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Symmetric Solutions to Symmetric Partial Difference Equations

Shiva Shankar

TL;DR

The paper addresses the existence and structure of symmetric solutions to linear partial difference equations on the lattice $\mathbb{Z}^n$ that are invariant under a finite symmetry group $G$. By identifying the solution space with the dual of a quotient $A^k/P$ and leveraging the representation theory of finite groups acting on $A$, it proves that nonzero $G$-invariant solutions exist whenever a nonzero solution exists and provides a precise description of the $G$-fixed subspace in terms of irreducible decompositions. The automorphism group $\mathrm{Aut}_{\mathbb{C}}(A)$ is shown to be the semidirect product $(\mathbb{C}^*)^n \rtimes GL_n(\mathbb{Z})$, enabling a detailed analysis of symmetric solutions for finite subgroups. The results are illustrated through explicit examples, highlighting how symmetry reduces the solution space and, in some cases, makes all solutions symmetric. This framework offers a rigorous approach to constructing symmetric solutions and understanding their dimensionality in discrete systems with symmetries.

Abstract

This paper studies systems of linear difference equations on the lattice $\Z^n$ that are invariant under a finite group of symmetries, and shows that there exist solutions to such systems that are also invariant under this group of symmetries.

Symmetric Solutions to Symmetric Partial Difference Equations

TL;DR

The paper addresses the existence and structure of symmetric solutions to linear partial difference equations on the lattice that are invariant under a finite symmetry group . By identifying the solution space with the dual of a quotient and leveraging the representation theory of finite groups acting on , it proves that nonzero -invariant solutions exist whenever a nonzero solution exists and provides a precise description of the -fixed subspace in terms of irreducible decompositions. The automorphism group is shown to be the semidirect product , enabling a detailed analysis of symmetric solutions for finite subgroups. The results are illustrated through explicit examples, highlighting how symmetry reduces the solution space and, in some cases, makes all solutions symmetric. This framework offers a rigorous approach to constructing symmetric solutions and understanding their dimensionality in discrete systems with symmetries.

Abstract

This paper studies systems of linear difference equations on the lattice that are invariant under a finite group of symmetries, and shows that there exist solutions to such systems that are also invariant under this group of symmetries.
Paper Structure (6 sections, 8 theorems, 6 equations)

This paper contains 6 sections, 8 theorems, 6 equations.

Table of Contents

  1. introduction
  2. solutions to a difference equation as a $G$-space
  3. finite subgroups of $\mathop{\sf Aut}_\mathbb{C}(A)$
  4. symmetric solutions
  5. examples
  6. acknowledgement

Key Result

Proposition 2.1

Let $G$ be a finite subgroup of $\mathop{\sf Aut}_\mathbb{C}(A)$ and let $V_1, \ldots, V_t$ be its irreducible representations ($V_1$ the trivial representation). Let $P$ be a submodule of $A^k$ which is invariant for the direct sum representation of $G$ on $A^k$. Then $A^k/P$ can be expressed uniqu

Theorems & Definitions (8)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 4.1
  • Theorem 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 4.2
  • Proposition 5.1