Questions tagged [involutions]
The involutions tag has no usage guidance.
35
questions with no upvoted or accepted answers
8
votes
0
answers
651
views
What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?
Groups
Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.
Question 1. What classifies involutive automorphisms on a given (non-...
7
votes
0
answers
149
views
Eigenspaces and covering relations of twisted involutions
Let $\theta:G\to G$ be an involution of a complex connected reductive Lie group, preserving a maximal torus $T$ (which, for me, lies inside a $\theta$-invariant Borel $B$). Let $K = G^\theta$ be the ...
7
votes
0
answers
132
views
Symmetric pairs of holomorphic type
Let $G$ be a real simple Lie group of Hermitian type; that is, $G/K$ carries a structure of a Hermitian symmetric space where $K$ is a maximal compact subgroup of $G$. Equivalently, the center $Z(\...
5
votes
0
answers
179
views
When is a Hermitian matrix of the form $g^*g$ for some matrix $g$
I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating.
I'm trying to figure out some properties of ...
4
votes
0
answers
133
views
Has the determinant of a involution of the first kind ever been considered as an invariant?
Let $k$ be a field of characteristic zero.
Let $A, B$ be central, simple algebras over $k$ of even degree $n,m > 1$.
Let $\sigma$ be an involution on $A$, which is either symplectic or orthogonal. ...
4
votes
0
answers
170
views
On 2-groups of exponent 4 and class 2
Suppose A is a 2-group with the following properties:
$\lvert A \rvert = t^3$ with $t$ some even power of $2$;
$A$ and $Z(A)$ (the center of $A$) are of exponent $4$;
$\lvert Z(A) \rvert = t$ and $[A,...
4
votes
0
answers
61
views
Color algebras and color involutions
If $A$ is a $G$-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined ...
4
votes
0
answers
390
views
Reference for classification of positive involutions
An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if $\mathrm{Tr}_{A/\...
4
votes
0
answers
816
views
Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
3
votes
1
answer
276
views
Unitary involutions on a simple central algebra after a scalar extension
$\DeclareMathOperator{id}{id}$
Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
3
votes
0
answers
35
views
Baer involutions fixing the same plane
Let $\mathbf{PG}(2,q^2)$ be the finite projective plane defined over the finite field $\mathbb{F}_{q^2}$. Then for each quadrangle, there is precisely one involution fixing it pointwise, and hence ...
3
votes
0
answers
149
views
Flatness of certain $R \subseteq \mathbb{C}[x,y]$
The two-dimensional Jacobian Conjecture over $\mathbb{C}$ says the following:
Let $p,q \in \mathbb{C}[x,y]$ satisfy $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Then $\mathbb{C}[p,q]=...
3
votes
0
answers
116
views
Comparison of length functions on Weyl groups
Let $G$ be a connected reductive group over an algebraically closed field $k$ (with nice enough characteristic), and let $\sigma:G\to G$ be a finite order automorphism of $G$. The connected component $...
2
votes
0
answers
170
views
Decomposition of finite abelian groups of even order if there is an involution
Let $G$ be a finite abelian group and $\sigma :G\rightarrow G$ an automorphism of order two ($\sigma\circ \sigma =id_G$). Denote by $F$ and $A$ the subgroups of fixed and anti-fixed points of $\sigma$ ...
2
votes
0
answers
112
views
Elementary abelian 2-subgroups of $\mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ (with and without choice)
Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$.
As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are ...
2
votes
0
answers
72
views
Filtration of norm-one elements of quaternion algebra over local field with respect to an involution
Let $K$ be a local non-archimedean field, with ring of integers $\mathcal{O}_K$, uniformizing element $\varpi_K$, and residue field $\mathcal{O}_K/\varpi_K\mathcal{O}_K \cong \mathbb{F}_q$. For ...
2
votes
0
answers
163
views
Involutory vs Involutary: Are both terms correct?
I have seen references for both terms, apparently referring to the same notion of a "self-inverse function".
Do both of these terms really mean the same thing? Is one a misspelling of the ...
2
votes
0
answers
68
views
Smooth involutions on homotopy 11-spheres or diffeomorphism classification of homotopy projective 11-space
Does anyone know if smooth fixed point free involutions on homotopy 11-spheres have been studied? Or equivalently, is something known about the diffeomorphism classification of homotopy $\mathbb{R}P^{...
