 This video is the second part of the Wienberg lecture on Wienberg's algorithm and Katz-Mudy algebras. There should be a link to the first part of the lecture in the description of the video below. So the first part of the lecture we covered Wienberg's original paper describing his algorithm and his calculation of some reflection groups. This lecture I'm going to discuss mainly Conwen Sloan's approach to Wienberg's results and in particular describe Conway's calculation of the Dinken diagram of the 26-dimensional even Lorentzian lattice. I'll start just by having a quick review of some of the things from the first lecture. First of all, we recall the lattice i n comma 1 is the lattice in Lorentzian space r n comma 1. It consists of all vectors m1 up to mn mn plus 1 with mi integers and the norm of this vector, the square of its length is given by m1 squared plus mn squared minus mn plus 1 squared. So this is a minus sign here as in special relativity. We recall that the even lattice is defined similarly except that all the mi are in z or all the mi are in z plus a half and also the sum of the mi is even. Here we take n congruent to 1 modulo 8 and this has the result that all vectors have even norm and what the first part of this lecture was doing was it was calculating the reflection groups of these lattices using Wienberg's algorithm and we saw that we could calculate the Dinken diagram and we found the Dinken diagram was finite for i n1 for n less than equal to 19. So here we have a finite Dinken diagram and the reflection group is a fundamental domain of finite volume but for i n comma 1 for n greater than or equal to 20, Wienberg showed that the Dinken diagram was infinite and similarly it's finite for the even lattices for i 91 and i i 17 1 but it's infinite for i i 25 1 so this is big jump in behavior when n goes from 19 to 20. So what we're going to do is to start off by looking at this case here. So first of all I want to describe the relation between norm zero vectors of this lattice and so we'd take norm zero vectors in this lattice i i 25 1 and nemia lattices so nemia lattices are the lattices in 24 dimensions they're even and they're unimodular that means that volume of a fundamental domain of the lattice and the translation is just one and there are 24 of these found by nemia and the most famous one is the leach lattice in which there's no norm two vectors so the norm two vectors are normally called roots because they correspond to reflections. Now if we take a nemia lattice so let's take a nemia lattice L and add on a little two-dimensional lattice with with inner product matrix that looks like this so it's Lorentzian it's it's got norm zero vectors and positive norm vectors and negative norm vectors then this is actually isomorphic to the 26th dimensional even Lorentzian lattice that's because there is only one such lattice up to isomorphism so this gives you a map from L to norm zero vectors because if we choose coordinates for this we can use coordinates lambda mn with lambda in L and mn integers and this thing has norm lambda squared plus 2 mn then we notice the vector w equals zero zero one has norm zero so w squared is zero so this is a way of going from lattices to orbits of norm zero vectors in this lattice here on the other hand if we've got a norm zero vector w then we can in this 26th dimensional even Lorentzian lattice what we can do is we take your orthogonal complements of w and this is 25 dimensional and it's not quite positive definite because it's got norm zero vector w in it but if we then quotient out by w this then becomes a nemire lattice so this gives us a bijection between me or isomorphism classes of nemire lattices and orbits of primitive norm zero vectors in this lattice here and you can use this in both directions you can either use the classification of nemire lattices to find the primitive norm zero vectors or you can run it backwards and use the classification of primitive norm zero vectors to classify the the nemire lattices so we can give an example of this for example we can ask can we find a norm zero vector in this lattice here corresponding to the leach lattice and this was solved by Conway and Sloane so we've got to find some vector m0 m1 up to m24 m25 and whose orthogonal complement modulo w itself is the leach lattice in particular that there can be no norm two vectors orthogonal to this and you notice if if two of these numbers were equal so if mn was equal to mn plus one then we would have a norm two vector orthogonal to this which just looks like zero one minus one zero zero zero and so on so mn is is not equal mn plus one for um for for any m i should say you may as well arrange these they're increasing order so we've got to find an increasing series of integers um such that um no two of them are equal up to sine and there's an obvious way of doing this which is we just take one two up to 24 now we've got to put something there and now there's another constraint which we must have omega squared equal zero um and that means we've got to find a number here whose square is equal to the sum of the squares of these numbers and by a weird coincidence there is such a number which is 70 because zero squared plus one squared so in all the work plus 24 squared is equal to 70 squared um incidentally 24 is the only non-trivial solution to this equation apart from you can have nought squared plus one squared equals one squared but apart from that um there are no other cases when the sum of the first n squares is a square um and now the vector omega per over omega is the leach lattice um it's not all that easy to show there are no other norm two vectors orthogonal to this i mean we've shown there are no obvious norm two vectors orthogonal to this