The eigenvalues of oppositeness graphs in
buildings of spherical type
Andries E. Brouwer
Department of Mathematics, Technological University Eindhoven,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2009-06-17
To Reza Khosrovshahi on the occasion of his 70th birthday
Abstract
Consider the graph Γ obtained by taking as vertices the flags in a finite
building of spherical type defined over Fq, where two flags are adjacent
when they are opposite. We show that the squares of the eigenvalues of
Γ are powers of q.
1 Introduction
Let G be a finite group of Lie type with Borel subgroup B and Weyl group
W , so that one has the Bruhat decomposition G =
w
BwB. Let (W, S) be
a Coxeter system, and let w0 be the longest element of W w.r.t. the set of
generators S. Then conjugation by w0 induces a diagram automorphism on the
Coxeter diagram of W (with vertex set S).
Let a type be a nonempty subset of S. Call two types J, K opposite when
K = J w0 (so that J = Kw0 ).
For J S, let WJ := J and PJ := BWJ B. Let an object of type S \ J, or
of cotype J, be a coset gPJ in G. Call two objects gPJ and hPK opposite when
their cotypes J, K are opposite, and moreover PK
h−1gPJ
= PK w0PJ .
Let ΓJ,K , with K = J
w0
, be the bipartite graph with as vertices in one part
the objects of cotype J and in the other part the objects of cotype K, where two
vertices in different parts are adjacent when they are opposite. If J = K, let ΓJ
be the graph with as vertices the objects of cotype J, adjacent when opposite.
Theorem 1.1 Let G be defined over Fq. Let J be a proper subset of S, and let
K = J
w0
. Let θ be an eigenvalue of ΓJ,K or, if J = K, of ΓJ . Then
θ2
=
qe
for some integer e.
The exponents e can be determined explicitly.
2010 Math Subject Classification: 05C50, 05Exx, 20E42, 20G40, 51N30
Contemporary Mathematics
Volume 531, 2010
1
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