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# The s-tame dimension vectors for stars [Elektronische Ressource] / vorgelegt von Angela Holtmann

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##### Mathematics

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Mathematik
The
Angela
s-tame
v
dimension
2003
v
ersit?t
ectors
v
for
Jan
stars
f?r
Dissertation
Univ
zur
Bielefeld
Erlangung
orgelegt
des
on
Holtmann
der
uar
F
akult?test?ndigem
1.
Pr?fung:
Gutac
der
h
Gedruc
ter:
ISO
Prof.
?ndlic
Dr.
M?rz
Claus
auf
Mic
apier
hael
ag
Ringel
m
2.
hen
Gutac
25.
h
2003
ter:
kt
Prof.
alterungsb
Dr.
P
Andreas
1
Dress
9706
Tectors
Con
9
ten
a
ts
Reections
1
tations
In
and
tro
.
duction
.
1
.
2
families
Subspace
.
represen
.
tations
of
9
.
3
.
Corresp
.
ondence
.
of
41
s-v
The
ectors
8.2
and
.
tuples
of
of
.
comp
.
ositions
yp
11
.
3.1
the
s-v
.
ectors
of
and
.
tuples
.
of
Reections
comp
.
ositions
.
.
.
.
of
.
of
.
er
.
represen
.
for
.
.
.
.
.
8.3
.
tations
.
dimension
.
.
.
.
.
.
.
s-tame
.
ectors
.
v
.
.
.
50
.
dimension
.
.
.
39
11
represen
3.2
ers
The
.
Tits
.
form
.
for
.
tuples
39
of
dimension
comp
.
ositions
.
.
.
.
.
.
.
.
.
.
F
.
osable
.
a
.
42
.
um
.
parameters
.
indecomp
.
.
.
ot
.
ers
.
of
.
.
.
.
.
.
.
.
13
of
4
osable
Ov
the
erview
yp
of
ectors
prop
.
erties
.
of
.
strict
.
tuples
.
of
.
comp
of
ositions
the
16
dimension
5
9.1
Classication
s-tame
of
.
the
.
s-h
.
yp
.
ercritical
Characterisation
and
yp
s-tame
ectors
v
.
ectors
.
27
i
5.1
7.1
Classication
for
of
tations
the
quiv
s-h
.
yp
.
ercritical
.
v
.
ectors
.
.
.
.
.
.
.
.
.
.
.
.
7.2
.
for
.
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
27
.
5.2
.
Classication
8
of
amilies
the
indecomp
s-tame
represen
v
and
ectors
Theorem
.
Kac
.
8.1
.
n
.
b
.
of
.
for
.
of
.
osable
.
tations
.
42
.
Ro
.
systems
.
quiv
.
and
.
Theorem
.
Kac
.
.
.
.
.
.
.
.
.
.
.
.
29
.
5.3
42
Pro
Existence
ofs
families
of
indecomp
Prop
represen
ositions
for
5.3
s-tame
and
s-h
5.4
ercritical
.
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
43
.
Characterisation
.
the
.
and
.
s-h
.
ercritical
.
v
.
50
.
Characterisation
.
the
31
dimension
6
ectors
Decomp
.
osition
.
prop
.
erties
.
of
.
the
.
s-tame
.
v
9.2
ectors
of
34
s-h
7
ercritical
Reection
v
functors,
.
Co
.
xeter
.
functors
.
and
.
the
51
Auslander-Reiten
translate.
ii
of
10
.
s-tame
ercritical
6
of
=
.
tame
.
52
.
10.1
73
An

example:
s-tame
not
of
all
stars)
s-tame
ectors
dimension
.
v
s-tame
ectors
.
are
represen
tame
the
.
.
.
of
.
.
.
.
.
form
.
Tits
.
.
.
for
.
for
.
.
.
.
52
12.2
11
.
Construction
.
metho
.
ds
of
for
Constructing
families
represen
of
v
indecomp
.
osable
.
represen
Constructing
tations
represen-
53
.
11.1
.
Construction
.
metho
116
ds
eness
for
cases
n
negativ
-parameter
tame
families
.
of
.
indecomp
12
osable
dimension
represen-
12.1
tations
s-h
with
v
n
.

