 Okay, welcome to the second part of the Schubert seminar with Alex Young telling us about new world illiteratures and numbers Okay, so the first half I reviewed the the grand history of illicit coefficients and the contributions that came in that subject I now the preview of of our results that The summary is and this is with she laying out good down a row Nicolaus Sarah myself is we give no illiterate generalizations of the Kelley actual results and Really conjectural version of the kitchen tower results for the classical groups And this is in the central dogma. What we're trying to say is that you have this red nl case That they interject between LR, which is the you know symmetric functions and the most general situation That's you know governed by the general test front Multiplicities and things like low-end path model that there's an interesting in between which I think deserves some independent investigation Okay, so remind Under what we mean by the classical groups Let's start with SOM. These are you fix them non degenerate symmetric bilinear form. Remember non degenerate just means the matrix form has non-zero determinant and We're looking at one Matrices which have this property that they preserve the form. Okay, and When you ask for such matrices the terms are your one or minus one and by definition we're making SOM the ones that determine plus one and similarly Define the subplectic group only for even to end Except they instead of using a symmetric bilinear form use a skew symmetric bilinear form and there the determinant is always one Okay For the classical groups the thing to understand is that a lot of the representation theory Looks very much like the case of GLM and this is emphasized in Vowell's book the classical groups in 1939 He says that for everyone these classical groups He's going to give you a construction and algorithm to give you Representations for every partition without most end rows just like last time just like in the in the first half and these are all irreducible except for one particular special case which Will be absolutely relevant to our conversation As you shall see Now if you take a classical group well really even more generally They're reductive and so The tensor products of your reducibles decompose and I'm going to call the tensor problem. Let's see T lambda me to T for tensor And one would like to understand These numbers and in the very first slide things like the Lohman path model and busy triangles and crystal based Crystal graphs are exactly a way to compute those sort of things even more generally than for classical groups Now here comes the definition this definition demands absolutely nothing about Representation theory. It is just a combinatorial definition I'm giving three partitions lambda mu new and I'm going to define numbers and lambda mu new you as a sum of product of LR coefficients and If you haven't seen this before that I want to just point out one of the features The you know, you're summing all these indices alpha beta gamma and what you're doing is go sort of a cyclic thing alpha The beta beta gamma gamma back to alpha Okay, so you're kind of you can if you had any structure cons is you could always create things like this And just to find such a thing and study it Okay, and Kiyuki and Toronto Proves the following things. Well, let me go back. What is the relationship between T and N? First point is that they are not equal in general. They're absolutely not equal in general However, what Kiyuki and Toronto showed in 1987 is that they are equal under a hypothesis and Hypothesis I put in the box here It's called a stable range. You assume the length of Lambda the number of rows of the young diagram plus the number of rows of mu is at most N where N is the same thing in your definition of the classical group An immediate corollary of this which is rather fantastic is That therefore in the stable range these numbers do not care which classical group you're working with which is very odd because in general these Tensors problem multiples is definitely will care which classical group you're working with Right, they're all in fact equal to this you a little bit number and then one final thing Which doesn't come up in this talk with but which you may find interesting is that Kiyuki and Toronto also define the basis of symmetric functions That behave precisely like the Sherpa on this dude They are the universal characters for the situation by which I mean if you multiply two of their symmetric functions and expand in the basis of their Symmetric functions that you'll get exactly and Lambda you I Ask about the symmetry Mm-hmm Maybe briefly like at a glance now N looks like Symmetric and Lambda mu nu it is I'm used to thinking of there being a duel in teas somewhere is or am I mistaken? No, you're right. So in this particular case they are told Symmetric and the point is the duel is black like Duel just means negative of the partition right in this particular case. I guess so Yeah, it ends up being symmetric it's surprising Okay, and and moreover, it's clear from the definition of symmetric Right for N. Yeah, okay. It's deed. Are they like a Toronto basis elements sure positive are the are the basis on sure positive If I oh actually, let's see Wait, I'm forgetting are they alternating in sign I'm forgetting at them Whether they're positive or alternate