Consider the family of spherical harmonics $Y_{n}^m$ on the sphere $\mathbb{S}^2$, with $n \geq 0$ and $-n \leq m \leq n$. The Wigner matrices are matrices $\mathrm{D}_{\mathrm{R},n} \in \mathbb{R}^{(2n+1)\times (2n+1)}$, defined for any $n \geq 0$ and any rotation $\mathrm{R} \in \mathrm{SO}(3)$...

In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid Lewin (Generalized Interval Systems and Transformations), Rudolf Wille (discussed also by Jan Thom...

In an unpublished paper by Ackerman-Freer-Patel (Forum Math. 2016; arXiv link), it is stated that if a relational structure has trivial definable closure, it has an invariant measure. My question is: What exactly do they mean by an invariant measure? If I had to explain to my uneducated (in gra...

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations. Theorem. The normal subgroups of $S_\infty$ are exactly the following four: $\{\mathrm{id}\}\subset A\subset S\subset S_\infty$ (see on Wikipedia...

Let $R$ denote the Rado graph, and let $c$ be a fixed vertex. Question 1. Is the structure obtained by extending $R$ by the constant $c$ interpretable in $R$ without parameters? By interpretable I mean first-order interpretable; see below for an equivalent formalism. A related group-theoreti...