Journal ArticleDOI

# On the Grassmann modules for the unitary groups

30 Mar 2010-Linear & Multilinear Algebra (Taylor & Francis Group)-Vol. 58, Iss: 7, pp 887-902

Abstract: Let V be 2n-dimensional vector space over a field equipped with a nondegenerate skew--Hermitian form f of Witt index n epsilon 1, let 0 be the fix field of and let G denote the group of isometries of (V, f). For every k {1, ..., 2n}, there exist natural representations of the groups G U(2n, /0) and H = G SL(V) SU(2n, /0) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.
Topics: Unitary group (56%), Group (mathematics) (53%), Vector space (52%), Field (mathematics) (50%), Linear algebra (50%)

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On the Grassmann modules for the unitary
groups
Bart De Bruyn
Ghent University, Department of Pure Mathematics and Computer Algebra,
Krijgslaan 281 (S22), B-9000 Gent, Belgium, E-mail: bdb@cage.ugent.be
Abstract
Let V be 2n-dimensional vector space over a ﬁeld K equipped with
a nondegenerate skew-ψ-Hermitian form f of Witt index n 1, let
K
0
K be the ﬁx ﬁeld of ψ and let G denote the group of isometries of
(V, f). For every k {1, . . . , 2n}, there exist natural representations
of the groups G
=
U(2n, K/K
0
) and H = GSL(V )
=
SU(2n, K/K
0
)
on the k-th exterior power of V . With the aid of linear algebra, we
prove some properties of these representations. We also discuss some
applications to projective embeddings and hyperplanes of Hermitian
dual polar spaces.
Keywords: Grassmann module, unitary group, Hermitian dual polar space, hy-
perplane
MSC2000: 15A75, 15A63, 20C33, 51A50
1 Introduction
This paper is an essay in which we will use methods based on linear algebra to
derive several facts regarding structures which are related to a 2n-dimensional
K-vector space V which is endowed with a nondegenerate skew-Hermitian
form f of maximal Witt index n. These methods allow us to give more
elegant proofs for some known results, and to state some known results in
a language which is more elegant and more suitable for future applications.
More precisely, we will do the following:
(1) If ψ denotes the involutary automorphism of K associated to f and
if K
0
K denotes the ﬁx ﬁeld of ψ, then we will prove the irreducibility of
certain modules for the groups U(2n, K/K
0
) and SU(2n, K/K
0
).
1

(2) We will give a more elegant description (and a more elegant proof
for the existence) of the Baer-K
0
-subgeometry PG(W
) of PG(
V
n
V ) which
aﬀords the Grassmann embedding of the dual polar space DH(2n 1, K, ψ)
associated to (V, f).
(3) Every hyperplane H of DH(2n1, K, ψ) which arises from the Grass-
mann embedding can be described by a certain vector of W
, a so-called
representative vector of H. De Bruyn and Pralle [9] proved that the ﬁnite
Hermitian dual polar space DH(5, q
2
) has 5 isomorphism classes of hyper-
planes arising from the Grassmann embedding. We determine a representa-
tive vector for each of these 5 isomorphism classes.
Remark. In [8], we used techniques based on linear algebra to derive several
facts regarding structures related to a 2n-dimensional vector space endowed
with a nondegenerate alternating bilinear form.
1.1 Certain representations of unitary groups
Let n be a strictly positive integer and let K
0
, K be two ﬁelds such that K is
a quadratic Galois extension of K
0
. Put K
:= K \ {0} and K
0
:= K
0
\ {0}.
Let ψ denote the unique nontrivial element in Gal(K/K
0
) and let V be a
2n-dimensional vector space over K equipped with a nondegenerate skew-ψ-
Hermitian form f of Witt index n.
An ordered basis (¯e
1
,
¯
f
1
, . . . , ¯e
n
,
¯
f
n
) of V is called a hyperbolic basis of V
if f(¯e
i
, ¯e
j
) = f(
¯
f
i
,
¯
f
j
) = 0 and f(¯e
i
,
¯
f
j
) = δ
ij
for all i, j {1, . . . , n}. Let
G denote the group of isometries of (V, f), i.e. the set of all θ GL(V )
satisfying f(θ(¯x), θ(¯y)) = f(¯x, ¯y) for all ¯x, ¯y V . Then G
=
U(2n, K/K
0
)
and H := GSL(V )
=
SU(2n, K/K
0
). The elements of G are precisely those
elements of GL(V ) which map hyperbolic bases of V to hyperbolic bases of V .
It can be proved (see Lemma 2.2) that if θ G, then there exists an η K
such that det(θ) =
η
ψ
η
. We denote by η
θ
any of the elements of K
satisfying
this property. The element η
θ
is uniquely determined up to a factor of K
0
.
If θ
1
, θ
2
G, then η
θ
2
θ
1
· η
1
θ
1
· η
1
θ
2
K
0
since det(θ
2
θ
1
) = det(θ
1
) · det(θ
2
).
For every k {0, . . . , 2n}, let
V
k
V be the k-th exterior power of V . Then
V
0
V = K and
V
1
V = V . If k {1, . . . , 2n}, then for every θ GL(V ),
there exists a unique
e
θ
k
GL(
V
k
V ) such that
e
θ
k
(¯v
1
¯v
2
· · · ¯v
k
) =
θ(¯v
1
) θ(¯v
2
) · · · θ(¯v
k
) for all vectors ¯v
1
, ¯v
2
, . . . , ¯v
k
V . The map θ 7→
e
θ
k
deﬁne representations R
k
and R
0
k
of the respective groups G
=
U(2n, K/K
0
)
and H
=
SU(2n, K/K
0
) on the
2n
k
-dimensional vector space
V
k
V . We
call the corresponding KG-modules (respectively KH-modules) Grassmann
modules for G (respectively H). We put
e
G
k
:= {
e
θ
k
| θ G} and
e
H
k
:=
2