but there you know how do you know there are some aren't some non-obvious ones um so um next we come on to the result by Conway Parker and Sloan which asks what is the covering radius of the leach lattice leach lattice is normally denoted by a capital lamb to stand for leach um so this means what what you do is you take spheres of radius r around lattice points um so so they just cover the whole of the the real vector space corresponding to the leach lattice so this shouldn't be confused with the packing radius and the packing radius asks what is the biggest sphere you can put around every point such that the spheres don't overlap the covering radius asks what is the smallest sphere you can put around every point so that the the insides of these spheres actually cover space so for example if i've got a um a triangular lattice in two dimensions um the the um the the packing radius you'd be looking for spheres that um don't overlap so so these would give you spheres of the packing radius the covering radius on the other hand you're looking for bigger spheres that cover the lattice so here you would have bigger spheres at this time they that they are going to be overlapping and you see that these slightly bigger spheres are actually covering r2 whereas these spheres are disjoint so this is the covering radius that we're going to be interested in and the points at maximal distance from all lattice points like this point here are going to be called deep holes they're sort of the in some sense the biggest holes you can find in the lattice that this are um as far as possible from lattice points um and um what they found is the covering radius of the leach lattice is the square root of two exactly this is a really amazing fact um their calculation of this is was rather long and difficult um in general calculating the covering radius of a lattice is rather hard you just have to um um um sort of get your hands dirty doing a lot of calculation considering all cases so most of the papers by Conway and Sloan um as i'm talking about this lecture been reproduced in the book sphere packings lattices and groups um in particular i just wanted to show you uh um what their calculation of the covering radius of the leach lattice looks like so here's their paper it sits chapter 23 of the book and if you look through it you see you're getting large numbers of case by case calculations here they're writing down vectors of the leach lattice um here they have a big table showing all the different ways of finding a certain an inside the leach lattice i explain that in a moment and it sort of goes on like this for many many pages there are there are huge numbers of cases to consider and the reason they had to consider so many cases is there are actually the answers rather complicated there are 23 orbits of deep hole so they had to actually find all these different orbits and show there are no other orbits and the amazing thing is these correspond to the 23 nemy lattices that are not the leach lattice 24 nemy lattices one of which is the leach lattice it's the 23 others that appear corresponding to the 23 holes um now how do the nemy lattices correspond to holes well um let's take a typical nemy lattice for instance we can look at the nemy lattice e8 cubed and you look at the dink and diagram of the reflection group of this nemy lattice so we draw three copies of e8 one two three four five six seven so here's one um and here's another and here's another and so so this is the dink and diagram of e8 cubed and now i'm going to make it into an affine dink and diagram and to make it into an affine dink and diagram for e8 you just add on these three points here and now i'm going to make these correspond to a configuration of vectors in the leach lattice as follows so if we've got two points in the dink and diagram not joined i want v to a distance four from w and if they are joined v w then v minus w squared should be six so um this looks absolutely bizarre you're taking an affine dink and diagram and using that to construct a configuration of vectors in the leach lattice and these um and if you take a configuration like this these turn out to be the vertices of a deep hole the vertices of a deep hole just mean the lattice vectors nearest to the deep hole and Coyne Sloan found that they got this exact correspondence between um um the affine dink and diagrams of the knee my lattice and the deep holes of the leach of the leach lattice this this correspondence was first suggested by Richard Parker who somehow noticed um i think he knew a couple of holes in the leach lattice and sort of noticed these seem to correspond to the knee my lattices a1 to the 24 and a2 to the 12 and pointed this out to Conway and Conway and Sloan then got rather excited about this and checked this work for all the other knee my lattices i've no idea how Richard Parker came up with this idea because the knee my lattices were a pretty obscure topic at the time the leach lattice was also pretty obscure so it was somehow putting together these two um very obscure um um objects so um so how can we explain this um well we we can explain this by recalling that knee my lattices correspond to norm zero vectors in in the 26 dimensional Laurentian lattice so here we take this is going to be a knee my lattice um and if we take um one of these norm zero vectors in the 26 dimensional even Laurentian lattice we can look at the roots of the 26 dimensional even Laurentian lattice which are um orthogonal to um w and these will be just the roots of the affine knee my lattice this is because the orthogonal complement of w is just the knee my lattice plus a little zero dimensional lattice and it's not difficult to check the root system of that is it's just the chorus you