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

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