CONFERENCE
Recording during the thematic meeting : «Current trends in representation theory, cluster algebras and geometry» the November 27, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France)
Filmmaker : Luca Recanzone
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So thank you so much uh so let me just give an outline of my three of my three talks so in lecture one I will explain what is the ampedrin and also what is a cluster Algebra I think the latter is going to be helpful for the two talks that follow
Mine in my second talk I will discuss in detail the m equals 2 amplitud hedrin and connections to Cluster algebras and in my third talk I will discuss the malal 4 ampedrin and connections to Cluster algebras so this first talk is meant to be um uh quite introductory so please feel
Free to interrupt if you have any questions um I arrived this morning after a red eye flight so if I’m not making any sense you’re also welcome to correct me okay so in order to talk about the ampid deedon I need to first talk about the grasm monian and and the positive grasm
Monian so the grasm monian grkn of R is the set of all subspaces in an N dimensional Vector space which have Dimension K and we will represent elements of of the grasm monian by full rank K byn matrices is C and then the row span of such a matrix represents a
Subspace so we want to talk about pler coordinates of the grass monan so if I have capital i a k ele subset of the numbers 1 through n [Applause] then the pler coordinate P subi of C is the minor of C in columns I so for
Example say my K is 2 and my n is 4 and my Matrix C is 1 0 – 3 012 1 then the row span of C gives me a two-dimensional Subspace of R4 and the pler coordinate say p13 of C is the determinant of the submatrix located in columns 1 and 3
Which is two okay so that’s the grasm monian and the pler coordinates and now the totally non- negative or TNN and sometimes called positive cronian grkn greater than equal to zero is the subset of gr knen where my pler coordinates P subi are non- negative for all I
Ranging over uh the K element subsets of 1 through n and so this object first arose in work of lustig where he defined the totally positive part of a real reductive group and the positive and non- negative parts of generalized partial flag varieties of which the grasan is a special
Case and then there was follow-up work by Connie reach in which she gave a cell decomposition for the non- negative part of Gmod p and then there was this very influential preprint of postnikov in 2006 where he discussed the special case of the grasm monian um and
Worked out a beautiful um indexing set indexing indexing sets for for the cells so okay so so let me so let me let me talk about the the cell decomposition of the TNN grasm monan so we can partition the TNN grasm monian into pieces based on which pler coordinates are strictly positive versus
Equal to zero so if we let M be a collection of K element subsets of 1 through n um then then we can define a corresponding subset of the TNN grasm monian consisting of all elements in the TNN grasm monian where our pler coordinate p subi is strictly
Positive if and only if I belongs to our collection okay so I’m just going to say okay you give me your favorite collection of pler coordinates and then we will Define the subset of the TNN grasm monian where my pluger coordinates are strictly positive if and only if um
They belong to the collection and otherwise the pler coordinates must be must be zero and um so by work of postnikov and in a slightly using slightly different definitions um we know that every non-empty U stratum SM is actually a cell so homeomorphic to an open ball And moreover the cells glue together nicely and um and this collection of cells is actually a CW complex okay so what we’ve done so far is just Define the TNN grasm monian and Define a stratification into cells uh they’re called positroid cells yes I should write [Applause]
That yes and um and positroid is um meant to sound like matroid so so this is related so one can talk about the matroid stratification of the entire grasm monan where you don’t impose the non- negativity and then um what you get when you when you um superimpose the
Matroid stratification onto the TNN grass monion is this decomposition into positroid cells okay and um and I mentioned that in postnikov preprint he classified all of these non-empty cells so the cell um they’ve been classified and they are in bijection with various combinatorial objects um one one nice one nice indexing set is
Um equivalence classes of Playbook [Applause] graphs uh which I’ll say more about in my in my third [Applause] talk question okay so now I want to Define this object called the amplitud hedrin and this object was was defined by physicists Aran Hamed and trinka in
2013 so okay so how do we Define the amplitud hedrin so first we’re going to fix positive integers and K&M with k+ M at most n and we’re also going to fix a matrix Z and this will be an N by k + m Matrix with maximal minor
