| Russian Mathematical Surveys |
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Brief Communications Roots of the characteristic equation for the symplectic groupoid L. O. Chekhovabc, M. Z. Shapirocb, H. Shibod a Steklov Mathematical Institute of Russian Academy of Sciences b National Research University Higher School of Economics c Michigan State University, East Lansing, USA d Xi'an Jiaotong University, Xi'an, Shaanxi, P.R. China
Russian version:
Uspekhi Matematicheskikh Nauk, 2022, Volume 77, Issue 3(465), DOI: https://doi.org/10.4213/rm9999 
Bibliographic databases:
    Morphisms on the set $\mathcal A_n\subseteq \operatorname{gl}_n$ of unipotent upper-triangular $ n\times n $ matrices are transformations $\mathbb A\mapsto B\mathbb AB^\top\in \mathcal A_n$. For $B=e^{\varepsilon g}$ linearized transformations $\delta_\varepsilon \mathbb A=g\mathbb A+\mathbb Ag^\top$ are analogous to Krichever–Novikov transformations [4]. The quantization of the Bondal Poisson structure on $\mathcal A_n$ (see [1]) obtained using the symplectic groupoid construction is the reflection equation with trigonometric $R$-matrix
$$
\begin{equation}
{\mathcal R}_{n}(q)\overset{1}{\mathbb A} {\mathcal R}_{n}^{t_1}(q)\overset{2}{\mathbb A}= \overset{2}{\mathbb A}{\mathcal R}_{n}^{t_1}(q) \overset{1}{\mathbb A}{\mathcal R}_{n}(q).
\end{equation}
\tag{1}
$$
Here $\mathbb A$ is an upper-triangular metric with diagonal entries $q^{-1/2}$ and self-adjoint operator entries $a_{i,j}$ for $i<j$. Then the combination $\mathbb A\mathbb A^{-\unicode{8224}}:=\mathbb A[\mathbb A^{\unicode{8224}}]^{-1}$ undergoes an adjoint transformation $\mathbb A\mathbb A^{-\unicode{8224}}\mapsto B \mathbb A\mathbb A^{-\unicode{8224}} B^{-1}$, and we are to find its eigenvalues $\lambda_i\in\mathbb C$ determining $[n/2]$ independent Casimir elements [1].
We express the entries of $\mathbb A$ using the quantum Fock–Goncharov variables $Z_\alpha=Z_{(i,j,k)}$ parameterized in terms of the barycentric coordinates ($i+j+k=n$) of the vertices of the $b_n$-quiver [3] (Fig. 1, left): a solid arrow from $\alpha$ to $\beta$ means that $Z_{\beta}Z_{\alpha}=q^{-2} Z_{\alpha}Z_{\beta}$, and a dashed arrow gives $Z_{\beta}Z_{\alpha}=q^{-1} Z_{\alpha}Z_{\beta}$. The directed network $N$ dual to the $b_n$-quiver is the directed graph of double arrows in Fig. 1. With any oriented path $P\colon j \rightsquigarrow i$ in any planar directed network $\mathcal N$ we associate the quantum weight $w(P)$, which is the Weyl ordered (denoted by the symbol In the case of the $b_n$-quiver we define the three quantum $(n\times n)$-transport matrices
$$
\begin{equation*}
\begin{gathered} \, (\mathcal M_1)_{i,j}= \sum_{\substack{P\colon j\rightsquigarrow i'}} w(P),\quad (\mathcal M_2)_{i,j}= \sum_{\substack{P\colon j\rightsquigarrow i'' }} w(P), \quad (\mathcal M_3)_{i,j}= \sum_{\substack{P\colon j' \rightsquigarrow i'' }} w(P), \end{gathered}
\end{equation*}
\notag
$$
where the paths contributing to $\mathcal M_3$ are obtained by reversing all horizontal double arrows in $N$. Note that ${\mathcal M}_1$ and $\mathcal M_3$ are lower-triangular matrices and ${\mathcal M}_2$ is an upper-triangular matrix. We define an antidiagonal matrix and let  denote the products of Fock–Goncharov variables along SE-diagonals of the $b_n$-quiver.
We have the groupoid condition [2] ${\mathcal M}_3 S {\mathcal M}_1= {\mathcal M}_2$, and Theorem 4.1 of [2] states that $\mathbb A:=\mathcal M_1^\top\mathcal M_3 S \mathcal M_1$ satisfies equation (1). Amalgamating the variables $Z_{(i,0,n-i)}$ and $Z_{(0,n-i,i)}$ pairwise we obtain the new Casimirs used to eliminate the variables $Z_{(l,n-l,0)}$, so that we obtain the $\mathcal A_n$-quiver (Fig. 1, right), whose Casimirs are the $[n/2]$ elements for $1\leqslant i<n/2$ and $C_{n/2}=T_{n/2}$ for integer $n/2$.
Theorem. The eigenvalues $\lambda_i\in \mathbb C$, $1\leqslant i\leqslant n$, of the operator $\mathbb A\mathbb A^{-\unicode{8224}}$ are Proof. For $\mathbb A=\mathcal M_1^\top\!\mathcal M_3 S \mathcal M_1$ we have $\mathcal M_i^\unicode{8224}=\mathcal M_i^\top$ and $\mathbb A^{\unicode{8224}}\!= \mathcal M_1^\top S^+\! \mathcal M_3^\top \mathcal M_1$, so that $\mathbb A-\lambda \mathbb A^{\unicode{8224}}= \mathcal M_1^\top\bigl(\mathcal M_3S- \lambda S^+ \mathcal M_3^\top\bigr)\mathcal M_1$, and the singularity equation becomes $\bigl(\mathcal M_3S- \lambda S^+ \mathcal M_3^\top\bigr)\psi=0$. The matrix $\mathcal M_3$ is lower triangular and its diagonal entries are $ m_1=Z_{(n,0,0)}$ and
 , $2\leqslant i \leqslant n$. The crucial observation is that both the matrices $\mathcal M_3S$ and $S^+\mathcal M_3^\top$ are upper antitriangular. The antidiagonal components of these matrices are respectively, and the singularity equation has a non-trivial solution if a combination of them contains the zero element, so that the admissible values are $\lambda_i=(-1)^{n-1}q^{-n} m_{n+1-i}/m_i$; the quotients of variables correspond to different cases in the theorem. $\Box$ 
Citation in format AMSBIB
\Bibitem{CheShaShi22} |
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