Sun-type determinant and permanent congruences

Abstract

Sun proposed a list of congruence and quadratic-residue conjectures for determinants and permanents over residue classes modulo a prime. This article gives a uniform treatment of Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list, while making explicit the overlap with two earlier contributions. Luo and Xia's Legendre-symbol formula for $D_p(b,1)$ already implies the non-vanishing assertion in Conjecture 4.6 when $p\equiv5\pmod {24}$; our determinant argument gives a root-quotient criterion for irreducible binary quadratic forms over $\Fp$ and also covers the remaining case $p\equiv19\pmod {24}$. For the Cauchy kernel $1/(x-y)$, we prove the derangement determinant and permanent congruences modulo $p^2$ and a polynomial fixed-point permanent congruence modulo $p$. For the Cayley kernel $(x+y)/(x-y)$, She, Sun and Xia's permanent identity supplies the structural input for the fixed-point permanent; combined with our Cauchy permanent congruence and Morley's congruence, it yields the congruence modulo $p^2$. Independent interpolation arguments give the signed fixed-point determinant congruences and the quadratic-residue assertion for the signed derangement determinant. Finally, a local expansion at the unique zero eigenvalue proves the half-size quadratic Cayley determinant divisibility by $p^2$, and by $p^3$ when $p\equiv7\pmod8$.