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

TL;DR

The paper addresses quantizing the Mishchenko–Fomenko argument-shift construction for $U\mathfrak{gl}_d$ by providing a complete, self-contained inductive proof that the second-order generators obey key combinatorial relations among polynomials with rational coefficients. The main method translates the problem into two equivalent forms—binomial-coefficient identities and polynomial identities—proved through two inductively established lemmas. It delivers a self-contained proof of Theorem 4 in Ikeda2024 and clarifies the combinatorial core underlying the quantum argument-shift construction. The results strengthen the algebraic foundation for natural quantizations and pave the way for further extensions in the theory of quantum shift algebras.

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.

Theorems & Definitions (13)

• Definition 1
• Proposition 1
• Definition 2
• Proposition 2
• proof
• Theorem 1
• Theorem 2
• Lemma 1
• proof : Proof of Theorem \ref{['thm:binomial']}
• proof : Proof of Lemma \ref{['lem:polynomial']}