Connections between conjectures of Alon–Tarsi, Hadamard–Howe, and integrals over the special unitary group

We show the Alon–Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some interesting integrals over SU(n)SU(n). Our investigations were motivated by geometric complexity theory.