Coxeter_graph_and_determinant
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From: GeometricGroup on 3 Jun 2010 01:02 I am doing a self-study of coxeter_systems. http://tartarus.org/gareth/maths/notes/ii/Groups_and_Representation_Theory_qs.pdf In problem 9 of page 4 in the link, I am having a hard time understanding the hint. In hint, "det2A = 2det2B - (m-2)det2C". Why? I think the formula is borrowed from the Humphrey's book "Reflection_groups_and_Coxeter_groups" page 33. For instance, the value of the determinant of 2A for D_n is 4, which is constant. I tried to use the formula, but it has not been successful for me. Any help will be appreciated. Thanks.
From: Christopher Henrich on 8 Jun 2010 00:07
In article <2100295647.270438.1275555797903.JavaMail.root(a)gallium.mathforum.org>, GeometricGroup <ggx213(a)gmail.com> wrote: > I am doing a self-study of coxeter_systems. > > http://tartarus.org/gareth/maths/notes/ii/Groups_and_Representation_Theory_qs. > > In problem 9 of page 4 in the link, I am having a hard time understanding the > hint. > > In hint, "det2A = 2det2B - (m-2)det2C". Why? > > I think the formula is borrowed from the Humphrey's book > "Reflection_groups_and_Coxeter_groups" page 33. > > For instance, the value of the determinant of 2A for D_n is 4, which is > constant. I tried to use the formula, but it has not been successful for me. > > Any help will be appreciated. > > Thanks. 2A is a matrix in which row n has only two nonzero entries. If you use standard determinant techniques to expand det(2A) in terms of those entries,you get det(2A) = (2A)_{n,n} det(2B) - (2A)_{n,n-1} det(M) where M is a submatrix of 2A. In M, column n-1 has only one nonzero entry. You can expand in terms of that column: det(M) = (2A)_{n-1,n} det(2C). Therefore det(2A) = (2A)_{n,n} det(2B) - (2A)_{n,n-1}(2A)_{n-n,n} det(2C). From context, I surmise that A_{n,n} = 1. So the first term is in agreement with the notes. In the second term, the coefficient of det(2C) is - 4 (cos(\pi/m))^2 . That this should equal - (m-2) for those two values of m is a tricksy coincidence. HTH -- Christopher J. Henrich chenrich(a)monmouth.com http://www.mathinteract.com "A bad analogy is like a leaky screwdriver." -- Boon |