В этой работе дано описание автоморфизмов матричного шара Б^П, ассоциированных с классическими областями второго типа, также изучены некоторые свойства матричного шара второго типа.
We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and
Θ
-reductivity ...of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
For a locally compact metrisable group G, we study the action of
${\rm Aut}(G)$
on
${\rm Sub}_G$
, the set of closed subgroups of G endowed with the Chabauty topology. Given an automorphism T of G, ...we relate the distality of the T-action on
${\rm Sub}_G$
with that of the T-action on G under a certain condition. If G is a connected Lie group, we characterise the distality of the T-action on
${\rm Sub}_G$
in terms of compactness of the closed subgroup generated by T in
${\rm Aut}(G)$
under certain conditions on the center of G or on T as follows: G has no compact central subgroup of positive dimension or T is unipotent or T is contained in the connected component of the identity in
${\rm Aut}(G)$
. Moreover, we also show that a connected Lie group G acts distally on
${\rm Sub}_G$
if and only if G is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on
${\rm Sub}^a_G$
, a subset of
${\rm Sub}_G$
consisting of closed abelian subgroups of G.
In the paper, we study the isotropy groups of
K
x
if D is a simple derivation. We first study the derivation of the form:
D
=
∂
x
1
+
∑
i
=
2
n
(
a
i
x
i
+
b
i
)
∂
x
i
with
a
i
,
b
i
∈
K
x
1
,
...
...,
x
i
−
1
for all
2
≤
i
≤
n
. We prove that
Aut
(
K
x
)
D
=
{
i
d
}
if D is simple with
deg
x
i
−
1
a
i
≥
1
for all
2
≤
i
≤
n
or
a
i
∈
K
*
for all
3
≤
i
≤
n
or
0
≤
deg
x
j
b
i
≤
deg
x
j
a
i
for all
2
≤
j
≤
i
−
1
,
3
≤
i
≤
n
. Thus, we conjecture that
Aut
(
K
x
)
D
=
{
i
d
}
if D is simple with
∏
i
=
2
n
a
i
≠
0
,
a
i
,
b
i
∈
K
x
1
,
...
,
x
i
−
1
for all
2
≤
i
≤
n
. Then we prove that
Aut
(
K
x
)
D
=
{
(
x
1
,
...
,
x
n
−
1
,
x
n
+
c
)
|
c
∈
K
}
if
D
=
(
x
1
s
x
2
+
g
)
∂
x
1
+
x
1
s
−
1
∂
x
2
+
g
3
∂
x
3
+
⋯
+
g
n
∂
x
n
or
D
=
(
x
1
s
x
2
t
+
c
)
∂
x
1
+
x
1
r
∂
x
2
+
g
3
∂
x
3
+
⋯
+
g
n
∂
x
n
with
g
∈
K
x
1
,
deg
(
g
)
≤
s
,
g
(
0
)
≠
0
and
g
i
∈
K
x
i
−
1
\
K
for all
3
≤
i
≤
n
.
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BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
Let F be an algebraically closed field of characteristic 0, L be a finite-dimensional simple Lie algebra of type Al (l≥1), Dl (l≥4), Ek (k=6,7,8) over F. A not necessarily linear map φ:L→L is called ...a 2-local automorphism if for every x,y∈L there is an automorphism φx,y:L→L, depending on x and y, such that φ(x)=φx,y(x), φ(y)=φx,y(y). In this paper, we prove that any 2-local automorphism of L is an automorphism.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Abstract We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$ . As a main step, we derive a labelling set for the irreducible ...$2$ -Brauer characters of the finite symplectic groups $\operatorname {Sp}_{2n}(q)$ (with odd q ), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.
We prove that the affine space of dimension $n\geq 1$ over an uncountable algebraically closed field ${\bf k}$ is determined, among connected affine varieties, by its automorphism group (viewed as an ...abstract group). The proof is based on a new result concerning algebraic families of pairwise commuting automorphisms.
We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group ...of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Let \(X\) be an integral affine scheme of characteristic \(p>0\), and \(\sigma \) a non-identity automorphism of \(X\). If \(\sigma \) is \(\textit{exponential}\), i.e., induced from a \({\bf ...G}_a\)-action on \(X\), then \(\sigma \) is obviously of order \(p\). It is easy to see that the converse is not true in general. In fact, there exists \(X\) which admits an automorphism of order \(p\), but admits no non-trivial \({\bf G}_a\)-actions. However, the situation is not clear in the case where \(X\) is the affine space \({\bf A}_R^n\), because \({\bf A}_R^n\) admits various \({\bf G}_a\)-actions as well as automorphisms of order \(p\). In this paper, we study exponentiality of automorphisms of \({\bf A}_R^n\) of order \(p\), where the difficulty stems from the non-uniqueness of \({\bf G}_a\)-actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of \({\bf A}_R^n\) of order \(p\) are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of \({\bf A}_R^n\) of order \(p\) for each \(n\ge 2\). Here, \(R\) is any UFD which is not a field. (3) We investigate the \({\bf G}_a\)-actions inducing an elementary automorphism of \({\bf A}_R^n\).
The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving ...Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let
be a finite group with a nilpotent subgroup
. Suppose that
acts faithfully on the center of each Sylow subgroup of
. Then it is proved that every class-preserving Coleman automorphism of
is an inner automorphism. In addition, if
is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of
is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.
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