig polynomials

Yes, welcome to second half of Christian Gates talk. Uh, please take it away. >> All right. Very good. So, I had just stated this theorem and then I was going to start explaining some of what some of the ideas that that go into the proof. Um so this idea uh so we we want to prove that this coefficient is a combinatorial and in fact we will give a combinatorial interpretation for it. So pimo was the first to use uh the definition I'm about to describe. So I'll write EUV for the set of cover relations of the interval U to V. And a subset of those edges is called diamond closed if whenever it contains two adjacent edges from some diamond, it also contains the other two. So bruha order has has lots of intervals of height two. They all look like diamonds like this. And what I'm saying is that if I contain these two in X then I must also contain these other two or if I contained the top two then I would have to contain the bottom two. So by adjacent I mean uh consecutive in the cyclic ordering of these uh of these edges. Uh so Xbar denotes the smallest diamond closed set containing X. So it's the diamond closure of X. And Pimo defined GUV as the smallest size of a diamond generating set for UV meaning a set whose closure whose diamond closure contains every edge. >> Question. >> Yes. So uh you say cyclic order on the edges as if you have a favored planer embedding of the hassa diagram. >> Um I just mean okay maybe that was uh not the right way of saying it. Two edges which share a vertex. If I have those two then I must contain the then I must have the other two of the diamond. Um, >> yeah. >> What what it does not include is if I contain if I have two opposite edges of the diamond that doesn't entail that I have the other two. Um, okay. So, in this interval, the three red edges are actually a diamond generating set for this interval. So, we can play a little game. So I start with these three. Now I can look for example at this diamond here. I have the bottom two edges of it. So I must also have those two. Um what can I do next? Um I have these two and there's this diamond. So I must have the top two. Now I have these two. So I must have the top two of this diamond. And it's not so hard to see that once I have a maximal chain, that is once I have every edge in some maximal chain, then because uh all of the maximal chains in a bruha interval are contained are connected under horizontal diamond flips that I will actually get everything. So once we have found a maximal chain uh we can stop and we know that we've diamond generated or we could play it out in this case. >> So in particular for your diamond if you have the two left edges then you must have the two right edges. >> If you have the two left then you have the two right and vice versa. Yep. >> Okay. So um this is certainly some finite number associated to any interval and okay a key an obvious but important feature of this thing is that it is cominatorily invariant. If I give you isomeorphic intervals this minimum is not so easy to compute but it's obviously an isomeorphism invariant just of the post. Okay. So um we were interested in this what is essentially the in closely related coefficients of the three families of the polinomials. So I will define duv to be the coefficient of the second highest power of q and r in ruv rather this coefficient is always negative. I want uh its absolute value is duv. Now it's possible just from the recurrences that I gave you to deduce uh these relations. So the coefficient of Q in P UV is the number of coatoms of the interval that is the number of elements uh covered by V which are in the interval from U to B minus Duv and the second highest power in the R tilda polinomial is the length of the interval minus DUV. The theorem which implies the combinatoral invariance result is that for any interval in any cox group in fact duv equals g of v. So this is some kind of combinatoral interpretation of these coefficients although it's not exactly counting something right it's this extreal problem of the minimum size of a diamond generating set and the reason that we get cominator variance for all of the coefficients of all of these polomials is that we have these relations and length is obviously cominatorily invariant that's just the height of my co-et number of co-atoms is obviously commonally invariant. So because this result implies that D only depends on the post set. So too only those these coefficients only depend on the post set. Okay, how do we prove this? So I should say this is the exact theorem that proved in finite type ad. So he also proved duv equals gu. Um it is known it was known prior to our work that pimo's proof method works only in finite types ad. So there is some version of the defining recurrences for the R polomial that allows certain special reflections to be used which are not simple reflections in such a way that you kind of always can um you can always be in in this case of the recurrence. So this is the case that takes you to sub intervals of the interval you started with. And so the recurrence kind of confines itself to the interval you started with. And he's able to use that to prove this fact that uh that property which is called the generalized lifting property that was introduced by Suckermanman and Williams that is known not to hold outside of finite