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Schaper Numbers, Palindrome Partitio...

Westrem, Karlee J.,

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Schaper Numbers, Palindrome Partitions, and Symmetric Functions, With Applications to Characters of the Symmetric Group /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Schaper Numbers, Palindrome Partitions, and Symmetric Functions, With Applications to Characters of the Symmetric Group // Karlee J Westrem.
作者:	Westrem, Karlee J.,
面頁冊數:	1 electronic resource (92 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-11, Section: B.
Contained By:	Dissertations Abstracts International86-11B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31847931
ISBN:	9798314867389

Schaper Numbers, Palindrome Partitions, and Symmetric Functions, With Applications to Characters of the Symmetric Group /
Westrem, Karlee J.,

Schaper Numbers, Palindrome Partitions, and Symmetric Functions, With Applications to Characters of the Symmetric Group /Karlee J Westrem. - 1 electronic resource (92 pages)

Source: Dissertations Abstracts International, Volume: 86-11, Section: B.

Chapter 2 is on the topic of representation theory of the symmetric group Σn. Finding the decomposition numbers for the symmetric group is a difficult problem and has led to many different research directions. For a given Specht module, the Schaper sum formula can be refined with knowledge of the Schaper number for the particular partition to give more precise information about the decomposition numbers. In Chapter 2, we give a combinatorial formula for the Schaper number when the partition λ has at most two columns. At the end, we provide a conjecture for the Schaper number for the partition λ = (3n).Chapter 3 covers the joint work with my advisor David Hemmer and is published in the Ramanujan Journal. There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r + 1. Motivated by work on rational numbers in the Calkin-Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions, and describe how to efficiently generate all of them. Atypically for partition generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when n = 3 or n + 1 is prime. We find an interesting new "branching diagram" for partitions, similar to Young's lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences.Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn characters sums over a new set called Ev(λ). In Chapter 4, I share my results from the investigation of an alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions. I found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.

English

ISBN: 9798314867389Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Schaper number

Schaper Numbers, Palindrome Partitions, and Symmetric Functions, With Applications to Characters of the Symmetric Group /
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245 1 0 $a Schaper Numbers, Palindrome Partitions, and Symmetric Functions, With Applications to Characters of the Symmetric Group / $c Karlee J Westrem.
264 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2025
300 $a 1 electronic resource (92 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertations Abstracts International, Volume: 86-11, Section: B.
500 $a Advisors: Hemmer, David Committee members: Keith, William; Schneider, Robert; El-Ganainy, Ramy.
502 $b Ph.D. $c Michigan Technological University $d 2025.
520 $a Chapter 2 is on the topic of representation theory of the symmetric group Σn. Finding the decomposition numbers for the symmetric group is a difficult problem and has led to many different research directions. For a given Specht module, the Schaper sum formula can be refined with knowledge of the Schaper number for the particular partition to give more precise information about the decomposition numbers. In Chapter 2, we give a combinatorial formula for the Schaper number when the partition λ has at most two columns. At the end, we provide a conjecture for the Schaper number for the partition λ = (3n).Chapter 3 covers the joint work with my advisor David Hemmer and is published in the Ramanujan Journal. There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r + 1. Motivated by work on rational numbers in the Calkin-Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions, and describe how to efficiently generate all of them. Atypically for partition generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when n = 3 or n + 1 is prime. We find an interesting new "branching diagram" for partitions, similar to Young's lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences.Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn characters sums over a new set called Ev(λ). In Chapter 4, I share my results from the investigation of an alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions. I found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.
546 $a English
590 $a School code: 0129
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Schaper number
653 $a Palindrome partitions
653 $a Symmetric functions
653 $a Symmetric group
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a Michigan Technological University. $b Mathematical Sciences. $e degree granting institution. $3 3823001
720 1 $a Hemmer, David $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-11B.
790 $a 0129
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31847931