Skip to main content

Questions tagged [involutions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
0 answers
26 views

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 $S^V$ ...
Megan's user avatar
  • 1,020
4 votes
1 answer
128 views

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 ...
DavidChi's user avatar
4 votes
1 answer
196 views

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 ...
Robert B's user avatar
  • 133
2 votes
0 answers
106 views

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 ...
scsnm's user avatar
  • 165
1 vote
0 answers
63 views

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 ...
THC's user avatar
  • 4,453
2 votes
0 answers
170 views

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 ...
THC's user avatar
  • 4,453
3 votes
1 answer
125 views

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 ...
THC's user avatar
  • 4,453
5 votes
1 answer
171 views

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\{\...
Mersn's user avatar
  • 51
0 votes
0 answers
61 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 ...
scsnm's user avatar
  • 165
0 votes
1 answer
100 views

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[...
scsnm's user avatar
  • 165
0 votes
1 answer
85 views

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[ \...
scsnm's user avatar
  • 165
1 vote
0 answers
97 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 $2$-...
THC's user avatar
  • 4,453
0 votes
0 answers
95 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 ...
wlad's user avatar
  • 4,873
0 votes
0 answers
93 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 ...
kindasorta's user avatar
  • 2,103
4 votes
0 answers
136 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. ...
nxir's user avatar
  • 1,429
2 votes
0 answers
175 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 $\sigma$ ...
Andrea Antinucci's user avatar
2 votes
0 answers
114 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 ...
THC's user avatar
  • 4,453
3 votes
1 answer
278 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 ...
Haowen Zhang's user avatar
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{...
user488802's user avatar
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 ...
THC's user avatar
  • 4,453
1 vote
0 answers
96 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 ...
Mira's user avatar
  • 129
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 ...
pbarron's user avatar
  • 71
0 votes
1 answer
111 views

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 $k$ ...
Martin Skilleter's user avatar
5 votes
1 answer
254 views

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 ...
Jérémy Blanc's user avatar
2 votes
0 answers
169 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 ...
Eduardo Reis's user avatar
25 votes
1 answer
902 views

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 $\...
Roland Bacher's user avatar
1 vote
0 answers
181 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 $\mathbb{Z}/2$-...
Emily's user avatar
  • 11.5k
3 votes
0 answers
150 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]=...
user237522's user avatar
  • 2,799
4 votes
0 answers
177 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,...
THC's user avatar
  • 4,453
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 ...
Alexander's user avatar
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 ...
Fabrizio's user avatar
9 votes
1 answer
525 views

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 ...
Hebe's user avatar
  • 841
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^{...
Kafka91's user avatar
  • 641
1 vote
1 answer
77 views

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 $E_7(q)$?
Yi Wang's user avatar
  • 271
1 vote
0 answers
82 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 ...
user237522's user avatar
  • 2,799
1 vote
0 answers
88 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) ...
user237522's user avatar
  • 2,799
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) \...
user237522's user avatar
  • 2,799
0 votes
0 answers
136 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 ...
user237522's user avatar
  • 2,799
3 votes
6 answers
2k views

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) ...
user144684's user avatar
4 votes
1 answer
413 views

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 ...
Mikhail Borovoi's user avatar
3 votes
0 answers
122 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 $...
Spencer Leslie's user avatar
5 votes
1 answer
2k views

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 ...
GA316's user avatar
  • 1,229
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 ...
Allen Knutson's user avatar
1 vote
0 answers
81 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 ...
Mr.Mysterious's user avatar
3 votes
1 answer
399 views

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 ...
Jana's user avatar
  • 2,022
1 vote
0 answers
159 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?
Thomas Nordahl's user avatar
8 votes
0 answers
663 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-...
Manuel Bärenz's user avatar
3 votes
1 answer
396 views

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 ...
Hebe's user avatar
  • 841
3 votes
1 answer
570 views

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 $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. ...
Hebe's user avatar
  • 841
7 votes
0 answers
133 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(\...
Hebe's user avatar
  • 841