 All right, great. So today it's a pleasure to introduce Logan Crew, who's speaking about chromatic symmetric function, the chromatic symmetric function with a K. So please go ahead. Thank you very much. It's a pleasure to be here. And yes indeed, we are going to be talking about the chromatic symmetric function both the usual one but also the subject of our work which is a K. Theoretic Analog. And this is joint work with Oliver Pachenec and Sophie Spirkel also both at the University of Waterloo. And let's get started. So we're going to be working with graphs. So just to be on the same page everyone a graph has a set of vertices V and a set of edges E that are pairs of vertices. Edges would be represented by going between two vertices. So here is an edge between Minneapolis and San Antonio in this map of a select number of US cities that have flights between them on Delta Airlines. And so this is the graph Delta. And this is just one example of a graph. It'll be the one we're using throughout this talk. So we're going to be working with graph colorings and in coloring of a graph is an assignment to each vertex of a positive integer one up through in. It's going to be a proper coloring if for every edge in the graph the endpoints get different colors. On the left I'm using actual colors to represent the integers. This is a proper three coloring because every edge connects two vertices that have received a different color. Over here on the right this is an improper three coloring because these two vertices have received the same color and they are connected by an edge. And people in general care a lot about proper colorings and counting them. So first there was the chromatic polynomial which was introduced by Birkhoff and has been the subject of a lot of study throughout the 1900s. We're going to start here with the original chromatic symmetric function which was a generalization proposed by Richard Stanley. So for the chromatic symmetric function we care not only about the number of colors used but we care about which ones in a certain sense. So this is a function in a countable number of variables. So a power series of sorts. And we sum overall proper colorings into each one we give a monomial for each vertex this monomial has an x indexed by the color it receives. So in particular the number the exponent of x sub i is the number of vertices that receive color i. So in particular this monomial is counting not just how many colors are used but also how many times each color is used. And that is the distinction here. And so first of all this function is a power series in with coefficients in in this particular case it's going to end up being in that being in natural numbers but generally it's over something. And the symmetric function because for every permutation of the positive integers it fixes the function. And this is because if we take a proper coloring of our graph and permute it, it's still going to be a proper coloring. And properly, we note that if you were to specialize by setting the first in of the variables equal to one. So we're going to recover Kaiji event because in this sum up here this product is going to be a one if and only if all of the x sub i have index less than or equal to in then meaning that it is a coloring with colors from the set what they're in. And it's zero otherwise. Also if anyone has any questions during the talk you're welcome to just say them in the chat or otherwise I don't know how that works. In terms of computing the chromatic symmetric function we're going to look at the specific case of the graph delta. So over here is a proper three coloring and we have three instances where the color one has been used represented by blue, and one time each that the colors two and three have been used. So we get a manual x one cubed x two x three. There's a different three coloring that uses one twice and uses two twice and uses three ones. And for that case we get an x one squared x two squared x three. So these are two different three colorings which would be counted the same by the chromatic polynomial but of different manumials assigned to them by the chromatic symmetric function. And the chromatic symmetric function itself is formally the sum of all of these things. There's another way to talk about computing x g to because right now it's in an infinite form and we want to put it in a finite form. So we're going to introduce a method of of taking care of the symmetric functions with finite terms. So in particular what's happening here is that the x one squared and a given monomial represents two vertices that get the same color, meaning that there's not an edge between them. So more generally, Xi to the in is in vertices all of which must be pairwise non adjacent in a proper coloring. So to kind of count this more easily let's let I denote any arbitrary partition of the vertex set of g into stable sets. And then we're going to let lambda of Pi be the multi set of integers arising from this by just counting the size of each block of Pi we call this an integer partition. So here's what are going to index our symmetric function basis elements. So here's a particular symmetric function. If you fix a given integer partition. So in this case, let's say we are looking at the partition that is two twos and one one, usually written by writing the parts and decreasing order left to right. m two to one, and that's going to be a specific sum of monomials in this case the sum of every monomial exactly once who that has three distinct variables two of which get exponent two and one of which gets exponent one. So m two to one has an x one squared x two squared x three. It also has all of the permutations of this like x one squared x two x three squared. It also has every other monomial you can make with exponents to twos and one one for across three distinct variables. This will be a symmetric function it's essentially one of the smallest ones we can get by symmetrizing with a given monomial. Now if we look at this graph, this is kind of where this is coming in is that this particular monomial here comes from a stable set of size three, and this two stable sets of size one. So what we're kind of hoping to say here is something along the lines of that. Symmetric symmetric function in general can be written as a sum overall stable set partitions of the vertex and the stable sets of this m lambda. Except this isn't quite right, because we need to account for the fact that we get the same monomial if we switch these two guys. And to get this graph over here, this is using the same stable set partition but as a different coloring that is also contributing the same monomial. So we have two parts of size one. In general, if we had five parts of size one, then that's going to contribute five different excise. And those five vertices could have those colors permuted and that's going to still give us the same monomial. So we do some overall stable sets but we have to also multiply by this factor for each I the number of I and lambda in the lambda of pi factorial. So we're placing m lambda with this overall function. This is a, this is an expansion of the chromatic symmetric function to these items. So for example, here there is one way to split the graph into five stable sets each of size one the trivial one. Now that there are four non edges of this graph so we get four m 2111. And there is an m 221 we just showed that earlier, I think, and one m 311. And those are all of the different ways you can partition the graph into