just get the corresponding affine um group um and Conway discovered that the dink and diagram of um the leach lattice so the dink and diagram of ii 25 comma one is the leach lattice and this doesn't even seem to make sense um because you know the dink and diagram of this lattice is going to be some sort of um graph with various distances between points and then leach lattice is a set of points in Euclidean space so how how on earth do you mean by saying they're the same well they're actually isometric so both of these are metric spaces the dink and diagram consists of simple roots and you can talk about the distance between two simple roots and of course the leach lattice is a subset of 24-dimensional space so it's a metric space um so let's see how they are actually isometric well um what Conway did was he applied Windberg's algorithm to the 26 dimensional even Laurentian lattice and you've got to choose coordinates for this and the obvious coordinates are to choose you know integers m1 up to m25 as i had before but Conway chose another coordinate system he chose to write it as the leach lattice plus 0 1 1 0 so um the vectors are going to be the form lambda mn where lambda is in the leach lattice and m and n are integers um and um this thing has norm lambda squared plus 2 mn and there are two conventions sometimes put a minus one there instead and i again this is something i sometimes get confused about so there may be a few sign errors now for windberg's algorithm you remember you have to start by choosing a sort of special controlling vector which i mean to take to be 0 0 1 so the orthogonal complement of p divided by p is is is just the leach lattice and now we're going to find the simple roots of this 26 dimensional lattice in in order of their inner product with p so we want rp to be 0 1 2 and so on and of course we want r squared equals 0 so if rp is 0 well then you see that um that the vector lambda mn would have the property that lambda squared was 2 and the leach lattice is no vectors of norm 2 so that should be a 2 not a 0 so um so there are no um no simple roots that have inner product 0 with p now let's look at the vectors with rp equals 1 well well here we get the vectors lambda 1 um lambda squared over 2 minus 1 for any vector lambda in the leach lattice so we've got an infinite number of roots of level 1 in accordance with windberg's discovery that the the root system is infinite so this is the first batch of simple roots and the amazing thing that conway noticed is there are no more and um this depends on the fact that lambda has covering radius root 2 and let's see why this is true so um suppose we've got some simple roots so suppose lambda ab is simple now this means that it has um inner product less than or equal to naught with all simple roots of the form mu m so sorry mu 1 um mu squared over 2 minus 1 and if you work out what this condition is you just work out the inner product of these two vectors and you find its equivalent to saying that lambda minus v over a squared is greater than or equal to 2 plus 2 over a squared sorry that should be a mu minus um lambda over a squared is greater than 2 plus 2 over a squared which is greater than 2 and this is not possible because the leach lattice is covering radius root 2 so you can't find any vectors mu that have any vectors lambda over a that have distance greater than root 2 from all all vectors mu um so um now this explains um the correspondence between neomyel lattices and deep holes because the deep hole is just the set of um simple roots in the leach lattice corresponding to the dink and diagram of the neomyel lattice and conway and slone now use this to reinterpret minberg's results on the reflection group of i n comma 1 and um i n comma 1 can almost be embedded in the leach lattice it can't quite be embedded because this has um vectors of odd norm and all vectors in in the 26 dimensional larynxian lattice of even norm so let's take the dink and diagram d n looking something like this and um we can embed this inside the leach lattice considered as the root system so it is as the dink and diagram of the 26 dimensional even larynxian lattice as i said conway showed that the leach lattice is this and then if we look at the orthogonal complement of d n this is going to be the even sub lattice of i 25 minus n comma 1 and um this consists of all vectors m 1 up to um m 25 minus minus n um m something such that the mi of our integral and the sum of the mi is even now this lattice usually has the same reflection group as i 25 minus n 1 usually does that there are a few cases when it doesn't but we will we will see later um so um all we have to do is to work out what is the dink and diagram of this lattice here and conway and slown showed that the dink and diagram of um the orthogonal complement of d n can be obtained as follows take all points um p of lambda so that p union the d n is a spherical dink and diagram now the reason for this is that this is the condition that um the projection of p in the lattice d and perp has norm greater than zero and if you want this projection to be a root it certainly has to have norm greater than zero there's also actually another condition that it has to satisfy that i'll be discussing a little bit later but for the moment um we're not going to worry about it um so let's see what happens as we go through the various possibilities for d n so let's take d n but n greater than or equal to eight um i'm just going to write out the case for n equals eight so we get one two three four five six seven eight so here's the dink and diagram d eight and there are just two ways we can add a point to this to get um a spherical dink and diagram first of all we can add a point here and then we get d n a one