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Combinatorial aspects of generalizat...

Drexel University.

Combinatorial aspects of generalizations of Schur functions.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Combinatorial aspects of generalizations of Schur functions.
Author:	Heilman, Derek.
Description:	65 p.
Notes:	Source: Dissertation Abstracts International, Volume: 74-12(E), Section: B.
Notes:	Adviser: Jennifer Morse.
Contained By:	Dissertation Abstracts International74-12B(E).
Subject:	Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3591019
ISBN:	9781303314209

Combinatorial aspects of generalizations of Schur functions.
Heilman, Derek.

Combinatorial aspects of generalizations of Schur functions. - 65 p.

Source: Dissertation Abstracts International, Volume: 74-12(E), Section: B.

Thesis (Ed.D.)--Drexel University, 2013.

The understanding of the space of symmetric functions is gained through the study of its bases. Certain bases can be defined by purely combinatorial methods, some- times enabling important properties of the functions to fall from carefully constructed combinatorial algorithms. A classic example is given by the Schur basis, made up of functions that can be defined using semi-standard Young tableaux. The Pieri rule for multiplying an important special case of Schur functions is proven using an insertion algorithm on tableaux that was defined by Robinson, Schensted, and Knuth. Further- more, the transition matrices between Schur functions and other symmetric function bases are often linked to representation theoretic multiplicities. The description of these matrices can sometimes be given combinatorially as the enumeration of a set of objects such as tableaux.

ISBN: 9781303314209Subjects--Topical Terms:

184409
Mathematics.

Combinatorial aspects of generalizations of Schur functions.
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100 1 $a Heilman, Derek. $3 660378
245 1 0 $a Combinatorial aspects of generalizations of Schur functions.
300 $a 65 p.
500 $a Source: Dissertation Abstracts International, Volume: 74-12(E), Section: B.
500 $a Adviser: Jennifer Morse.
502 $a Thesis (Ed.D.)--Drexel University, 2013.
520 $a The understanding of the space of symmetric functions is gained through the study of its bases. Certain bases can be defined by purely combinatorial methods, some- times enabling important properties of the functions to fall from carefully constructed combinatorial algorithms. A classic example is given by the Schur basis, made up of functions that can be defined using semi-standard Young tableaux. The Pieri rule for multiplying an important special case of Schur functions is proven using an insertion algorithm on tableaux that was defined by Robinson, Schensted, and Knuth. Further- more, the transition matrices between Schur functions and other symmetric function bases are often linked to representation theoretic multiplicities. The description of these matrices can sometimes be given combinatorially as the enumeration of a set of objects such as tableaux.
520 $a A similar combinatorial approach is applied here to a basis for the symmetric function space that is dual to the Grothendieck polynomial basis. These polynomials are defined combinatorially using reverse plane partitions. Bijecting reverse plane partitions with a subset of semi-standard Young tableaux over a doubly-sized alphabet enables the extension of RSK-insertion to reverse plane partitions. This insertion, paired with a sign changing involution, is used to give the desired combinatorial proof of the Pieri rule for this basis. Another basis of symmetric functions is given by the set of factorial Schur functions. While their expansion into Schur functions can be described combinatorially, the reverse change of basis had no such formulation. A new set of combinatorial objects is introduced to describe the expansion coefficients, and another sign changing involution is used to prove that these do in fact encode the transition matrices.
590 $a School code: 0065.
650 4 $a Mathematics. $3 184409
650 4 $a Applied Mathematics. $3 530992
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710 2 $a Drexel University. $b Mathematics (College of Arts and Sciences). $3 660379
773 0 $t Dissertation Abstracts International $g 74-12B(E).
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791 $a Ed.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3591019

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