# Partitions of unity in $$\mathrm {SL}(2,\mathbb Z)$$ SL ( 2 , Z ) , negative continued fractions, and dissections of polygons

Partitions of unity in $$\mathrm {SL}(2,\mathbb Z)$$ SL ( 2 , Z ) , negative continued fractions,... We characterize sequences of positive integers $$(a_1,a_2,\ldots ,a_n)$$ ( a 1 , a 2 , … , a n ) for which the $$2\times 2$$ 2 × 2 matrix $$\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right)$$ a n - 1 1 0 a n - 1 - 1 1 0 ⋯ a 1 - 1 1 0 is either the identity matrix $$\mathrm {Id}$$ Id , its negative $$-\,\mathrm {Id}$$ - Id , or square root of $$-\,\mathrm {Id}$$ - Id . This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# Partitions of unity in $$\mathrm {SL}(2,\mathbb Z)$$ SL ( 2 , Z ) , negative continued fractions, and dissections of polygons

, Volume 5 (2) – Apr 9, 2018
25 pages

/lp/springer-journals/partitions-of-unity-in-mathrm-sl-2-mathbb-z-sl-2-z-negative-continued-Heo55vd4rW

# References (28)

Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
DOI
10.1007/s40687-018-0139-z
Publisher site
See Article on Publisher Site

### Abstract

We characterize sequences of positive integers $$(a_1,a_2,\ldots ,a_n)$$ ( a 1 , a 2 , … , a n ) for which the $$2\times 2$$ 2 × 2 matrix $$\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right)$$ a n - 1 1 0 a n - 1 - 1 1 0 ⋯ a 1 - 1 1 0 is either the identity matrix $$\mathrm {Id}$$ Id , its negative $$-\,\mathrm {Id}$$ - Id , or square root of $$-\,\mathrm {Id}$$ - Id . This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Apr 9, 2018