Fix method is_real_root() in root_lattice_realizations.py

[sagetrac-jonathan-judge]

Currently, the element method is_real_root() of root_lattice_realizations.py is not quite correct. It checks only if norm_squared() is positive. This is correct in a root system, but not when in the root lattice. We can add an additional check in this situation: see Proposition 5.10b in Kac's book.

sage: R = RootSystem(['A',1,1])
sage: Q = R.root_lattice()
sage: alpha = Q.simple_roots()
sage: beta = 3*alpha[0] + alpha[1] # not a root
sage: beta.norm_squared()
8
sage: beta.is_real_root()
True

sage: R = RootSystem("A2")
sage: Q = R.root_lattice()
sage: alpha = Q.simple_roots()
sage: beta = 2*alpha[1] + alpha[2] # not a root
sage: beta.norm_squared()
6
sage: beta.is_real_root()
True

Updated: Kac's Proposition 5.10b applies to finite, affine, and hyperbolic types. Ticket #18645 implements the checks for these. More generally, the correct check for a real root is to see if its norm_squared() is positive and if its multiplicity (#18000) is positive.

Depends on #18000
Depends on #18645

CC: @bsalisbury1 @tscrim @sagetrac-sage-combinat

Component: combinatorics

Keywords: days65

Branch/Commit: u/jonathan.judge/fix_method_is_real_root___in_root_lattice_realizations_py @ 55d2f52

Issue created by migration from https://trac.sagemath.org/ticket/18644

[sagetrac-jonathan-judge]

Changed keywords from none to days65

[sagetrac-jonathan-judge]

Branch: u/jonathan.judge/fix_method_is_real_root___in_root_lattice_realizations_py

[sagetrac-jonathan-judge]

Dependencies: #18000, #18645

[sagetrac-jonathan-judge]

Commit: c7a1b37

[sagetrac-jonathan-judge]

New commits:

c7a1b37

Fixed element method is_real_root() in root_lattice_realizations.py

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4958ee0	Added a function multiplicity() on elements of a root lattice. The computation is done via Peterson's recurrent formula
d700a67	Merge branch 'develop' into t/18000/root_multiplicities_for_symmetrizable_kac_moody_algebras
77f1bf6	Added methods is_indefinite(), is_hyperbolic(), is_indecomposable(), and principal_submatrices() to the class CartanMatrix
2560229	Merge branch 't/18645/add_some_methods_to_cartanmatrix' into t/18000/root_multiplicities_for_symmetrizable_kac_moody_algebras
3cc6587	Merge branch 't/18645/add_some_methods_to_cartanmatrix' into t/18644/fix_method_is_real_root___in_root_lattice_realizations_py
6784229	Merge branch 't/18644/fix_method_is_real_root___in_root_lattice_realizations_py' into t/18000/root_multiplicities_for_symmetrizable_kac_moody_algebras
55d2f52	Rewrote multiplicity computation to take advantage of new and corrected methods in 18644 and 18645

[sagetrac-git]

Changed commit from c7a1b37 to 55d2f52