so we just add an a one and this turns out to give you a norm two root in um i uh 25 minus n one the other thing we can do is we can add on a new root here this gives us a d n plus one dink and diagram and this gives us a norm one root so this uh um explains where we're getting norm two norm one roots from there from the two ways of extending d n to a bigger spherical dink and diagram so now let's look at d seven where something a little uh um little extra happens so let's take d seven one two three four five six seven and now and there are three things we can do because as well as we can add a point there or we can add a point here just as before and these give us norm two and norm one roots but there's a third thing we can do we can add on a root here and we get an e eight dink and diagram and these give us norm one roots well um this also gives us norm one roots so what's the difference between these sorts of norm one roots well the difference is that these norm one roots are parity vectors so you remember a parity vector is a root r such that the inner product of r with v is congruent to v v mod two for all vectors v um so you remember um this corresponds to the case of um i um 18 one and you remember i commented in the previous lecture that in the case of i 18 one vinberg and kaplan sky i found that they actually got three sorts of roots and the reason for the third sort of root is coming from this e eight dink and diagram and it's it's giving you this this funny extra parity vector um you can't do this for i 17 one because that would be an e nine dink and diagram and the e nine dink and diagram doesn't give you a a spherical or positive definite root system and we can do the same thing for d six so so this is the d seven case d six is kind of similar one two three four five six and again we can add roots there or we can add a root here and this again gives us norm one um this gives us norm two and this time we can add in a root here and this gives us an e seven and this gives us now gives us norm two roots and again that these norm two roots are parity vectors so we we we get a second batch of norm two roots for the case of i 19 one um so um this accounts for vinberg's diagrams um also it makes it rather easier to calculate the dink and diagrams because you just have to search for certain vectors in the leach lattice which turns out to be rather easier than running vinberg's algorithm um so what happens for the case d five so let's take a look so so here just as before um the possibilities are we could add a point then get a norm two root or we could add a vector here and get a norm one root or we could add a vector i think i'm using purple for these we'd add a vector here and this time we get an e six root well except it's not a root because this gives us norm three vectors and these are not roots um and this accounts for why um why if you take the lattice i um 21 the reflection the dink and diagram is actually infinite i mean roughly speaking this was a root then then then we'd find a finite fundamental domain for the reflection root but because this gives us something isn't a root we we only get an infinite fundamental domain the question is why is this not a root um so if you take the vector here we get a perfectly good positive norm vector in the lattice i 21 and it turns out there's an extra condition we need to satisfy for this to be a root um dink and diagrams have something called an opposition involution um and what this does is um it's an auto it you if you if you take the root system contained in um euclidean space rn then um rn has an automorphism minus one you can just multiply all vectors by minus one and this can be written as a product of reflections times an automorphism of the dink and diagram um and this automorphism turns out to be of order one or two and we can see what it is in various cases if you take a n then the opposition involution just does the obvious thing so it's non trivial unless n equals one if you look at d n it's a bit subtle it depends on whether n is odd or even so if for d n n even it does nothing for d n n odd it flips these two vectors here and for e six it flips um the two endpoints and for e seven it does nothing it can't do anything because e seven has a trivial automorphism group and for e eight it does nothing and now the problem is that is the fact that the this opposition involution for e six um has norm is is non trivial and it turns out that um for in order to get a root what you would need is if you take the d five wave that we were looking at and extend it to an e six we notice the opposition involution of this e six acts like this so the opposition involution of e six does not preserve um the d five sub diagram and this is what is causing the problem if you look at all the other cases we had we see that the opposition involution of the bigger dink and diagram you know preserves the smaller dink and diagram i mean you know that if we take d n inside d n plus one and the opposition involution of d n plus one is always preserving d n so this is this is what is causing um the dink and diagram of the Lorentzian lattice to become infinite and that we can now see why i twenty one has an infinite dink and diagram and the reason is that twenty is equal to the dimension of i i twenty five one minus the dimension of e six um now vinberg had a different reason for why the dink and diagram becomes infinite so vinberg pointed out there were two lattices a eleven d seven and e six cubed as i mentioned in the previous video such that the rank of the root system is only 18 whereas the lattice is dimension 19 so we seem to have two completely different explanations for why things break down one says that e six has this funny extra um opposition