Questions tagged [involutions]

For problems related to involutions , that is functions that are their own inverses .

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Involutions in PCO

In the algebraic group $G=\operatorname {PCO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there in $G \setminus \...
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Recurrence differential equations arising from the Normal PDF

Let $a,b,c:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable functions, with $a(t)\rightarrow -\infty$ and $c(t)\rightarrow -\infty$ as $t\rightarrow -\infty$, and with $a(t)\rightarrow\infty$ and $c(...
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Centralizer generators

This is a question posted on overflow but no reply has been received. In the algebraic group $G=\operatorname {PSO}(4,K)<\operatorname {PCGO}(4,K)$ where $K$ is an algebraically closed field of an ...
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Passing an involution to a quotient algebra

This question is inspired by the discussion under this MO answer. I hope I have captured correctly what is going on in the below. Let $A_0$ be a finitely generated universal unital complex algebra $...
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Every continuous involution on R^n has a fixed point [duplicate]

I'm looking for a reference which supports the claim that every continuous involution on $\mathbb{R}^n$ has a fixed point. This fact is discussed here, but I'd like to see this as a standalone claim ...
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Could any "incoherently involutive" endofunctor be made "coherently involutive"?

Given a category $C$ with an endofunctor $F:C \to C$ and a natural isomorphism $\epsilon:FF \cong 1_C$, call the pair $(F, \epsilon)$ an involutive endofunctor. Also, call the pair $(F, \epsilon)$ a ...
Geoffrey Trang's user avatar
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Solving the Functional Equation $f(f(x))=x$ gone wrong

For a function to be its inverse (i.e. an involution), it needs to satisfy the functional equation $f(x)=f^{-1}(x)$ or $f(f(x))=x$. I expressed $f^{\circ n}(x)$ (the composition of $f(x)$ to itself $n$...
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$L^1(G)$ is a Banach *-Algebra [duplicate]

Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution ...
Pedro Lourenço's user avatar
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Does every complex involutive algebra admit at least one non-trivial C*-seminorm?

Let $A$ be a unital involutive algebra over $\mathbb{C}$. A $C^*$-seminorm is a seminorm $p$ such that $p(x^*x) = p(x)^2, \forall x \in A$. I understand that if the spectral radius of $x \in A$ given ...
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Determine the involution when given two pairs of point on a line

I'm studying about involution on a projective line (line with point at infinity). An involution is a map from a projective line $l$ to itself that satisfied $f \circ f =$ is the identity map. ...
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In a finite reflection group, an involution is a product of commuting reflections

I am working through the book Reflection Groups and Coxeter Groups by Humphreys. I got stuck while trying Exercise 1.12.3: If $w \in W$ is an involution, prove that $w$ can be written as a product of ...
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Solving a Certain Combinatorial Game

I've been trying to come up with a combinatorial game even simpler than Hex with non-trivial gameplay and been failing dismally. Currently, my idea is that players sequentially lay pieces on a ...
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What is precisely an anti-linear anti-involution?

Given an arbitrary $\mathbb{C}$-algebra $A$, what should be required from a function $f:A\to A$ for it to be an anti-linear anti-involution? As I understand β€œanti-linear” for $A$ as a vector space ...
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A weird construction transforming a semigroup with action to just semigroup

Let $(S,E,\circ,\dagger,\lambda)$ be a semigroup $(S,\circ)$ with involution $f\mapsto f^\dagger$ and action $E\to E$ of element $f$ defined as $x\mapsto \lambda_f x$. Consider the semigroup on $S\cup ...
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How can I naturally derive the Legendre transform formula? Why is the term $px$ there, and not just $f(x(p))$?

A few months back I managed to find an article1 presenting the Legendre transform in a satisfactory way, in essence it being The Legendre transform $f^*$ of a (strictly) convex $C^2$ function $f$ is ...
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Involute of circle: Get point by distance

Given is an involute of a circle. The basic circle radius $a$ and a point $A$ on the involute defined by the involute angle $\phi$ and the radius of the involute $\rho$ (and thus the arc length $l$) ...
B Roberts's user avatar
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Show the Banach algebra is not a $C^*$-algebra

Working with the Banach algebra $\ell^1(\mathbb{Z})$ and the involution $f^* (n)=\bar{f(-n)}$, I want to show this is not a $C^*$ algebra. I know I need to find some function $f$ such that $||f^* f||...
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Involution on monic cubic polynomials related to nesting/denesting of cubic radicals

Consider the involutive transformation $$\mathbb{R}^3 \ni (a,b,c) \overset{\phi}{\mapsto} \left( \frac{a + 2 c}{\sqrt{3}}, \frac{a^2 + a c + c^2}{3} - b , \frac{a - c}{\sqrt{3}}\right)$$ Show that if $...
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The entries of diagonal matrix $D$ are only $1$ or $-1$ then $D=D^{-1}$

The entries of diagonal matrix $D$ are only $1$ or $-1$ then $D=D^{-1}$ My attempt: First I thought maybe determinant works but it does not because it is not certain that determinant is equal to $1$ ...
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Involutions and matrices.