ad. So we had some other proof method. So it follows from work of Dyer that duv is less than or equal to GV. uh Dyier had um some vector space some formally defined vector space and he proved it had dimension duv and um it's clear that any diamond generating set is a spanning set of dyier's vector space and so this inequality is clear but in order to prove uh the reverse inequality We need to construct an a diamond generating set of size duv thereby proving the reverse inequality. And it turns out that the construction of this diamond generating set is motivated by the recently discovered cluster structure on the coordinate ranging of the open Richardson. Although we do not actually rely on any results about cluster algebbras because for example the cluster structure is only known in finite type but we are able to just write down this diamond generating set an arbitrary type and show that it does what we need it to do. Okay. So in this example we already looked at uh in fact the example I gave you was a minimal dime range set. So GV equals DUV is three for this interval. So computing the various polomials P polomial should be the number of coatoms which in this case is four minus GV. So this recovers the 1 plus Q that we saw. Our tilda is always monic of degree equal to the length. But the second highest coefficient is what we learn about. This is the length of the interval minus GUV. So this is 3 - 3. So that goes away. And the R polomial um GUV is just uh this coefficient and also the second highest coefficient because the R polinomial has this um symmetric coefficients up to possible negation. Okay. Um next I also want to mention some results of ours related to the Gabri Joseph conjecture. So just recall that the R polinomials are the point counts of these open Richardson varieties. Um so if G is a complex semicol algebra I'll write m subv for the verma module with this this highest weight for any anti-dominant regular integral weight lambda. So uh yeah the uh the vermma modules naturally come together in uh groups indexed by the val group. These are just these are the blocks of category O and there are going to be no Xs between V modules from from different blocks. So in fact I might as well just index this by the element V rather than by this weight here. Um so these are infinite dimensional modules for le algebra and the subm module relation uh is exactly bruh order. Um so Gavin Joseph defined the pol this polomial which they called R prime just to be the alternating uh you know graded dimension count of these X groups in category O between two ver modules. Now the Gabbert Joseph conject uh paper was definitely trying to draw parallels between the RR prime polomials and the R polomials. In fact they prove that the R primes satisfy most of the cases of the recurrence for the R polomials. So people uh started calling the fact that our polomial that ru was equal to r prime the gab joseph conjecture although gav Joseph did not explicitly state that as a conjecture. um this would be a desirable thing because we have this nice uh recursive way of computing the R pol R polinomials just by this definition but the R primes there's actually no known good way to compute to compute these X groups in general sort of algorithmically like the R polinomials um so it follows from paras work that the linear coefficients agree that is that um they're one if if they make sense if u is less than v in order uh car in 86 showed that the top degree coefficients agree that is that the r prime polinomial is also monic of that degree um but bo found a counter example to the conjecture in 92. So it is not true that r equals r prime in general. However, we prove that uh the next highest coefficient cubed the second highest q to the length minus one uh we still do have equality between the coefficients of the r prime and the r polinomials. So our our name for this coefficient of the r polinomial was duv. So in other words dimension of the second highest X group between these var modules is duv and as a consequence we not only have this relation between these coefficients but also since we prove that duv is a common variant that also implies that this x group is a common variant that is if you have isomeorphic bruhai intervals and potentially even different groups these x groups will have the same rank. as long as the BR intervals are. Okay. Um so I should say really what we uh prove in order to get this is we prove that duv is also the dimension of h1 of the open Richardson. Um this follows because we showed that some of the mixed hodge groups of the open Richardson vanish which so so in general I can I can relate a point count which is the R polomial. Uh it can be expressed uh in terms of uh dimensions of mixed spaces. But because these particular ones vanish uh the only contribution to this coefficient of the point count is actually the remaining piece of h1 which is this h1 equals the entire h1 comma 111 mixed hodge space and that's why we get this um so also this betty number of open ridges and varieties is common trariant um by the same reasoning. So it is it is known in general that the x dimension should be equal to the um the betting number of the open Richardson. Okay. Questions about this? Okay. Uh so lastly I'll say a little bit about this connection to the cluster structure which motivated our