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Indian Institute of Science

1. Verma, Abhinav. Irreducible Representations Of The Symmetric Group And The General Linear Group.

Degree: MS, Faculty of Science, 2013, Indian Institute of Science

URL: http://etd.iisc.ac.in/handle/2005/1909

Representation theory is the study of abstract algebraic structures by representing their elements as linear transformations or matrices. It provides a bridge between the abstract symbolic mathematics and its explicit applications in nearly every branch of mathematics. Combinatorial representation theory aims to use combinatorial objects to model representations, thus answering questions in this ﬁeld combinatorially. Combinatorial objects are used to help describe, count and generate representations. This has led to a rich symbiotic relationship where combinatorics has helped answer algebraic questions and algebraic techniques have helped answer combinatorial questions.
In this thesis we discuss the representation theory of the symmetric group and the general linear group. The theory of these two families of groups is often considered the corner stone of combinatorial representation theory. Results and techniques arising from the study of these groups have been successfully generalized to a very wide class of groups. An overview of some of the generalizations can be found in [BR99]. There are also many avenues for further generalizations which are currently being explored.
The constructions of the Specht and Schur modules that we discuss here use the concept of Young tableaux. Young tableaux are combinatorial objects that were introduced by the Reverend Alfred Young, a mathematician at Cambridge University, in 1901. In 1903, Georg Frobenius applied them to the study of the symmetric group. Since then, they have been found to play an important role in the study of symmetric functions, representation theory of the symmetric and complex general linear groups and Schubert calculus of Grassmannians. Applications of Young tableaux to other branches of mathematics are still being discovered.
When drawing and labelling Young tableaux there are a few conﬂicting conventions in the literature, throughout this thesis we shall be following the English notation. In chapter 1 we shall make a few deﬁnitions and state some results which will be used in this thesis.
In chapter 2 we discuss the representations of the symmetric group. In this chapter we deﬁne the Specht modules and prove that they describe all the irreducible representations of Sn. We conclude with a discussion about the ring of Sn representations which is used to prove some identities of Specht modules.
In chapter 3 we discuss the representations of the general linear group. In this chapter we deﬁne the Schur modules and prove that they describe all the irreducible rational representations of GLmC. We also show that the set of tableaux forms an indexing set for a basis of the Schur modules.
In chapter 4 we describe a relation between the Specht and Schur modules. This is a corollary to the more general Schur-Weyl duality, an overview of which can be found in [BR99].
The appendix contains the code and screen-shots of two computer programs that were written as part of this thesis. The programs have been written in C++ and the data structures have been…
*Advisors/Committee Members: Viswanath, Sankaran (advisor).*

Subjects/Keywords: Linear Group; Symmetric Group; Specht Modules; Schur Modules; Combinatorics; Algebra

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APA (6^{th} Edition):

Verma, A. (2013). Irreducible Representations Of The Symmetric Group And The General Linear Group. (Masters Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/1909

Chicago Manual of Style (16^{th} Edition):

Verma, Abhinav. “Irreducible Representations Of The Symmetric Group And The General Linear Group.” 2013. Masters Thesis, Indian Institute of Science. Accessed September 28, 2020. http://etd.iisc.ac.in/handle/2005/1909.

MLA Handbook (7^{th} Edition):

Verma, Abhinav. “Irreducible Representations Of The Symmetric Group And The General Linear Group.” 2013. Web. 28 Sep 2020.

Vancouver:

Verma A. Irreducible Representations Of The Symmetric Group And The General Linear Group. [Internet] [Masters thesis]. Indian Institute of Science; 2013. [cited 2020 Sep 28]. Available from: http://etd.iisc.ac.in/handle/2005/1909.

Council of Science Editors:

Verma A. Irreducible Representations Of The Symmetric Group And The General Linear Group. [Masters Thesis]. Indian Institute of Science; 2013. Available from: http://etd.iisc.ac.in/handle/2005/1909

2.
Loubert, Joseph.
Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between *Specht* *Modules* for KLR Algebras in Affine Type A.

Degree: PhD, Department of Mathematics, 2015, University of Oregon

URL: http://hdl.handle.net/1794/19255

This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras R_α of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in R_α are generated by idempotents. This in particular implies the (known) result that the global dimension of R_α is finite.
In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James.
This dissertation includes previously published coauthored material.
*Advisors/Committee Members: Kleshchev, Alexander (advisor).*

Subjects/Keywords: Affine cellularity; KLR algebras; Specht modules

…and Ram (20) provide a presentation for the *Specht* *modules* as *modules* over the… …between Certain *Specht* *Modules*
Let Hd be an Iwahori-Hecke algebra for the symmetric group Sd… …The *Specht* *modules* are described in (4) as *modules* over the cyclotomic
KLR… …*Specht* *modules* for the (full) KLR algebras Rd are explicitly defined in terms
of… …corresponds to
k = 0. Our *Specht* *modules* are the dual of the *Specht* *modules* defined in (11)…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Loubert, J. (2015). Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/19255

Chicago Manual of Style (16^{th} Edition):

Loubert, Joseph. “Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A.” 2015. Doctoral Dissertation, University of Oregon. Accessed September 28, 2020. http://hdl.handle.net/1794/19255.

MLA Handbook (7^{th} Edition):

Loubert, Joseph. “Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A.” 2015. Web. 28 Sep 2020.

Vancouver:

Loubert J. Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A. [Internet] [Doctoral dissertation]. University of Oregon; 2015. [cited 2020 Sep 28]. Available from: http://hdl.handle.net/1794/19255.

Council of Science Editors:

Loubert J. Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A. [Doctoral Dissertation]. University of Oregon; 2015. Available from: http://hdl.handle.net/1794/19255

University of Oregon

3. Nash, David A., 1982-. Graded representation theory of Hecke algebras.

Degree: 2010, University of Oregon

URL: http://hdl.handle.net/1794/10871

We study the graded representation theory of the Iwahori-Hecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory.
We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of two-column partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th -root of unity to those of the group algebra of the symmetric group over a field of characteristic p.

Subjects/Keywords: Symmetric groups; Specht modules; Irreducible representation; Graded representation; Hecke algebras; Mathematics; Theoretical mathematics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Nash, David A., 1. (2010). Graded representation theory of Hecke algebras. (Thesis). University of Oregon. Retrieved from http://hdl.handle.net/1794/10871

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Nash, David A., 1982-. “Graded representation theory of Hecke algebras.” 2010. Thesis, University of Oregon. Accessed September 28, 2020. http://hdl.handle.net/1794/10871.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Nash, David A., 1982-. “Graded representation theory of Hecke algebras.” 2010. Web. 28 Sep 2020.

Vancouver:

Nash, David A. 1. Graded representation theory of Hecke algebras. [Internet] [Thesis]. University of Oregon; 2010. [cited 2020 Sep 28]. Available from: http://hdl.handle.net/1794/10871.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Nash, David A. 1. Graded representation theory of Hecke algebras. [Thesis]. University of Oregon; 2010. Available from: http://hdl.handle.net/1794/10871

Not specified: Masters Thesis or Doctoral Dissertation