Positive so this is an N by K plus n Matrix okay so this Matrix is going to be a sort of a a tall and skinny Matrix because of this inequality and so the maximum minors will be the k+ M by k+ m minors and we’re requiring that these are
Positive and now I’m going to define a map Z Tilda so this is going to be a map from the n grasm monan grkn to another grasan of K PLS and K plus m space and this is a map that takes the K byn Matrix C to the matrix product
CZ so Z is n by K plus plus M so this matrix product CZ is a k by k + m Matrix um and again so so remember I’m always identifying matrices with their row span so this Matrix C represents an element in the TNN grason of KL and n
Space and I I will identify this K by K plus n Matrix with its row span and um in order to convince you that this map is well defined let me just point out that one can show that the fact that Z has maximal minors positive implies that this Matrix has
Full range Rank and hence represents a point in this grass monan okay so so what I’ve done so far is Fix N K&M and a matrix Z and then I’ve defined this map Z Tilda from this TNN grasm monan to a related grasm monian and now what’s the amplitud hedrin
It’s going to be the image of the TNN grasm monian under this map so the amplitud hedrin which depends on these three numbers n k n km together with the Matrix Z is by definition the image of the TNN grasm monian under this map and so this is going to be a
Subset of the grassian of K planes in k+ m space okay so maybe I should say a few words about motivation here so the motivation is coming from Nal 4 super Yang Mills Theory and um so there was this famous recurrence bcfw recurrence bcfw stands for brido Kazo
Fang and Witten and this bcfw recurrence uh which is from like 2005 expresses scattering amplitudes as sums of rational [Applause] functions of momenta and in this recurrence so This recurrence has a whole bunch of terms and the individual terms have spous pulls [Applause] meaning singularities not present in the
Amplitude so when you sum all all of the terms up you get something where these poles have have disappeared but the bcfw recurrence itself is this sum of terms where where you get some some spous spous poles and um and then in 2009 PES observed that in some
Cases the amplitude is actually the volume of a polytope with these spous poles arising from internal facets of a triangulation okay and um and so motivated by these examples that he looked at he asked is the scattering amplitude the volume of some geometric object [Applause]
And um and then in this 2013 work of Aran Hamed and trinka they said yes and the geometric object is the ampedrin and then in this paper um uh which was which had all these constructions but the actual statements were largely conjectural they interpreted the bcfw recurrence as giving a quote unquote
Triangulation of the amplitud hedrin okay all right so um that’s all I was going to say about the physics C motivation but if you have any questions I can try to answer them so the amplitud hedin um is full dimensional inside of this grass monan
And so it has Dimension K * m but um for the for the um the application they had in mind which is these scattering amplitudes um the relevant for for the scattering amplitudes in n equals 4 super Mills and for the bcfw recurrence the relevant case is where little m
Equals 4 and so then it will be a 4K dimensional object uhhuh we getu yes yes so you get actual poles from the boundary yeah okay so now let me um talk about a few examples so um I guess one point I want to make is that the ampedrin generalizes lots of nice
Objects so first of all if Z is a square Matrix then the map Z Tilda which is basically given by matrix multiplication is an isomorphism so Z Tilda will be injective and then the ampedrin just recovers the TNN grass mononon okay so that’s sort of a boring example um the
Next example which is more interesting is when say k is 1 and M is 2 and then the claim is that this amplitud hedrin which will lie inside gr K comma k+ M which here is gr1 comma 3 which is the same as P2 is a polygon
In projective space so let me just say why let me just convince you that this is true so in this k equals 1 example so n will be General um but K is 1 then my map Z Tilda is going from gr K comma n which is gr1 comma
N so Z Tilda will be a map taking points in in the non- negative part of gr1 comma n so all of these coordinates are non- negative and it Maps this element of the T TNN grasm monan to the matrix product of this row Vector times my Matrix Z which is an N
By k + m Matrix or in other words an N by3 Matrix and now notice that if EI if I use EI to just denote the 01 Vector with a one in position I then Z Tia of EI is just going to be my I row
Vector of Z so let me actually write out the rows of Z I’ll call my rows Z1 through ZN so now if I apply Z Tilda to this 01 Vector EI then Z Tilda of EI just just gives me Zu subi and um okay and now if I draw so so remember
I’ve required my Matrix Z to have maximum minors positive and so what that will mean is if I draw my vectors Z1 through ZN and P2 then they must be in convex position that’s what the that’s what the positivity of the minors tells me um so let me just write that so
