4d $\mathcal{N}=1$ quiver gauge theories and the ${\mathrm{A}}_{\mathrm{n}}$ Bailey lemma

We study the integral Bailey lemma associated with the ${\mathrm{A}}_{\mathrm{n}}$-root system and identities for elliptic hypergeometric integrals generated thereby. Interpreting integrals as superconformal indices of four-dimensional $\mathcal{N}=1$ quiver gauge theories with the gauge groups being copies of $\mathrm{S}\mathrm{U}(\mathrm{n}+1)$, we provide evidence for various new dualities. Further confirmation is achieved by explicitly checking that the `t Hooft anomaly matching conditions holds. We discuss a flavour symmetry breaking phenomenon for supersymmetric quantum chromodynamics (SQCD), and by making use of the Bailey lemma we indicate its manifestation in a web of linear quivers dual to SQCD that exhibits full s-confinement.