**7**

votes

**0**answers

477 views

### Can one give a “nice” expression for this determinant?

I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.
...

**1**

vote

**0**answers

483 views

### Equivalence classes induced on binary strings by set of permutations

Let $\mathbb{F}_2^{n}$ be the set of binary strings of length $n$ and let $f: \mathbb{F}_2^{n} \rightarrow \mathbb{R}$ be a function from the set of binary strings of length $n$ to the reals.
Let's ...

**0**

votes

**0**answers

167 views

### Invariant Polynomes under group action - given the invariants looking for the group. algorithmic solution?

I have given a finite set $S$ of polynomes in the ring $R = C[x_1,\dots,x_n]$. I need to know the minimal group $G$ wich acts on $R$ such that $C[S]$ is the ring of invariants of $R$ under the action ...

**6**

votes

**3**answers

525 views

### What is the ring of invariants of GL acting on quaternary cubic forms?

Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute ...

**9**

votes

**2**answers

592 views

### On the field of invariants of a finite group

So let $G$ be a finite group and let $\iota:G\rightarrow S_n$ be an embedding of $G$ in a symmetric group of degree $n$ for some fixed integer $n$. Let $K$ be a fixed field of characteristic $0$. The ...

**13**

votes

**2**answers

468 views

### Invariants and orbits of $n$-tensors

My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.
Let $V_1, \ldots V_n$ be finite dimensional ...

**5**

votes

**0**answers

201 views

### Modular reduction of exceptional complex reflection groups

I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that ...

**4**

votes

**1**answer

294 views

### Monotonicity of complete homogeneous symmetric polynomials

The complete homogeneous symmetric polynomials are defined as
$$
h_k (x_1, \dots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq n} x_{i_1} x_{i_2} \cdots x_{i_k}.
$$
For example,
$$
...

**4**

votes

**2**answers

398 views

### Adem-Wu relations from Bullett-Macdonald identities

Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...

**1**

vote

**2**answers

195 views

### Invariant space of lifted Chevalley automorphisms of the tensor algebra

Question. Let $k$ be a field of characteristic $0$. Let $L$ be a $k$-vector space. Consider the subspace $S$ of $L\otimes L\otimes L\otimes L$ spanned by all tensors of the form
$\left[a,\left\lbrace ...

**3**

votes

**1**answer

358 views

### The ring of SL_2 invariants in sums of conjugation and tautological modules

Rings of Invariants
Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free ...

**3**

votes

**0**answers

280 views

### Ring of invariants of finite subgroup of $GL_2(\mathbb{C})$

In the paper 'FINITE LINEAR GROUPS WHOSE RING OF INVARIANTS IS A COMPLETE INTERSECTION' by VICTOR KAC AND KEI-ICHI WATANABE published in BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY ...

**6**

votes

**1**answer

241 views

### Orbits for homogenous complex polynomials under unitary rotation of variables

Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$.
...

**4**

votes

**3**answers

396 views

### Molien for modular representations?

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or ...

**2**

votes

**1**answer

257 views

### Characterization of the weight orbit in the projective space via second order Casimir.

This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...

**8**

votes

**0**answers

181 views

### Characteristic classes from moduli of alternating forms

Suppose, just for example, that you have a smooth manifold $M$ of dimension greater than $8$,and a cohomology class $q$ in $H^3(M,{\Bbb R})$. Suppose further that one can represent $q$ with a ...

**2**

votes

**3**answers

312 views

### Is the space of polynomial functions on M_n a faithful U(gl_n)-module?

We are over some field $k$ of characteristic $0$. The general linear group $\mathrm{GL}_n$ canonically acts from the left and from the right on the space $\mathrm{M}_n$, and thus also acts from the ...

**6**

votes

**4**answers

2k views

### Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$

One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...

**4**

votes

**1**answer

660 views

### Invariant theory

My question might be an easy or could be a bit complicate and classic. Actually I am trying to understand why the discriminant of a binary quadratic form is a "the fundamental invariant" under ...

