Questions tagged [involutions]

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

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1answer
24 views

Is there an involutive automorphism mapping two given elements of a poset or lattice?

$S$ is a finite poset or lattice ; $A$ and $B$ two distinct elements. If there is at least one automorphism that maps $A$ to $B$, can I find one such automorphism that is an involution? The set of ...
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1answer
26 views

Trace of involution is the dimension of the space implies it is the identity.

I was trying to write explicitly a linear automorphism $\phi$ on a 2-dimensional vector space $V$. I knew that $\phi^2=Id$ and $\text{Tr } \phi=2$. Using the matrix representation $\phi=\begin{pmatrix}...
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20 views

Positive Elements in Real Vector Space with a Linear Involution

Let $V$ be a real vector space equipped with a linear involution. Let $V^h := \{v \in V \mid v^* = v \}$ be the set of hermitian elements. I am reading a paper and the author claims that there is a ...
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1answer
39 views

If $n > 1$, there are no non-zero $*$-homomorphisms $M_n(\Bbb{C}) \to \Bbb{C}$

If $n > 1$, there are no non-zero $*$-homomorphisms $M_n(\Bbb{C}) \to \Bbb{C}$. A $*$-homomorphism is an algebra morphism $\varphi: M_n(\Bbb{C}) \to \Bbb{C}$ with $\varphi(\overline{A}^T) = \...
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1answer
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Need help in understanding that any permutation can be written as a product of two involutions.

I have seen this proof Is any permutation the product of two involutions? but that is unclear to me This proves it for only the cycle $(1,2,3...n)$ How does proving it only for the cycle $(1, 2, 3...n)...
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1answer
31 views

After applying a sequence of involutory real matrices to a vector, is the norm of this vector bounded from below?

For $n,N \in \mathbb{N}$, let $A_1, \ldots, A_n$ be a finite sequence of involutory $(N \times N)$-matrices over $\mathbb{R}$, i.e. $A = A^{-1}$. We know, that the eigenvalues of any involutory matrix ...
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3answers
64 views

Does there exist a non-symmetric involutory matrix?

Let $A$ be a real involutory matrix i.e. $$A^2 = I.$$ Is it necessarily symmetric? Any help will be highly appreciated. Thank you very much.
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38 views

Involutive automorphism of simple Lie algebra

Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan decomposition for a noncompact real simple Lie algebra $\mathfrak{g}$ corresponding to a Cartan involution $\theta$, where $\mathfrak{k}$ is ...
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1answer
124 views

How many involutions does $\Bbb Z/999$ have?

How many involutions does $\Bbb Z/999$ have? Order of $g$ is donate by $o(g)$. And $o(g)=2$ iff $g$ is an involution. I know that order of $\Bbb Z/999$ is $999$ but how can I check $999$ elements ...
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15 views

Singularities of the quotient minimal surface with an involution

I am reading Guletskii Paper "Bloch Conjecture for surfaces with involution and of p_g=0" and I would like to have any advice to prove the following affirmation (see page 13 of this paper): If S is a ...
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1answer
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If $p^{2}=p^{*}=p$ and $pa^{*}a=paa^{*}=0$, then $a^{*}p+pa=0$.

Suppose that $p$ is a projection (i.e. $p^{2}=p^{*}=p$) in a C*-algebra $A$. Let $a\in A$ be an element such that $pa^{*}a=paa^{*}=0$. I want to prove that $$a^{*}p+pa=0.$$ I tried to express $a$ in ...
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2answers
60 views

Can the property $z^*z\in [0,\infty)$ be generalized to other fields?

Let $K$ be a field equipped with the following structure: Let $*$ be an involution $*:K\to K,z\mapsto z^*,$ where $*$ preserves field structure and $(z^*)^* = z$ for all $z\in K$. Assume that $z^*z\...
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1answer
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A theorem about the anti-derivation of the algebra

Sadri Hassani wrote down a theorem about the anti-derivation in his Mathematical Physics: A Modern Introduction to Its Foundations - Second Edition. Theorem 3.4.10 Let $\Omega_1$ and $\Omega_2$ be ...
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1answer
29 views

Does a fixpoint-free, self inverse homeomorphism define a smooth manifold?

I was wondering if the following theorem still holds if one only requires $\tau$ to be a homeomorphism and not a diffeomorphism: "Let M be a smooth manifold and $\tau:M\to M$ a differentiable function ...
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1answer
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what are the eigenvalues of a matrix A of m-involution

"The set of eigenvalues of a matrix A of m-involution (for which $A^m=I$ for an integer m>1) belongs to the set of m-th roots of unity." How do I prove this? It can be shown for m=2, the eigenvalues ...
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71 views

When $\mathbb{C}[u,v]$ is a UFD, for special $u,v \in \mathbb{C}[t]$?

