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all right it's my pleasure to introduce gabor uh from the university of north carolina at charlotte and he'll be talking about the dual the type b permetohedron as a chebyshev triangulation thank you very much i feel very honored in this great company of speakers and this is a wonderfully organized conference the only thing i am missing is the trip to jerusalem which i was really counting on but maybe next year okay so i have posted a full paper on the archive and it's more detailed than the extended extract there are some new results there and the references are also more extensive okay so the plan is the following i will talk about a lot of stuff you may already know and then so about some of my new results and by that time hopefully i run out of time okay that's not really the plan but anyway first i will talk about chebyshev triangulations this is the part i feel most important about then degraded post set of intervals which has a large literature then about the dual of the type of permittahedron and then about flag number formulas okay and since we don't want to lose time is i want to give a visual definition of the chebyshev triangulations and if your eyes can are fast i can do it really fast okay so the idea is you have first of all a simply short complex this is my simplicial complex here it has one empty set so the negative one-dimensional face is only one it has four vertices it has five edges and it has two phases and what we are going to do is we will put a vertex in the middle of each edge and we will connect it to all the other vertices inside the largest phase that contains it okay so first we put this midpoint here that contacted it connected it and we put this this this this okay so once again it goes like this okay and at the end we still have one empty set we have now nine vertices 16 edges and eight triangles but of course you can do this in many ways here is another way of doing this and once again you get one empty set nine vertices sixteen edges and eight triangles same as before i observed this but i didn't know how to prove it so i took aaron navo as my co-author and together mostly him actually only him i proved this that what we should have triangulations of the same simplicial complex have the same face numbers actually we proved this in a greater generality but this is what we need okay now you may wonder why the name chebyshev the reason is because when you do this triangulation what happens to the face numbers is most easily defined described in terms of the chebyshev polynomials first of all you need to take a somewhat unusual f polynomial where you multiply the number of j element faces j minus one dimensional phases by x minus one over two to the j so for example our original complex had one empty set four vertices five edges two triangles and after simplification i'm getting x plus two x squared plus x cubed over four okay and then for the chebyshev triangulation one empty set nine vertices 16 edges and so on i get this other polynomial and what's the relation now watch it look here the coefficients were one fourth two fourths one fourth and you replace x x squared and x cubed with the first championship polynomial the second chebyshev polynomial and the search which have polynomial and you get this f polynomial of the triangulation okay so where this is one way of defining of the chebyshev polynomials of the first kind that you take cosine and x and you write it as a polynomial of cosine x okay so this is one thing we want to know about the other thing is the chebyshev triangulations of the second kind so these are a little trickier because we will talk about multi-sets of simplicial complexes okay we look at the multi-set of links of the original vertices so here the original vertices are black the midpoints of the edges are white and if i look at any vertex here it is contained in three edges which become three vertices and it is contained in two triangles which become two edges and we have four times the same picture so now i will say that the f negative one is four because i have four complexes number of uh vertices is twelve number of edges is eight and if you look at the other picture miracle of miracles i get the same numbers again i had no idea why but aaron never was with me and so he proved that which i wish have triangulations of the second kind have this same face numbers okay now uh why are these chebyshev triangulations of the second kind because we can play the same game remember we had this f polynomial for the original complex okay you look at this chebyshev triangulation of the second kind you get this and actually what you get is that you are supposed to replace the powers of x by the chebyshev polynomials of the second higher uh kind and you just need to take half of it so there is a factor of one half just to annoy everybody but that's it okay and so actually actually this is pretty interesting because for example if you plug in a rule a unit complex number into a polynomial with real coefficients and you want to get the real part and imagine every part and actually you are doing this kind of transformations that you send x to the n into the chebyshev polynomials okay now next i would like to mention that originally i came up with this idea by defining the chebyshev transform of a postset okay the easiest way to do this is the following that you think of the intervals if the proposed as a half open intervals okay and then you say that one interval is less than equal to the other if uh either the lower interval is an initial in segment of the uh larger one or it's entirely below it i don't want to talk about this posit today i just wanted to mention that yeah so one thing that that if you look at the order complex of the chebyshev triangulation of the chebyshev transform of a poset it is the chebyshev triangulation of the suspension of the original order complex if you don't know the order complex of a pusset is a simply a complex whose faces are the chains okay and irenborg and reddy have a great paper about this where they look at this transformation which takes greater graded post sets into graded process and so i don't want to talk about this today too much except i will be stealing ideas and failing by trying to do so so i am like a bad student trying to copy in a new setting okay now the next topic is the post set of intervals this has been widely studied before it's simply the intervals of a poset ordered by inclusion and there is an old result of walker saying that if you take the opposite of intervals uh its order complex is a triangulation of the order complex and i have a new proof of this which shows that this is actually a chebychev triangulation okay so here is a simple poset it has two chains and