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Traces and Their Combinatorial Inter...

Scutt, Caleb,

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Traces and Their Combinatorial Interpretations: A Comparative Study of the Symmetric Group Algebra, Hecke Algebra, and Immanants of Totally Nonnegative Matrices /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Traces and Their Combinatorial Interpretations: A Comparative Study of the Symmetric Group Algebra, Hecke Algebra, and Immanants of Totally Nonnegative Matrices // Caleb Scutt.
作者:	Scutt, Caleb,
面頁冊數:	1 electronic resource (94 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31997138
ISBN:	9798283140582

Traces and Their Combinatorial Interpretations: A Comparative Study of the Symmetric Group Algebra, Hecke Algebra, and Immanants of Totally Nonnegative Matrices /
Scutt, Caleb,

Traces and Their Combinatorial Interpretations: A Comparative Study of the Symmetric Group Algebra, Hecke Algebra, and Immanants of Totally Nonnegative Matrices /Caleb Scutt. - 1 electronic resource (94 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

In this dissertation, we study how separate fields of algebraic combinatorics can be viewed through the lens of Young tableaux. Young tableaux were invented by Alfred Young to study representations of the symmetric group algebra, the algebra generated by the permutations on n-elements. The trace space of the symmetric group algebra is the module of linear maps from the symmetric group algebra to the integers, and bases of this space can be formed in connection with bases of the space of homogeneous symmetric polynomials. There are six well known bases of the latter space: the elementary basis, the homogeneous basis, the Schur basis, the power sum basis, the monomial basis, and the forgotten basis. These bases, as well as their analogue trace space bases, have been extensively studied by algebraic combinatorialists, including Richard P. Stanley in his book Enumerative Combinatorics: Volume 2. A deformation of the symmetric group algebra is the Hecke algebra. A common object of study in the Hecke algebra is the Kazhdan-Lusztig basis. Hecke algebra traces are defined similarly to symmetric group traces, and their actions on these elements will be studied in this paper. Symmetric group traces can be used to define immanants, which generalize the concepts of the determinant and permanent of a matrix. Calculating immanants is computationally complex, but the process can be simplified in the context of totally nonnegative matrices, matrices whose minors are all non-negative.These three branches, the representation theory of the symmetric group algebra, the representation theory of the Hecke algebra, and immanants of totally nonnegative matrices, can all be connected through combinatorially interpreting different varieties of tableaux. In Section 2, we introduce new combinatorial proofs of symmetric group traces acting on elements associated with the chromatic polynomial in terms of Young tableaux. In Section 3, we then translate some of the results from section 2 into the context of q-analogue traces through the use of P-tableaux. In Section 4, we propose a conjecture which, if true, will allow us to generalize these results to the context of LLT-analogue traces. In Section 5, we introduce the concept of G-tableaux in the context of planar networks and then translate the results from Section 2 into this new context.

English

ISBN: 9798283140582Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Algebraic combinatorics

Traces and Their Combinatorial Interpretations: A Comparative Study of the Symmetric Group Algebra, Hecke Algebra, and Immanants of Totally Nonnegative Matrices /
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520 $a In this dissertation, we study how separate fields of algebraic combinatorics can be viewed through the lens of Young tableaux. Young tableaux were invented by Alfred Young to study representations of the symmetric group algebra, the algebra generated by the permutations on n-elements. The trace space of the symmetric group algebra is the module of linear maps from the symmetric group algebra to the integers, and bases of this space can be formed in connection with bases of the space of homogeneous symmetric polynomials. There are six well known bases of the latter space: the elementary basis, the homogeneous basis, the Schur basis, the power sum basis, the monomial basis, and the forgotten basis. These bases, as well as their analogue trace space bases, have been extensively studied by algebraic combinatorialists, including Richard P. Stanley in his book Enumerative Combinatorics: Volume 2. A deformation of the symmetric group algebra is the Hecke algebra. A common object of study in the Hecke algebra is the Kazhdan-Lusztig basis. Hecke algebra traces are defined similarly to symmetric group traces, and their actions on these elements will be studied in this paper. Symmetric group traces can be used to define immanants, which generalize the concepts of the determinant and permanent of a matrix. Calculating immanants is computationally complex, but the process can be simplified in the context of totally nonnegative matrices, matrices whose minors are all non-negative.These three branches, the representation theory of the symmetric group algebra, the representation theory of the Hecke algebra, and immanants of totally nonnegative matrices, can all be connected through combinatorially interpreting different varieties of tableaux. In Section 2, we introduce new combinatorial proofs of symmetric group traces acting on elements associated with the chromatic polynomial in terms of Young tableaux. In Section 3, we then translate some of the results from section 2 into the context of q-analogue traces through the use of P-tableaux. In Section 4, we propose a conjecture which, if true, will allow us to generalize these results to the context of LLT-analogue traces. In Section 5, we introduce the concept of G-tableaux in the context of planar networks and then translate the results from Section 2 into this new context.
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