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Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index $5\frac{1}{4}$, which includes $3+\sqrt{5}$, the first interesting composite index. 

Abstract For a braided fusion category $$\mathcal{V}$$, a $$\mathcal{V}$$-fusion category is a fusion category $$\mathcal{C}$$ equipped with a braided monoidal functor $$\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$$. Given a fixed $$\mathcal{V}$$-fusion category $$(\mathcal{C}, \mathcal{F})$$ and a fixed $$G$$-graded extension $$\mathcal{C}\subseteq \mathcal{D}$$ as an ordinary fusion category, we characterize the enrichments $$\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$$ of $$\mathcal{D}$$ that are compatible with the enrichment of $$\mathcal{C}$$. We show that G-crossed extensions of a braided fusion category $$\mathcal{C}$$ are G-extensions of the canonical enrichment of $$\mathcal{C}$$ over itself. As an application, we parameterize the set of $$G$$-crossed braidings on a fixed $$G$$-graded fusion category in terms of certain subcategories of its center, extending Nikshych’s classification of the braidings on a fusion category. 

Abstract We investigated the potential mechanisms driving habitat‐linked genetic divergence within a bird species endemic to a single 250‐km²island. The island scrub‐jay (Aphelocoma insularis) exhibits microgeographic divergence in bill morphology across pine–oak ecotones on Santa Cruz Island, California (USA), similar to adaptive differences described in mainland congeners over much larger geographic scales. To test whether individuals exhibit genetic differentiation related to habitat type and divergence in bill length, we genotyped over 3000 single nucleotide polymorphisms in 123 adult island scrub‐jay males from across Santa Cruz Island using restriction site‐associated DNA sequencing. Neutral landscape genomic analyses revealed that genome‐wide genetic differentiation was primarily related to geographic distance and differences in habitat composition. We also found 168 putatively adaptive loci associated with habitat type using multivariate redundancy analysis while controlling for spatial effects. Finally, two genome‐wide association analyses revealed a polygenic basis to variation in bill length with multiple loci detected in or near genes known to affect bill morphology in other birds. Our findings support the hypothesis that divergent selection at microgeographic scales can cause adaptive divergence in the presence of ongoing gene flow.