Let $f=f(t) \in \mathbb{C}[t]$ be a non-scalar polynomial. Write $f=a_{2n}t^{2n}+a_{2n-1}t^{2n-1}+\cdots+a_2t^2+a_1t+a_0$, where $a_j \in \mathbb{C}$. The map $\iota: t \mapsto -t$ is an involution ...
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35 views

$\mathbb{C}(s_1,s_2,k)=\mathbb{C}(x,y)$, where $s_1,s_2$ are symmetric and $k$ is skew-symmetric

Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution (= $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of degree two) defined by $(x,y) \mapsto (x,-y)$. It is not difficult to ...
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1answer
61 views

Locus of fixed points of an involution in a surface

I am reading Guletskii Paper "Bloch Conjecture for surfaces with involution and of p_g=0" and I do not know why the following is true. If S is a minimal smooth projective surface with an involution i....
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0answers
53 views

Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric

Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution on $\mathbb{C}[x,y]$ defined by $(x,y) \mapsto (x,-y)$. Let $s_1,s_2,s_3 \in \mathbb{C}[x,y]$ be three symmetric elements with ...
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48 views

The form of $w \in \mathbb{C}(x,y)$ satisfying $\beta(w)=w$, $\beta$ an involution on $\mathbb{C}[x,y]$

The symmetric case: Let $w \in \mathbb{C}(x,y)$, and write $w=\frac{u}{v}$, where $u,v \in \mathbb{C}[x,y]$. Let $\beta: (x,y) \mapsto (x,-y)$; $\beta$ is an involution (= an automorphism of order ...
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Given an involution *, does xx* = x*x? If not, what is a counterexample?

Let A be an algebra with involution. An involution $*$ is a unary operation that satisfies the following properties: $\forall a \in A, (a^*)^* = a$ $\forall a, b \in A, (a + b)^* = a^* + b^*$ $\...
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Form of $u,v \in \mathbb{C}[x,y]$ satisfying $\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(u,v)$

Let $\beta$ be an involution on $\mathbb{C}[x,y]$, namely, $\beta$ is a $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of order two. Denote the set of symmetric elements with respect to $\beta$...
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56 views

A subfield of $\mathbb{C}(x,y)$ invariant under an involution

Let $u,v \in \mathbb{C}[x,y]$. Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution (= $\mathbb{C}$-algebra automorphism of order two) defined by $(x,y) \mapsto (x,-y)$. Denote the ...
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26 views

Does every finite Coxeter group have a unique longest element?

If $W$ is a finite Coxeter group, then it has a unique longest element $w_0$. In particular, $w_0$ is an involution. This is a (paraphrased) quote from Fayers' paper $0$-Hecke Algebras of finite ...
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Are involutions in infinite vector spaces always diagonalisable?

As the title says, I'm wondering if involutions in infinite dimensional vector spaces always have the entire space as its eigenspace? This is of course true for finite vector spaces, but I've never ...
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30 views

Is every involution (on an orientable manifold) orientation reversing?

I am sorry if this comes accros as a low effort question, but i couldn't find any reliable source and i guess it's true but i would appreciate any confirmation. Let $M$ be a orientable $m$-Manifold. ...
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1answer
97 views

An infinite group generated by all its order two elements

Let $G$ be an infinite group generated by all its order two elements. Is there something interesting that can be said about such $G$? The group $G$ I had in mind is the group of automorphisms of $...
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0answers
34 views

Banach subalgebras and * subalgebras of a C* algebra

Given a unital C* algebra $A$, or more specifically, $A:=\mathcal L(H)$ be the bounded linear operators of a certain Hilbert space $H$. How do its subalgebras look like if one only considers the pure ...
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1answer
63 views

Why Exactly 3 Involutions Are Used in The Proof of Roger Heath-Brown?

In the book "Proofs from THE BOOK" (click here, on page 20)by Martin Aigner and Günter M. Ziegler, a proof of representing number as sum of two square is given. The proof is due to Roger Heath-Brown (...
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1answer
72 views

Proof of Roger Heath-Brown: Interchanging Solutions of Involution

On page 20 of "Proofs from THE BOOK" (click here) a proof of representing number as sum of two square is given. The proof is due to Roger Heath-Brown (1971,appeared in 1984). The first part of the ...
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1answer
69 views

Relationship between trace and determinant of a non-diagonal $3 \times 3$ involutory matrix [closed]

Let $A$ be a $3 \times 3$ real, non-diagonal matrix with $A^{-1}=A$. Show that $\mbox{tr}(A) = -\det(A) = \pm 1$.
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1answer
48 views

Conjugacy class of involutions in finite simple group

I've been working on this problem, but I can't seem to do it. For a finite simple group $G$, is there a unique conjugacy class consisting of involutions? Clearly, a conjugacy class with an ...
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1answer
18 views

Is there a term for a function that is like an involutory function, but generalized to more recursions?

