Questions tagged [invariant-theory]
Invariant theory deals with an algebraic, geometric or analytic structure $X$, submitted to the action of an (algebraic) group $G$. It studies $G$-invariant elements of $X$ as well as the set of $G$-orbits.
400
questions
5
votes
0
answers
203
views
Stability of nodal hypersurfaces
We denote by $\Pi_{n,d}$ the space of homogeneous polynomials of degree $d$ in $n+1$ variables $x_0,\ldots,x_n$, i.e. $\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group $G=SL(n+1)$ ...
3
votes
1
answer
440
views
Why is the largest invariant set the following?
Consider this paper:
Hai-Feng Huo and Li-Xiang Feng, "Global stability for an HIV/AIDS epidemic model with different latent stages and treatment", Applied Mathematical Modelling, Volume 37, ...
- 185
3
votes
0
answers
160
views
Nef cone of a GIT quotient
I want to know how to calculate nef cone of a GIT quotient. In particular let $X$ be a projective variety and $L$ be an ample line bundle on $X$ and $G$ be a reductive algebraic group and chosen a $G$ ...
- 61
11
votes
2
answers
636
views
Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 569
4
votes
0
answers
144
views
Polynomial invariants of infinite reflection groups
It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...
- 41
1
vote
1
answer
610
views
Polynomial invariant — from product formula to monomial expansion
Context
This question deals with the polynomial invariant denoted by $ H_{n} $ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....
- 149
1
vote
0
answers
81
views
Ring structure of coinvariant of $W(U(4))$
I want to know the ring structure of the coinvariant of $W(U(4))$, where $W(G)$ is the Weyl group of G.
I know that the ring structure of the coinvariant of $W(U(3))$ is $\mathbb{Z}[x_1,x_2,x_3]$ with ...
2
votes
0
answers
79
views
A question about fundamental invariants in the context of neural networks
I'm reading in depth the first part of the following paper: https://arxiv.org/pdf/1804.10306.pdf, paying specific attention to the following result, that I re-write here for the sake of convenience:
[...
- 133
1
vote
1
answer
100
views
A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$
If $H = S_n$ then then the fundamental symmetric polynomials allow to write any $S_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\...
- 133
4
votes
0
answers
107
views
Finding special form for integral binary cubic form
Let $f(x,y)=px^3+3qx^2y+3rxy^2+sy^3$, $p,q,r,s\in\mathbb{Z}$, be an integral binary cubic form. Under what conditions is $f$ equivalent to a form $g(x,y)=t x^3+3u x^2y+3v xy^2+w y^3$ with $u=v$?
Here ...
- 291
7
votes
1
answer
199
views
$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve
Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
- 384
6
votes
0
answers
106
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,184
9
votes
2
answers
899
views
Character variety of the free group
A classical result of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi_\mathbb{C}$ associated to the free group $F_2$ and the algebraic group $\mathrm{SL}_2(\mathbb{C})$...
- 2,681
2
votes
0
answers
73
views
Periodic functions on groups
I'll give you some context: We can say that a (classic) periodic function $f:\mathbb{R}\rightarrow\mathbb{R}$ with period $\omega$ is an invariant function under the cyclic subgroup $G=\langle\omega\...
- 171
2
votes
1
answer
213
views
Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring
Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...
- 33.2k
5
votes
0
answers
347
views
What representation theoretic properties does the semi-invariant ring tell us?
I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
I have been studying about semi-...
- 767
2
votes
0
answers
92
views
Names of quotients of affine varieties by reducible group actions
Let $X$ be an affine variety over $\mathbb C$ and let $G$ be a reductive group (over $\mathbb C$) acting on $X$. It is well known then that the ring of invariants $\mathbb C[X]^G$ is finitely ...
- 14k
5
votes
1
answer
532
views
Invariant section of a linearized sheaf
I am struggling to understand what an invariant section with respect to a linearization of a line sheaf is. In Geometric Invariant Theory, given a $k$-scheme $X$ (being $k$ an algebraically closed ...
