The Deligne-Simpson Problem

Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks to determine whether or not one can find matrices in these classes whose product is the identity and with no common invariant subspace. The first author conjectured an answer in terms of an associated root system, and proved one implication in joint work with Shaw. In this paper we prove the other implication, thus confirming the conjecture.

💡 Research Summary

The paper addresses the classical Deligne‑Simpson problem, which asks for the existence of matrices A₁,…,A_k belonging to prescribed similarity (conjugacy) classes C₁,…,C_k ⊂ GL_n(ℂ) such that their product equals the identity and the collection is irreducible (i.e., the matrices have no non‑trivial common invariant subspace). This problem originates from the classification of linear ordinary differential equations on the complex plane via their local monodromy data.

The authors begin by encoding each conjugacy class C_i through three pieces of combinatorial data: a weight w_i (the length of the Jordan block chain), a list of eigenvalues ξ_{i,1},…,ξ_{i,w_i}, and an integer sequence n_{i,0}≥n_{i,1}≥…≥n_{i,w_i}=0 describing the ranks of successive partial products (A_i−ξ_{i,p})…(A_i−ξ_{i,1}). Collecting all these numbers yields a weight sequence w = (w₁,…,w_k) and a vector α_C in the root lattice Γ_w associated to a star‑shaped quiver Q_w whose vertices are a central node ∗ together with vertices