Positivity of the minors of Z implies the Z are in convex position and we have that um EI always maps to zi but now I’m more I’m what I’m really interested in is the image of of my Vector A1 through a an under this map
And A1 through a an is just a positive combination of the E and so a general Point A1 through a n is going to get mapped to a positive combination of these zis so as the AI vary over non- negative reals the image gives all points All Points of this polygon viewed
In projective space Okay so so this is the example when K is 1 and M is 2 and more generally if K is one and M is arbitrary then the amplitud hedrin which lies in gr1 M +1 I.E projective space of Dimension m is combinator equivalent to a cyclic polytope with n
Vertices in PM okay now any questions so far oh and I’ll add I’ll add one more example to this list so if we take instead the case where m equals 1 then the m the tedron is homeomorphic to the bounded complex of a cyclic hyperplane arrangement in PK and this was a case
That was worked out in joint work with Steven karp okay now from the point of view of physics m 2 and m equal 4 are the most interesting so that’s what I’ll be talking about um in my next couple of lectures it’s been a while since I’ve
Used these um squeegee things so if you have any pointers on technique you’re also welcome to let me know okay now um let me just um say a little bit about our notion of triangulation for the amplitud hedrin because that’s going to take us back to the physics applications okay now the amplitud
Hedrin despite the name it’s not a polytope um and uh yeah so we we can’t really talk about triangulations but we can talk about something that is a bit analogous so recall that we have this um expression we have this decomposition of the TNN Gras molan into cells and we also
Have the surjective map from the TNN grasm monian onto the amplitud hedron and um the TNN grasm monian has Dimension K * n minus K and the amplitud hedrin which sits inside the grasm monion of K planes in K plus m space has full Dimension here so
This has Dimension K * m and in general by our choice of N and K plus M right we we required that K plus m is at most n in general this number is going to be bigger than this number but what we would like to do is understand this
Object the amplitud hedin using cells of the correct dimension on this side so this is dimension K * m so what we want to do is find a collection of K * m dimensional cells here such that Z Tilda is injective on those cells and such that the images of those cells fit
Together nicely to tile the amplitud hedrin so let me make some definitions so if Z Tilda is injective on a km dimensional cell SM then we’ll say that ZM which I’m defining to be the closure of the image of that cell is a tile for the amped toed so yeah so we
Are interested in particular in the km dimensional cells over here such that Z Tilda is injective on them and when we find such a cell then we look at its image and we take the closure and we call that a tile and now a tiling of the amplitud hedrin is a collection
ZM of tiles such that their Union is the whole amplitud [Applause] hedrin and their Interiors are pairwise disjoint so this is going to be our notion of triangulation but there’s nothing you know there’s nothing polytopal here and there’s nothing to do with simplicity here but this is this is how we would
Like to understand the amplitud hedron yeah yeah so sometimes we ask for something stronger um so we Define a good tiling to be a tiling where if two tiles do intersect you want them to intersect at something that’s like a common face of both so in general we
Won’t require that for a tiling but in some of the things we do we talk about good tilings and then we care about how things intersect yeah okay sorry ah good question the question was do tilings always exist and um and uh they don’t they don’t so for
So for m equal 1 for malal 2 and for malal 4 there are many such examples um and for for for a particular value for yeah for a fairly small n and K and for m equal 6 we already have an example where a tiling doesn’t literally exist yeah
Yeah this so there’s some kind of weakening of the notion where where you can yeah there there’s something weaker that happens but in general tiling as I’ve defined it doesn’t exist but for m 1 2 and4 it does say it again oh um I forget I forget the
Details but I could try to say more after if you like okay all right so now just to go back to the motivation of the amplitud hedrin the statement the claim that the volume of the ampedrin [Applause] computes scattering amplitudes is very closely related to the statement that certain collections of bcfw
Cells in the TNN grasm monian give a tiling of the malal 4 amplitud hedrin okay so so somehow it was basically um more or less the statement appeared in the first paper of Aran Hamed and trinka basically their their sort of reason for the definition of the
