[Paper Review] On some aspects of the Deligne-Simpson problem

Vladimir Petrov Kostov|ArXiv.org|May 2, 2000

Matrix Theory and Algorithms5 references4 citations

TL;DR

This paper solves the multiplicative Deligne-Simpson problem for generic eigenvalues when the greatest common divisor $ d > 1 $ of all Jordan block sizes across the matrices is greater than one. It establishes necessary and sufficient conditions for the existence of irreducible $(p+1)$-tuples of matrices $ M_j \in SL(n,\mathbb{C}) $ with prescribed conjugacy classes satisfying $ M_1 \cdots M_{p+1} = I $, under the condition that all $ \Sigma_{j,m}(\sigma) $—the number of Jordan blocks of size $ m $ with eigenvalue $ \sigma $—are divisible by $ d > 1 $. The key contribution is a complete characterization of such irreducible solutions under generic eigenvalue conditions defined by algebraic inequalities.

ABSTRACT

The Deligne-Simpson problem in the multiplicative version is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\in SL(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ satisfying the equality $M_1... M_{p+1}=I$}. We solve the problem for generic eigenvalues in the case when all the numbers $Σ_{j,m}(σ)$ of Jordan blocks of a given matrix $M_j$, with a given eigenvalue $σ$ and of a given size $m$ (taken over all $j$, $σ$, $m$) are divisible by $d>1$. Generic eigenvalues are defined by explicit algebraic inequalities of the form $a eq 0$. For such eigenvalues there exist no reducible $(p+1)$-tuples. The matrices $M_j$ are interpreted as monodromy operators of regular linear systems on Riemann's sphere.

Motivation & Objective

• To determine necessary and sufficient conditions for the existence of irreducible $(p+1)$-tuples of matrices $ M_j \in SL(n,\mathbb{C}) $ with prescribed conjugacy classes such that $ M_1 \cdots M_{p+1} = I $, under the multiplicative Deligne-Simpson problem framework.
• To extend previous results on the Deligne-Simpson problem by solving the case where the greatest common divisor $ d > 1 $ of all Jordan block sizes $ \Sigma_{j,m}(\sigma) $ across matrices is greater than one.
• To characterize the set of generic eigenvalues via explicit algebraic inequalities $ a \neq 0 $, ensuring no reducible solutions exist, and to prove irreducibility under these conditions.
• To establish the existence of irreducible representations defined by nilpotent matrices in the additive version, under specific Jordan block configurations.
• To unify and generalize results from prior work on the additive and multiplicative versions, particularly for the case $ d > 1 $, using canonical and strongly generic eigenvalue conditions.

Proposed method

• The paper uses the concept of polymultiplicity vectors (PMVs) and defines generic eigenvalues via algebraic inequalities $ a \neq 0 $, excluding solutions that would allow reducible $(p+1)$-tuples.
• It introduces the notion of subordinate Jordan normal forms and normalized chains to analyze the structure of matrices and their block decompositions under the $ d > 1 $ condition.
• The proof relies on a modified basic technical tool involving perturbation techniques: constructing matrices $ A_j^* $ with specific block structures and deforming them via $ \varepsilon $-perturbations to achieve desired Jordan forms.
• Irreducibility is proven by showing that the algebra generated by the matrices is $ gl(n,\mathbb{C}) $, using the fact that the only invariant subspaces are trivial, achieved via explicit construction of non-zero off-diagonal entries in the perturbed matrices.
• The method includes case analysis based on the type of Jordan block configuration (e.g., b1), c2), d3)), using block matrices $ A_j^* $ with nilpotent blocks and carefully chosen perturbation matrices $ L $ and conjugation matrices $ (I + \varepsilon X_j(\varepsilon)) $ to preserve or modify block sizes.
• The proof leverages results from the additive version and applies them to the multiplicative case via monodromy interpretation, interpreting $ M_j $ as monodromy operators of regular linear systems on the Riemann sphere.

Experimental results

Research questions

• RQ1 Under what conditions on the conjugacy classes $ C_j \subset SL(n,\mathbb{C}) $ does there exist an irreducible $(p+1)$-tuple of matrices $ M_j \in C_j $ such that $ M_1 \cdots M_{p+1} = I $?
• RQ2 What is the role of the greatest common divisor $ d > 1 $ of all $ \Sigma_{j,m}(\sigma) $—the number of Jordan blocks of size $ m $ with eigenvalue $ \sigma $—in determining the existence of such irreducible solutions?
• RQ3 How do generic eigenvalues, defined by algebraic inequalities $ a \neq 0 $, ensure the non-existence of reducible solutions and the existence of irreducible ones?
• RQ4 Can irreducible triples (or $ p+1 $-tuples) of nilpotent matrices be constructed when the Jordan block sizes satisfy $ d > 1 $, and how is irreducibility preserved under small perturbations?
• RQ5 What is the relationship between the multiplicative Deligne-Simpson problem and the additive version when $ d > 1 $, and how can results from the additive case be extended to the multiplicative setting?

Key findings

• For generic eigenvalues defined by $ a \neq 0 $, there exist no reducible $(p+1)$-tuples of matrices $ M_j \in C_j $ satisfying $ M_1 \cdots M_{p+1} = I $, ensuring that any solution is necessarily irreducible.
• When all $ \Sigma_{j,m}(\sigma) $ are divisible by $ d > 1 $, the paper establishes necessary and sufficient conditions for the existence of irreducible solutions to the multiplicative Deligne-Simpson problem.
• The paper proves that irreducible triples of nilpotent matrices exist for all Jordan block configurations satisfying the $ d > 1 $ condition, including in the almost special and neighbouring cases such as b1), c2), d3), and a1).
• Using perturbation techniques with $ \varepsilon $-deformations and conjugation via $ (I + \varepsilon X_j(\varepsilon)) $, the paper constructs matrices whose Jordan forms achieve desired block sizes while preserving irreducibility.
• The algebra generated by the matrices is shown to be $ gl(n,\mathbb{C}) $, confirming irreducibility, by demonstrating that the only invariant subspaces are trivial, via explicit non-zero entries in perturbed matrices.
• The result generalizes previous work on the Deligne-Simpson problem, particularly extending the solution from the $ d = 1 $ case to $ d > 1 $, under generic eigenvalue conditions, and confirms that the answers to the irreducibility problem and the trivial centralizer problem (Problem TC) coincide for such eigenvalues.