stable sets so this is the complete expansion of x delta in this basis. I haven't explicitly said yet this is a basis. It is a basis for space of symmetric functions. Now, one thing that Sophie Spirkel and I did a few years ago was look at this on vertex weighted graphs. This is something that's been considered in a similar form by other people before noble and welsh and the W polynomial and other people in the context of the silly have not invariance. And what we want to do is we want to give an integer weight to each vertex. And what we do is that for each vertex we take X of its color and raise it to the weight. So now in this context, the exponent of Xi is the total weight of all vertices receiving color I. There's a couple of reasons why this is a nice thing to do. The reason that we did it was to make it that so that there's a very direct and simple usage of the deletion contraction relation for the chromatic polynomial in the sense that we get that the chromatic symmetric function of a vertex weighted graph is equal to that graph with the edge deleted minus that of the graph with the edge contracted where we also need to contract the weights. So when we take two vertices of weight and weight in and we contract we need the new vertex to have weight and plus in. And so as an illustration here, this is a proper coloring. And it's also a proper coloring of the graphics G. So this is so this is rewriting the formula in this form. And this is G delete E. And the coloring of this where these get different colors is also coloring when you add the edge back in and has the same monomial. On the other hand, if these vertices get the same color, then it's instead going to be a coloring over here where I give that color to the contracted vertex and I still have the right term because this green to the fifth is maintained across contraction. And to say a bit more about the algebra symmetric functions that we're going to be using those M lambda form of basis. Some other bases that we're going to use the elementary symmetric functions, which are defined by ease of in is good for in the partition that only has the integer in is just the sum over all monomials within distinct variables. So E 3 2 multiplies across parts. So if you have an integer partition with multiple integers you just take each one separately. So E 3 2 for example is going to be this product. And the power some symmetric functions are defined in the similar way except that we take P in to be the sum of every variable to be in power, and then we multiply across parts so P 3 2 is this. So another thing that's nice about vertex weighted graphs is that we can represent individual symmetric functions by a graph. The m basis is represented by a complete graph with the corresponding with its parts as corresponding vertex weights. The power some basis is the chromatic symmetric function of this graph where it has the vertex weights but there are no edges. The E basis can be represented as a disjoint union of cliques. So E 3 2 1 up to a constant is going to be in the clique of size three clique of size two and a clique of size one. That should be up to a constant. And another symmetric function basis that we're going to be talking about but that doesn't admit a simple graph interpretation is the shore basis. The shore functions are defined among other ways in terms of filling boxes of a diagram with positive integers and that is the approach I'm going to discuss here. So a shore function first for a integer partition lambda is formed by first taking a young diagram of shape lambda, which is going to consist of length of lambda here is the number of distinct parts of lambda number of numbers used. And it's going to have that many rows that are justified to the top and left here, and the I throw from the top contains lambda I boxes so this is the diagram shape two, two, and one. A semi standard young tableau of shape lambda is going to be a filling of this diagram so that you put in each box one positive integer, so that the rows are weekly increasing from left to right, and the columns are strictly decrease strictly increasing from top to bottom should be changed. So this is a semi standard young tableau of shape two to one, we have one is less than or equal to one to is less than three. So two three going down and one three going down. Then the shore function of type lambda is given by summing over all of these fillings, and we multiply across all I Xi to the number of times I occurs and T. So in the shore function of type two to one we have from this tableau and X one squared X two X three squared. We care about sure functions, especially given that they're their definitions is a bit out there and they don't have a simple graphical interpretation. Sure functions are some of the nicest symmetric functions in a couple of relevant ways. For one, we have a connection to representation theory, one can map the sure function of type lambda to the irreducible summit and symmetric group character of type lambda. There's being a graded ring isomorphism and actually an isometry with respect to the right inner products of the space of symmetric functions with the space of class functions over permutation groups. On the other hand there's a natural connection with topology. This also represents in a natural sense the co homology class of the Schubert variety indexed by lambda. So there's at least two different senses, depending on what you might know, or what you might care about in which sure functions are very fundamental. So we're going to get into the K chromatic symmetric function soon. We're going to do one more topic before the break I think, which is, we're going to motivate first of all why we ended up to find the function the way we did. So we're going to talk about Groten-Diek functions, a K theoretic function that for our purposes we're going to think of as being a lift of the sure functions in a natural sense, whereas the sure functions represent co homology classes of Schubert varieties Groten-Diek functions are representing the structure sheaf of those same varieties. So the symmetric Groten-Diek function work for our purposes we're looking at it in terms of multi valued young tableau. So we're looking at the same young tableau that we did for sure functions but instead of filling each box with a single positive integer we fill it with as many positive integers as we want, subject to the constraint that we still want them to be semi standard tableau when we pick one from each for each box. Equivalently, the maximum number here should be less than or equal to the minimum number here, and every number here should be less than every number here. So we have that valid multi value tableau of shape two to one, since two is less than or equal to two, four is less than or equal to five, and going down here we get one two four seven, and going down here we get two and then five eight. So these stable Groten-Diek functions and represent them by an S lambda with a bar over it to indicate this relationship as a sum over these multi valued tableaus of this product with the added constraint that we usually will give it a sign of minus one to the total number of numbers used minus the number of boxes used. So in this case it would be there's two more numbers than there are boxes. So the sign would be positive for this particular function. I think this is a good place to take a five minute break. Very good. Thanks very much. Any quick questions for Logan before we take our five minute break?