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Questions tagged [involutions]

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

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Archimedean spiral $\oplus$ sawtooth = circles

Archimedean spiral $\oplus$ sawtooth = circles But what about the involute of the circle instead of the Archimedean spiral? Let the unit circle be given by $$\gamma_\text{circ}(t) = (x_\text{circ}(...
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A concrete example of involution in group theory

I am reading the textbook "Introduction to Modern Algebra, Joyce 2017" and in the Cyclic groups and subgroups section, there is a following sentence about involution. An involution $a$ is an ...
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Nilpotent, Idempotent and Involutory Matrix

With exception of the zero matrix, can a matrix be nilpotent $(A^k=0)$ and idempotent $(A^2=A)$ at the same time? and With exception of the identity matrix, can a matrix be idempotent and involutory ...
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Find the involutions in the indefinite orthogonal group O(2,1)

I would like to find the (linear) involutions that conserve the quadratic form $x^2+y^2-z^2$. Finding reasonable equations for the entries of the matrix is possible, but not particularly nice or ...
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Involution action on $H^3(S^1\times S^2)$

I am studying about involution action $I^*$ on de Rham cohomology group $H^3(S^1\times S^2)$ induced from an action $I\cdot (z,x)=(\overline{z},-x) $ where $S^1\times S^2\subset \mathbb{C}\times \...
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If $f$ has no non trivial fixed points and $f\circ f$ is the identity then $f(x)=x^{-1}$ and $G$ is abelian for $f$ an automorphism of $G$ [duplicate]

Let $f$ be an automorphism of the finite group $G$ such that $f\circ f=id$ and $f(x)=x\implies x=e$ Prove that $f(x)=x^{-1}~\forall x\in G$ If we can prove that $f(x)$ and $x$ commute for any ...
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Find $f,g$ s.t. $f\circ g=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ 10 & 4 & 5 & 7 & 8 & 9 & 2 & 6 & 3 & 1\end{pmatrix}.$

Let $f$ and $g$ be permutations such that $$f \circ f = id,$$ $$g \circ g = id,$$ and $$f\circ g =\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &...
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Recursion $I_{n+1} = I_n + nI_{n-1}$ for the amount $I_n$ of the involutions in $S_n$

A permutation $\pi \in S_n$ is an involution, when $\pi^2 = \text{id}$. How can one show for the amount $I_n$ of the involutions in $S_n$ the following recursion: $$I_{n+1} = I_n + nI_{n-1}$$ ...
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Show that $1= \sum_{k=0}^{m} (-1)^k {m \choose k}2^{m-k}$ using sign reversing involution

Using the sign reversing involution, how can I show that $$1= \sum_{k=0}^{m} (-1)^k {m \choose k}2^{m-k}.$$ I have been trying to figure out the what the signed sets are namely $S^{ +}$ and $S^{ -}$. ...
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Proof that the Lascoux-Schützenberger involutions satisfies the braid-relations

I am interested in the Lascoux-Schützenbereger involutions $\theta_i$, defined on semistandard Young tableaux. See this paper for definitions, page 4. These involutions satisfy the braid relations: (...
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An involution on a pair of pants fixing one boundary component and permuting other two?

Let $S$ be a pair of pants (i.e. sphere with 3 boundary components). Let $\gamma_1, \gamma_2, \gamma_3$ denote the 3 boundary components of $S$. Does there exist an orientation-preserving ...
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positive matrices diagonalised by 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. ie. $P^2 = I$. If there is any characterization ...
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What does it mean for a ring to have an involution? Are there any examples?

A ring $R$ is said to be a ring with an involution if there exists a mapping $*\colon R \to R$ such that for every $a, b \in R$: $a^{**} = a$, $(a + b)^* = b^* + a^*$, $(ab)^* = b^*a^*$. Can ...
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If a $2 \times 2$ matrix $A$ satisfies $A^2=I$, then is $A$ necessarily Hermitian?

I cannot find an appropriate counterexample. Is there a counterexample? Or is $A$ indeed Hermitian?
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About the exponential generating function of the involutions of $\mathbb{S}_n$

I'm trying to construct the exponential generating function of the sequence $(a_n)_{n\in \mathbb{N}_0}$, with $a_n$ the amount of involutions in $\mathbb{S}_n$. So far I've made a combinatorial ...
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Are $f(x)=x$ and $f(x)=-x$ the only odd bijective involutions from $\mathbb{R}$ to $\mathbb{R}$?