Let $a$,$b$, $c$ be members of a field. (?) I noticed that $$f:\left(-\infty, \frac{a}{c}\right)\cup \left(\frac{a}{c}, \infty \right)\to \left(-\infty, \frac{a}{c}\right)\cup \left(\frac{a}{c}, \...
Chris Christopherson's user avatar
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If there is a $2\times 2$ matrix $A^2 = I$, what will the complete solution set for $A$ be?

So the basic thing I started with was equating the diagonal and the other other two entries of matrix $A$ respectively which gives, $$a=\pm d, \\b=c$$ But there is this matrix $\begin{bmatrix}0 & ...
Chatrapal Singh Rathore's user avatar
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Is it always true that matrix representation of a skew-symmetric non-degenerate bilinear form is orthogonal with respect to some basis?

I am going through the definition of symplectic matrices. While searching through various properties of those class of matrices I found that every symplectic matrix is necessarily invertible and the ...
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Is it true that all matrices of this type are involutory?

I came across a problem that suggested the following for a $3 \times 3$ matrix. $$\text{Let } H = I - \frac{2}{3}A$$ $$\text{Then } H^2 = I$$ For $H, A \in M_{3,3}$ and $A$ being a matrix of all ones. ...
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Unitary group modulo the Jacobson radical

Let $F$ be any field, and let $A$ be an associative unital finite dimensional $F$-algebra, with an $F$-linear involution $*$ (possibly trivial), that is an isomorphism of the $F$-vector space $A$ of ...
GreginGre's user avatar
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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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Involution-like functional equation $f(f(x))=ax+b$

Consider the following functional equation: $$f(f(x))=ax+b$$ where $f$ is defined on the whole $\mathbb{R}$. Questions: If $b \neq 0$, what are the solutions? If $a=1, b =0$ (ie. $f$ is an involution ...
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Multistep involutions

I'm familiar with the concept of an involution. For example, two simple single-variable functions are $f(x) = 1-x$ and $g(x) = 1/x$. If we apply the function twice, we get back to the identity ...
Doug's user avatar
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Name for functions that are anti-symmetric about $y=x$

Even functions are functions that are symmetric about the $Y$-axis, and odd functions are functions that are symmetric about the origin. Functions that are symmetric about $y=x$ ($y=f(x)$ implies $x=...
Caleb Kisby's user avatar
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Polynomial injection between: $\mathbb{N}^{2n}/\phi$ and $\mathbb{N}$.

Let $\phi$ an involution defined as follows: $$\phi:\mathbb{N}^{2n}\longrightarrow \mathbb{N}^{2n}$$ $$(a_1,b_1,a_2,b_2,\dots,a_n,b_n)\longrightarrow (b_1,a_1,b_2,a_2,\dots,b_n,a_n).$$ Is there any ...
Tio Miserias's user avatar
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What is an invariant embedding?

Let $S$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $0$, equipped with a regular involution $\iota$, i.e., with an action of the order two (cyclic) group $...
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Group action on monoid with involution - Laws relating involution of element (from monoid) with inverse function (from group)

Preliminaries (Remarks on notation: To denote function application, I will use the Haskell notation $f\ x$, rather than the traditional mathematical notation $f (x)$. The expression $x^-$ denotes an ...
Antonielly's user avatar
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representatives of conjugacy classes of involutions

I'm aware that the representatives of conjugacy classes of involutions of $G = PGL(4,\mathbb{C})$ which have a conjugate in a fixed maximal torus of $G$ are $\begin{bmatrix} -1 & 0 & 0\\ 0 &...
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Let $x$ be an involution in a group $G$ and $K$ be a component of $C_G(x)$. Show that $K$ normalizes every component of $G$

The following is an exercise in Kurzweil and Stellmacher: Let $x$ be an involution in a finite group $G$, $K$ be a component of $C_G(x)$ and $L$ be a component of $G$. Show that $K$ normalizes $L$. ...
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Let $x$ be an involution in a finite group $G$ and $K$ be a component of $C_G(x)$. Show that $K\subseteq E(G)$ or $[C_{E(G)}(x),K]=1$.