construction and from what from which you can get a few more corlarius of our result. Um okay so reflections t are in bjection with positive roots alpha t a reflection order this was defined by a di is a total order on the reflections so that if I have any triple of positive roots such that gamma is in the positive cone spanned by alpha and the beta then gamma must appear between alpha and beta in my total order. I guess the way I phrase this, this should be an order on the positive roots which are injection with the reflections. Um, and what Dyier proved is that if we fix a reflection order, then the coefficient of Q to the K in the art polomial is the number of increasing paths in the bruha graph from U to V of length K. length k meaning they take k steps. Remember that the bruha graph included these longer edges that weren't necessarily cover relations. Increasing means that as I take steps uh in my bruha graph each of those edges is labeled by some reflection. So I want that the root labels of those reflections are increasing with respect to my fixed reflection order. So it is not at all obvious that this is independent of which reflection order you've chosen but that's true and in fact I proved this. So this is a nice in some sense this is a cominatorial interpretation for the artera polinomial but it's not useful for combinatorian variance at least directly because we are not given the information of which reflections which roots are labeling our edges and so we cannot hope to count increasing paths directly just given the information of the post set. Uh so just to give an example of this here's my bruh graph for S3. There's two choices of uh reflection order. This one or its reverse. In fact in finite type the reflection orders are going to correspond to reduced words of w knot. So there are there are two reduced words of w kn in this case. So I told you that our total polinomials are always monic that corresponds to the fact which di proves that there is always among the saturated chains that is among the chains that use only uh length one steps there's always exactly one of those between U and V which is increasing with respect to our fixed reflection order. In this case it's this one. And then there's also one path of length one which is uh trivially increasing because it has only one step. So this dire tells us that this arta polinomial is q cubed + one in this case. Um remember this coefficient here which is one or this was supposed to be this one was supposed to be the length minus GUV. The length is three here. So we expect that GUV is two in this example. And I'll explain how to get a diamond generating set of that size from this information about increasing paths. So here is the theorem. So fix a reflection order such that the inversions of V are prefix of the reflection order. So the inversions of V are set of roots. Those need to be the first so many roots of my reflection order. Uh and let gamma 0 be the unique increasing maximal chain from U to V. So that's the one in red that we saw. We prove that at most one increasing path of the second longest possible length diverges from this one at each point. So that's actually special to this chain and these reflection orders. In general, it's possible to have bad reflection orders for which this is not true. But this means that if we let FUV be the set of edges of gamma 0 with no such divergence, let FUV be that number. Then FUV is equal to DUV. The the size of it is equal to DUV. And this FUV is a diamond generating set. So in our example, I have these three steps of gamma 0. At the first step there is some second longest increasing path diverging from gamma zero at that point namely the green path. But at this step and at this step there is no diverging increasing path. So the second and third edges of gamma zero I keep and those are the elements of fuv and there are indeed two of them. And we can check that these two things do diamond generate the entire interval. So I can flip this diamond and I can look at these two and flip it down and flip down again. So how would one come up with uh the definition of FUV I just gave? So recent work of two sets of authors. So one casal gorski gorski le shet and simmonthal and the other galian lamb sharan bennett inspire they both um prove that the coordinate rang of the open variety is a cluster algebra. So it has clusters which consist of some frozen variables which are in every cluster together with some set of mutable variables in each cluster. Um so really what's going on is that under the natural bjection between increasing paths which are the objects I just talked about and distinguished sub expressions which are the David style object that uh these authors use to index their cluster data. The set fuv maps exactly to the frozen variables in their cluster structure. So we learned that there are duv frozen and length minus duv mutable variables in each of their clusters and also because we have proven that d is a common respon variant those numbers are also commonly invariant and depend only on the isomorphism type of the so we don't need their cluster structure but certainly this correspondence is how we were able to write down the definition that we used which in fact works in any cox and uh I'll end there. Thank you. >> Thank you very much. Let's thank let's thank our speaker