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http://hdl.handle.net/1903/10792

Title:  Combinatorics of KTheoretic 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 combinatorics jeu de taquin tableaux 
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 Ktheory of (type A) Grassmannians using combinatorial machinery similar to that for cohomology. This rule naturally generalizes to give a conjectural rootsystem uniform rule for any minuscule flag variety G/P.
In this dissertation, we see that the rootsystem uniform rule is welldefined 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 Ktheory, 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 Ktheory of cominuscule Grassmannians, arXiv:1005.2605, 2010.
[2] C. Lenart: Combinatorial aspects of Ktheory of Grassmannians. Ann. Combin. 4 (2000), 6782.
[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. SpringerVerlag Berlin, Lec. Notes in Math. 579 (1977), 59113.
[5] H. Thomas and A. Yong: A jeu de taquin theory for increasing tableaux, with applications to Ktheoretic Schubert calculus. Algebra Number Theory 3 (2009), no. 2, 121148. 
URI:  http://hdl.handle.net/1903/10792 
Appears in Collections:  UMD Theses and Dissertations Mathematics Theses and Dissertations

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