# Questions tagged [involutions]

The involutions tag has no usage guidance.

78
questions

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### Equivariant KR-theory of representation sphere

I would like to say my question first.
Let $G$ be a compact Lie group acting on a good space $X$ in a good way. Let $V$ be a $G$-representation whose real dimension may be less than 8, and let $S^V$ ...

4
votes

1
answer

128
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### LS category of the quotient of a manifold by its involution

$\DeclareMathOperator\cat{cat}$Let $\cat(X)$ denote the Lusternik–Schnirelmann (LS) category of $X$. This homotopy invariant has been well-studied for CW complexes: it is $0$ if and only if $X$ is ...

4
votes

1
answer

196
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### Quotient of the plane by the standard Cremona involution

Consider the standard Cremona involution $i:\mathbb{P}^2\dashrightarrow \mathbb{P}^2$, $[x:y:z]\rightarrow [yz:xz:xy]$.
Let $Y$ be the blow-up of $\mathbb{P}^2$ in the three base points of $i$, so ...

2
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0
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### Involution centralizers in $\mathrm{PCSO}^{+}(8,3)$

According to Table 4.5.1 of [1], there should be 10 classes of involutions of type "p" and "e" in $\operatorname{Aut}(K)$ where $K=K_a=P\Omega^{+}(8,3)$.
And Table 4.5.1 also gives ...

1
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0
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### Doubly transitive groups in which a one point stabilizer has a normal subgroup of even size

In 1972, Hering classified the finite doubly transitive permutation groups $(G,X)$ ($G$ acting faithfully on $X$) in which $G_x$, with $x \in X$, contains a normal subgroup $N_x$ of even order which ...

2
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0
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170
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### Doubly transitive groups in which a point stabilizer has an abelian normal subgroup

Let $G$ be a finite doubly transitive group in its action on the set $X$, such that a point stabilizer $G_x$ ($x \in X$) has an abelian normal subgroup $N_x$.
I have read that if $\vert N_x \vert$ is ...

3
votes

1
answer

125
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### Holt's Theorem on doubly transitive groups with $2$-central involutions fixing only one letter

In D. F. Holt, Transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes a unique point, Proc. London Math. Soc. 37 (1978), 165–192, Derek Holt classifies finite doubly ...

5
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1
answer

171
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### Rational functions of order $3$

Let $p$ be a prime, let $\mathbb K$ a field with $\operatorname{char}(\mathbb K)=p$ and $\mathbb{K}(x)$ be the field of fractions of the polynomial ring $\mathbb{K}[x]$, i.e. $$\mathbb{K}(x)=\left\{\...

0
votes

0
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61
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### 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
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1
answer

100
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### An explicit matrix form in the symplectic group

In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[...

0
votes

1
answer

85
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### An explicit matrix form

In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[
\...

1
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0
answers

97
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### 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 $2$-...

0
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0
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### 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
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0
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93
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### 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 ...

4
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0
answers

136
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### 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. ...

2
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0
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175
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### 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
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### 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 ...

3
votes

1
answer

278
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### 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 ...

1
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0
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### 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{...

3
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0
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35
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### 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 ...

1
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0
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96
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### 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 ...

2
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### 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 ...

0
votes

1
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111
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### Order 2 matrices with entries in the polynomial ring over a field are diagonalisable

This is a variant on the question posed here, in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}_n(A)$, where $A$ is a $k$-algebra for some field $k$ ...

5
votes

1
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254
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### Is every matrix involution over a UFD diagonalisable?

Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}_n(A)$ diagonalisable?
This is of ...

2
votes

0
answers

169
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### 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 ...

25
votes

1
answer

902
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### Reference request for a proof of the two-square Theorem

One can show (see below for a sketch of a proof) that every odd prime number $p$
can be written in exactly $(p+1)/2$ different ways as
$$p=a\cdot b+c\cdot d$$
with $a,b,c,d\in\mathbb N$ satisfying $\...

1
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0
answers

181
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### 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 $\mathbb{Z}/2$-...

3
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0
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150
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### 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]=...

4
votes

0
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177
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### 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,...

0
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0
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97
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### 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 ...

4
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0
answers

61
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### 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 ...

9
votes

1
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### Involutive automorphism of simple Lie algebra

I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan ...

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### 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

1
answer

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### What do conjugacy classes of involutions like in finite simple group $E_7(q)$?

Are there any refences for conjugacy classes of involutions in finite simple group $E_7(q)$?

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0
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### 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 ...

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0
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### 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) ...

0
votes

0
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### 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
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0
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### 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 ...

3
votes

6
answers

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### Finite field special functions

I am looking for special class of involutive functions (f) with exactly one (zero) fixed point over finite field $F_q$ with properties:
1) $f(f(x))=x$ for any $x \in F_q$ , $f(x)=x$ iff $x=0$
2) ...

4
votes

1
answer

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### Symmetric subgroups of simple algebraic groups over finite fields

Let $G$ be a simply connected simple algebraic group over a field $k$.
Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2).
Let $H=(G^\theta)^0$, the identity ...

3
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0
answers

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### 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 $...

5
votes

1
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### Positive definite matrices diagonalised by orthogonal matrices that are also involutions

Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$.
I want to know when this matrix is also an involution, i.e., $P^2 = I$.
If there is any ...

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votes

0
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### 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 ...

1
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0
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### 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 ...

3
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1
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399
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### Descent of coherent sheaves on finite coverings

Let $X$ be a non-singular hyperelliptic curve (over $\mathbb{C}$) and $\pi:X \to \mathbb{P}^1$ be a $2:1$ covering. Let $\sigma:X \to X$ be the hyperelliptic involution and $E$ be a locally free sheaf ...

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0
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### More on finite groups generated by involutions

Can finite groups, which are generated by involutions, be represented as a quotient of a Coxeter group?

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663
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### 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-...

3
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1
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396
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### Irreducible Symmetric Pairs

Let $\mathfrak{g}$ be a simple Lie algebra with a compact subalgebra $\mathfrak{k}$ such that $(\mathfrak{g},\mathfrak{k})$ corresponds to an irreducible Riemann symmetric space. Denote by $\sigma$ be ...

3
votes

1
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570
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### Compact dual of a noncompact Lie group

Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra, and fix a Cartan involution $\theta$ of $\mathfrak{g}_0$, which gives a Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. ...

7
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### 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(\...