## USF Tampa Graduate Theses and Dissertations

#### Title

Leonard Systems and their Friends

2016

Thesis

M.A.

#### Degree Name

Master of Arts (M.A.)

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Brian Curtin, Ph.D.

#### Committee Member

Brendan Nagle, Ph.D.

#### Committee Member

Jean-Francois Biasse, Ph.D.

#### Keywords

Equitable sl_2 basis, Leonard Pairs, Linear Algebra

#### Abstract

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let

\text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$.

A {\em Leonard system} on $V$ is a sequence

$(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),$

where

$\A$ and $\B$ are multiplicity-free elements of \text{End}$(V)$;

$\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$

are orderings of the primitive idempotents of $\A$ and $\B$, respectively; and

for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and

nonzero when $\vert i-j \vert = 1$.

%

Leonard systems arise in connection with orthogonal polynomials, representations of many nice algebras, and the study of some highly regular combinatorial objects. We shall use the construction of Leonard pairs of classical type from finite-dimensional modules of $\mathit{sl}_2$ and the construction of Leonard pairs of basic type from finite-dimensional modules of $U_q(\mathit{sl}_2)$.

Suppose $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ is a Leonard system.

For $0 \leq i \leq d$, let

$U_i = (E^*_0V+E^*_1V+\cdots + E^*_iV)\cap (E_iV+E_{i+1}V+\cdots + E_dV).$

Then $U_0$, $U_1$, \ldots, $U_d$ is the {\em split decomposition of $V$ for $\Phi$}.

%

The split decomposition of $V$ for $\Phi$ gives rise to canonical matrix representations of $\A$ and $\B$ in terms of useful parameters for the Leonard system. %These canonical matrix representations for $\A$, $\B$ are respectively lower bidiagonal and upper bidiagonal.

In this thesis, we consider when certain Leonard systems share a split decomposition.

We say that Leonard systems

$\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$

and

$\hat{\Phi}:=(\hat{\A} ;\hat{\B}; \lbrace \hat{E}_i\rbrace_{i=0}^d; \lbrace \hat{E^*}_i\rbrace_{i=0}^d)$

are {\em friends} when $\A = \hat{\A}$ and $\Phi$, $\hat{\Phi}$ have the same split decomposition.

%

We obtain Leonard systems which share a split decomposition by constructing them from closely related module structures for either $\mathit{sl}_2$ or $U_q(\mathit{sl}_2)$ on $V$. We then describe friends by a parametric classification.

In this manner we describe all pairs of friends of classical and basic types.

In particular, friendship is not entirely a property of isomorphism classes.

COinS