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research
Transformations of polar Grassmannians preserving certain intersecting relations
Authors
Wen Liu
Mark Pankov
Kaishun Wang
Publication date
8 July 2013
Publisher
View
on
arXiv
Abstract
Let
Ξ
\Pi
Ξ
be a polar space of rank
n
β₯
3
n\ge 3
n
β₯
3
. Denote by
G
k
(
Ξ
)
{\mathcal G}_{k}(\Pi)
G
k
β
(
Ξ
)
the polar Grassmannian formed by singular subspaces of
Ξ
\Pi
Ξ
whose projective dimension is equal to
k
k
k
. Suppose that
k
k
k
is an integer not greater than
n
β
2
n-2
n
β
2
and consider the relation
R
i
,
j
{\mathfrak R}_{i,j}
R
i
,
j
β
,
0
β€
i
β€
j
β€
k
+
1
0\le i\le j\le k+1
0
β€
i
β€
j
β€
k
+
1
formed by all pairs
(
X
,
Y
)
β
G
k
(
Ξ
)
Γ
G
k
(
Ξ
)
(X,Y)\in {\mathcal G}_{k}(\Pi)\times {\mathcal G}_{k}(\Pi)
(
X
,
Y
)
β
G
k
β
(
Ξ
)
Γ
G
k
β
(
Ξ
)
such that
dim
β‘
p
(
X
β₯
β©
Y
)
=
k
β
i
\dim_{p}(X^{\perp}\cap Y)=k-i
dim
p
β
(
X
β₯
β©
Y
)
=
k
β
i
and
dim
β‘
p
(
X
β©
Y
)
=
k
β
j
\dim_{p} (X\cap Y)=k-j
dim
p
β
(
X
β©
Y
)
=
k
β
j
(
X
β₯
X^{\perp}
X
β₯
consists of all points of
Ξ
\Pi
Ξ
collinear to every point of
X
X
X
). We show that every bijective transformation of
G
k
(
Ξ
)
{\mathcal G}_{k}(\Pi)
G
k
β
(
Ξ
)
preserving
R
1
,
1
{\mathfrak R}_{1,1}
R
1
,
1
β
is induced by an automorphism of
Ξ
\Pi
Ξ
and the same holds for the relation
R
0
,
t
{\mathfrak R}_{0,t}
R
0
,
t
β
if
n
β₯
2
t
β₯
4
n\ge 2t\ge 4
n
β₯
2
t
β₯
4
and
k
=
n
β
t
β
1
k=n-t-1
k
=
n
β
t
β
1
. In the case when
Ξ
\Pi
Ξ
is a finite classical polar space, we establish that the valencies of
R
i
,
j
{\mathfrak R}_{i,j}
R
i
,
j
β
and
R
i
β²
,
j
β²
{\mathfrak R}_{i',j'}
R
i
β²
,
j
β²
β
are distinct if
(
i
,
j
)
β
(
i
β²
,
j
β²
)
(i,j)\ne (i',j')
(
i
,
j
)
ξ
=
(
i
β²
,
j
β²
)
.Comment: 13 page
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