Questions tagged [involutions]

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24 votes
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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 $\...
Roland Bacher's user avatar
20 votes
1 answer
2k views

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 ...
smyrlis's user avatar
  • 2,873
15 votes
2 answers
2k views

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 ...
Pete L. Clark's user avatar
10 votes
2 answers
261 views

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 ...
Noam Zeilberger's user avatar
9 votes
3 answers
664 views

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-...
probably's user avatar
  • 403
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 ...
James Propp's user avatar
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8 votes
1 answer
488 views

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 ...
Hebe's user avatar
  • 821
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-...
Manuel Bärenz's user avatar
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 ...
Allen Knutson's user avatar
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(\...
Hebe's user avatar
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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 ...
Scott Van Thuong's user avatar
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 ...
kelly's user avatar
  • 127
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 ...
Jérémy Blanc's user avatar
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 ...
sunspots's user avatar
  • 161
5 votes
1 answer
2k views

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 ...
GA316's user avatar
  • 1,219
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\{\...
Mersn's user avatar
  • 51
5 votes
0 answers
179 views

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 ...
kneidell's user avatar
  • 993
4 votes
1 answer
408 views

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 ...
Mikhail Borovoi's user avatar
4 votes
1 answer
656 views

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. ...
Anurag's user avatar
  • 1,157
4 votes
1 answer
650 views

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 $\mathbb{F}_q$. ...
Max Horn's user avatar
  • 5,112
4 votes
1 answer
256 views

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 $x\mapsto\overline{x}$ ...
Andrei Smolensky's user avatar
4 votes
0 answers
133 views

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. ...
nxir's user avatar
  • 1,409
4 votes
0 answers
170 views

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,...
THC's user avatar
  • 4,313
4 votes
0 answers
61 views

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 ...
Fabrizio's user avatar
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/\...
Aurel's user avatar
  • 4,878
4 votes
0 answers
813 views

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)\...
user40597's user avatar
3 votes
3 answers
215 views

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 ...
Howei's user avatar
  • 31
3 votes
6 answers
2k views

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) ...
user144684's user avatar
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 $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. ...
Hebe's user avatar
  • 821
3 votes
1 answer
715 views

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 ...
THC's user avatar
  • 4,313
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 ...
Hebe's user avatar
  • 821
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 ...
Jana's user avatar
  • 2,022
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 $g$ ...
Hebe's user avatar
  • 821
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 ...
Haowen Zhang's user avatar
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 ...
THC's user avatar
  • 4,313
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]=...
user237522's user avatar
  • 2,783
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 $...
Spencer Leslie's user avatar
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\...
ariel's user avatar
  • 21
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 $(\...
Captain Lama's user avatar
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(...
user avatar
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-...
Max Horn's user avatar
  • 5,112
2 votes
2 answers
296 views

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)...
M Pandhari's user avatar
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 $\sigma$ ...
Andrea Antinucci's user avatar
2 votes
0 answers
112 views

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 ...
THC's user avatar
  • 4,313
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 ...
pbarron's user avatar
  • 71
2 votes
0 answers
161 views

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 ...
Eduardo Reis's user avatar
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^{...
Kafka91's user avatar
  • 641
1 vote
1 answer
77 views

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 $E_7(q)$?
Yi Wang's user avatar
  • 261
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 $2$-...
THC's user avatar
  • 4,313
1 vote
0 answers
60 views

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{...
user488802's user avatar