On the Combinatorial Core of Second-Order Quantum Argument Shifts in $U{\mathfrak{gl}}_d$

Abstract

We provide a complete, self-contained proof that reduces second-order generators of the quantum argument-shift algebra in the universal enveloping algebra $U{\mathfrak{gl}}_d$. We prove the necessary combinatorial identities -- expressed as relations among polynomials with rational coefficients -- by induction.