.
2
.
.
.
.
for
.
v
.
.
.
.
.
.
.
.
.
.
.
Constructing
.
osable
.
111
.
-parameter
.
osable
.
tions
.
yp
.
with
.
.
.
.
.
.
.
.
.
114
.
parameter
.
osable
.
for
.
ectors
.
.
.
.
.
.
.
.
.
.
.
.
54
Pro
11.2
p
Construction
the
metho
the
ds
B
for
the
one
of
parameter
in
families
(only
of
References
indecomp
.
osable
.
repre-
.
sen
69
tations
Orbits
.
the
.
v
.
71
.
Orbits
.
the
.
yp
.
dimension
.
ectors
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
71
.
Orbits
.
the
.
dimension
.
ectors
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
13
.
families
.
indecomp
.
subspace
.
tations
.
13.1
58
n
11.3
families
Construction
indecomp
metho
subspace
ds
ta-
for
for
indecomp
s-h
osable
ercritical
represen
ectors
tations
n
.
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
13.2
67
one
11.4
families
Another
indecomp
construction
subspace
metho
tations
d
the
for
v
one
.
parameter
.
families
.
of
.
indecomp
.
osable
.
represen
.
tations
.
.
.
.
.
.
.
.
.
.
.
.
A
.
of
.
the
.
ositiv
.
of
.
Tits
.
in
.
nite
.
136
.
Pro
.
of
.
non
.
eness
.
the
.
form
.
the
.
cases
.
for
.
137
.
138
.tation
iii
all
A
opp
c
mem
kno
diploma
wledgemen
m
ts
in
First
guests)
of
Bielefeld
all
for
I
tell
w
w
ould
h
lik
all
e
ers
to
the
thank
group
m
m
y
Ph.D.
sup
me
ervisor,
y
Prof.
ab
C.
curren
M.
and
Ringel,
questions
for
ed
his
this
con
other
tin
b
ued
(and
of
monitoring
represen
and
theory
all
in
his
during
encouragemen
y
ts
and
on
studies
m
giving
y
the
represen
ortunit
tation
to
theoretic
them
w
out
a
y
y
t
.
ork
I
for
w
their
ould
whic
also
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lik
me
e
writing
to
thesis.
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er
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osable
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it
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classes
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osable
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v
but
for
one
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c
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haracterise
y
the
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b
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eha
the
viour
of
of
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for
represen
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tations
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in
inj
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nice
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wing
a
b
y:
sho
There
(see
are
2):
no
the
t
tral
w
of
o
dimension
parameter
ector
families
a
of
is
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osable
w
represen
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tations,
e
and
indecomp
the
represen
dimension
with
v
dimension
ectors
ector,
of
this
the
one
subspace
parameter
tation.
families
the
of
osable
indecomp
tations
osable
s-v
represen
are
tations
a
are
subspace
exactly
tations.
the
s-de
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omp
tegral
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m
s-v
ultiples
d
of
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the
osition
critical
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dimension
r
v
=1
ector
i
corresp
onding
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quiv
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er
that
(see
h
T
the
able
d
4).
is
The
s-v
classication
The
(up
ectors
to
compared
isomorphism)
follo
of
If
all
is
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osable
d
represen
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tations
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of
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tame
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ers
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1
as
called
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ler
b
d
y
and
Dlab
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and
bigger
Ringel
d
in
and
1976
2
(see
The
[5]).
n
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b
dimension
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v
for
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tations
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osable
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an
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ctor
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ev
In
classes
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conditions
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(
3
all
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:
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er
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e
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2
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erties:
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d
,
2
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2

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from
Kac
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y
sho
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wn
is
the
um
follo
parameters
wing:
of
If
with
d
ector?
is
to
a
ectors
p
follo
ositiv
o
e
There
ro
parameter
ot
osable
(see
for
Section
(ii)
8.2),
s-decomp
the
nev
n
-parameter
um
osable
b
with
er
for
1
summands.
q
d
(
er
d
if
)
d
is
,
exactly
ev
the
s-decomp
maximal
d
n
2
um
(
b

er
(
of

parameters
These
needed
to
for
minimal
families
(w.
of
ositions)
indecomp
there
osable
-parameter
represen
represen
tations

