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New combinatorial formulae for nested Bethe vectors II

M. Kosmakov, V. Tarasov

TL;DR

The paper develops new combinatorial formulae for vector-valued weight functions, also known as off-shell nested Bethe vectors, for evaluation modules over the Yangian $Y(\mathfrak{gl}_n)$, generalizing previous work at $n=4$. It introduces a splitting framework that decomposes weight functions along a chosen $m$-split into contributions from $Y(\mathfrak{gl}_m)$ and $Y(\mathfrak{gl}_{n-m})$, interconnected by $R$-matrices and a rich network of symmetric weight functions. The main contribution is an explicit formula (Theorem mainth) expressing $\mathbb{B}_{\boldsymbol{\xi}}(\boldsymbol{t})v$ as a structured sum over partitions and combinatorial data, built from lower-rank weight functions and auxiliary functions, with reductions to known results in boundary cases. This provides concrete, higher-rank tools for constructing weight functions, with potential applications in integrable models, hypergeometric $q$KZ solutions, and geometric representation theory.

Abstract

We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian Y(gl_n). This paper extends the result for the Yangian Y(gl_4) established earlier in arXiv:2312.00980.

New combinatorial formulae for nested Bethe vectors II

TL;DR

The paper develops new combinatorial formulae for vector-valued weight functions, also known as off-shell nested Bethe vectors, for evaluation modules over the Yangian , generalizing previous work at . It introduces a splitting framework that decomposes weight functions along a chosen -split into contributions from and , interconnected by -matrices and a rich network of symmetric weight functions. The main contribution is an explicit formula (Theorem mainth) expressing as a structured sum over partitions and combinatorial data, built from lower-rank weight functions and auxiliary functions, with reductions to known results in boundary cases. This provides concrete, higher-rank tools for constructing weight functions, with potential applications in integrable models, hypergeometric KZ solutions, and geometric representation theory.

Abstract

We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian Y(gl_n). This paper extends the result for the Yangian Y(gl_4) established earlier in arXiv:2312.00980.
Paper Structure (7 sections, 10 theorems, 80 equations)

This paper contains 7 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Notations
  4. Splitting property
  5. Weight functions
  6. Main theorem
  7. Proof of Theorem \ref{['mainth']}

Key Result

Proposition 3.1

Let $v\in V(x)$ be a weight singular vector and $\,\boldsymbol\xi=(\xi_1,\ldots,\xi_{n-1})$ be a collection nonnegative integers. Then where the sum is taken over all sequences $\boldsymbol{a}=(a_1,a_2,\ldots,a_{\xi_m})$, $\boldsymbol{b}=(b_1,b_2,\ldots,b_{\xi_m})$, such that $a_i\in \{1,2,\ldots, m\}$, $b_i\in \{m+1,m+2,\ldots,n\}$ for all $i=1,\ldots,\xi_m$, and

Theorems & Definitions (25)

  • Example
  • Proposition 3.1
  • proof
  • Remark
  • Remark
  • Lemma 4.1
  • Definition 1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 15 more