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

144 pages
MathematikTheAngelas-tamevdimension2003versit?tectorsvforJanstarsf?rDissertationUnivzurBielefeldErlangungorgelegtdesonDoktorgradesHoltmannderuarFakult?test?ndigem1.Pr?fung:GutacderhGedructer:ISOProf.?ndlicDr.M?rzClausaufMicapierhaelagRingelm2.henGutac25.h2003ter:ktProf.alterungsbDr.PAndreas1Dress9706TectorsCon9tenatsReections1tationsInandtro.duction.1.2familiesSubspace.represen.tationsof9.3.Corresp.ondence.of41s-vTheectors8.2and.tuplesofof.comp.ositionsyp11.3.1thes-v.ectorsofand.tuples.ofReectionscomp.ositions....of.of.er.represen.for.....8.3.tations.dimension.......s-tame.ectors.v...50.dimension...3911represen3.2ersThe.Tits.form.for.tuples39ofdimensioncomp.ositions..........F.osable.a.42.um.parameters.indecomp...ot.ers.of........13of4osableOvtheerviewypofectorsprop.erties.of.strict.tuples.of.compofositionsthe16dimension59.1Classications-tameof.the.s-h.yp.ercriticalCharacterisationandyps-tameectorsv.ectors.27i5.17.1Classicationforoftationsthequivs-h.yp.ercritical.v.ectors............7.2.for.v....................27.5.2.Classication8ofamiliestheindecomps-tamerepresenvandectorsTheorem.Kac.8.1.n.b.of.for.
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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
Doktorgrades
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
advice,
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
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also
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lik
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osable
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for
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tations
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in
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nice
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are
2):
no
the
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tral
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of
o
dimension
parameter
ector
families
a
of
is
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osable
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e
and
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v
dimension
ectors
ector,
of
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the
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tation.
families
the
of
osable
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osable
s-v
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tations
a
are
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exactly
tations.
the
s-de
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omp
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m
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the
osition
critical
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dimension
r
v
=1
ector
i
corresp
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onding
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quiv
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h
T
the
able
d
4).
is
The
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ectors
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of
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osable
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ers
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ler
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y
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in
and
1976
2
(see
The
[5]).
n
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b
dimension
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tations
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ector
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of
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osable
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are
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ctor
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if1.
1
ev
In
classes
tro
conditions
duction
(
3
all
The
called
Tits
t.
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the
:
for
Z
a
Q
represen
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2.
!
are
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of
and
a
(
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er
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t
Q
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an
;
of
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q
1
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s;
e
t
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)
w
is
)
giv
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en
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b

y

q
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question
d
innitely
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tations,
:=
er
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i
aim
2
with
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erties:
0
of
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Bernstein,
2
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tations
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ectors
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])
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of
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parameter
Chapter
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tations
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ectors
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tuples
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ercritical
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of
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text
4
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close
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classications
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ar
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5.
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In
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osable
6
ac
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see
with
ho

w
for
in
of
the
summands.
cases

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