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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$SL(n) \times SL(n)$-invariants of $m$-tuples of matrices
I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows
$$
(A,B) \cdot (M_1, \ldots, M_m) ...
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1answer
94 views
Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?
In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...
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function invariants under integral transforms
Some integral transforms have various invariants (Fourier transform, etc.), while others don't, such as Hilbert transform. I am wondering if there is any general method to find the invariant functions ...
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58 views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
6
votes
2answers
338 views
Representations of $SL(2)$ in characteristic 2
In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of SL(2)-modules. In characteristic $p$ things are more complicated.
I am interested in the special case ...
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1answer
171 views
Schur-Weyl duality for arbitrary tensor products of simple finite-dimensional $GL_n$-modules
Let $M$ and $N$ be two simple finite dimensional $GL_n$ modules. Is there a way of expressing the heighest weight vectors of the simple submodules of $M\otimes N$ in terms of the heighest weight ...
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151 views
Earliest use of the term “linearly reductive”?
Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...
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78 views
First Fundamental Theorem for Alternating Group
I know it fails but is there an answer?
More precisely, let $V$ be the standard complex $n$-dimensional representation of the alternating group $A_n$, $kV$ the direct sum of its $k$ copies, $S(kV)$ ...
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109 views
Invariants of the Determinant Form
Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...
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77 views
The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request
All cohomology and homology will be $Z/2$ coefficient. The restriction map
$H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of
the Dickson invariant $Z/2[w_2,w_3]$ into the ...
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71 views
Galois correspondence for action of general linear group on purely transcendental extension
For a fixed positive integer $n$, the group $G=GL_n(\mathbb{C})$ acts on the field $K=\mathbb{C}(t_1,\ldots,t_n)$ by linear change of variables. I would like to know if there is something like a ...
4
votes
1answer
209 views
The existence of a finite dimensional Lie algebra with a given symmetric invariant metric
The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
4
votes
1answer
192 views
Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$
Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
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1answer
101 views
Specht polynomials for dihedral groups
The representation theory of the symmetric group is best understood via the Specht polynomials. In wonder how this works for other finite reflection groups, such as dihedral groups. Are the similarly ...
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votes
0answers
67 views
Weyl group invariants of the representation ring of a split torus
Let $G$ be a semisimple split algebraic group, $T$ its split maximal torus and $W$ corresponding Weyl group. Let $T^*$ denote the character lattice of $T$ and $\Lambda$ denote the weight lattice, so ...
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votes
1answer
160 views
Chevalley restriction theorem for exterior algebras
Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem ...
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2answers
361 views
Equivariant normalization?
Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...
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2answers
138 views
Decomposition of $G$-harmonic polynomials
Let $G$ be a finite group and let $H_G$ denote the $G$-harmonic polynomials. What is the structure of $H_G$ as a $G$-module? Is it isomorphic to the regular representation?
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votes
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48 views
different and discriminant for finite invariants
Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...
4
votes
1answer
241 views
quotient by finite group actions that are smooth
Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that ...
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126 views
determine if a toric variety is Gorenstein
Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...
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0answers
129 views
Multiplicative groups of fields
Let $G$ be a finite group, and let be a field on which $G$ acts
faithfully. Let $C$, be a $\mathbb{Z}G$-lattice, and let $\alpha$
be a one-cocycle from $G$ to $Hom(C,L^{*})$. Let $L[C]$ the group
...
9
votes
2answers
282 views
Extension of the Weyl dimension formula
Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
5
votes
1answer
322 views
Where does the name “Reynolds operator” come from?
I always found it strange that, in the context of invariant and representation theory,
averaging over the group is called the "Reynolds operator". As far as I know the work of Reynolds was in fluid ...
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vote
1answer
62 views
Jacobian of primary invariants
Let $G$ be a finite group and consider $R=K[V]^G$ be the invariant ring of the group G over the field K of char 0. Let $f_1,\ldots f_n$ be a set of primary invariants. Is there a nice geometric ...
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27 views
The invariant field for the dual of the augmentation lattice
Let $G$ be some finite group and $J_G$ be $ZG$ modulo the ideal $\{n\sum_{g} g | n\in\mathbb N\}$, or equivalently the dual of the augmentation ideal which is the kernel of $ZG\to Z$ sending $g$ to ...
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189 views
How to determine there exists a unique invariant subspace for a set of matrices
Hi everyone,
Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...
1
vote
1answer
508 views
What is the “fundamental theorem of invariant theory” ?
The basic question I guess can be formulated as - given two integers $N_f$ and $N_c$ what are the ways in which the fundamental and the anti-fundamental representations of $U(N_f)$ be combined to get ...
4
votes
3answers
175 views
Generate a higher degree symmetric polynomial from an existing one
Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as
$q(x_1, x_2, \cdots, x_{n+1}) = ...
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1answer
112 views
Ideal membership (concerning polynomial invariants of orthogonal groups)
Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials
$$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$
I'm ...
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vote
1answer
93 views
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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votes
2answers
224 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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0answers
58 views
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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votes
2answers
274 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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0answers
72 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
...
5
votes
0answers
141 views
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 ...
2
votes
1answer
206 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*) ...
2
votes
1answer
154 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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vote
1answer
227 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 ...
3
votes
2answers
200 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 ...
2
votes
1answer
130 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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votes
1answer
411 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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votes
2answers
238 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}^ * ) ...
6
votes
3answers
302 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 , ...
5
votes
1answer
546 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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votes
1answer
182 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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0answers
222 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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votes
2answers
966 views
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 ...
12
votes
2answers
379 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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votes
3answers
468 views
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 ...
