Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. (Def: http://en.m.wikipedia.org/wiki/Invariant_theory)

learn more… | top users | synonyms

3
votes
2answers
82 views

Ring of Invariants of symmetric group

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
0
votes
2answers
138 views

Is it possible to find a 2D distribution function such that the higher order moments always exist?

Is it possible to generate a 2D distribution function $f(x,y)$ with function supports specified as $[-a,a]$ and $[-b,b]$ for $x$ and $y$ respectively, such that it always has moments which are NON ...
2
votes
1answer
40 views

Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
2
votes
1answer
62 views

Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of $G$....
2
votes
1answer
42 views

If $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, find $t$ operators such that $T=T_1\oplus\dots\oplus T_t$

Be $T\in \mathscr{L}(V)$ a linear operator with characteristic polynomial $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, $n_i\geq 1$ and $\lambda_i\neq\lambda_j$ if $i\neq j$. Show that $T$ can ...
2
votes
1answer
59 views

Quotients of curves

Magma (link) has a lot of functionality for computing quotients of curves by group actions. I am interested to know how one does this in general and I am finding it oddly difficult to find literature ...
1
vote
1answer
24 views

Steenrod Operations an algebraic Approach

Assume that $q=p^{r}$, where $p$ is a prime either 2 or odd and $\mathbb{F}_{q}$ is a Galois field and $V$ a finite dimensional $\mathbb{F}_{q}$-vector space. Then due to Larry Smith in this http://...
1
vote
1answer
65 views

Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
1
vote
1answer
24 views

What is known about rational points on the ideal of relations / syzygy ideal?

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation (...
1
vote
1answer
117 views

Affine transformation invariants and lie groups

Is it possible to generate geometric properties which are invariant under affine transformations? I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine ...
1
vote
1answer
117 views

What is an example of a syzygy in invariant theory or pre-abstract algebra

I need an example involving the old invariant theory that pre-dated abstract algebra.
0
votes
1answer
23 views

Number of eigenvalues for this operator

Say I have a F - vector space V and a subspace U given by U={va : a is in F}. Now suppose I have an operator defined by $Tv=av$. Clearly, U is invariant under T, since for any element of U, say bv, I ...
0
votes
1answer
48 views

What is a trivial linear operator?

The question asks to find a non-trivial linear operator T to make a subspace T-invariant. I'm thinking of $T(x)=2x$, since $2x$ clearly stays in the subspace by scalar multiplication closure, but not ...
1
vote
0answers
72 views

How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$

I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on. (...
1
vote
0answers
166 views

What was Cayley's formula for the number of invariants? (Lost Formula!?)

I need to find Cayley's formula for the number of linearly independent invariants of homogenous polynomials. This is a combinatorial formula. He is believed to have discovered it in 1854. ...
0
votes
0answers
45 views

Group action dimension of orbits

I am working on the following problem right now: Let $V,W,H$ be finite dimensional vector spaces. I have a group action $Gl(V)\times Gl(W) \curvearrowright Hom(V\otimes H,W)$ in the obvious way i.e. ...
0
votes
0answers
19 views

Primary and Secondary invariants for finite groups

For a finite group G and complex representation V of degree n, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants ...
0
votes
0answers
34 views

How t0 show $U+W$ and $U\cap W$ are invariant under $T$ if $U$ and $W$ are invariant under $T$?

Suppose $T:V\to V$ is a linear transformation, how t0 show $U+W$ and $U\cap W$ are invariant under $T$ if $U$ and $W$ are invariant under $T$ ? My try: $1.$ Let $u\in U,w\in W$. Then $T(u+w)=T(u)+T(...
0
votes
0answers
20 views

Integer matrix invariant factors

Have question about finding invariant factors of integer matrix: $\begin{pmatrix} 6 & 2 \\ -2 & 6 \end{pmatrix}$ Was sick during lectures, and not completely that what I do is right. What I ...
0
votes
0answers
14 views

Independence of a binary form and its Hessian

Let $f\equiv f(X,Y):=\sum_{i=0}^{n}a_iX^{n-i}Y^i$ be a binary form of degree $n\geq3$ with coefficients over $\mathbb{C}$ and no repeated roots in $\mathbb{C}^2$ (up to scaling). The Hessian of $f$ is ...
0
votes
0answers
45 views