**3**

votes

**0**answers

190 views

### Invariant Subvarieties of Variety of Quiver Representations

I'd like to understand a special case of the following rather general algebraic geometry question:
Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...

**13**

votes

**3**answers

948 views

### Noether's bound for anticommutative invariant theory (diff. forms instead of polynomials)?

EDIT: Now with a concrete request to CAS experts (see the end of the post).
Let $G$ be a finite group, and $V$ a finite-dimensional representation of $G$. The classical invariant theory of $G$ and ...

**8**

votes

**4**answers

2k views

### “Why” is every polynomial representation of SL(2) selfdual?

Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...

**2**

votes

**2**answers

353 views

### Is there an invariant theory explanation of the orbit structure of GL₂ acting on second-diagonal symmetric matrices by g∙X = gXJg^tJ ?

Statement of the Specific Result
Let $J$ denote the matrices with ones on the "second diagonal", meaning the diagonal between the (1,n) and (n,1) entry, and zeros elsewhere. So in the case $n=2$, ...

**10**

votes

**6**answers

2k views

### Polynomial invariants of the exceptional Weyl groups

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let ...

**4**

votes

**0**answers

327 views

### A relative Noether number for invariants

EDIT: Wrong definition of $\beta\left(G,H\right)$ fixed. One of the results is open (i. e., I cannot prove it).
In "Finite Groups and invariant theory" (a paper in Malliavin's LNM #1478 which can ...

**4**

votes

**0**answers

219 views

### When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?

Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements ...

**3**

votes

**2**answers

259 views

### How to compute the ring of invariants of SO_3(k) acting on a polynomial ring

Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in ...

**44**

votes

**1**answer

3k views

### Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...

**10**

votes

**2**answers

1k views

### Chevalley–Shephard–Todd theorem

The wikipedia article claims that the theorem "was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof". I ...

**5**

votes

**2**answers

395 views

### Left U_n-invariants of SL_n - an exercise in Kraft-Procesi

I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S_n$, and I don't want these to get ...

**6**

votes

**5**answers

943 views

### Explicit invariants (under change of basis) of maps $V \to V \otimes V$.

It is standard to construct numbers associated to a linear transformation $f: V \to V$ of a finite-dimensional vector space which are invariant under change of basis. The coefficients of the ...

**2**

votes

**2**answers

238 views

### Subrings of rational functions invariant under change of sign

Let $R$ be a ring generated by $k$ rational functions in the
variables $x_1,...,x_n$ over the real numbers.
Is there an algorithm that computes a set of rational functions
$f_1,...,f_l \in R$ which ...

**4**

votes

**1**answer

233 views

### Invariant forms

Is there a classification of pairs $(\mathfrak g,V)$, with $\mathfrak g$ a Lie algebra and $V$ a $\mathfrak g$-module, such that $(\Lambda^3V)^{\mathfrak g}\neq0$?

**2**

votes

**1**answer

640 views

### Weyl group Invariants

What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group
of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra
and the action is the diagonal action?
Is ...

**10**

votes

**1**answer

561 views

### When Are Quotients Complete Intersections?

Let $S_{n}$ denote the permutation group on $n$ letters and $G\subset S_{n}$ a transitive subgroup. The inclusion of $G$ in $S_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a ...

**4**

votes

**3**answers

301 views

### Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.

Start with the really well-known fact that $R[x_1, \ldots, x_n]^{S_n}$, where $R$ is any commutative ring, is polynomial on elementary symmetric polynomials. Now consider the slight generalization of ...

**6**

votes

**1**answer

459 views

### Generalizing cosine rule to symmetric spaces

The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...

**7**

votes

**3**answers

977 views

### Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?

Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...

**2**

votes

**1**answer

985 views

### What is affine invariant used in computer vision?

Affine invariant for 4 coplanar points ABCD is said to be Area(ACD)/Area(ABC). Can somebody provide the proof of this means why is this invariant under affine ...

**9**

votes

**8**answers

2k views

### Resources on Invariant Theory

Hi,
So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd ...