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Diagrams of affine permutations and their labellings

Author(s)
Yun, Taedong
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Massachusetts Institute of Technology. Department of Mathematics.
Advisor
Richard P. Stanley.
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M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582
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Abstract
We study affine permutation diagrams and their labellings with positive integers. Balanced labellings of a Rothe diagram of a finite permutation were defined by Fomin- Greene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict balanced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define set-valued balanced labellings in which the labels are sets of positive integers, and we investigate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings is shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian. Moreover, for finite permutations, we show that the usual Grothendieck polynomial of Lascoux-Schiitzenberger can be obtained by flagged column-strict set-valued balanced labellings. Using the theory of balanced labellings, we give a necessary and sufficient condition for a diagram to be a permutation diagram. An affine diagram is an affine permutation diagram if and only if it is North-West and admits a special content map. We also characterize and enumerate the patterns of permutation diagrams.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
 
Cataloged from PDF version of thesis.
 
Includes bibliographical references (pages 63-64).
 
Date issued
2013
URI
http://hdl.handle.net/1721.1/83702
Department
Massachusetts Institute of Technology. Department of Mathematics.
Publisher
Massachusetts Institute of Technology
Keywords
Mathematics.

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  • Mathematics - Ph.D. / Sc.D.
  • Mathematics - Ph.D. / Sc.D.

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