New Combinatorial Formulae for Nested Bethe Vectors

Technical Summary: New Combinatorial Formulae for Nested Bethe Vectors

Problem Statement
The paper addresses the problem of constructing explicit combinatorial formulae for vector-valued weight functions (also known as off-shell nested Bethe vectors) associated with evaluation modules over the Yangian $Y(\mathfrak{gl}_n)$. While previous work, specifically reference [25], provided formulae based on recursions induced by standard Lie algebra embeddings ($\mathfrak{gl}_1 \oplus \mathfrak{gl}_{n-1} \subset \mathfrak{gl}_n$ and $\mathfrak{gl}_{n-1} \oplus \mathfrak{gl}_1 \subset \mathfrak{gl}_n$), these approaches yield specific orderings of non-commuting factors in the basis expansion. The authors identify that certain natural orderings, crucial for applications such as those found in reference [8], are not covered by the existing formulae. Specifically, for $n=4$, the basis ordering involving the product $e_{32}e_{31}e_{42}e_{41}e_{21}e_{43}$ (where $e_{ij}$ are standard generators of $\mathfrak{gl}_n$) cannot be derived from the recursive structures used in [25]. This ordering arises from more general embeddings of the form $\mathfrak{gl}_m \oplus \mathfrak{gl}_{n-m} \subset \mathfrak{gl}_n$ with $1 < m < n-1$.

Methodology
The authors focus on the case $n=4$ to provide a clear, detailed exposition of the calculations without introducing overly cumbersome notation, noting that the results generalize to arbitrary $n$ (as detailed in a separate work, reference [5]). The methodology proceeds as follows:

1. Definitions and Setup: The paper utilizes the standard rational $R$-matrix and the generating series $T^a_b(u)$ for the Yangian $Y(\mathfrak{gl}_n)$. Vector-valued weight functions are defined as specific matrix entries of products of $T$-operators and $R$-matrices acting on a singular vector $v$.
2. Splitting Property: A central tool is the "splitting property" of the weight functions (Proposition 4.1). This property allows the decomposition of the $Y(\mathfrak{gl}_4)$ weight function $B_\xi(t)v$ into a sum involving weight functions of $Y(\mathfrak{gl}_2)$ modules. This decomposition relies on specific embeddings $\phi_2$ and $\psi_2$ of $Y(\mathfrak{gl}_2)$ into $Y(\mathfrak{gl}_4)$ and the use of evaluation modules $L(x)$ and $\bar{L}(x)$ on the vector space $\mathbb{C}^2$.
3. Recursive Reduction: The problem is reduced to computing weight functions for $Y(\mathfrak{gl}_2)$ acting on tensor products of evaluation modules. Propositions 5.2 and 5.3 provide explicit formulae for these $Y(\mathfrak{gl}_2)$ weight functions acting on singular vectors, utilizing symmetrization operators and specific rational functions $F_I(t, z)$.
4. Combinatorial Algebra: The core technical challenge involves summing over subsets of indices to match the desired basis ordering. The authors employ an algebraic framework involving two commuting copies of the symmetric group $S_k$ and rational functions (Section 7). They define elements $\hat{s}_a$ satisfying braid relations and utilize lemmas regarding the coefficients $X_{\sigma, \tau}(z)$ of these elements to prove that the sum over all pairs of subsets $(I, J)$ with fixed cardinalities and intersection size yields a symmetric function independent of the specific choice of subsets.
5. Proof of Proposition 5.1: The authors provide a detailed proof of a key proposition regarding the coproduct of the $Y(\mathfrak{gl}_2)$ weight function (Appendix A), which is essential for the recursive steps.

Key Contributions and Results
The primary result is Theorem 5.7, which provides a new closed-form combinatorial formula for the expansion of the vector-valued weight function $B_\xi(t)v$ in the evaluation module $V(x)$ over $Y(\mathfrak{gl}_4)$.

• New Basis Ordering: The formula expands the weight function in the basis:
  $$\{ e_{32}^{m_{32}} e_{31}^{m_{31}} e_{42}^{m_{42}} e_{41}^{m_{41}} e_{21}^{m_{21}} e_{43}^{m_{43}} v \mid m_{ij} \in \mathbb{Z}_{\ge 0} \}$$
  This ordering corresponds to the embedding $\mathfrak{gl}_2 \oplus \mathfrak{gl}_2 \subset \mathfrak{gl}_4$, which was inaccessible via the methods in [25].
• Explicit Coefficients: The coefficients $F_{\vec{m}}(t)$ in the expansion are given explicitly in terms of symmetrized rational functions $G_{i,j}(t)$ (Equation 5.13). These functions involve products of terms depending on the spectral parameters $t$, the evaluation point $x$, and the weights $\Lambda_i$ of the singular vector.
• Independence of Choice: The authors prove that the resulting symmetrized expression is independent of the specific choice of index sequences $i$ and $j$ used to define the subsets, ensuring the well-definedness of the formula.
• Rational vs. Trigonometric: The paper restricts its analysis to the rational case (Yangian modules). The authors explicitly note that extending these specific combinatorial formulae to the trigonometric case (quantum loop algebra $U_q(\widehat{\mathfrak{gl}}_n)$) encounters an obstacle due to the essential non-commutativity of $q$-analogues of generators $e_{ij}$ for $i > j$ under the general embeddings considered.

Significance
The paper claims significance by providing a necessary generalization of existing combinatorial formulae for nested Bethe vectors. By addressing the case of general embeddings $\mathfrak{gl}_m \oplus \mathfrak{gl}_{n-m} \subset \mathfrak{gl}_n$, the authors enable the construction of weight functions in basis orderings that are required for certain applications in quantum integrable models and representation theory, which were previously out of reach. The detailed derivation for the $n=4$ case serves as a transparent model for the general $n$ case, which is treated in reference [5]. The work also highlights the limitations of current methods when applied to the trigonometric case, identifying a specific algebraic obstruction that prevents a direct $q$-deformation of these new combinatorial expressions.