So, an involutory function is a function that is its own inverse. In other words, a recursive call to this function would return the original input: $$f(f(x)) \equiv x$$ Suppose we had a function ...
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60 views

Does anyone know what $H'_{m}$ means in this group theory paper by Miller?

https://www.jstage.jst.go.jp/article/tmj1911/17/0/17_0_88/_pdf Page 90, third paragraph, where it says "By continuing this process we arrive at a group $H_m$ containing the central of G and each ...
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Classification of Pseudo-Involutory Matrices

I'm interested in solving the following problem from Strang's Linear Algebra and Learning from Data: Which matrices have $A^+=A$? Why are they square? Look at $A^+A$. My work so far: If $A$ is $m \...
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1answer
24 views

Are there a set of $3\times3$ involutory matrices that form a basis?

My understanding is that any $2\times2$ matrix can be decomposed into the Pauli matrices and the $2\times2$ identity matrix, which are all involutory. Is there a set of $3\times3$ matrices which are ...
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0answers
61 views

Subspaces and submodules of $k[x,y]$

Let $k$ be a field of characteristic zero, $\beta$ an involution on $k[x,y]$ (a $k$-algebra automorphism of $k[x,y]$ of order two), and $U \subseteq k[x,y]$ an ideal of $k[x,y]$ (a $k[x,y]$-submodule ...
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Concerning an ideal invariant under an involution

The following is a variation on this question (or this question). Let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \mapsto (x,-y)$, namely, $\beta$ is an automorphism of order ...
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0answers
59 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 $\mathbb{C}[x,y]$ (= an automorphism of order two): $\beta: (x,y) \mapsto (x,-y)$. Let $s_1,s_2 \in S_{\beta}$ and $k \in K_{\beta}$. Assume that the ...
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1answer
44 views

Number of Non-equivalent Generating Triples of Involutions in Finite Simple Groups

Let $G$ be a group. We say $(x,y,z)\in G\times G\times G$ is a generating triple of involutions if $|x|=|y|=|z|=2$ and $\langle x,y,z\rangle = G$. A generating triple $(x,y,z)$ of $G$ is said to be ...
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0answers
19 views

Involution of $L^1$ reversing composition order

Let $L^1_{\mu}$ be a separable $L^1$ space for a Borel measure $\mu$. Does there exists a (continuous) involution $i$ on $L^1_{\mu}$, such that $$ i(f \circ g) = g\circ f; $$ whenever composition is ...
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1answer
142 views

How can I find all involutions whose reciprocals are also involutions?

How can I solve the following functional equation? $$\frac{1}{f(x)} = f\left( \frac{1}{x} \right)$$ This functional equation amounts to finding all involutions whose reciprocals are also involutions....
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Does $k(s_1,s_2)=k(x,y^2)$ imply that $k[s_1,s_2]=k[x,y^2]$, where $s_1,s_2$ are symmetric w.r.t. $(x,y) \mapsto (x,-y)$?

Let $k$ be a field of characteristic zero. Let $\beta: k[x,y] \to k[x,y]$ be the involution (a $k$-algebra automorphism of order two) defined by $(x,y) \mapsto (x,-y)$. The set of symmetric elements ...
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0answers
155 views

Ideal of $k[x,y]$ invariant under an involution

Let $k$ be a field of characteristic zero. The two-dimensional case: Let $p,q \in k[x,y]$, $I=\langle p,q \rangle$ a proper ideal of $k[x,y]$ and $\delta$ an involution on $k[x,y]$, namely, a $k$-...
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34 views

$p=s_1+k_1,q=s_2+k_2 \in k[x,y]$ such that $s_1,k_1,s_2,k_2 \in \langle p,q \rangle$.

Let $k$ be a field of characteristic zero, $k[x,y]$ be the polynomial ring and $\beta$ the following involution (= automorphism of order two) $\beta: (x,y) \mapsto (x,-y)$. Denote the set of symmetric ...
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2answers
77 views

Solutions to an Involution Functional Equations

I'm reading functional equations and how to solve them and I am very confused by one of the statements made Let $$f(x) = f(\frac{x}{x-1})$$ then a solution would be $$f(x) = cos log (x-1)$$ but ...
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1answer
50 views

Properties of an unique involution for $2$ points in $\mathbb{R}P^1$

Consider a projective line $\mathbb{R}P^1$. An involution $\phi$ is defined as a projective transformation such that $\phi^2 = I_d \neq \phi$. I want to proof that there exist for $2$ points $A,B \in \...
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0answers
72 views

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}(t),...
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4answers
265 views

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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1answer
298 views

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 ...