the sign in a predictable way, but I believe the answer is basically yes She leung says no, okay. Well, I Thought they were yeah, I just we were kind of example. I just simply forgot it Yeah, well, actually one of the point is that these polynomials were in fact studied Some time ago by people like rock-follower and they worked out all kinds of terminal expressions I mean in QQ it's rather there's a jacobi-truity like determinant and there was some comforts worked out about these quite some time ago And but we're not examining that in this particular talk okay Here's the definition of the new little numbers again and at this point. I'd like to make a shout out to Hikki Young Han who was a number theorist at Duke because Yidan, Chi-ling and I were originally stimulated by her paper, which indeed considered these exact numbers and she did so at Interpaper motivated by language beyond endoscopy proposal In our first paper of what we tried to do is think about all from First principles, what can you say about these numbers just essentially knowing the definition? And I'm not going to review the results from that first paper here, but rather I'm just going to talk about a few facts That give you a sense of the things that one would think about in the first hour of meeting this definition and may give you a sense of what we were up to well Besides what Dave already noticed that they're symmetric Another fact that's clear from the definition is that if you impose that size of mu equal size of lambda plus size of mu then You can easily check that the L and L numbers are equal to LR numbers So therefore this is in fact not only an analog of the LR numbers. It's a straight-up generalization The second thing you can prove using only the things I've told you in this talk is that these things are in fact the semi group so if you Have a triple lambda mu nu with a positive NL you can sum up another triple and get a third triple And then the third thing is something that was already appeared in Hans paper It's this interesting parody condition You can a necessary condition for the NL number to be positive is that the sum Of the sizes of the partitions is even Okay, so from these facts let's again in by analogy of the LR case to find the semi group and L semi group is just going to be triples for which the NL coefficient is positive and similarly, let's define Saturate NL semi group, which is this defective form of the NL semi group with this extra existence parameter condition What we conjecture She-Lan Gao, Guidonarela Witz and I is an analog of Kutzen-Tau's Theorem and the statement is the following. It's not true That the NL coefficients are saturated in any way that you normally think that you would know to think about but the only Flaw is precisely the parody constraint That is the lattice points of this intersection that the the convex hull even of these things is exactly NL And that's precisely The only fix you need to get a statement Precisely analogous to Kutzen-Tau Okay Now what about this Connection to eigencones or eigenvalues of matrices the following result together with Nicolae Rousseir really is a corollary of Our of our theorems that I'll state later But since I didn't tell you what the theorem is I it's kind of hard to call this a corollary yet So let me just state it as a theorem You have three partitions lending you new you're an NL sat if and only if there exists matrices m1 m2 m3 of this weird form Who sum to zero and with and have eigenvalues essentially lending you new by which essentially? I mean that rather than lending you new you take lambda hat Which is okay is written, okay? So that's that's the statement of the analog to sums of Hermitian matrices in this context Now this is not a magically created a set of matrices from our perspective really it's derived from the theorem of Bataille and Kumar and In the context of their theorem, we're looking at the well the algebra of some practically algebra intersect We're really anti's Hermitian matrices, but we set up the Hermitian matrices here And I'll say more about that later when we get to our many theorems All right, what about the Horn-Cliaccio inequalities? In the second paper they wrote with the gallon or Elvis we conjectured a list of inequalities for both NL and NL sat well really NL To be absolutely honest with you. We didn't think about NL sat there's a separate thing at that point And the general form of an inequalities has some similar flavor Instead of well one less equal to two you have three less equal to three and you're summing over subsets of the eigenvalues But with a whole bunch of crazy conditions On them, and it's the third condition here that I want to emphasize It looks awful because you're summing over sums of products of six LR cold Fish in what what is that? However, I want you to notice the form of summation. It's alpha one goes to alpha two alpha two goes to beta one One beta one goes to beta two beta two goes to gamma one gamma one to gamma two and gamma two back to alpha one It's in fact exactly the same formulation as the NL coefficients themselves only more of it and And therefore we call these multiple NL coefficients So we're trying to talk about NL coefficients and what our inequalities are talking about our multiple NL coefficients So it's not exactly a recursive, but it's suggestive And To formulate our