Centralizers of the complex orthogonal and symplectic group

Technical Summary: Centralizers of the Complex Orthogonal and Symplectic Groups

Problem Statement
The paper addresses the computation of centralizers (stabilizer groups) for the complex orthogonal group $O_n(\mathbb{C})$ and the complex symplectic group $Sp_{2n}(\mathbb{C})$ under the similarity transformation action. Specifically, it seeks to determine the structure of the group $C_G(R) = \{Q \in G \mid Q^{-1}RQ = R\}$ for an arbitrary element $R$ in these groups.

The problem is approached through the lens of Lie theory and matrix equations. By utilizing the Jordan-Chevalley decomposition $R = SU$ (where $S$ is semisimple and $U$ is unipotent), the centralizer $C_G(R)$ is reduced to the intersection $C_G(S) \cap C_G(U)$. While $C_G(S)$ is well-understood as a direct sum of classical groups, the structure of the unipotent part $C_G(U)$, particularly its unipotent radical $Ru(C_G(U))$, requires detailed analysis. Furthermore, the paper establishes an equivalence between computing centralizers of $H$-orthogonal/$H$-symplectic matrices and computing isotropy groups of $H$-skew-symmetric/$H$-Hamiltonian matrices under similarity, leveraging the exponential map.

Methodology
The author employs a recursive algorithmic approach grounded in the normal forms of matrices under $H$-orthogonal or $H$-symplectic similarity. The methodology proceeds through the following steps:

1. Reduction to Normal Forms: The problem is reduced to analyzing matrices in specific canonical forms (Jordan blocks $J_m(\lambda)$ and exchange matrices $\Gamma_m$) under the action of the group. The isotropy group of a matrix $A$ in these normal forms is analyzed.
2. Transformation to Sylvester Equations: The condition $Q^{-1}AQ = A$ is transformed into a Sylvester equation $J(A)X = XJ(A)$ via a transition matrix $U$ (where $A = U^{-1}J(A)U$). The solutions $X$ to this equation are known to be block matrices with rectangular upper triangular Toeplitz blocks.
3. Imposing Group Constraints: The constraint that $Q$ must belong to the orthogonal or symplectic group ($Q^T HQ = H$) is translated into a matrix equation involving the transformed variable $X$. This results in a specialized equation of the form $B = F X^T F B X$ (or variations thereof), where $F$ is a block exchange matrix and $B$ is a block diagonal matrix with specific symmetry properties.
4. Recursive Solution of Block Equations: The core technical contribution lies in solving the equation $C = F X^T F B X$ for $X$ within the space of block upper triangular Toeplitz matrices ($T_{\alpha, \mu}$). The author derives a recursive algorithm (Lemma 3.1) that determines the entries of $X$ block by block. This involves:
   • Solving quadratic matrix equations for the diagonal blocks.
   • Expressing off-diagonal blocks and higher-order diagonal entries as polynomial functions of previously determined blocks and arbitrary parameters (subject to symmetry constraints).
   • Utilizing generating functions and Catalan numbers to resolve specific recurrence relations arising in the off-diagonal blocks (Example 2.4, Lemma 3.3).

Key Contributions and Results
The paper provides a complete structural description and a computational algorithm for these centralizers:

• Structural Characterization: The centralizers (and isotropy groups) are shown to be conjugate to subgroups of nonsingular block matrices with rectangular block Toeplitz blocks. These subgroups admit a semidirect product decomposition $L \ltimes U$, where $L$ is a reductive part (a direct sum of orthogonal or symplectic groups) and $U$ is a unipotent radical.
• Explicit Algorithm: A recursive algorithm is presented to compute the centralizer for any element $R$. The algorithm explicitly constructs the matrix $X$ satisfying the defining equations, determining which entries are free parameters and which are polynomially dependent on others.
• Dimension Formulas: The paper derives precise formulas for the dimension of the centralizer/isotropy group over the field $\mathbb{C}$ (and conditions for the real case).
  • For the unipotent case (Theorem 2.5), the dimension is given by:
    $$\dim_{\mathbb{F}}(X) = \sum_{r=1}^N \left( \frac{1}{2}\alpha_r \mu_r^2 + \sum_{s=1}^{r-1} \alpha_s \mu_r \mu_s \right) + (-1)^c \sum_{\alpha_r \text{ odd}} \frac{1}{2}\mu_r$$
  • For the non-unipotent case (Theorem 2.6), the dimension is:
    $$\dim_{\mathbb{C}}(X) = \sum_{r=1}^N m_r \left( \alpha_r m_r + 2 \sum_{s=1}^{r-1} \alpha_s m_s \right)$$
• Generalized Groups: The results extend to indefinite generalized orthogonal and symplectic groups $O_n(\mathbb{C}, H)$ and $Sp_{2n}(\mathbb{C}, H)$, showing that their centralizers are conjugate to those of the standard groups via specific transition matrices.

Significance and Claims
The paper claims to provide a "detailed description" of the unipotent radical of these centralizers, a component previously less explicitly characterized in terms of block Toeplitz structures. The author notes that while the general approach relates to previous work on Hermitian and symmetric matrices (citing [17], [18]), the current problem involves "several quite subtle differences" that make the analysis "considerably more involved." Specifically, the interaction between the symmetry constraints of the orthogonal/symplectic groups and the Toeplitz structure of the solutions leads to new phenomena in the group structure, particularly in the recursive determination of the off-diagonal blocks.

The work is presented as a foundational step for solving simultaneous reduction problems under congruence for quadruples of matrices, with potential applications to systems of transpose-Sylvester matrix equations. The author explicitly states that the real case is "more involved" and is reserved for future work, maintaining a modest scope regarding the current paper's applicability to real fields.