Questions tagged [involutions]
The involutions tag has no usage guidance.
71
questions
24
votes
1
answer
888
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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 $\...
20
votes
1
answer
2k
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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 ...
15
votes
2
answers
2k
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What is the subgroup generated by involutions?
I was recently taking some notes on the Cartan-Dieudonné theorem: if $(V,q)$ is a nondegenerate quadratic space of finite dimension $n$ over a field of characteristic not $2$, then every element of ...
10
votes
2
answers
261
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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 ...
9
votes
3
answers
664
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Algebraic axiomatization for AB+BA^T operation on matrices
Let us consider a matrix algebra $\operatorname{Mat}_{n\times n}(K)$, where $K$ is a field, $\operatorname{char} K \neq 2$.
It is well-known that the axiomatization of commutator operation $[A,B]=AB-...
9
votes
0
answers
299
views
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 ...
8
votes
1
answer
488
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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 ...
8
votes
0
answers
648
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-...
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 ...
7
votes
0
answers
131
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(\...
6
votes
1
answer
675
views
Does a fixed-point free "homotopy involution" imply that a manifold bounds?
Let $M^n$ be a closed (compact, connected, without boundary) smooth manifold. It is known that if there exists a fixed point free involution $\tau:M \rightarrow M$, then M bounds. That is, there ...
5
votes
2
answers
691
views
Does a manifold which bounds always admit a free involution?
If a closed smooth manifold $M$ admits a smooth free involution $T$, then it bounds. In fact, the mapping cylinder of the quotient map $M \to M/T$ is the manifold whose boundary is $M$.
Is the ...
5
votes
1
answer
247
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 ...
5
votes
1
answer
475
views
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 ...
5
votes
1
answer
2k
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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 ...
5
votes
1
answer
157
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\{\...
5
votes
0
answers
179
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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 ...
4
votes
1
answer
408
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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 ...
4
votes
1
answer
656
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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.
...
4
votes
1
answer
650
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Infinite dimensional division algebras with finite center, and their involutions
Let $q$ be a prime power, and $D$ a non-commutative division algebra (skew field) over $\mathbb{F}_q$ (the finite field with $q$ elements) such that the center $C(D)$ equals $\mathbb{F}_q$.
...
4
votes
1
answer
256
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Heisenberg-type groups over rings with involution
Hello everyone!
In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:
Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...
4
votes
0
answers
133
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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. ...
4
votes
0
answers
170
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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,...
4
votes
0
answers
61
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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 ...
4
votes
0
answers
389
views
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/\...
4
votes
0
answers
813
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Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
3
votes
3
answers
215
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A question on an involution of $E_8$ lattice
There exits an involution $\iota$ of the $E_8$ lattice such that $(E_8)^{\pm} \cong D_4$, where $(E_8)^{\pm}$ denotes the $\pm$ eigen-lattice of the involution $\iota$. Could someone kindly give me an ...
3
votes
6
answers
2k
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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) ...
3
votes
1
answer
555
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 $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. ...
3
votes
1
answer
715
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Involution centralizers in simple groups
I often see lower bounds on the size of centralizers of involutions
in finite (nonabelian) simple groups, but is there a general upper bound
for the size of an involution centralizer in such a ...
3
votes
1
answer
378
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 ...
3
votes
1
answer
395
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 ...
3
votes
1
answer
200
views
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 $g$ ...
3
votes
1
answer
271
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 ...
3
votes
0
answers
34
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 ...
3
votes
0
answers
148
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]=...
3
votes
0
answers
116
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 $...
2
votes
1
answer
591
views
Complete De Morgan algebra
Recall that an algebra $(A,\sim)$ is a De Morgan algebra if $A$ is a bounded distributive lattice and $\sim$ is a unary operation which satisfies:
${\sim} (x\vee y)={\sim} x\wedge {\sim} y$ and ${\sim\...
2
votes
1
answer
436
views
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 $(\...
2
votes
1
answer
98
views
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(...
2
votes
1
answer
273
views
(Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity)
Let $K$ be a skew-field, infinite dimensional over its center $F$.
From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-...
2
votes
2
answers
296
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Involution of $E_{8}$ lattice
Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are isomorphic)...
2
votes
0
answers
170
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 $\sigma$ ...
2
votes
0
answers
112
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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 ...
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 ...
2
votes
0
answers
161
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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 ...
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^{...
1
vote
1
answer
77
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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 $E_7(q)$?
1
vote
0
answers
92
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 $2$-...
1
vote
0
answers
60
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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{...