this one here has three elements it becomes a triangle and this one here just becomes an edge and here is it's both set of intervals ordered by inclusion and all you need to go is you need to go downwards you know you go downwards and you get a chebyshev triangulation okay this seems like little more than walker's result the only difference is that now you know that every time this happens you know how to count the faces because you just substitute the chebyshev polynomials okay and we can make this into a graded process by just adding a unique minimum element which we call empty set so for example if we have a a chain like this here with four elements it becomes a fairly symmetric nice posit like this please note that at the bottom level you have the single tones which are like the original vertices in the order complex and after that you have the midpoints of the edges okay and i would like just to point out that if you look at the other chebyshev transform that i originally defined and that was studied extensively by air and border and ready then for a chain you get something like this so it's very very different this picture doesn't look like this picture you could prove it but maybe you don't have to just look at it and you see it's different okay by the way here the original vertices are up on top right any element to the maximum element at the end and to the maximum that's it okay and so in analogy analogy to the situation of the chebyshev transform of a postset one can show that the order complex of uh deposit of intervals with the minimum and maximum element removed is the chebyshev triangulation of the suspension of the original order complex okay let me talk about a minute about this suspension thing because i am going too fast okay so usually when you have a green leaf booster the greatest opposite is opposite with a rank function with a unique minimum and maximum element okay if you if you take just the order complex of that then the minimum element and the maximum length then belongs to every phase so you get like a complex cone twice people like to remove that okay and when we are talking about suspension it means we are putting them back but we are not putting back the the chains containing both the minimum and maximum element okay so you get like the boundary of a bicone okay and now comes the type b permutation where all that happened was that i realized at some point that the numbers for the phase numbers of the type b permutation and type a permutation look familiar to me from earlier work where i was working with chebyshev triangulation so i started getting suspicions okay now first of all the type a perimeter hatred is something i don't want to define it's a simple polytope and its dual is a simply shell complex it's uh it's the uh boolean algebra okay and then for the type of b permitoidion again it's a simple polytope it's dual is simplicial and there is a little more complicated description of its spaces okay i this is a mouthful to read out i'm not going to read this out all i would like to point out is that the facet inequalities of the type b permetoheteron look like this some of certain coordinates let's call this set like k one plus minus sum of other coordinates let's call them k one minus are less than equal to some magic number okay so you can describe every facet by a pair of subsets of the n element set some of them correspond to the positive coordinates the others the negative coordinates and then you have some compatibility condition and if those are satisfied then the intersection of these facets is not empty okay and so all i i realized it that in this magic recording you need to replace the negative sets with their complements and then suddenly you have intervals and the condition becomes just that you need to have a chain of intervals in the whole set of intervals of the boolean algebra okay so what we obtain that way is that the dual of the time b permutation is a simply shell polytope whose boundary complex is combinatorial equilibrium to the chebychev triangulation of the suspension of the order complex of the boolean algebra so here is a picture actually here are two pictures okay so this is how i would draw half of the three dimensional type b uh perimeter hedron okay first of all at the corners you have just the elements of an element set one two three okay then uh you take the barycentric subdivision of the boundary did i do that yeah i did that okay okay so these are still intervals but non-trivial sorry these are still one element intervals but uh not one element sets anymore okay and then here this is the upper half of the picture you put in the empty set and then the other intervals become midpoints and you do a chebychev triangulation so you should imagine that this is a triangle and this empty set is above it you know it's like a cone here okay and you subdivide it and this is the upper half and the lower half is similar except that here you have the one two three the inclusion relations will be different so the triangulation will be different you glue these two pieces uh pieces together and that's how you get a drawing of the type the dual of the type b per meter hetero okay if it looks familiar you shouldn't be surprised because the opposite of intervals of the boolean algebra have been studied by athanasiadis and sabidu studied the type b derangement polynomials and also very recently amwar and nazir published a paper in the in gcta about interval subdivisions okay and it is a a consequence of their results result that the each polynomial of the typical center complex sorry for using this word without introduction that simply the dual of the type b permetohedron has real roots and now i realize that i have proved this years ago this statement without realizing it okay because years ago i have proved that for for another chebyshev triangulation of the bullying algebra that uh the the phase polynomials they're actually the so-called derivative polynomials for the secant okay the derivative polynomials for the secant are defined by this formula you take the nth derivative of the secant and it becomes a degree n polynomial of tangent multiplied by secant okay and so the exact relation is this that if you look at the whole set of intervals of the boolean algebra with the minimum and maximum element removed and you count the faces in the order complex and you multiply by powers of x minus 1 over 2 then you need to plug in square root of negative 1 to get to get alternating signs so i could also say actually i'm looking at the derivative polynomial for the hyperbolic secant okay and and that's what's going on and many years ago i proved that all roots of the derivative polynomials for the hyperbolic tangent and secant are interlaced reals and they belong to the interval