[Paper Review] Invariants of commuting matrices

Claudio Procesi|arXiv (Cornell University)|Jan 21, 2015

Advanced Algebra and Geometry7 references3 citations

TL;DR

This paper establishes that the ring of invariants for $ m $-tuples of commuting $ n \times n $ matrices under conjugation is isomorphic to the ring of symmetric polynomials in $ n $ variables under the action of the symmetric group $ S_n $. The key result, proven via classical invariant theory and multilinear reduction, confirms that the restriction map from matrix invariants to symmetric polynomials is an isomorphism, resolving a long-standing question for commuting matrices and extending to all $ m \geq 2 $. The proof relies on the structure of trace invariants and combinatorial data from set partitions.

ABSTRACT

We comment two papers of Domokos and Vaccarino proving that the restriction to diagonal matrices of the scheme of commuting matrices is an isomorphism when restricted to invariants to the symmetric group invariants.

Motivation & Objective

• To resolve the open problem of whether the ideal generated by the commutator relations $[X_i, X_j] = 0$ is prime in the polynomial ring of $ n \times n $ matrices.
• To establish a structural isomorphism between the invariants of $ m $-tuples of commuting matrices under $ GL(n,F) $-conjugation and the invariants of $ m $ diagonal matrices under $ S_n $-action.
• To provide a unified framework for understanding the ring of invariants of commuting matrices using classical invariant theory and polynomial maps.
• To generalize prior results for $ m=2 $ to arbitrary $ m $, including $ m = \infty $, using multilinear reduction techniques.

Proposed method

• Use of the restriction map $ \tilde{\pi}: A_{n,m}^{GL(n,F)} \to B_{n,m}^{S_n} $, which maps invariants of commuting matrices to symmetric polynomials.
• Application of Roby's theorem on polynomial maps to factor the determinant map $ D: F[x_1,\dots,x_m] \to A_{n,m}^{GL(n,F)} $ through the $ n $-th symmetric power.
• Reduction to multilinear invariants via Arnold's method, showing it suffices to verify the isomorphism on multilinear elements.
• Identification of multilinear invariants in $ B_{n,m}^{S_n} $ with orbits of functions $ f: [m] \to [n] $, parametrized by set partitions of $ [m] $ into at most $ n $ parts.
• Construction of invariants in $ A_{n,m}^{GL(n,F)} $ as products of traces $ tr(M) $ for monomials $ M $, with the number of factors bounded by $ n $.
• Proof of injectivity and surjectivity of the map $ \bar{D} $, showing that the inverse correspondence holds via lexicographic ordering of monomials.

Experimental results

Research questions

• RQ1 Is the ideal generated by the commutator relations $[X_i, X_j] = 0$ prime in the polynomial ring of $ n \times n $ matrices over a field of characteristic 0?
• RQ2 Can the ring of invariants of $ m $-tuples of commuting $ n \times n $ matrices under $ GL(n,F) $-conjugation be described as a symmetric algebra?
• RQ3 Does the restriction map from matrix invariants to symmetric polynomials in $ n $ variables induce an isomorphism of invariant rings for all $ m \geq 2 $?
• RQ4 How do multilinear invariants in the commuting matrix setting correspond to combinatorial data such as set partitions of $ [m] $?
• RQ5 What is the structure of the ring of invariants when $ m = \infty $, and how does it relate to symmetric polynomials?

Key findings

• The restriction map $ \tilde{\pi}: A_{n,m}^{GL(n,F)} \to B_{n,m}^{S_n} $ is an isomorphism, establishing a complete correspondence between invariants of commuting matrices and symmetric polynomials.
• The ring of invariants $ A_{n,m}^{GL(n,F)} $ is generated by products of traces $ tr(M) $, with at most $ n $ such factors, due to the Cayley–Hamilton identity.
• Multilinear invariants in $ A_{n,m}^{GL(n,F)} $ are spanned by elements $ t_\Lambda = \prod_{i=1}^k tr_{S_i} $, where $ \Lambda = \{S_1, \dots, S_k\} $ is a partition of $ [m] $ into at most $ n $ subsets.
• The leading monomial of each such invariant $ t_\Lambda $ corresponds to the lexicographically largest representative of its orbit under $ S_n $, ensuring linear independence.
• The nilradical $ J $ of $ A_n $ contains no nonzero $ GL(n,F) $-invariant elements, implying that the invariant ring is reduced.
• The homogeneous components of the invariant ring decompose into Schur functors $ S_\lambda(F^m) $, with the multilinear part corresponding to permutation representations of $ S_m $ on cosets of Young subgroups.