involution whereas vinberg's explanation says you get these two extra lattices and i mentioned 19 with not enough roots well it turns out these two explanations are really more or less the same the question is how do you get these two lattices well there are two nemy lattices a eleven d seven e six and e six four or at least that's their root systems and you can get these two lattices by taking the orthogonal complement of a d five which is contained in some e six which is contained in inside some nemy lattice so we're taking the nemy lattices that contain an e six component taking the d five inside them and then taking the orthogonal complement of that and that gives you the even sub lattice of these unimodular lattices so in both cases the problem is caused by this d five being embedded in e six in a funny way such that the orthogonal complement of the d five is not a root um so that sort of explains exactly why vinberg's algorithm breaks down at um at a particular dimension well we can go further um so um let's look at the case of um d four so this is going to correspond to the lattice i 21 comma one and dink improved the reflection group again has infinite dink and diagram and let's take a look at the possibilities well first of all we can have normed two roots and then we can add in a root here so this has norm one um and this is giving us a d five dink and diagram and then there are two ways of extending this d four to an e five we can either add a point here or we can add a point here so what's e five well well e five looks like this and you see it's actually the same as d five so um these two ways of adding extra roots are really the same as this way of adding a root um and in fact if you count how many ways of doing this there are there are 56 roots like this and 56 like this and 56 like this and for that matter there are 42 times that um and if you look at the opposition involution of d five you see it's actually fixing this d four i mean the opposition involution of this d five flips these two vertices but that's okay that maps d four to itself so this seems to be prediction that we should get a finite dink and diagram um with um 56 plus 56 plus 42 um um points comes out to 210 so um um we we we we see that first sight of a contradiction so so vinberg's algorithm is saying that the dink and diagram of this lattice is infinite but um here we see in an argument saying that the dink and diagram with a complement of this d four is finite well um the the complements of this d four is the even sub lattice of i 21 1 and the even sub lattice usually have the same automorphism group but in this particular case these things have different automorphism groups in fact the automorphism group of this lattice is three times as large as the automorphism group of that lattice in some sense well i mean they're both infinite groups one one has index three in the other um and um in order to understand what's going on it's easier to look inside a small dimension so this is in 22 dimensions which is a bit hard to figure out but we can see the same thing going on with d four so d four is contained in the lattice i four so i four is just all vectors m one not m four with m i in z and d four actually can is contained in three different copies of i four um and we can see this as follows so d four is the set of all vectors like this with the sum of the mi even and we can um extend d four in three different ways so we can either add the points m one up to m four with m i odd and this just gives us the lattice i four or we can add the points um m one up to m four with mi in z plus half and the sum even or we can add the points m one up to m four with mi in z plus half and the sum is odd um and if we do this we notice the vector a half a half a half a half actually has norm one and in fact we can find enough other vectors of norm one that we're actually getting that this lattice is actually isomorphic to i four so we've actually got three different copies of i four um you notice this is something funny that happens whenever the dimension is congruent to four modulo eight except in general we won't get norm one vectors but in dimension four we happen to get norm vectors um and the same sort of thing is happening for um if we take i 21 one and take its even sub lattice is contained in three copies of i 21 one um so um what's going on is that um um the even sub lattice and this lattice here have different automorphism groups because this has many more roots of norm four um so these aren't roots of norm four of this unimodular lattice but they are roots of norm four if it's even sub lattice so this is how how the um how the how the um dink and diagram of the even sub lattice of this lattice can actually be finite even though the dink and diagram of this lattice is infinite we now have a sort of summary table as follows so let's do a summary so let's take the lattices d eight perp d seven perp d six perp d five perp d four perp d three perp d two perp so um you notice d three is just the same as a three and d two is the same as a one squared so i'm going to take these are all going to be sub lattices of rent in lattice um in 26 dimensions so here we're getting the lattice i 17 one rather we're getting the even sub lattices of these lattices here octa i um 23 one um and um we can get three different sorts of roots we can get the roots corresponding to a one d n which are all norm two roots or we can get the roots corresponding to d n plus one which are all norm one or we can get the roots corresponding to e something e n so so um um here that there aren't any um so this should be e n plus one because e nine isn't a spherical dink and diagram but here the roots of norm one um two three four five and six and there's