- 439
1
vote
0
answers
350
views
Invariant ring of linear algebraic groups
Let $G$ be a connected linear algebraic group. This question concerns Hilbert's 14th Problem for the adjoint action of $G$ on itself. Let $k[G]^G$ denote the algebra of regular functions on $G$ ...
- 2,681
3
votes
0
answers
169
views
Polynomial invariant relating the circumradius and sides of a cyclic polygon
This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon.
This invariant is discussed briefly in the seminal paper On the Areas ...
- 149
7
votes
0
answers
228
views
Chern-Weil theory on some noncompact groups, and characteristic classes in differential cohomology
$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups,
...
- 6,766
5
votes
2
answers
392
views
Invariant theory over $\mathbb R$
$\DeclareMathOperator\SO{SO}$Suppose we have a (continuous) linear action of $\SO(n,\mathbb R)$ on a vector space $\mathbb R^N$. Consider the ring of invariants $A\subset \mathbb R[x_1,\ldots, x_N]$, ...
- 14k
2
votes
2
answers
381
views
$O(n)$ Polynomial invariant of symmetric tensors
I apologize in advance if this question is not up to the level of research level questions on Math overflow. I am a complete outsider to invariant theory/representation theory and would like someone ...
- 1,025
0
votes
1
answer
232
views
Simplex invariants?
Let $k=s$ be a positive definite symmetric function and a simililarity over the natural numbers such that $k(a,a) = 1 $ for all natural numbers $a$.
A similarity $s:X\times X \rightarrow \mathbb{R}$ ...
- 2,462
6
votes
0
answers
101
views
subalgebra of invariants for a reductive subgroup
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Spec{Spec}$Trying to understand some tannakian reconstruction, I've stumbled about the following problem in invariant theory. I guess it's something ...
- 867
3
votes
1
answer
152
views
Are there any results on an upper bound for the number of secondary invariants needed to generate the invariant ring of a finite group?
If $ G $ is a finite cyclic group, $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ is a linear $ n $ dimensional representation of $ G $, and $ \{x_{1},\dots,x_{n}\} $ is a basis of $ \mathbf{V}^{\ast} ...
- 782
2
votes
1
answer
262
views
How to express this hyperbolic extension of the cross-ratio in terms of hyperbolic distances and volumes?
Given $4$ distinct points on the Riemann sphere, thought of as the sphere at infinity of hyperbolic $3$-space $H^3$, one may define the cross-ratio in the usual way. Note that the cross-ratio is the ...
- 5,011
1
vote
0
answers
80
views
Are these maps, associated to finite simple graphs, interesting?
Given a finite simple graph on $n$ vertices, say $G = (V,\, E)$, where
$$ V = \{ v_1, \ldots , \, v_n \} $$
and
$$ E \subseteq \{ (v_a, \, v_b) \, | \, 1 \leq a < b \leq n \},$$
does there exist a ...
- 5,011
14
votes
0
answers
811
views
What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
3
votes
0
answers
134
views
A good stratification of a variety on which an algebraic group acts
Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0
(a reduced separated scheme of finite type over $k$).
Let $G$ be a connected linear algebraic group over $k$ (...
- 13.6k
3
votes
1
answer
456
views
Projective invariants of the plane and cross ratio
I am looking for a reference for the following admittedly imprecise statement:
Any projective invariant of n points in the projective plane may be
expressed as a function of well-chosen cross-ratios.
...
- 18.5k
4
votes
2
answers
315
views
Ring of invariants for $n$-tuples of Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...
- 142
2
votes
0
answers
160
views
Calculating the polynomials which are invariant under the action of a simple finite group
Let $G$ be a simple, finite group. In general, $G$ is not abelian.
Let $\rho$ be a representation of this group, where each $\rho(g)$ for $g\in G$ is a unitary, complex, $d$-dimensional matrix, $\rho(...
- 129
5
votes
1
answer
389
views
Invariant ideal generated by invariant elements
Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions?
