Digital Repository at the University of Maryland (DRUM)  >
Theses and Dissertations from UMD  >
UMD Theses and Dissertations 

Please use this identifier to cite or link to this item:

Title: Combinatorics of K-Theoretic Jeu de Taquin
Authors: Clifford, Edward Grant
Advisors: Tamvakis, Harry
Department/Program: Mathematics
Type: Dissertation
Sponsors: Digital Repository at the University of Maryland
University of Maryland (College Park, Md.)
Subjects: Mathematics
Keywords: algebraic geometry
jeu de taquin
Issue Date: 2010
Abstract: Thomas and Yong [5] introduced a theory of jeu de taquin which extended Schutzenberger's [4] for Young tableaux. The extended theory computes structure constants for the K-theory of (type A) Grassmannians using combinatorial machinery similar to that for cohomology. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P. In this dissertation, we see that the root-system uniform rule is well-defined for certain G/P other than the Grassmannian. This gives rise to combinatorially defined rings which are conjecturally isomorphic to K(G/P). Although we do not prove that these rings are isomorphic to K(G/P), we do produce a ``Pieri rule" for computing the product of a general class with a generating class in the type B combinatorial case. We also investigate some symmetries which support the conjectural isomorphism. Moreover, our results combined with recent work of Buch and Ravikumar [1] imply that this conjecture is in fact true. Lenart [2] gave a Pieri rule for the type A K-theory, demonstrating that the Pieri structure constants are binomial coefficients. In contrast, using techniques of [3], we show that type B Pieri structure constants have no such simple closed forms. References: [1] A. Buch and V. Ravikumar: Pieri rules for the K-theory of cominuscule Grassmannians, arXiv:1005.2605, 2010. [2] C. Lenart: Combinatorial aspects of K-theory of Grassmannians. Ann. Combin. 4 (2000), 67--82. [3] M. Petkovsek and H. Wilf and D. Zeilberger: A=B. A K Peters, Ltd. (1996). [4] M.-P. Schutzenberger: Combinatoire et representation du groupe symetrique. Springer-Verlag Berlin, Lec. Notes in Math. 579 (1977), 59--113. [5] H. Thomas and A. Yong: A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus. Algebra Number Theory 3 (2009), no. 2, 121--148.
Appears in Collections:Mathematics Theses and Dissertations
UMD Theses and Dissertations

Files in This Item:

File Description SizeFormatNo. of Downloads
Clifford_umd_0117E_11435.pdf306.6 kBAdobe PDF257View/Open

All items in DRUM are protected by copyright, with all rights reserved.


DRUM is brought to you by the University of Maryland Libraries
University of Maryland, College Park, MD 20742-7011 (301)314-1328.
Please send us your comments