#### Graduation Year

2009

#### Document Type

Thesis

#### Degree

M.A.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Brian Curtin, Ph.D.

#### Committee Member

Mohamed Elhamdadi, Ph.D.

#### Committee Member

Xiang-dong Hou, Ph.D.

#### Committee Member

Stephen Suen, Ph.D.

#### Keywords

Leonard pairs, Antiautomorphisms, Valid sequence, Simplicial complex, Hamming association scheme

#### Abstract

Let *V* denote a vector space of finite positive dimension. An ordered triple of linear operators on *V* is said to be a Leonard triple whenever for each choice of element of the triple there exists a basis of *V* with respect to which the matrix representing the chosen element is diagonal and the matrices representing the other two elements are irreducible tridiagonal. A Leonard triple is said to be modular whenever for each choice of element there exists an antiautomorphism of End(*V*) which fixes the chosen element and swaps the other two elements. We study combinatorial structures associated with Leonard triples and modular Leonard triples. In the first part we construct a simplicial complex of Leonard triples. The simplicial complex of a Leonard triple is the smallest set of linear operators which contains the given Leonard triple with the property that if two elements of the set are part of a Leonard triple, then the third element of the triple is also in the set. In the second part we construct a Hamming association scheme from modular Leonard triples using a method used previously in the context of Grassmanian codes.

#### Scholar Commons Citation

Sobkowiak, Jessica, "Some Combinatorial Structures Constructed from Modular Leonard Triples" (2009). *Graduate Theses and Dissertations.*

https://scholarcommons.usf.edu/etd/27