From 9a86d6e3f547246988a83414881d4bdcf8d04389 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 8 Mar 2026 22:10:30 -0400 Subject: [PATCH] mjo/clan/vinberg_clan.py: cite references --- mjo/clan/vinberg_clan.py | 52 ++++++++++++++++++++++++++++++++++------ 1 file changed, 45 insertions(+), 7 deletions(-) diff --git a/mjo/clan/vinberg_clan.py b/mjo/clan/vinberg_clan.py index 86ab8a8..9770dc3 100644 --- a/mjo/clan/vinberg_clan.py +++ b/mjo/clan/vinberg_clan.py @@ -1,3 +1,26 @@ +""" +The (up to isomorphism) clan associated with the Vinberg cone. + +REFERENCES: + + .. [Herrington2021] Elliot Michael Herrington. + Highly symmetric homogeneous Kobayashi-hyperbolic manifolds. + Ph.D. Thesis, University of Adelaide, School of Mathematical + Sciences, 2021. https://hdl.handle.net/2440/133439 + + .. [Ishi2013] Hideyuki Ishi. + On a class of homogeneous cones consisting of real + symmetric matrices. + Josai Mathematical Monographs, 6(1)71-80, 2013. + :doi:`10.20566/13447777_6_71`. + + .. [IshiKoufany2021] Hideyuki Ishi and Khalid Koufany. + The Compression Semigroup of the Dual Vinberg Cone. + Springer Proceedings in Mathematics & Statistics, vol 366. + Springer, Cham. :doi:`10.1007/978-3-030-78346-4_8`. + +""" + from mjo.clan.normal_decomposition import NormalDecomposition class VinbergClan(NormalDecomposition): @@ -241,11 +264,14 @@ class VinbergClan(NormalDecomposition): r""" Generate a random triangular automorphism of the Vinberg cone. - Elliot Herrington in his thesis "Highly symmetric homogeneous - Kobayashi-hyperbolic manifolds" gives a formula for the - connected component of the identity in the group of triangular - automorphisms. This won't generate the whole group, but it's - a good start. + We use the formula in [Herrington2021]_ for the connected + component of the identity in the group of triangular + automorphisms. This triangular group is simply connected, so + we obtain the whole thing. + + Another option would be to use Lemma 4 in [Ishi2013]_ which + applies more generally to homogeneous cones arising from + chordal graphs. """ from sage.matrix.matrix_space import MatrixSpace R = self.base_ring() @@ -273,8 +299,8 @@ class VinbergClan(NormalDecomposition): Generate a random automorphism of the Vinberg cone that fixes the unit element. - This is effectively a guess, based on the work done by Ishi - and Koufany for the **dual** Vinberg cone. + This is ultimately a guess based on the results in + [IshiKoufany2021]_ for the dual Vinberg cone. SETUP:: @@ -333,6 +359,18 @@ class VinbergClan(NormalDecomposition): def random_cone_automorphism(self): r""" Generate a random automorphism of the Vinberg cone. + + We have two options here. The current implementation combines + a random triangular automorphism from [Herrington2021]_ with a + random isotropy. + + The introduction to [Ishi2013]_ claims that the full + automorphism group is generated by "two" transformations. His + clan differs from ours, but only in that the first two + idempotents are swapped, so we could easily translate between + the two. Option (b) would be to generate a random product of + these transformations, and then pre- and post-compose the + result with the e11 <-> e22 swap. """ T = self.random_triangular_cone_automorphism() K = self.random_isotropy_cone_automorphism() -- 2.51.0