Question

I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a "diagonal" coefficient of a unitary representation of $G$.

For a definition and discussion of positive definite function see here.

I've often wished I had a collection of diverse examples of positive definite functions on groups, for the purpose of testing various conjectures. I hope the diverse experience of the participants of this forum can help me collect a list of such examples.

To clarify what I'd like to see:

What is an example of a positive definite function on a group $G$ that is not easily seen to be a coefficient of a unitary representation of $G$? What are some positive definite functions that arise in contexts sufficiently removed from studying the coefficients of unitary representations?

Also, the weirder the group $G$ the better. I'd like a collection of quirky beasts...

Answer 1

For me, often positive-definite functions arise as kernel functions in machine learning

A small list can be found at this link

Also, I would also add one of the classic books on this subject:

Harmonic analysis on semigroups by Christian Berg, Jens Peter Reus Christensen, Paul Ressel.

Answer 2

I guess the example below provides one answer to your first question.

A famous positive-definite function** is the one in the Bessis-Moussa-Villani conjecture:

Let $A$ and $B$ be $n \times n$ Hermitian matrices. Then the function $$\phi(t) = \mbox{trace}(e^{A+i t B}),$$ is a positive-definite function.

** Conjectured to be positive-definite, though apparently it has been proved very recently; however, until that has been verified independently, I will adhere to the safety of the word "conjecture"

Answer 3

Perhaps you are already aware of this, but I thought I'd mention it for other interested google-enabled readers.

1. Infinitely divisible distributions are one place where positive-definite functions come up (Lévy processes, Lévy-Khintchine formula, etc., are also relevant keywords)

2. Infinite divisibility in Free Probability is another related place.