Let k be any field and let G be a connected reductive algebraic k-group. Associated to G is an invariant first studied in the 1960s by Satake Ann. of Math. (2) 71 (1960), 77–110 and Tits Théorie des ...Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62 that is called the index of G (a Dynkin diagram along with some additional combinatorial information). Tits Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62 showed that the k-isogeny class of G is uniquely determined by its index and the k-isogeny class of its anisotropic kernel G_a. For the cases where G is absolutely simple, all possibilities for the index of G have been classified in by Tits Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62. Let H be a connected reductive k-subgroup of maximal rank in G. We introduce an invariant of the G(k)-conjugacy class of H in G called the embedding of indices of H \subset G. This consists of the index of H and the index of G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k-subgroups of G, and observe that the G(k)-conjugacy class of H in G is determined by its index-conjugacy class and the G(k)-conjugacy class of H_a in G. We show that the index-conjugacy class of H in G is uniquely determined by its embedding of indices. For the cases where G is absolutely simple of exceptional type and H is maximal connected in G, we classify all possibilities for the embedding of indices of H \subset G. Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k has cohomological dimension 1 (resp. k=\mathbb {R}, k is \mathfrak {p}-adic).
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Let
k
k
be any field and let
G
G
be a connected reductive algebraic
k
k
-group. Associated to
G
G
is an invariant first studied in the 1960s by Satake Ann. of Math. (2) 71 (1960), 77–110 and Tits ...Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62 that is called the index of
G
G
(a Dynkin diagram along with some additional combinatorial information). Tits Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62 showed that the
k
k
-isogeny class of
G
G
is uniquely determined by its index and the
k
k
-isogeny class of its anisotropic kernel
G
a
G_a
. For the cases where
G
G
is absolutely simple, all possibilities for the index of
G
G
have been classified in by Tits Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62. Let
H
H
be a connected reductive
k
k
-subgroup of maximal rank in
G
G
. We introduce an invariant of the
G
(
k
)
G(k)
-conjugacy class of
H
H
in
G
G
called the embedding of indices of
H
⊂
G
H \subset G
. This consists of the index of
H
H
and the index of
G
G
along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of
k
k
-subgroups of
G
G
, and observe that the
G
(
k
)
G(k)
-conjugacy class of
H
H
in
G
G
is determined by its index-conjugacy class and the
G
(
k
)
G(k)
-conjugacy class of
H
a
H_a
in
G
G
. We show that the index-conjugacy class of
H
H
in
G
G
is uniquely determined by its embedding of indices. For the cases where
G
G
is absolutely simple of exceptional type and
H
H
is maximal connected in
G
G
, we classify all possibilities for the embedding of indices of
H
⊂
G
H \subset G
. Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when
k
k
has cohomological dimension 1 (resp.
k
=
R
k=\mathbb {R}
,
k
k
is
p
\mathfrak {p}
-adic).
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3.
Random generation of associative algebras Sercombe, Damian; Shalev, Aner
Journal of the London Mathematical Society,
January 2024, 2024-01-00, Volume:
109, Issue:
1
Journal Article
Peer reviewed
Open access
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions ...for finite and profinite associative algebras. Let k=Fq$k=\mathbb {F}_q$ be a finite field. Let A$A$ be a finite‐dimensional, associative, unital algebra over k$k$. Let P(A)$P(A)$ be the probability that two elements of A$A$ chosen (uniformly and independently) at random will generate A$A$ as a unital k$k$‐algebra. It is known that if A$A$ is simple, then P(A)→1$P(A) \rightarrow 1$ as |A|→∞$|A| \rightarrow \infty$. We extend this result to a large class of finite associative algebras. For A$A$ simple, we find the optimal lower bound for P(A)$P(A)$ and we estimate the growth rate of P(A)$P(A)$ in terms of the minimal index m(A)$m(A)$ of any proper subalgebra of A$A$. We also study the random generation of simple algebras A$A$ by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let A$A$ be a profinite algebra over k$k$. We show that A$A$ is positively finitely generated if and only if A$A$ has polynomial maximal subalgebra growth. Related quantitative results are also established.
