Degree complexity of birational maps related to matrix inversion: Symmetric case

TL;DR

This paper calculates the degree complexity of a birational map related to matrix inversion and entry-wise reciprocal on symmetric matrices, confirming a conjecture relevant to statistical mechanics.

Contribution

It provides the first computation of the degree complexity for this class of birational maps, verifying a previously conjectured value.

Findings

Degree complexity computed for the map $K| ext{Sym}_q$

Confirmation of the conjecture by Angles d'Auriac et al.

Insights into the algebraic structure of matrix inversion-related maps

Abstract

For $q\geq 3$, we let $\mathcal{S}_q$ denote the projectivization of the set of symmetric $q\times q$ matrices with coefficients in $\mathbb{C}$. We let $I(x)=(x_{i,j})^{-1}$ denote the matrix inversion, and we let $J(x)=(x_{i,j}^{-1})$ be the matrix whose entries are the reciprocals of the entries of $x$. We let $K|\mathcal{S}_q=I\circ J:\mathcal{S}_q\rightarrow \mathcal{S}_q$ denote the restriction of the composition $I\circ J$ to $\mathcal{S}_q$. This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity of $K|\mathcal{S}_q$, thus confirming a conjecture of Angles d'Auriac, Maillard, and Viallet in [J. Phys. A: Math. Gen. 39 (2006), 3641--3654].