2015/06/17 by Carbone, Lisa, Kownacki, Matt, Murray, Scott H. +1
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1506.05405
Let Δ be a rank 2 hyperbolic root system. Then Δ has generalized Cartan matrix H(a,b)= (\beginsmallmatrix ~2 & -b -a & ~2 \endsmallmatrix) indexed by a,b∈ℤ with ab≥ 5. If a≠ b, then Δ is non-symmetric and is generated by one long simple root and one short simple root; whereas if a= b, Δ is symmetric and is generated by two long simple roots. We prove that if a≠ b, then Δ contains an infinite family of symmetric rank 2 hyperbolic root subsystems H(k,k) for certain k≥ 3, generated by either two short or two long simple roots. We also prove that Δ contains non-symmetric rank 2 hyperbolic root subsystems H(a',b'), for certain a',b'∈ℤ with a'b'≥ 5. One of our tools is a characterization of the types of root subsystems that are generated by a subset of roots. We classify these types of subsystems in rank 2 hyperbolic root systems.