conjecture the statement that well, let me state it this way NL sat a triple Doesn't NL sat if and only if they satisfy these particular inequalities and this would be analog of Kiliacco's theorem All right now I want to talk about our main results The first theorem is that the conjecture. I just stated is in fact true and that's with gout or a low it's everything about the state is with with the four of us and The proof is based on the following Let me call observation The NL sat is equal to this tensor cone. What this tensor cone is the following I'm interested in arbitrary tensor work multiplicities without the Hypothesis of stable range for the symplectic group that is to say I'm interested in triples of partitions with this weird quantified Such that this tensor product multiplicity is positive If M isn't Mary is extremely large compared to N is a Nancy Then this equality is trivial because Joaquin try to tell you that in stable range the NL and the tensor problem with these are Same what is interesting with this term in my opinion is that this equality is true for Arbitrarily arbitrary and gringled and even Free the stable range once you know that is true, and this is just to restate what I said Then then we will get all our consequences Okay, so how do we prove the R&B really let me talk about about Cali Kumar inequalities right now but Cali Kumar Think about these sorts of questions of knowing about tensor the saturated tensor cones things like like this thing over here in general without any constraint Complex you know for some I guess complex reductively groups and and what What they do is they give an answer a very beautiful answer in terms of Schubert calculus They define a certain deformed cup product and they say that something is an inequality if I satisfy a certain deformed Schubert calculus question. So there's where Schubert calculus comes up. That's just a two second explanation of their results. I Apologize to the experts. What was Sarah's doing? Doing in a follow-up paper is he obtains the same results But actually removes the homological description. He replaces it at least in the classical groups with just a condition on little enriching coefficient The literates and coefficients. So we're using those sorts of results as part of our arguments Now what about Cali Kumar do is they in fact give minimal Inequalities for these saturated tensor cones in general in absolute generality and they also prove an eigencone description Therefore since we know That NL is equal is in fact equal to the saturated tensor cone We immediately get our eigencone description by just writing out what they mean in that particular case and we get to choose the Plectic group because according to Kiyokin, Toronto, it doesn't matter which group you choose among the classical groups So we just choose one go with it and also We obtain the first minimal set of inequalities for NL sat that corollary C depends on fear and B because even though you You know that the NL coefficients are tensor product multiple is in a stable range When you impose the stable range condition, you may affect in principle what the minimal qualities are in the but Cali Kumar theorem and to go back to the Calliartical horn inequalities, we have to do some pretty heavy lifting in Tableau combinatorics and I Don't want to get into that but it's a it's a connection RSK and something called demotion in one of our earlier papers and this is to prove some particular non vanishing result of the non vanishing result about these multiple NL numbers Okay, another corollary or theorem that comes from this This analysis We have a surf factorization of the NL numbers on the boundary of NL sat boundary means that you assume that the inner Poles are met with the quality and then and you get That the NL number is equal to an LR number times some smaller NL number I don't want to state the exact theorem because I would have to then state what the boundary is precisely But for those in the know, this is an NL analog of certain factorization theorems of King told me to say and Dirksen Bayman and possibly others I Think you had such a result exactly about this and maybe was there as well Okay, so is that all There's a do well, no in fact where we are right now is we are exactly in the condition of Cousin and Tao in the late 90s Once we know theorem a It now follows that the saturation ejector that we had for NL would imply that the NL itself Not just NL sat is described by these horn and to the actual inequalities But I don't know we don't know yet how to prove what would seemingly be just a common troll question What is some evidence for? For the saturation ejector. Well, I we know for N equals to that the inner Poles are correct for NL from our first paper and in our second paper This is with Gavin or Ella. What's we know that it's true Well, if you add an extra constraint that landing you knew as a roller column, so we know it in those special cases However, I think the strongest evidence for this comes from our most recent paper It's that we prove computationally that NL saturation holds for all n less than the five This means that so you're fixing the number of rows But there's still infinitely many partitions land in your new and we're proving for all those infinitely many triples of partitions the NL saturation is is true and With that, I thank you. Thank you