This is motivated by a question that was posted last night, but was deleted (I think by the author) before any answers to it appeared. I don't think it has been re-posted since then. What bijective ...
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1answer
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$\mathbb{Z}_2$-grading of a vector space by an involution

Assume we have a complex vector space $V$ and an involution $\omega:V\to V$, that is $\omega²=id$. Does this give a $\mathbb{Z}_2$-grading of $V$ somehow?
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Reverse operation on Quaternions

I'm currently studying Clifford algebras and I came across a concept called reverse. It has the following properties: $(AB)^† = B^†A^†$ for all $A$ and $B$. $v^† = v$ for all vectors $v$. I was ...
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Involution of the 3 and 4-holed torus and its effects on some knots and links

I've been working with the intersection between the topology of knots and surfaces and I have some specific questions about the involution of torus with multiple holes and the effect on the knots ...
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Extension and restriction of involutions

Let $A \subseteq B$ be two associative commutative $k$-algebras with $1$. Let $f$ be an involution on $A$, namely, a $k$-algebra automorphism of $A$ of order two. Can one tell when $f$ can be ...
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Conjugacy classes in subgroup of index 2

Let $G$ be a finite group, $H$ a subgroup of index 2. It is well-known how to count the conjugacy classes of $H$ once we know those of $G$: look at which ones get split into two classes, based on ...
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Order of commutator in core-free group

I was reading the paper "Central elements in core-free groups" of Glauberman: http://www.sciencedirect.com/science/article/pii/0021869366900305 He defines the core of a finite group $G$ to be the ...
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Generating Symmetric Group with Transpositions

There is a theorem in our book which I'm trying to prove, It says: Theorem: Number of Transpositions which generate $\mathbb{S}n$ can not be lower than $n-1$. I've already proved that $\mathbb{S}...
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Counting involutions

Suppose we have a group $G = H \rtimes K$, where $K = \langle \alpha \rangle$ is of order 2. We suppose that $K$ centralizes some 2-Sylow subgroup. Then every involution is of one of these forms: 1) $...
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What is the asymptotic behavior of the number of involutions?

Sequence A000085 in the On-Line Encyclopedia of Integer Sequences counts the number $A_n$ of involutions on $n$ letters, and also, the number of Young tableaux with $n$ cells. I am curious, what is ...
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Functional Square Root of Piecewise Functions

Let $s:\mathbb R\to\mathbb R$ be the function defined as $$s(x)=x+(-1)^{\lfloor x\rfloor}$$ Find a function $t:\mathbb R\to \mathbb R$ such that $t^{\circ 2}=s$, or prove that no such function ...
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involutions on the circle $S^1$

I am studying involutions on $S^1$; namely, continous functions $f:S^1 \rightarrow S^1$ satisfying $f\circ f = id$. Examples of such maps are reflections along a given diameter, or map induced by ...
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If $w$ is an involution on an n-dimensional space, $det(w) = (-1)^r$ where $r$ is the rank of the map $i - w$

In the book of Linear Algebra by Werner Greub, at page 109, it is asked that Let $w$ be a linear transformation of $E$ such that $w^2 = i$.Show that $det (w) = (-1)^r$ where $r$ is the rank of ...
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Determinant of a matrix that is its own inverse [duplicate]

I need help in showing that when computing the determinant the inverse of an $n \times n$ matrix with the property $$M=M^{-1}$$ that is $$M^2 = I$$ the determinant is either $1$ or $-1$. I'...
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Is every involution a palindrome of transpositions?

An involution is a permutation $P$ which is its own inverse: $P\cdot P = \text{id}$. Every permutation can be written (in various ways) as a sequence of single-element swaps (transpositions). ...
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Strange Examples of Involutory Functions?

I am very interested in functions $\gamma:\mathbb R\to\mathbb R$ with the following property: $$\gamma^2(x)=x$$ One form of a function satisfying this is $$f(x)=\frac{a-x}{1+bx}$$ Which has the ...
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Eigenvalues after Involution

Consider the product of two $n \times n$ matrices $UA$, where $U$ is involutory and $A$ has eigenvalues $\lambda_i$ with $i = 1, \dots, n$. What can I say about the eigenvalues of $UA$?
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Involution that brings sets to disjoint sets

Let $A$ be a collection of subsets of $\{1,2,\dots,n\}$ that is closed under taking subsets (that is, if $U\in A$ and $V\subseteq U$ then $V\in A$). Is there always an involution $f:A\to A$ such that $...
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Local galois involution

Let $x(z):=-\frac{(z+1)^3}{z}$ is a meromorphic function from $CP^1\rightarrow \mathbb{C}$. At the point $z=1/2$ and $z=-1$, $dx=0$ hence these are ramifications point. I want to study the Galois ...
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Confusing question in Conceptual Mathematics on reflexive, symmetric, transitive, and jointly monomorphic maps