negative 1 1. and by the way there is this magic transformation between the f polynomial and the h polynomial and so if these guys have real roots and they are not one because that would be just uh the sum of positive things then then it's uh if these have real roots then these have real roots this transformation sending t into one plus t over one minus t is called the mobius transformation in complex analysis and it comes up in stability theory anyway so so i didn't know this because i at the time i wrote that paper i didn't realize that the dual of the type b permutahedron has the same h polynomial than the object i was looking at but now i know because both archibishop triangulations okay okay now flag numbers okay so uh if you start with the graded posit and you take deposit of intervals you get again a graded profit and i'm going to dash through this okay so when you have a graded boost that you have the so-called flag f vector where you count chains hitting a certain set of runs and you associate to them and a b a polynomial in non-computing variables richard alves said that a is absent b is belong and then okay uh now if you take the interval transform it turns out to be a linear map yogi urich proved this and he used the so-called ironborg radical product which simply takes a a word of length u n and it skips a letter letter the ice letter in any every possible ways and the initial segment is before the tensor sign and the rest is after and the left end uh came up with this formula for which he has a proof and now in my paper i have another proof which is a little shorter but i'm not going to talk about this today instead of that i would like to talk about the interval transform of the second kind remember that taking deposit of intervals corresponded to a chebyshev triangulation but then there is the chebyshev triangulation of the second kind where you look at the links of the vertices the original vertices now this corresponds to the following transformation that you take the whole set of intervals and then the original vertices correspond to the singleton intervals so you look at just the set of intervals containing a given singleton and you take the multiset of these that would be the interval transform of the second kind and there is a beautiful formula to what happens to the a b index okay using the magical operator m is for magical no m is for mixing okay irene morgan ready introduce this operator if you want to compute the a b index of the direct product of two posets then you need to take apply this mixing operator to this to their a b and this is okay and so you do that uh for you and you take you reverse this star means write it in reverse and then in the first coordinate you plug in the reverse of u1 and you get this formula every talk needs to have a joke and the proof here is the pro here is the proof which is like a joke okay uh it turns out that directly if you look at what are the intervals y z that are contained in this interval it means that y needs to be less than x so it needs to be in 0x and z needs to be in x1 okay and the order is reversed because if you go down with the y it becomes larger the interval becomes larger and that's the proof okay now for eulerian process asanasia this is already observed based on walker's result that you get an eulerian process so there are cd index formula cd indices were introduced by buyer and clubber okay now the ladder postset here is famous because it see the index is c to the n and uh already computed what happens if you apply the interval transform to this and uh got the coefficient which is the same as the ironborg ready formula for the chebyshev transform of the simple set okay and i also wanted to do something so what i did i did the same calculation for the interval transform of the second kind and i found this and the proof involves expressing this mixing operating value values as total weights of lattice path okay and then okay so deposit of intervals of the mood and algebra becomes a cubical that is and irenborg and reddy and myself had formulas for the cdn devs of death in terms of the under permutations and signed variations of them and just like in the iran burgrady case the the uh cd index of the boolean algebra becomes like an eigenvector of the interval transform of the second kind irenborg and reddy found all the eigenvectors of derchabishav operator of the second kind i tried to cheat copy their ideas but i didn't succeed completely the lifting operator needs to be changed what they had and then they had this pyramid operator and so the problem i run into is that we have a non-trivial kernel if an expression is anti-symmetric it's in the kernel okay so i only have conjectures namely that there is nothing more in the kernel than this anti-symmetric expression and in the symmetric expressions you can still copy the air and already construction except that you will get a generating set and not a basis of eigenvectors but this is only conjecture and i ran out of time as promised let's thank god more thank you for a very lovely talk are there any questions i don't see anything in the chat beautiful results so i would like to mention if i have a moment here that in my original paper i mean original in the paper where i was working with the derivative polynomials for tangent and secant i was trying to look at stability which asked whether the h polynomial has only roots with negative real parts or the f polynomial has roots only inside the unit circle these are called sure stability and and uh horrible stability in the literature and i made the contracture that if you take the direct product of heart stable posets it will be very stable process it will be orbit stable i didn't know how to do it but now maybe with intervals transforms the whole changes but the phase numbers are the same it could be worth looking at at any rate so the point is just that with this chebyshev triangulations you can save a lot of work computing the same formulas over and over again i believe okay uh actually richard stanley has a question um he says what about a q analog involving intervals of a subspace lattice over f q well i haven't looked at that it sounds like an interesting question i mean already irenborg and reddy and then yoichen might have were looking at flag vectors so yes there should be refinements to this we have one more question i have is that so uh in a way maybe you can also truly turn the other chebyshev that i defined before or the jebus of my travisha transform of the bull and algebra into some sort of a type b polytope and what's that what's that i don't know what's that it should be a polytope with the same face numbers if it's a polytope yes richard do you have a question okay all right if there are no more questions let's thank gabor again