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Entries in canonical_matrix for Coxeter groups

asked 2019-11-06 11:05:59 -0600

Bob67846 gravatar image

Can anyone please tell me what a is doing in the following output? (Depending on the ordering of the Dynkin diagram, I believe it should be either 1 or 3.)

sage: W = WeylGroup(["G",2]) ; s = W.simple_reflections() ; s[2].canonical_matrix()

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answered 2019-11-06 13:23:30 -0600

tmonteil gravatar image

updated 2019-11-06 14:12:14 -0600

a is the square root of 3, as the coefficients of this matrix are defined on the number field with defining polynomial x^2 - 3, see:

sage: M = s[2].canonical_matrix()
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
sage: M
[ 1  0]
[ a -1]
sage: M[1,0]
a
sage: M[1,0]^2
3
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Woops, thanks!

Bob67846 gravatar imageBob67846 ( 2019-11-17 07:43:19 -0600 )edit

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Asked: 2019-11-06 11:05:59 -0600

Seen: 19 times

Last updated: Nov 06