On p.294 of Conceptual Mathematics 2nd ed., Exercise 12 states: For a given map $h: Y \rightarrow Z$, consider all parallel pairs $f, g: X \rightarrow Y$ (for various $X$) such that $hf = hg$. ...
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n-Involutory Rational Functions

Lately, I've been toying with involutory functions (functions that are their own inverses), especially involutory rational functions with linear numerator and denominator. Then I noticed that some ...
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Average number of inversions in an involution

I'm working through some exercises in Sedgewick's Analysis of Algorithms, but I'm stuck on 7.45: “Find the CGF for the total number of inversions in all involutions of length N. Use this to find the ...
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Involutions in which all transpositions have the same distance

I am wondering whether there exist a name in the literature for the following special type of permutation $\pi$ over $\{1, \dots n\}$. $\pi$ is an involution, i.e., it is composed of nonoverlapping ...
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Existence of a quadratic residue $r$ mod $p$ such that $-r-1$ is a quadratic nonresidue.

I would be able to simplify a proof about sums of squares significantly if I were able to prove that the involution $$\Bbb{F}_p\ \longrightarrow\ \Bbb{F}_p:\ x\ \longmapsto\ -x-1,$$ does not preserve ...
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Diagonalizable matrix with eigenvalues $\pm 1$

Let $A \in \mathbb{R}^{n \times n}$ be a diagonalizable matrix with $1$ or $-1$ as its only eigenvalues. Prove that $A^{2} = I_{n}$. Could someone help me on this one? I have no idea how to start.
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Bijections as the composition of 2 involutions [duplicate]

How can we prove that any bijection from any set to itself is a composition of 2 involutions ? So I know that any involution is a bijection, and so it has an inverse (which happens to be itself here)....
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Composition of 2 involutions

How can we prove that any bijection on any set is a composition of 2 involutions ? Since involutions are bijections mapping elements of a set to elements of the same set, I find it weird that this ...
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1answer
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A step in the proof that the Legendre transform is involutive without differentiaiblity

Let $f : \mathbb{R} \to \mathbb{R}$ be a convex function, and define the Legendre transform of $f$ by $$L(a) := \sup_{x \in \mathbb{R}} (ax - f(x)) \ .$$ I wish to show that $$f(a) = \sup_{x \in \...
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Question about involutions and bijections [duplicate]

I have seen a few times now that, any bijection on any set( finite, or not) can be written as a composition of two involution. However, it is usually said that this is a well known result. However, ...
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tensor product of *-algebra as a *-algebra: well-definedness of involution

I have a question regarding the proof of the following proposition: Proposition: Let $A$ and $B$ two $\mathbb{K}$-algebras with involution. Let $A\otimes B$ be the algebraic tensor product of $A$ and ...
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$y=x/(1+a(x))$, $\quad$ $x=y/(1+b(y))$. What is known about $a\mapsto b$?

\begin{align} y & = f(x) = \frac x {\displaystyle 1 + \sum_{n=1}^\infty a_n \frac{x^n}{n!}} \\[15pt] x & = f^{-1}(y) = \frac y {\displaystyle 1 + \sum_{n=1}^\infty b_n \frac{y^n}{n!}} \end{...
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Definition of C$^\ast$-algebra: which conditions can be deduced from the others?

A C$^\ast$-algebra is a Banach algebra $A$ with an involution, i.e. a map $\ast$ such that: $(x^\ast)^\ast=x$ for all $x\in A$; $(x+y)^\ast=x^\ast+y^\ast$ for all $x,y\in A$; $(ax)^\ast=\overline ax^\...
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On products of involutions

Is it possible to write $$M = \begin{bmatrix}x&0\\0&-\dfrac{1}{x}\end{bmatrix}, x \in \mathbb{R} \backslash \{0\}$$ into a product of two involutory real matrices? I know that it can be ...
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Involutions in $\mathbb{Q}$.

Define involution in an associative ring $k$ with identity as a map $k\rightarrow k$ mapping each $\alpha \in k$ to $\bar{\alpha}\in k$ such that i) $\overline{\bar{\alpha}} = \alpha$ ii) $\overline{...
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1answer
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On cardinality of involutions

Denote $S_n$ symmetric group on $n$ letters. Denote $C_n$ to be set of cycles in $S_n$ of length $n$. Denote $I_n$ to be set of involutions in $S_n$. What are the cardinalities of $C_n$, $I_n$ and $...