Combinatorics of determinantal identities

In this thesis, we apply combinatorial means for proving and generalizing classical determinantal identities. In Chapter 1, we present some historical background and discuss the algebraic framework we employ throughout the thesis. In Chapter 2, we construct a fundamental bijection between certain monomials that proves crucial for most of the results that follow. Chapter 3 studies the first, and possibly the best-known, determinantal identity, the matrix inverse formula, both in the commutative case and in some non-commutative settings (Cartier-Foata variables, right-quantum variables, and their weighted generalizations). We give linear-algebraic and (new) bijective proofs; the latter also give an extension of the Jacobi ratio theorem. Chapter 4 is dedicated to the celebrated MacMahon master theorem. We present numerous generalizations and applications. In Chapter 5, we study another important result, Sylvester's determinantal identity. We not only generalize it to non-commutative cases, we also find a surprising extension that also generalizes the master theorem. Chapter 6 has a slightly different, representation theory flavor; it involves representations of the symmetric group, and also Hecke algebras and their characters. We extend a result on immanants due to Goulden and Jackson to a quantum setting, and reprove certain combinatorial interpretations of the characters of Hecke algebras due to Ram and Remmel.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2008

Author dc:creator

• Konvalinka, Matjaž

Advisor dc:contributor.advisor

• Igor Pak.

• Mathematics.

Statement dc:rights

• 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.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/43790
• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Identifier URI

http://dspace.mit.edu/handle/1721.1/43790

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/43790