Ampedrin or the reason the ampedrin was the right thing boils down to um the conjectural statement that certain distinguished collections of cells in the TN and grasm monian give a tiling of the edron and um and actually in that first paper um they conjecture that there are many different tilings of the amplitud
Hedrin and one such tiling was proved for the first time in a paper of Evan Zohar lre and Tesla so one such bcfw tiling was given in their paper from 2021 and then in recent work with them together with matate paresi and Melissa sh Bennett um we showed that all
Conjectural bcfw collections of cells in the TNN cmon actually do give tilings of the amp hedron so I’ll I’ll say more about this in my third lecture but not not much more for not any more about it today oh wait that’s not what I meant to do
Okay all all right so now what I want to do is shift gears and give a quick introduction to the to Cluster algebras oh question yes yes um so let’s see Um yeah yeah we don’t yeah the things that we get here don’t need to be regular yeah yeah the notion of TI the notion of tiling is fairly relaxed it’s fairly weak um yes yes yes okay so next I want to Define cluster algebra because cluster algebras and
Cluster coordinates wind up being very closely connected to the study of the ampedrin and in particular the m equals 2 amplitud hedrin is very closely connected to the cluster structure on the grasson of two planes in n space and the ampedrin for m equal 4 its structure is very closely
Connected to the cluster structure of the grasm monian of four planes in n space so basically the the idea is that the The Little M parameter in the definition of the grasm monian in the definition of the ampl hedron that M uh yeah maybe let me just uh say let me just write
Something so so this amplitud hedron is closely connected to the cluster structure on the grasm monian of M ples in n space okay so that’s that’s a a theme that we’ll see in the next two two lectures but first I should just Define cluster algebra so apologies to those of
You who have seen this definition many times before okay so cluster algebra were defined by fian and zinsky around the year 2000 a cluster algebra is a particular kind of a commutative ring with some distinguished generators and um and uh and these cluster albas have a have a very rich combinatorial
Structure um there’s also a geometric analog notion of cluster variety which is roughly a variety whose coordinate ring homogeneous coordinate ring has a cluster algebra structure okay um so so cluster algebras are defined using Quivers so a quiver is a finite directed graph multiple edges are allowed oriented
Cycles of length one and two are forbidden and there are two types of vertices there are the mutable vertices and there are the Frozen vertices so let me draw a picture here so I will denote mutable vertices by dots and I will denote Frozen variables by little squares to remind you of
Frozen ice cubes so here are some arrows multiple edges are Fine okay and um okay and uh all right n will be my number of mutable ver well okay fmin and salinsky use use n and M for the number of mutable vertices and the total number of [Applause] vertices this should not be confused with the n and the M of the amplitud
Hedrin I could try switching notation right now but I’m not sure if I could succeed in carrying it through without a mistake uh okay so all right now um now I want to Define quiver mutation okay so quiver mutation is an operation that I can perform if I choose a quiver
And I choose a particular mutable ver so okay so here’s quer mutation let’s let K be a mutable vertex of Q then we can mutate at K to get a new quiver that will denote by mu subk of Q and we perform quiver mutation quiver mutation is obtained by three steps
First for each path of length two whose Central node is K we introduce the edge J to L then we reverse the direction of all edges incident to K and finally we remove oriented two cycles okay so let’s take Q to be this quiver and let me
Draw let me draw another copy of it down below and then we’ll start doing this uh we’ll start mutating at it so let’s say that this is my vertex k then the first thing I’m supposed to do is look for all Paths of length two
Whose middle node is K so I have two Paths of length two that do this and for each one I need to add a new Arrow from the start to the finish so I should add two arrows like this and then I also have a path of length two that does this
And so I should add a new Arrow going from the start to the Finish okay so I just performed step one now I reverse reverse the direction of all edges incident to K so this one will go the other way these will go the other way this goes the other
Way and now finally I remove oriented two cycles and I have an oriented two cycle here so I can cancel this white arrow with one of the orange ones and this is what I have left so this is Mu subk of Q oh and um it’s important to note that mutation is an
Involution so if I mutate twice at the same node I get back to this the original quiver okay and um and then we say that two Quivers are mutation equivalent if you can go from one to the other by a sequence of mutations okay now um there’s also an algebraic counterpart