with
t
dimension
o
v
hold:
ector
q
d
d
and
=
there
,
is
2.
a
d
unique
d
indecomp
+
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2
represen
an
tation
osition,
with
q
dimension
d
v
)
ector
1
d
q
if
d
and
)
only
1
if
where
q
is
(
Tits
d
corresp
)
to
=
star.
1
next
.
arising
And
this
if
If
d
are
is
man
not
isomorphism
a
of
ro
represen
ot,
what
there
the
is
n
no
b
indecomp
of
osable
needed
represen
families
tation
subspace
with
tations
this
this
dimension
v
v
Our
ector.
is
By
nd
an
s-v
s-nite
d
dimension
the
v
wing
ector
w
w
prop
e
(i)
mean
is
an
one
s-v
family
ector
indecomp
with
subspace
the
tations
prop
d
ert
and
y
for
that
ery
there
osition
are
is
only
er
nitely
n
man
family
y
indecomp
isomorphism
subspace
classes
tations
of
n
subspace
2
represen
either
tations
the
with
An
this
ector
s-v
is
ector.
s-hyp
The
critic
classication
,
of
1.
the
(
s-nite
)
dimension
0
v
and
ectors
for
has
ery
b
trivial
een
osition
done
=
b
1
y
d
Magy
the
ar,
q
W
d
eyman,
)
and
0
Zelevinsky
q
,
d
and
)
can
0
b
fullled.
e
will
found
out
in
b
[13].
the
In
dimension
particular,
ectors
an
r.
s-v
s-decomp
ector
for
d
h
is
exists
s-nite
n
if
family
and
subspace
only
tations
if
n
the
2
follo
winge
4
(in
THE
sho
S-T
the
AME
e
DIMENSION
of
VECTORS
[9],
F
Chapter
OR
esentations
ST
osable
ARS
is
So
o
if
of
w
families
e
w
w
b
an
a
t
is
to
of
nd
the
the
the
s-v
s-tame
ectors
Chapter
with
duced
the
one
prop
nd
erties
pro
(i)
s-tame
and
2
(ii),
dimension
w
is
e
d
ha
ctor
v
c
e
for
to
of
searc
n
h
ers
them
b
among
ectors.
the
in
s-tame
quiv
ones,
to
where
in
w
s-tame
e
an
sa
reection
y
Gel'fand
that
sho
an
problem
s-v
Chapter
ector
results
d
[11
is
existence
s-tame
represen
,
n
if
ones
the
one
follo
and
wing
indeed
t
result
w
Theorem.
o
aic
conditions
fol
hold:

1.

q
ameter
(
ac
d
for
)
omp
=
-p
0
p
,
(over
and
either
2.
tame
if
of
d
tations
=
constructed
d
dimension
1
it
+
that
d
trast
2
of
is

an
p
s-decomp
ose
osition,
v
then
the
q
w
(
v
d
giv
1
v
)
er

in
0
y
and
P
q
])
(
ho
d
reduce
2
nding
)
tations.

one
0
summary
.
Kac
It
[12]
will
and
turn
for
out
one
that
indecomp
the
for
s-tame
ectors
dimension
families
v
yp
ectors
n
are
(Basically
exactly
to
the
the
s-v
yp
ectors
ectors
with
ots.)
the
the
prop-
this
erties
v
(i)
et
and
an
(ii).
ly
W
Then
e
assertions
see
quiv-
that
dimension
(b
is
y
e
denition)
p
ev
of
ery
osable
s-v
r
ector
K
whic
,
h
s-de
is
ther
smaller
an
than
ameter
an
c
s-tame
subsp
one
epr
has
)
either
2
Tits
the
form
non

quiv
1
families
or
decomp
is
represen
also
can
s-tame.
e
s-nite,
for
s-tame
s-tame
and
v
s-h
Here
yp
is
ercritical
wn
dimension

v
con
ectors
to
will
situation
alw
tame
a
ers
ys
it
b
not
e
ossible
dimension
decomp
v
an
ectors
dimension
of
ector
subsp
to
ac
sum
e
t
r
o
epr
dimension
esentations
ectors.
.
7
This
es
text
o
is
erview
organised
v
as
the
follo
functors
ws:
tro
In
b
Chapter
Bernstein,
2
and
w
onomarev
e
[2
are
and
going
ws
to
w
sho
can
w
the
that
of
all
families
indecomp
represen
osable
In
represen
8
tations
can
of
a
the
of
s-v
of
ectors
(from
are
[10],
and
subspace
])
represen
the
tations.
of
Then
the

of
in
parameter
Chapter
of
3
osable

tations
w
the
e
v
rewrite
and
the
-parameter
s-v
for
ectors
s-h
in
ercritical
terms
with
of

tuples
.
of
,
comp
has
ositions
sho
and
that
calculate
s-tame
the
s-h
Tits
ercritical
form
v
for
are
the
ro
tuples
In
of
9
comp
main
ositions.
of
Chapter
text
4
pro
sho
en:
ws
L
the
K
main
e
prop
algebr
erties
al
of
close
the
eld.
Tits
the
form
lowing
for
ar
the
e
tuples
alent:
of
A
comp
ve
ositions
d
and
s-tame.
giv
Ther
es
is
the
one
basics
ar
for
family
the
inde
classications
omp
of
subsp
all
e
s-h
epr
yp
(over
ercritical
)
and
d
s-tame
but
v
every
ectors
c
whic
osition
h
e
are
never
done
n
in
ar
Chapter
family
5.
inde
In
om-
Chapter
osable
6
ac
w
r
e
esentations
will
K
see
with
ho

w
for
in
of
the
summands.
cases