{
e
θ
k
| θ H}. The following result might be known (during the course of
writing this paper, the author observed that a group-theoretical proof of this
fact is also contained in the preprint [2]). Anyhow, we will prove it in Section
3 with the aid of elementary linear algebra.
Theorem 1.1 For every k {1, . . . , 2n}, the representation R
0
k
is irre-
ducible.
Theorem 1.1 has the following corollary:
Corollary 1.2 (1) For every k {1, . . . , 2n}, the representation R
k
is irre-
ducible.
(2) For every k {1, . . . , n}, the subspace of
V
k
V generated by all vectors
of the form ¯v
1
· · · ¯v
k
such that h¯v
1
, . . . , ¯v
k
i is totally isotropic with respect
to f coincides with
V
k
V .
Proof. Claim (1) follows from the fact that H is a subgroup of G.
Now, let k {1, . . . , n}. Obviously, the subspace of
V
k
V generated by
all vectors of the form ¯v
1
· · · ¯v
k
such that h¯v
1
, . . . , ¯v
k
i is totally isotropic
with respect to f is stabilized by
e
G
k
. Claim (2) then follows from Claim (1).
In Section 4, we prove the following:
Theorem 1.3 There exists a set W
of vectors of
V
n
V satisfying the fol-
lowing properties:
(1) The set W
is a
2n
n
-dimensional vector space over K
0
of vectors and multiplication with scalars inherited from
V
n
V ).
(2) For every θ G,
e
θ
n
(W
) = {
α
η
θ
| α W
}.
If θ H, then η
θ
K
0
and we have
Corollary 1.4 If θ H, then
e
θ
n
(W
) = W
.
Now, for every map θ H, let
b
θ be the element of GL(W
) mapping α W
to
e
θ
n
(α) W
. Then the map θ 7→
b
θ deﬁnes a representation
b
R of the group
H
=
SU(2n, K/K
0
) on the
2n
n
-dimensional K
0
-vector space W
. The corre-
sponding K
0
H-module is also called a Grassmann module for SU(2n, K/K
0
).
Put
b
H := {
b
θ | θ H}. As a consequence of Theorem 1.1, we have
Corollary 1.5 The representation
b
R is irreducible.
3

Proof. Suppose U is a subspace of W
which is stabilized by
b
H. The
subspace U is contained in a unique subspace U of
V
n
V with the same
dimension as U. Obviously, U is stabilized by
e
H
n
. So by Theorem 1.1, either
U = 0 or U =
V
n
V . Hence, either U = 0 or U = W
.
1.2 The Grassmann embedding of the dual polar space
DH(2n 1, K, ψ)
A full (projective) embedding of a point-line geometry S is an injective map-
ping e from the point-set P of S to the point-set of a projective space Σ
satisfying (i) he(P)i
Σ
= Σ and (ii) e(L) is a line of Σ for every line L of S.
Let Π be a polar space (Tits [12], Veldkamp [13]) of rank n 2. With Π
there is associated a point-line geometry which is called a dual polar space,
see Cameron [3]. The points of are the maximal singular subspaces of Π,
the lines of are the next-to-maximal singular subspaces of Π, and incidence
is reverse containment. If ω
1
and ω
2
are two maximal singular subspaces of
Π, then d(ω
1
, ω
2
) denotes the distance between ω
1
and ω
2
in the collinearity
graph of ∆. We have d(ω
1
, ω
2
) = n 1 dim(ω
1
ω
2
). The points ω
1
and
ω
2
of are called opposite if they lie at maximal distance n from each other.
The dual polar space is a near polygon, which means that for every point
x and every line L there exists a unique point on L nearest to x. If x is
a point of ∆, then x
denotes the set of points of equal to or collinear
with x. There exists a bijective correspondence between the possibly empty
singular subspaces of Π and the nonempty convex subspaces of ∆. If ω is an
(n 1 k)-dimensional singular subspace of Π, then the set of all maximal
singular subspaces of Π containing ω is a convex subspace of of diameter k.
These convex subspaces are called quads if k = 2. Any two points x
1
and x
2
of at distance k from each other are contained in a unique convex subspace
hx
1
, x
2
i of diameter k. If x is a point and S is a convex subspace, then there
exists a unique point π
S
(x) S such that d(x, y) = d(x, π
S
(x)) + d(π
S
(x), y)
for every point y S. The convex subspaces through a given point x of
deﬁne an (n 1)-dimensional projective space which we will denote by
Res(x).
As in Section 1.1, let V be a 2n-dimensional vector space over K equipped
with a nondegenerate skew-ψ-Hermitian form f of Witt index n 2. With
the nondegenerate skew-ψ-Hermitian form f, there is associated a Hermitian
polar space H(2n 1, K, ψ) and a Hermitian dual polar space DH(2n
1, K, ψ). The singular subspaces of H(2n 1, K, ψ) are the subspaces of
PG(2n 1, K) which are totally isotropic with respect to the Hermitian
4