Let $x$ be an involution in a finite group $G$ and $K$ be a component of $C_G(x)$. Show that $K\subseteq E(G)$ or $[C_{E(G)}(x),K]=1$. Here, $E(G)$ is the layer of the group $G$. My attempt: Suppose $...
Guest's user avatar
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Question about involutions and projections

In my course of Linear Algebra we were assigned the following problem: Let $V$ be a vector space over the field $\mathbb F = \mathbb R$ or $\mathbb F=\mathbb C$ with $\dim V<\infty$ and let $X \in \...
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Non-trivial smooth involutions of the reals

I'm interested in certain involutions of the real numbers, i.e functions $f$ such that $f\circ f = \text{id}_{\mathbb{R}}$. It has been shown here that $\text{id}_{\mathbb{R}}$ is the only increasing ...
Syst3ms's user avatar
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Extension of an involution on $G$ to an involution on $G_\mathbb{C}$

Let $G$ be a compact connected Lie group and $ \sigma :G \rightarrow G $ be an involution on $G$. Let $G^\sigma :=\lbrace g \in G, \sigma(g)=g \rbrace$. Denote by $G_\mathbb{C}$ the complexification ...
Mira's user avatar
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1 answer
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Complete intersection Calabi-Yau of dimension $3$ with anti-holomorphic map such that each component of fixed point set has $b^1 \neq 0$

I am looking for a complete intersection Calabi-Yau manifold $X$ of complex dimension $3$ that admits an anti-holomorphic involution $\sigma: X \rightarrow X$ such that $L:=\operatorname{fix}(\sigma)$ ...
user505117's user avatar
5 votes
2 answers
220 views

Does the Nullity Theorem hold in fields of characteristic 2?

I'm playing around with involutory ($M^2 = I$) matrices over finite fields with characteristic 2 ($\mathbb{F}_{2^m}$). I came across the nullity theorem, which seems very useful to check if ...
DasArchive's user avatar
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Find fixed point of involution of complex projective line

Question: Find fixed points of involution g: $P_1(C) -> P_1(C)$, $g^2$ = Id, if g(2/3) = 3 and g(-2/3)= 1/4 My ideas: to use cross-ratio, maybe we can say g(3) = 2/3 and g(1/4) = -2/3 so we can ...
Joseph Jordan's user avatar
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A query on the eigenvalues of anti-involution matrix

An involution matrix $A$ is defined by the condition $$ A^2 = I \tag{1} $$ The eigenvalues of an involution matrix $A$ are the roots of unity. Generalizing, an $m$-involution matrix $A$ is defined by ...
Dr. Sundar's user avatar
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Determinant of involutory matrix

If $A$ is an $n \times n$ involutory matrix, then show that $$\det (A) = (-1)^{n - \text{tr}(A) \over 2}$$ A matrix is involutory if it is its own inverse, $A^{-1} = A$. Thus, the eigenvalues of an ...
Saurabh Rana's user avatar
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Fields with involution whose fixed field is ordered?

Let $K$ be a field with an involution $*$, meaning $*:K\to K$ is an automorphism and $(x^*)^*=x$ for all $x\in K$. Suppose further that the fixed field of $*$ is ordered (i.e., it can be given an ...
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Exercise about involutions [closed]

I have the exercise below and I can't do the last item. Remember that an operator $S$ is an involution if $S^2=\text{Id}$. Let $T:\mathbb R^n \rightarrow \mathbb R^n$ be a linear operator. Show that: ...
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1 answer
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Showing that an involution of the projective line with a fixed point fixes exactly two points

My idea to show this is to notice that a projective transformation is in general given by the form $\phi(z)=\frac{az+b}{cz+d}$, then $\phi(z)=z$ gives a second order equation with at most two ...
Gokimo's user avatar
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1 answer
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Examples of Banach $^*$-algebras where $^*$ is not an isometry

I'm reading up on $C^*$-algebras at the moment, and in the Wiki article (for instance) on the topic they note that the condition $||a^*a||=||a||^2$ implies that the $^*$-involution is an isometry. But ...
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Is $f(a)$ normal in $A\ $?

Fact $:$ Let $K$ be a compact subset of $\mathbb C$ and $\Omega$ be an open set containing $K.$ Then there exists a cycle $\Gamma : = \sum\limits_{j=1}^{n} n_{j} \gamma_{j}$ in $\Omega \setminus K$ ...
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Number of involutions by combinatorial

Let $𝑖_𝑛$ be the number of involutions in $\sigma_n$. Show that $𝑖_0 = 𝑖_1 = 1$ and for $𝑛 \geq 2$ $𝑖_𝑛 = 𝑖_{π‘›βˆ’1} + (𝑛 βˆ’ 1)𝑖_{π‘›βˆ’2}.$
Panupong Daengpradap's user avatar
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1 answer
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Involution elements in infinite groups

In this, involutions in a finite group are either conjugate or have an involution centralizing both of them. I wonder if there are similar results for an infinite group. I think and look for it but I ...
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Determining all real functions $g=\frac{y}{x}$, where $y$ is an involution

Suppose that we have the following functional equation $$y(x)=x \,g(x)$$ where both $y$ and $g$ are functions over the reals. In addition, we want $y(x)$ to be an involution, i.e. $y(y(x))=x$, for ...
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