Invariant theory deals with an algebraic, geometric or analytic structure X , submited to the action of an (algebraic) group G . It studies G-invariant elements of X as well as the set of G-orbits.

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FFT like theorems for tensor product

The fundamental theorem of symmetric functions states that $\mathbb C[V]^{S_n}$ is generated by the elementary symmetric polynomials, where $V$ is the natural representation of $S_n$. Then a theorem ...
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Surjectivity of Invariants

Suppose $V, W, U$ are $Z_p$ module over a field $F$ of characteristic $p$ and $V=W \oplus U$. Is there a degree preserving surjective map from $F[V]^{Z_p}$ to $F[W]^{Z_p}$ ? In non-modular case the ...
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Alternating multilinear invariants of GL(n) on End (k^n)

Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...
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Diagonal invariants of the symmetric group on $k[X_1,X_2,…,X_n,Y_1,Y_2,…,Y_n]$

This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...
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Structure of $[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ for semisimple $\mathfrak{g}$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. $S(\mathfrak{g})^G$ is a polynomial algebra with rank $\mathfrak{g}$ generators. Call them $c_i(x)$, where $x\in \mathfrak{g}$ and ...
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presentations for complex involutory reflection groups

It's a well-known result due to J.Tits that a finite-dimensional real reflection group has a faithful presentation, given by its Coxeter diagram (i.e. the linear group in question is isomorphic to the ...
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area of triangle from coefficients of its cubic?

Three points $z_1$, $z_2$, $z_3$ on the complex plane are given by the coefficients $a_k$'s of the cubic polynomial $f(z)=(z-z_1)(z-z_2)(z-z_3)=\sum_{k=0}^3 a_k z^k$. How does one express the ...
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When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same?

Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly. It is easy too see, that a different ...
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308 views

Invariants of co-diagonalizability in real symmetric matrices

This question has been mentionned to me by U. Frisch. He wanders whether it has ever been considered by algebraists. In the vector space ${\bf Sym}_n({\mathbb R})$, two elements commutte to each ...
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243 views

Invariant forms of a binary quintic

Does anyone have a reference for the invariant binary forms of a quintic? That is, what are the $SL_2(C)$ invariant polynomial functions on the space of binary quintics.
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Dimension of spaces of invariants/tableaux functions

The Hook lenght formula gives the number of standard Young tableaux on a given diagram. A variant gives the number of semistandard tableuax. Does there exist a formula for counting "weighted ...
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310 views

GL(n+1)- vs. GL(n)-orbits

Let $f,g \in \mathbf {C}[x_1, \ldots, x_n]\subseteq \mathbf {C}[x_1, \ldots, x_{n+1}]$ be two polynomials with complex coefficients and suppose that there exists $h_1 \in \mathop{GL}_{n+1}(\mathbf C)$ ...
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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. ...
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406 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 ...
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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 ...
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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 ...
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515 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 ...
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436 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 ...
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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 ...
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274 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, $$ ...
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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; ...
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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 ...
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315 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 ...
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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 ...
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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 $U\in SU(n)$. ...
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377 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 ...
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1answer
216 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) ...
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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 ...
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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 ...
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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 ...
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627 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 ...
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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 ...
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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 ...
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“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, ...
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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 $n=2$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $(\Lambda^3V)^{\mathfrak g}\neq0$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...