Normalisation of an invariant measure

Is there an example of a Markov chain with invariant measure $\pi$ and $\sum_{i \in I}\pi(i) = \infty$ that can be normalised so that we can consider an invariant distribution instead? This is a ...
0
votes
0answers
31 views

Invariant factors. Prove that $q_i|q_{i+1}$

How could I show that $q_i|q_{i+1}$ without using the theorem of Smith Normal Form ? $q_i$ is the invariant factor. It defines how $q_i = f_i/f_{i-1}$, being $f_i$ the determinantal divisor. I ...
0
votes
0answers
25 views

Lie group action and Lie algebra action.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $r \in g \otimes g$ and $b \in g$. Consider the adjoint action $g \times g \otimes g \to g \otimes g $ given by $(b, x, y) \mapsto b.(x \otimes y) = ...
0
votes
0answers
77 views

Difference between invariant and contractive sets

I came across this particular notion of contractive sets. I know what an invariant set is, but can anyone explain what a contractive set is and the difference between invariant and contractive sets?
0
votes
0answers
24 views

Does Magma let you specify primary invariants?

I am cross-posting this question from scicomp.SE. The computer algebra system Magma can calculate primary invariants (i.e. a homogeneous system of parameters) in an invariant ring of a finite group ...
0
votes
0answers
52 views

Matrix representation of a 6-dimensional Lie algebra

The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and ...
0
votes
0answers
16 views

G-invariant polynomials for a concrete G

The collection of matrices$$ \left( \begin{array}{lll} a^2 & 2ab &b^2 \\ ac & bc+ad & bd \\ c^2 & 2cd & d^2 \end{array} \right)$$ indexed by $a,b,c,d \in \mathbb{R}$ is a ...
0
votes
0answers
73 views

4D representation of finite group and projection operator

Suppose I have a finite group, say $C_2$, and its Cartesian representation $ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \...
0
votes
0answers
21 views

Geometric Invariants of a conic section

There are three independent invariants for every conic section, viz., $$[I_1,I_2,I_3]= [ (a + b + c), (a b -h^2), Det(( a,h,g), (h,b.f), (g,f,c) )] $$ How are they related to the known geometric ...
0
votes
0answers
47 views

equivalent definition of a good categorical quotient?

Working solely with varieties, as you may know, a pair ($Y$, $\pi$) is a "good categorical quotient" for the $G$-variety $X$ if: 1) $\pi$ is surjective and constant along orbits 2) for any open $U \...
0
votes
0answers
67 views

Why is a graded $\mathbb{C}$-algebra free as a module over the subring generated by a regular sequence?

I am reading Bernd Sturmfels' book Algorithms in Invariant Theory. On p. 38 he makes the following assertion: "If $\theta_1,\dots,\theta_n$ are algebraically independent over $\mathbb{C}$, then the ...
0
votes
0answers
32 views

Invariant theory and irreducible representation of a group

I am now reading the book 'Invariant theory' by Neusel. It seems that the logic is like this: You have a representation of a group $G$ on some vector space $V$, this might be a reducible ...
0
votes
0answers
24 views

Algebra of invariants finitely generated

If algebraic torus $T = \mathbb G_m^n$ over an algebraically closed field $k$ acts on finitely generated algebra $A$ over $k$, is the algebra $A^T$ of invariants finitely generated?
0
votes
0answers
25 views

Invariant subspace under representation $\phi$

Let $\phi: \Bbb{Z}/n\Bbb{Z}\to GL_{2}(\Bbb{C})$ be the representation which takes $\bar{m} \to \begin{bmatrix} \cos (\theta) & -\sin (\theta) \\ \sin (\theta) & \cos (\theta) \end{bmatrix}$ ...
0
votes
0answers
37 views

Regular sequence in rings of invariants

When I was reading a book in invariant theory, I've come across this assumption: "Let $f_1, f_2 \in \mathbb{F}[V]^G$ be a regular sequence in $\mathbb{F}[V]$". I know that we cannot assume anything ...
0
votes
0answers
54 views

Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
0
votes
0answers
27 views

Existence of dense subsets $G$-invariant.

Let $G$ be a group that acts on a manifold $X$. It is well know that the orbir space $X/G$ isn't in general a manifold. But how can I prove that there is always a dense open $U \subset X$ that is $G$-...