For Quivers and quiver mutation and that’s going to be the notion of seeds and Seed mutation so we let F be a field of rational functions in M independent variables over C and then a seed in F is a pair consisting of a quiver together with what’s called a cluster [Applause]
X where Q is a quiver with n mutable vertices and M minus n Frozen [Applause] vertices and this xar is an extended [Applause] cluster which means an M Tuple of algebraically independent elements of f indexed by the vertices of Q so a very um boring example would be Perhaps Perhaps our
Quiver is this type A3 dinkin quiver and uh and our extended cluster could must be the labeling of vertices by X1 X2 and X3 and we’d consider this AED in the field of rational functions in algebraically independent variables X1 X2 and X3 okay so a seed is basically a
Labeling of a of a of the nodes of a quiver by algebraically independent variables algebraically independent rational functions okay and just as we could we had this operation of quiver mutation there also an operation of seed mutation so let’s let K be a mutable vertex in Q and let’s let XK be the
Corresponding cluster variable oh maybe I should say yeah the elements in xar are called cluster variables so in this example each of X1 X2 and X3 are called cluster variables so then the seed mutation UK from QX to Q prime xar prime is defined as follows so our new quiver is what you
Get when you mutate the old quiver at vertex K and then this new extended cluster is nearly the same as the old cluster but we’re going to throw in one new cluster variable indexed by the vertex K and we’re going to remove the old cluster variable associated with node
K where x k Prime is defined by the following equation so XK * XK Prime so the old times the new cluster variable labeling the cith node is equal to the sum of two monomials and one monomial is associated to the set of all arrows in the quiver pointing to
K the other monomial is associated to all arrows in the quiver pointing out from K and we’re just multiplying together the cluster variables on the other side of that edge okay okay so seed mutation is a new seed where the new quiver is obtained by mutating the old quiver node k
And our new extended cluster is a is obtained from the old extended cluster by swapping out XK for XK Prime according to this [Applause] relation okay and just as before seed mutation is an involution okay so um yeah so remark C mutation is an involution okay so now I can finally Define cluster
Algebra so we can let Sigma be a seed with my extended cluster equal to some set of of mutable variables called initial cluster variables together with the Frozen cluster variables and then we let Kai be the set of all cluster variables obtained from X by all possible sequences of
Mutations and we’ll let R be our ground ring consisting of laurant polinomial in the cluster in the Frozen variables and then finally the cluster algebra Associated to our seed is the ring generated by all cluster variables with coefficients in the ground ring so this is going to be a sub ring
Of the field of all rational functions F so this is the r sub algebra generated by all of our possibly infinite cluster variables okay and um let’s see I guess I have three minutes left okay so I’ll just mention then let me just give myself a little more more space right here so
Um so everyone’s favorite example is that of the grasm monan of two planes in n space and uh and a way that we can get a seed for this um yeah so we can talk about the grasm monian of two planes in n space and then we can talk about the
Coordinate ring of the Aline cone over the grasm monian so this is the homogeneous coordinate ring and this is generated by the pler coordinates P sub i j where I is less than J and so we can get an initial seed for this cluster algebra by taking a triangulation of the
Enon and then what I do to get a quiver is I put a mutable node on every diagonal I put a frozen node on every boundary segment and then I inscribe directed triangles say oriented clockwise in all of my white triangles of my triangulation and then my cluster variables and Frozen variables get
Labeled by pler coordinates so since since each vertex lies on a diagonal or a side I can label it by the corresponding PL coordinate and this one is p24 and then the claim is that this gives a seed for a cluster algebra which can be identified with the
Homogeneous coordinate ring of the grasm monian and it’s it’s a nice exercise to check that mutation of a triangulation so mutation of a mutation of a mutation at a node in our in my orange quiver corresponds to flipping a diagonal in the Triangular relation and the exchange relations you know this algebraic
Relation here exactly corresponds to a three- term pler relation and uh and then we will see so I guess I didn’t write down the definition but if I have a collection of cluster variables that appear in a given cluster they are called compatible and um and compatible collections of cluster variables exactly
Correspond to collections of non-crossing diagonals in my polygon okay so I think I should stop [Applause] there