On the SO(n + 3) to SO(n) branching multiplicity space

We study the decomposition as an $\text{SO}(3)$-module of the multiplicity space corresponding to the branching from $\text{SO}(n+3)$ to $\text{SO}(n)$. Here, $\text{SO}(n)$ (resp. $\text{SO}(3)$) is considered embedded in $\text{SO}(n+3)$ in the upper left-hand block (resp. lower right-hand block). We show that when the highest weight of the irreducible representation of $\text{SO}(n)$ interlaces the highest weight of the irreducible representation of $\text{SO}(n+3)$, then the multiplicity space decomposes as a tensor product of $?(n+2)/2?$ reducible representations of $\text{SO}(3)$.     

Odd values of the Rogers–Ramanujan functions

Let $g(n)$ and $h(n)$ be the coefficients of the Rogers–Ramanujan identities. We obtain asymptotic formulas for the number of odd values of $g(n)$ for odd n, and $h(n)$ for even n, which improve Gordon's results. We also obtain lower bounds for the number of odd values of $g(n)$ for even n, and $h(n)$ for odd n.     

A topological nullstellensatz for tensor-triangulated categories

Let $\mathrm{Spec}(\mathbb{T})$ be the spectrum of a tensor-triangulated category $(\mathbb{T},?,\mathbf{1})$. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(\mathbb{T},?,\mathbf{1})$ and the collection of open subsets of $\mathrm{Spec}(\mathbb{T})$ in inverse topology. In fact, we prove a more general result in terms of supports on $(\mathbb{T},?,\mathbf{1})$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.