another special property that these roots have which that they are parity vectors so um let's see what happens uh well here the extra roots of norm one or two so they're they're just roots and this gives bin bergen kaplan skyer's example when the um when when when when the um reflection group is still infinite well here um a norm three vector is not a root so we get the dividing line between the cases when this lattice is as finite dink and diagram the cases when it's an infinite dink and diagram but this case is a bit funny because these vectors are still roots of the even sub lattice the point is they have norm four but they're also parity vectors and um this means if you take the roots and take the inner product with v um this is always even and this means that um true r v over r r is now always an integer because this is four and can and we've got a factor of two here and a factor of two here so so in this particular case we still get roots and in these cases it's still we still get knock um they're still not roots so there's this funny um extra thing going on in in in in 22 dimensions where we unexpectedly get some roots um well we can also um make a closer look at the um case of d five so the case of d five um perp was the even sublattice of i 21 um and this still has um infinite dink and diagram um but we can ask can we describe the dink and diagram even though it's infinite and the answer is we can sort of and the point is the fundamental domain of the of the reflection group is tree like so what do i mean by this well let's draw um hybolic space as the unit disk and here i i'm really talking about hybolic space in 20 dimensions but i'm drawing it as if it were two-dimensional and if we draw the fundamental domain then you can think of it as being something like this what's happening is um we get a lot of roots um and these are kind of dividing up the um fundamental domain and the fundamental domain is is um looks a bit like this we get a sort of tree like structure and the reason we get a tree like structure is um we can add in the hypo planes of norms of the norm three vectors and these are not roots but they still divide up the fundamental domain into pieces of finite volume they were roots then then we would get a finite volume fundamental domain and what what we end up with is we get a sort of tree here um we can check that we're getting a sort of nice structure looking like this except there aren't three things coming from each node there are quite a lot of things coming from each node but um and um if you've got a tree you can quite often describe its automorphism group as an amalgamated product um uh seh has a nice book on this called trees and there are a few pictures of this so here's a picture of a tree and here's a picture of the upper half plane model of um hypovolic space where he's drawn a sort of tree here and this tree is acted on by sl2z acting on the upper half plane in the usual way and seh points out the fact that sl2z is acting on this tree means you can write sl2z as an amalgamated product of two groups in fact it's an amalgamated product of a cyclic group of order four and six amalgamated over a cyclic group of order two and since this is a bit hard to read and the same thing happens here but because the fundamental domain has a sort of tree-like structure we can actually describe the automorphism group of the fundamental domain as an amalgamated product so the automorphism group of the dinking diagram can be written as follows it's g e6 amalgamated over g d5 containing e6 with g d5 so what does this mean well this means that the sub the subgroup of the group of automorphisms of each lattice that preserves this e6 dinking diagram and this means the sub subgroup that preserves both the d5 and the e6 contained in it and similarly for this one and we can work out the orders of these groups this is a little group of order 72 this group has ordered 36 and this group has ordered 1440 so this gives us a description of the automorphism group of the dinking diagram it's just an amalgamated product of these three groups here and you notice that these groups have indexed two and this is kind of because this reflection is sort of trying to be a this sorry this vector it's trying to be a root giving you a reflection and if it was a reflection then this group would be this group times a cyclic group of order two and it would have indexed two and this still has indexed two but it's not quite a product um it's I mean it turns out what you're getting is not a reflection but a kind of reflection twiddled by another non-trivial automorphism so um in some sense if if if we if if the vector here was a root then we would sort of delete this bit of the dinking diagram and the automorphism group of the dinking diagram would just be this but but we have this extra bit because the vectors we're getting here are not roots um so um um what what um should meant them that um in in the case of the even sub lattice of 21 comma 1 which had a finite dinking diagram um Dolgachev and Kondo found that the this group would say the even vectors of this actually turns up naturally and algebraic geometry um in fact there's a super singular A3 with a picard group of it so sorry super singular K3 and characteristic two with a picard group that corresponds to this lattice here and um as a result you get the this um this dinking diagram with finite reflection group turning up naturally and algebraic geometry um um you can ask what happens if you go to um higher dimensions so we've done the case of ii 25 1 well then you can ask what happens if you go to the next case which is ii 33 1 and here you seem to get a horrendous mess there seems to be no analog of the leach lattice there's nothing with covering