1
vote
0
answers
92
views
Almost simple groups and their involutions without CFSG
Suppose $A$ is a finite almost simple group (meaning, by definition, that there exists a finite simple group $P$ such that $P \leq A \leq \mathrm{Aut}(P)$). Suppose furthermore that $A$ acts $2$-...
1
vote
0
answers
60
views
Choice of generators to make the centralisers connected
In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...
1
vote
0
answers
91
views
Extension of an involution on $G$ to an involution on $G_\mathbb{C}$
I asked this question on MSE https://math.stackexchange.com/questions/4475382/extension-of-an-involution-on-g-to-an-involution-on-g-mathbbc but didn't receive any answer so far. My question is the ...
1
vote
0
answers
176
views
Involutions in $\infty$-categories
$\newcommand{\id}{\mathrm{id}}$An involution in a category is a functor $\mathbf{B}\mathbb{Z}/2\to\mathcal{C}$, corresponding precisely to an object $X$ of $\mathcal{C}$ together with a $\mathbb{Z}/2$-...
1
vote
0
answers
81
views
Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric
Perhaps the following question is not in the level of MO questions, but it has not received comments in MSE, so I ask it here also:
Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution ...
1
vote
0
answers
87
views
Characterizing subfields $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$ invariant under an involution
Let $\iota$ be an involution on $\mathbb{C}(x,y)$, namely, a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ of order two.
Examples of involutions: $\alpha: (x,y) \mapsto (y,x)$, $\beta: (x,y) ...
1
vote
0
answers
80
views
Isotropy of skew-Hermitian forms over division algebras
Assume char(F) $\neq$ 2.
Let $D$ be a central division algebra over a field $F$ and $h: V \rightarrow D$ be an anisotropic skew-Hermitian form. We can easily see that $h_{\bar{F}}$ is totally ...
1
vote
0
answers
156
views
More on finite groups generated by involutions
Can finite groups, which are generated by involutions, be represented as a quotient of a Coxeter group?
1
vote
0
answers
73
views
Generators of fixed function fields under involutions
I am trying to prove something for function fields in two generators and only one involution but I don't know if it is true, if true, I would like to generalize, here it is.
Let $K=k(\eta_1,\eta_2)$ ...
0
votes
0
answers
58
views
Involutions in $\operatorname {PSO}(4,K)$
In the algebraic group $G=\operatorname {PSO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there and what are the ...
0
votes
0
answers
94
views
Is there a "cohomology theory" for involutive algebras?
I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
0
votes
0
answers
86
views
Automorphism group of symmetric square
Say I have a hyperelliptic curve without any automorphism beyond the hyperelliptic involution.
Is it possible for its symmetric square to obtain new automorphisms beyond the one induced by the ...
0
votes
0
answers
97
views
Related involutions
Let's say that we have finite field $\mathbb F_q$ and we have a couple of involutions $g,f$ with exactly one fixed point (zero).
Let's take any element $\alpha \in \mathbb F_q$
Let's start applying ...
0
votes
0
answers
102
views
An ideal invariant under an automorphism
The following question appears here; hopefully, it is appropriate for MO.
Let $k$ be a field of characteristic zero, and let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \...
0
votes
0
answers
125
views
Certain $u,v,w \in \mathbb{C}[x,y]$ such that $\mathbb{C}(u,v,w)=\mathbb{C}(x,y)$
Let $\beta$ be the following involution on $\mathbb{C}[x,y]$,
$\beta: (x,y) \mapsto (x,-y)$.
Assume that $s_1,s_2 \in S_{\beta}$ and $k \in K_{\beta}$ satisfy:
(i) $s_1,s_2$ are algebraically ...
0
votes
0
answers
56
views
On generating an explicit formula for an involution
I apologize for the very specific question I am asking. Define the relative entropy $D:[0,1]\times[0,1]\mapsto[0,\infty]$ by
$$D(x,y) = x\log_e\frac{x}{y}+(1-x)\log_e\frac{1-x}{1-y}.$$
Note $D(x,y)\...
0
votes
0
answers
181
views
$T^2$-fibered K3 surface with involution
Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...