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4.
Agency, potential and contagion Newton, Jonathan; Sercombe, Damian
Games and economic behavior,
January 2020, 2020-01-00, Volume:
119
Journal Article
Peer reviewed
We consider two fundamental forces that can drive the diffusion of an innovation on a network. The first of these forces is potential maximization, a method of aggregating payoff incentives of ...players under individual agency. Potential maximization is related to the graph theoretic property of close-knittedness (Young, 2011). The second force is collective agency, under which sets of players decide together on whether to adjust their strategies. Collective agency is shown to be related to the graph theoretic property of cohesion (Morris, 2000). We compare the relative strengths of these forces under (i) different payoff specifications in coordination games and (ii) different network structures.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Let \(G\) be an affine algebraic group scheme over a field \(k\). We show there exists a unipotent normal subgroup of \(G\) which contains all other such subgroups; we call it the restricted ...unipotent radical \(\mathrm{Rad}_u(G)\) of \(G\). We investigate some properties of \(\mathrm{Rad}_u(G)\), and study those \(G\) for which \(\mathrm{Rad}_u(G)\) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine \(k\)-groups.
Let \(k\) be any field and let \(G\) be a connected reductive algebraic \(k\)-group. Associated to \(G\) is an invariant first studied by Satake and Tits that is called the index of \(G\) (a Dynkin ...diagram along with some additional combinatorial information). Tits showed that the \(k\)-isogeny class of \(G\) is uniquely determined by its index and the \(k\)-isogeny class of its anisotropic kernel \(G_a\). For the cases where \(G\) is absolutely simple, Satake and Tits classified all possibilities for the index of \(G\). Let \(H\) be a connected reductive \(k\)-subgroup of maximal rank in \(G\). We introduce an invariant of the \(G(k)\)-conjugacy class of \(H\) in \(G\) called the embedding of indices of \(H\) in \(G\). This consists of the index of \(H\) and the index of \(G\) along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of \(k\)-subgroups of \(G\), and observe that the \(G(k)\)-conjugacy class of \(H\) in \(G\) is determined by its index-conjugacy class and the \(G(k)\)-conjugacy class of \(H_a\) in \(G\). We show that the index-conjugacy class of \(H\) in \(G\) is uniquely determined by its embedding of indices. For the cases where \(G\) is absolutely simple of exceptional type and \(H\) is maximal connected in \(G\), we classify all possibilities for the embedding of indices of \(H\) in \(G\). Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when \(k\) has cohomological dimension \(1\) (resp. \(k=R\), \(k\) is \(p\)-adic).
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, and finite simple groups in particular. In this paper we study similar notions ...for finite and profinite associative algebras. Let \(k=F_q\) be a finite field. Let \(A\) be a finite dimensional, associative, unital algebra over \(k\). Let \(P(A)\) be the probability that two elements of \(A\) chosen (uniformly and independently) at random will generate \(A\) as a unital \(k\)-algebra. It is known that, if \(A\) is simple, then \(P(A) \to 1\) as \(|A| \to \infty\). We extend this result to a large class of finite associative algebras. For \(A\) simple, we find the optimal lower bound for \(P(A)\) and we estimate the growth rate of \(P(A)\) in terms of the minimal index \(m(A)\) of any proper subalgebra of \(A\). We also study the random generation of simple algebras \(A\) by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let \(A\) be a profinite algebra over \(k\). We show that \(A\) is positively finitely generated if and only if \(A\) has polynomial maximal subalgebra growth. Related quantitative results are also established.
Let \(G\) be a connected real algebraic group. An unrefinable chain of \(G\) is a chain of subgroups \(G=G_0>G_1>...>G_t=1\) where each \(G_i\) is a maximal connected real subgroup of \(G_{i-1}\). ...The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of \(G\). We give a precise formula for the length of \(G\), which generalises results of Burness, Liebeck and Shalev on complex algebraic groups and also on compact Lie groups. If \(G\) is simple then we bound the depth of \(G\) above and below, and in many cases we compute the exact value. In particular, the depth of any simple \(G\) is at most \(9\).
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