radius root 2 the dinking diagram is absolutely huge um you can see roughly how complicated it is by by looking at the number of orbits of norm zero vectors in the following case so for ii 9 1 there's one orbit of norm primitive norm zero vector for ii 17 1 there are two orbits here we get 24 orbits found by Niemeyer um this one we get more than a billion orbits this is an estimate due to oliver king who used a refined form of the minkowski zeagle mass formula to find a um a very large lower bound for the number of orbits of norm zero vectors here so these correspond to 32-dimensional even unimodern lattices so they're at least a billion of those and classifying them probably would be within reach of a big computer but i mean the the main obstruction is rather hard to think of any point in classifying them i mean there are huge numbers of lattices they all look very similar and classifying them it's a bit like trying to classify all grains of sand on a beach i mean the grains of sand are all different but who cares um so what seems to be happening for vinberg's algorithm is that for small dimension less than or equal to about 25 things are kind of tame and we can quite often classify them for dimension greater than about 25 the reflection groups all seem to become rather wild that they're very large and rather difficult to describe and they all have infinite dinkin diagrams and so on and the thing that seems to be causing this difference is the leech lattice so the leech lattice is sort of on the boundary the leech lattice is the dink and diagonal of the 26-dimensional Lorentzian lattice and what seems to be going on is that um you can use the 26 dimensional lattice to control things in dimension less than 26 and and many of the reflection groups are reasonably nice at least if they're small discriminant but once you go beyond this things just go crazy um people have of course um um done a lot of work on classifying lattices with finite reflection groups in dimension up to 25 um the classification is a bit complicated because there are all sorts of different things you can look at so for instance the fundamental domain can be compact or it can be finite volume but not compact or it can be infinite volume and the infinite volume case is still interesting for example um Conway's example of the 26-dimensional even Lorentzian lattice is perhaps the most interesting of all um reflection groups and that has a fundamental domain of infinite volume so you you can't even restrict a finite volume you can also ask um should the um should you take a lattice or or you can ask for um could be a lattice over z or over um some sort of number field um or it could be um a non arithmetic um group and lattices over z are what we've been discussing um there are some quite striking examples over number fields in particular the biggest known examples um over um um the biggest known examples with compact fundamental domain were found by Boogayenko and he was working over a number field um generated by a square root of five um his examples are still very mysterious um they're in about eight dimensions and I don't know how to relate them to um this group here although I suspect there might be some relation um also um we can talk about whether um the reflection groups are maximal or non-maximal and I just summarise a few of um the examples um um for more about this there's a very nice survey article um written by um Benelopetsky and I'll try and put a link to that in the video description below um so the most striking result of all is by Wienberg which says there are no finite volume in large dimensions and I find this result really amazing because um for Euclidean and spherical groups there are finite volume or compact examples in all dimensions and if you look at hyperbolic groups in small dimensions there seem to be vastly more hyperbolic reflection groups than than for Euclidean and spherical reflection groups and and you'd expect this to get even worse as dimension increases and it doesn't first but when you start hitting 20 dimensions the number suddenly plummets down to almost zero um so that the largest known example is in um hyperbolic space of dimension 21 um so Esselman showed that if you look at reflection groups of lattices then this is indeed the biggest possible dimension but whether you can find examples over other number fields or non-arithmetic example feels seems to be a bit unclear um um nickel in has classified many of the examples in dimensions in in small dimensions and there are really hundreds of them you there are just too many to do by hand and you you need to get a computer running Vinberg's algorithm in order to find them um um and um there are some finiteness examples for instance there are only a finite number of maximal reflection groups in each dimension so altogether there are only a and as Vinberg bounded the possible dimensions this shows that there are a finite number of maximal hyperbolic reflection groups Daniel Allcott showed that um up to dimension 19 there are actually an infinite number of non-maximal reflection groups so so if you want to find out this you you need to restrict the maximal ones um well um what else can you do with these reflection groups well there's one very obvious thing you can do with them these reflection groups are all giving you dink and diagrams now for the finite and euclidean dink and diagrams um these are all dink and diagrams of lee algebras you get the finite dimensional semi-simple lee algebras and you get the affine cat's moody algebras so an obvious question is do you get interesting lee algebras corresponding to the hyperbolic reflection groups um so this will be the subject of the third part of this talk which will appear in the next video