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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Can one pick generators for the ring of invariants of binary n-ic forms which have rational coefficients?

The problem of determining a set of generators of the ring of invariants of the group $\textrm{SL}_2$ acting on the complex $n+1$-dimensional vector space of binary $n$-ic forms is known to be very ...
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287 views

Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)? E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...
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An almost permutation G-lattice

I've been trying to determine the rationality of certain fields of invariants coming from G-lattices. More precisely, letting $G$ be a finite group, $L=\mathbb{Z}^n$ a free abelian group with a $G$ ...
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323 views

Quotient of affine space by cyclic permutation

The quotient of the affine space $\mathbb{A}^n$ by the symmetric group $Sym_n$ is again an affine space of the same dimension, and invariants are given by elementary symmetric polynomials. What ...
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79 views

A slightly odd (integral of Whittaker functions / sum of characters of $GL_n(\mathbb C)$ / sum of Schur functions)

Let $W$ be the normalized spherical Whittaker function attached to a spherical representation $\pi$ on $GL_n(k)$, where $k$ is a $p$-adic field and $n\ge 3$. I'm faced with the slightly-odd integral ...
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algebra of endomorphisms over the diagonal invariants

Let $k$ be a field of characteristic 0 (say $\mathbb{C}$). Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$. It is known that the ...
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1answer
208 views

Invariants of a set of real unit vectors in 3d space, under SO(3)

I have a set of $n$ real unit vectors, in 3-dimensional space. (It is a follow up of Sets of vectors related by a rotation.) Is there a construction providing a complete set of independent*) ...
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163 views

Modules of invariants?

Let $G \subset SL(V, \mathbb{C})$ be a finite group and $R=(\operatorname{Sym}\[V\])^G$ is the ring of polynomial invariants, $W$ some irreducible complex representation of $G$. I want to know is ...
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1answer
234 views

What does this particular geometric quotient locally look like?

Let $k$ be a field and consider the algebraic group $GL_n$ over $Spec(k)$. It has as a closed (but not normal) algebraic subgroup the group $M$ of monomial matrices, i.e. matrices having exactly one ...
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221 views

Intersection theory for $G$-varieties - an action on the chow ring?

Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed ...
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1answer
140 views

Invariance group of Morse charts

Suppose I have a smooth function $\varphi$ that vanishes at $p$ and has a positive definite Hessian at that point (suppose that we are on a smooth manifold of dimension $M$). Then the Morse lemma ...
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478 views

A question on invariant theory of $GL_n(\mathbb{C})$.

Let $\rho$ denote the irreducible algebraic representation of $GL_n(\mathbb{C})$ with the highest weight $(2,2,\underset{n-2}{\underbrace{0,\dots,0}})$. Let $k\leq n/2$ be a non-negative integer. ...
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253 views

About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$

Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) ...
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3answers
311 views

Invariants of group action: SL_n acts simultaneously on m symmetric matrices

Let $\rm{SL}_n$ be the special linear group and let $\rm{Sym}_n$ be the set of all symmetric matrices of size n. $\rm{SL}_n$ acts on $(\rm{Sym}_n)^m$ by $g(A_1, \ldots , A_m)=(gA_1 g^{\rm T}, \ldots , ...
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683 views

Is $k[X]^G$ integral closed in $k[X]$.

May assume field $k=\mathbb{C}$. Let $X$ be an affine variety and $G$ be a reductive group (may assume connected). Is the ring of invariants $k[X]^G$ integral closed in $k[X]$? The claim may not ...
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1answer
202 views

When the affine quotient is faithfully flat?

It may be easy for the expert. Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$. My question is when this map is ...
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246 views

Group action on Grassmannian: Intersection of two special invariant rings

Let $K$ be a field with characteristic $0$. Let $G:=G(d,nd)$ the Grassmannian of all $d-$dimensional subspaces of $K^{nd}$ and let $H:=O_d(K)^n$ the n-fold direct product of the orthogonal group. $H$ ...
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Invariants for the exceptional complex simple Lie algebra $F_4$

This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise. Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...
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411 views

Explicit generators for matrix invariants of the symmetric group

Let $V$ be the space of $n$ by $n$ complex matrices with the conjugate action of the symmetric group $G=S_n$. Is any explicit set of generators for the invariant ring $C[V]^G$ known?
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Why can I divide an affine variety by the action of the general linear group?

Let $G\subseteq\mathrm{Gl}_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine ...
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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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766 views

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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4answers
485 views

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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217 views

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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5answers
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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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331 views

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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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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1answer
271 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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172 views

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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1answer
315 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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429 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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555 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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2answers
460 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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1answer
282 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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379 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; ...
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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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1answer
335 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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1answer
238 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 $U\in SU(n)$. ...
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383 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
227 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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180 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 ...
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3answers
306 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 ...
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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 ...