polarity of P G(V ) deﬁned by f. In Section 4, we will prove the following
regarding the vector space W
alluded to in Theorem 1.3.
Theorem 1.6 (1) For every maximal singular subspace ω = h¯v
1
, ¯v
2
, . . . , ¯v
n
i
of H(2n 1, K, ψ), there exists a unique point e
gr
(ω) = hβi in PG(W
) such
that β W
and ¯v
1
¯v
2
· · · ¯v
n
are linearly dependent vectors of
V
n
V .
(2) The map ω 7→ e
gr
(ω) deﬁnes a full embedding of DH(2n 1, K, ψ)
into the Baer-K
0
-subgeometry PG(W
) of PG(
V
n
V ).
The projective embedding e
gr
mentioned in Theorem 1.6(2) is called the
Grassmann embedding of DH(2n 1, K, ψ).
Remark. Another description of the Baer-K
0
-subgeometry of PG(
V
n
V )
which aﬀords the Grassmann embedding of DH(2n 1, K, ψ) was given in
[7]. The description and proof which we will give in Section 4 seem more
elegant. In [5, Proposition 5.1], there was given a description of a K
0
-vector
space W
V
n
V stabilized by
e
H
n
such that PG(W ) aﬀords the Grassmann
embedding of DH(2n1, K, ψ). The proof given in [5] is however not correct
as was already pointed out in [7]. Also some corrections must be performed
in [5] in order to get the right equation for W (e.g., observe the coeﬃcient
(1)
l
in the formula at the beginning of Section 4).
A set of points of DH(2n1, K, ψ) distinct from the whole point-set is called
a hyperplane of DH(2n1, K, ψ) if it intersects every line in either a singleton
or the whole line. If π is a hyperplane of PG(W
), then the set of all points p
of DH(2n1, K, ψ) such that e
gr
(p) π is a hyperplane of DH(2n1, K, ψ).
Any hyperplane of DH(2n 1, K, ψ) which can be obtained in this way is
said to arise from e
gr
.
If K is the ﬁnite ﬁeld F
q
2
with q
2
elements (so, K
0
=
F
q
and ψ : K K :
x 7→ x
q
), then we will denote H(2n 1, K, ψ) and DH(2n 1, K, ψ) also by
H(2n 1, q
2
) and DH(2n 1, q
2
).
Consider now the special case n = 3, K = F
q
2
and let A denote the group
of automorphisms of DH(5, q
2
). For every ϕ A, there exists a unique
collineation eϕ of PG(W
) such that e
gr
(ϕ(p)) = eϕ(e
gr
(p)) for every point p
of DH(5, q
2
). By De Bruyn and Pralle [9], the group A has 5 orbits on the
set of hyperplanes of DH(5, q
2
) arising from e
gr
. In Section 5, we will show
that this implies that
e
A := { eϕ | ϕ A} has 5 orbits on the set of points of
PG(W
), and we will determine an explicit description of a point of each of
these ﬁve orbits.
5

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Abstract: Complexes.- Coxeter complexes.- Buildings.- Reduction.- The building of a semi-simple algebraic group.- Buildings of type An, Dn, En.- Buildings of type Cn. I. Polar spaces.- Buildings of type Cn. II. Projective embeddings of polar spaces.- Buildings of type Cn. III. Non-embeddable polar spaces.- Buildings of type F4.- Finite BN-pairs of irreducible type and rank ? 3.- Appendix 1. Shadows.- Appendix 2. Generators and relations.

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• ...Let Π be a polar space (Tits [12], Veldkamp [13]) of rank n ≥ 2....

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• ...The generalized quadrangle Q−(5, q) admits subquadrangles isomorphic to Q(4, q), see Payne and Thas [10]....

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• ...With Π there is associated a point-line geometry ∆ which is called a dual polar space, see Cameron [3]....

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43 citations

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