Student: | Farid Salazar |
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Office: | Hill 323 |
School: | Florida International University |
E-mail: | fsala021 [at] fiu [dot] edu |
Project: | Combinatorial K-Theory |
Mentor: | Anders Buch |
In the realm of algebraic geometry, intersection theory aims to understand features of intersecting closed algebraic subvarieties. One of the main tools to attack problems in intersection theory is to study K-theory. On this perspective K-theory is to algebraic geometry what Cohomology is to complex differential geometry, it encodes very succinctly geometric and topological data. In K-theory, two important constructions are associated to an algebraic variety X: the Grothendieck group K^0(X) of vector vector bundles on X and the Grothendieck Group K_0(X) of coherent sheaves on X. Given a closed algebraic subvariety Y, we define its Grothendieck class as the class of its structure sheaf. The multiplicative structure of the Grothendieck Ring (multiplication given by tensor product) has an interesting property that carries information about the intersection, more specifically given two subvarieties Y and Z that intersect transversally the product of its corresponding Grothendieck classes satisfies [Y]*[Z]=[Y int Z] where [Y int Z] is the Grothendieck class of the intersection of the subvarieties Y and Z.
In this project, we will focus our attention to the combinatorics of K-Theory of Grassmannians. We will study Stable Grothendieck Polynomials that are combinatorial representations of Schubert classes (which form a basis of the K-cohomology of the Grassmanians). In particular, we are interested in the composition of two Stable Grothendieck Polynomials and in formulas expressing their expansion as linear combinations of other Grothendieck Polynomials.
Met twice with my adviser Dr. Anders Buch. Spent some time acquiring basic background in Algebraic Geometry (affine and projective varieties) and the construction of the Grothendieck Group for the category of finitely generated and projective R-modules. Compute a few examples of the Grothendieck Group for R a field and R a PID. Read about representation theory of the algebraic torus and basic operations on representations. Looked at Equivariant K-theory; in particular, compute a few examples of T-stable affine subvarieties and verify the properties of product of two Grothendieck classes. Finally, gave the first presentation of my project to my colleagues.
Two meetings with Dr. Buch and two meetings with Sjuvon Chung (PhD student). Spent several hours understanding the underlying connection between the geometry and the algebra. Read about vector bundles, coherent sheaves, K-cohomology and K-homology and their relationship. Starting to realize how representations fit into Grothendieck's construction (equivariant vector bundles and equivariant sheaves). Computed classes of determinatal subvarieties and notice that the results do not depend on the chosen representation.
Met twice with Dr. Buch and once with Sjuvon. Spent some more time clarifying my understanding of the geometry, trying to keep track of what the constructions capture. Read about morphisms of vector bundles and coherent sheaves. Learned about the Koszul complex and tensor product of complexes as a tool to find resolutions. Furthermore, I started reading the paper Combinatorial K-Theory by Dr. Buch where I learned about basic combinatorial tools such as Young Tableaux, Stable Grothendieck polynomials, Littlewood Richardson and splitting coefficients, and finally the Thom-Porteous formula.
Met once with Dr. Buch and discussed the direction of our project and the necessity to use a computer program to obtain data and make conjectures. Read about the connection of Stable Grothendieck Polynomials and Schubert classes. Understood the relevance of the Littlewood-Richardson coefficients in K-Theory of Grassmannians. Attempted to write a computer code in MATLAB to compute Stable Single Grothendieck Polynomials, LR and splitting coefficients in order to obtain Double Grothendieck polynomials from definition in terms of Tableaux. Read about a recursive definition of Stable Double Grothendieck Polynomials using permutations and isobaric divided differences. Wrote a computer program in MATLAB and computed 720 Stable Single Grothendieck Polynomials and 24 Stable Double Grothendieck Polynomials. Applied Thom-Porteous formula to verify results from determinatal varieties.
Improved previous code to compute Stable Double Grothendieck Polynomials. Read about Grassmannians and how these are realized as projective varieties (via Plucker embedding). Attempted to study composition of Stable Grothendieck polynomials and attempted to come up with systematic way to decompose the result as linear combination of other Stable Grothendieck polynomials. In particular, found a total order of these polynomials with nice independence relations. Computed a few examples of decomposition of products to obtain familiarity with the process for expressing results as linear combinations. Got Maple installed on my computer and started to get familiar with it (this software might be more suitable for the calculations involving Grothendieck Polynomials). In particular, looked at Dr. Buch’s code in Maple and understood the details.
Wrote a code in Maple that expands products of Stable Grothendieck polynomials into linear combination of these polynomials and recover the structure constants. Met twice with Dr. Buch and discussed my progress as well as the specialization to composition of Stable Grothendieck polynomials. Finally, wrote a code that expands composition of Grothendieck polynomials and observed that in some cases it seems to be possible to obtain a finite linear combination in terms of "dual" Stable Grothendieck polynomials and in other cases simply as Stable Grothendieck polynomials. Observed some interesting patterns in the expansion and made two conjectures for particular cases of composition that have some unexpected behaviour.
Met three times with Dr. Buch and discussed the patterns observed in the expansion of "dual" Grothendieck polynomials. Came up with a conjecture for the coefficients of this expansion in terms of combinations. Read about Bialgebra, Hopf Algebra and the antipode map, and found a connection between "dual" Stable Grothendieck polynomials and the image of antipode of Stable Grothendieck polynomials. Read a paper by Lenart about expansion of Grothendieck polynomials in Schur functions and viceversa. Wrote a possible proof for my conjecture using Lenart's formulas and the previously found connection of the antipode map. Presented the progress of the research to my fellows.
Revisited the proof of the conjecture for 'dual' Grothendieck polynomials and obtained further understanding of the Hopf Algebra of the completion of the ring of Stable Grothendieck Polynomials. Revisited exterior product and found more familiar formulas for a few Grothendieck Polynomials in terms of these. Applied these results to prove conjectures of compositions. Sent my proofs to Dr. Buch for revision before including them in final report.
Finished writing final report. End of the program.