Introduction

Many sequences of combinatorial interest are known to be unimodal or log-concave and there

has been a considerable amount of interest and research devoted to this topic in recent years.

Even though these two properties have "one-line" definitions it has now become apparent

that to prove the unimodality or log-concavity of a sequence is often a very difficult task that

requires the use of refined and sophisticated mathematical tools. The number and variety

of these tools has been constantly increasing and is quite bewildering and surprising. They

include, for example, classical analysis (see, e.g., [2], [56], [68]), linear algebra (see, e.g., [47]),

the representation theory of Lie algebras and superalgebras (see, e.g., [47], [49], [61], [62]),

algebraic geometry (see, e.g., [65]) as well as the use of bisections and injections (see, e.g.,

[53]). An excellent survey of all these techniques is [65] which also contains many further

references.

The main object of this work is to point out another branch of mathematics that can

be successfully used to attack the kind of problems above referred to, namely, the theory of

total positivity. This theory comes about in unimodality and log-concavity questions in a

quite natural way, in fact, whenever we have in mathematics a set of "objects" that we want

to study we immediately look for "morphisms" between these objects that "preserve" their

structure. In the case at hand (unimodal and log-concave sequences) it is therefore natural

to look for transformations of Rrf into itself that preserve the unimodality or log-concavity

property. Now, the simplest kind of transformations that we can think of are, of course,

the linear transformations. Simple as this idea is it seems that no systematic study of these

kind of linear transformations has ever been made. One possible reason for this is that

examples show that there are very few linear transformations that preserve the unimodal

or log-concave property. However, a closer look reveals that a (very natural) strengthening

of these concepts, that of a Polya frequency sequence (a concept closely connected to that

of total positivity), is much better behaved. The fact that many of the unimodal and log-

concave sequences arising in combinatorics (as well as in algebra, geometry and probability)

turn out to be actually Polya frequency sequences was enough motivation for this author

to begin the study of linear transformations that preserve the Polya frequency (or PF, for

short) property.

This study has turned out to be a very rich subject that can easily fill a mathematician's

life. For this reason, and because the author's interest in these matters stemmed from

combinatorial motivations, only the most fundamental of these linear transformations are

vn