Questions tagged [involutions]
The involutions tag has no usage guidance.
64
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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. ...
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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 $\sigma$ ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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 $k$ ...
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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 ...
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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 ...
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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 $\...
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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 $\mathbb{Z}/2$-...
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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]=...
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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,...
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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 ...
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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 ...
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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^{...
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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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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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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) ...
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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) \...
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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 ...
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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) ...
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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 ...
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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 $...
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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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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 ...
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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 ...
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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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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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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-...
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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 ...
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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 $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. ...
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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(\...
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A question about involutions and polynomials
Let $x = (a,b) \in \mathbb{Q}^2$ and let $p(x,t) = t^2-at+b$. Does there exist an involution $\tau$ of $\mathbb{Q}^2$ such that for all $\tau(x) \neq x$, $x \in \mathbb{Q}^2$ one of the polynomials $p(...
user6671
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equivalence classes of arch diagrams in bijection with permutations
By an arch diagram of size $n$, I mean a diagram consisting of $n$ arches matching $2n$ points, where the points are ordered on a line running from left to right. An arch diagram is basically just a ...
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Number of fixed points in Zagier's involution (Fermat's Theorem) [closed]
Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
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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/\...
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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 $K=k(\eta_1,\eta_2)$ ...
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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 ...
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Stinespring's dilation without $C^{\ast}$-algebras
Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra?
I will now state the version of Stinespring's dilation ...
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Involution on the components of a group algebra
If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\...
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Extension of an involutive automorphism
Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra.
If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ ...
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An angle-doubling trick of Kirillov and Berenstein [closed]
Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a ...
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A question on conjugacy classes of central involutions in a finite group
An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$, or equivalently if the centralizer of $a$ contains a sylow $2$-subgroup.
...
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Why are all involutions conjugate in the special linear group of degree 2?
It appears to be standard that the set of non-identity involutions in $SL(2, 2^n) = PSL(2, 2^n)$ forms a single conjugacy class. What is the best reference for this?
I note that
https://math....
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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)\...
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A function composed with itself produces the identity
Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...
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