...
- 1,373
10
votes
1
answer
657
views
Basis of invariant tensors of rank n in three dimensions
[This is a question motivated by theoretical physics, so apologies if the language is rough...]
In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...
- 1,038
2
votes
0
answers
73
views
Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind
Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension,
and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$.
Then $^\...
- 13.6k
4
votes
1
answer
487
views
Do all orbits have the same dimension?
Well, I've already asked this question at math.SE — but no-one's answered or commented. So now I'm posting it here (it's about the research paper — I think that it isn't an off-topic for this forum) — ...
- 41
8
votes
1
answer
354
views
Invariant ring of $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ under $\textrm{SO}(4)$
Consider the representation of $\textrm{SO}(4)$ on $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ induced by the standard representation of $\textrm{SO}(4)$ on $\mathbb{R}^4$. I am interested in the ring of ...
- 2,883
0
votes
0
answers
77
views
On four non-cocyclic integral points on ellipse
Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...
- 13.7k
8
votes
1
answer
235
views
Decomposition of $\bigotimes^{m} \mathbb{C}^{n}$ under the action of $\operatorname{GL}_{n}\times \operatorname{S}_{m}$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\S{S}$I want to know the proof of the following theorem. It is stated somewhere that, a proof can be found in: "Roger Howe, Perspectives on ...
- 109
10
votes
0
answers
281
views
Quotients by algebraic group actions at the level of the Grothendieck ring
$\DeclareMathOperator\SGro{SGro}\DeclareMathOperator\Gro{Gro}\DeclareMathOperator\GL{GL}$For an algebraically closed field $K$, the Grothendieck semiring of $K$ consists of, say, quasi-projective $K$-...
- 60.2k
5
votes
1
answer
195
views
Coordinate-free description of an alternating trilinear form on pure octonions
Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$.
The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$,
and ...
- 13.6k
1
vote
0
answers
83
views
invariant theory for $G\times \text{O}(n)$ [closed]
We know that the invariants of the orthogonal group $\text {O}(n)$ gives us the Brauer algebra. Is there any known results for the invariants of $G\times \text{O}(n)$ acting on the tensor space where $...
- 109
11
votes
2
answers
588
views
To describe an invariant trivector in dimension 8 geometrically
$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R_p=S^p R$;...
- 13.6k
3
votes
0
answers
59
views
Kernel of the map $\mathbb{C}[G]^U \to \mathbb{C}[U^+]$
$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $G=\SL_k$ be the special linear group, $U$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $U^+$ the unipotent subgroup ...
- 6,121
8
votes
1
answer
247
views
$\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$
Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves ...
- 83
4
votes
1
answer
263
views
Nilpotent orbits in representations of exceptional groups
The first nontrivial irreducible representation of $G_2$ is of 7-dimensional, and the first nontrivial representation of $F_4$ is of 26-dimensional.
My question is: how much is known about the ...
- 1,447
2
votes
0
answers
112
views
invariant theory for non-polynomial functions (eg Hilbert spaces)
I am looking for references regarding the study of group invariant functions that are not polynomials. In particular, I have a (nice) group $G$ and I am interested in what can be said about the $G$-...
- 315
2
votes
0
answers
101
views
In char zero $ \operatorname{Cox}(\operatorname{Bl}_{[1:1:1]}(\mathbb{P}(a:b:c))) $ is finitely generated, but not in char p. How?
Let $ X(a_{1}:a_{2}:a_{3}) $ be the blow-up of $ \mathbb{P}(a_{1}:a_{2}:a_{3}) $ at $ [1:1:1] $, the identity of the torus. In Steven Dale Cutkosky's paper Symbolic Algebras of Monomial Primes ...
- 782
4
votes
0
answers
112
views
Cover by $K$-invariant affine open sets
Let $X$ be a non-singular complex algebraic variety (quasi-projective if necessary) and $K$ a connected compact Lie group acting on $X$ real algebraically, i.e. the action map $K \times X \to X$ is ...
- 1,373