# 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.

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### 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 $k$ (...
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### 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. ...
For a classical 3d conicoid with coefficients $(a,b,c,f,g,h)$ the first two invariants are $I_1= (a+b)/2, I_2= (a b -h^2)$ and the third one is $$I_3= \begin{pmatrix} a & h & g \\ h & b &... 2answers 234 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}... 0answers 91 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(... 1answer 161 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? ... 1answer 343 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, ... 0answers 37 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 ^\... 1answer 329 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) — ... 1answer 305 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 ... 0answers 66 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 ... 1answer 192 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 ... 0answers 180 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-... 1answer 142 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 ... 0answers 75 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 ... 2answers 566 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;... 0answers 49 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 ... 1answer 185 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 ... 1answer 243 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 ... 0answers 90 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-... 0answers 86 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 ... 0answers 81 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 ... 1answer 413 views ### GIT and singularities Let G be a complex reductive group acting on a complex affine variety X and let X // G = \operatorname{Spec}\mathbb{C}[X]^G be the GIT quotient. Is there a relationship between the singular ... 0answers 49 views ### Invariants of a kG-module via its comoposition series, when does M^P \supsetneq N^P hold for a p-group for N\subseteq M maximal? Let G be a finite group, k a field, M a kG-module, M^G the invariants of M under G, P a Sylow p-subgroup of G where p = \text{char}(k), N a maximal submodule of M and S the ... 0answers 118 views ### How to find the polynomials that define a compact Matrix Lie group from its Lie algebra? Consider a compact (connected) Lie group, or more generally, a linear algebraic Lie group. Suppose we are given the Lie algebra corresponding to the Lie group. How can we find a set of polynomial ... 2answers 291 views ### What are all invariant polynomials on the space of algebraic curvature tensors? Let V = (\mathbb{R}^n, g), where g is the Euclidean inner product on V. Denote by G the orthogonal group O(V) = O(n) and by \mathfrak{g} the Lie algebra of G. Let W \subset \Lambda^2V^* ... 0answers 172 views ### Invariant theory in universal algebra Let \mathcal{L} be a finite first-order language with no relation symbols, and \mathcal{K}:=\mathcal{V}(\Theta) a variety in this language definited by a set of identities \Theta. My questions ... 1answer 224 views ### An explicit negative solution to the Lüroth problem for non-algebraically closed fields Let \mathsf{k} be a field of characteristic 0, and consider \mathsf{k}(x,y). If \mathsf{k} is algebraically closed, then every field L such that the inclusion \mathsf{k} \subset L \subset \... 0answers 120 views ### An analogue of Noether's Problem for non-rational varieties For the sake of simplicity, let \mathsf{k} be algebraically closed and of zero characteristic. Varieties are irreducible. The (linear) Noether's Problem (which goes back to the early 1910's in ... 1answer 117 views ### Behavior of invariants under reduction mod p Let R be a finitely generated \mathbb{Z}-algebra with an [edit: linear algebraic] action of G(\mathbb{Z}) where G is a split simply-connected semisimple group. Then for any prime p we have a ... 0answers 82 views ### Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix Let \Sigma\subseteq\mathrm{Sym}(n) be a permutation group on N:=\{1,...,n\}. My goal is to determine the irreducible invariant subspaces of the permutation action of \Sigma on \Bbb R^n, and I ... 2answers 382 views ### Continuous version of the fundamental theorem of invariant theory for the orthogonal group A standard result in the invariant theory of the orthogonal group states the following. Theorem Let (E, \langle .,. \rangle) be an n-dimensional euclidean vector space, let f : E^m \rightarrow {\bf ... 0answers 88 views ### Invariant theory for the orthogonal group and Clifford algebras The first fundamental theorem of invariant theory for the orthogonal group O_n(k) asserts that the ring of invariants is generated by the scalar products: a polynomial function of m vectors v_1,..... 0answers 15 views ### Bivariate basis functions with span that is closed under rotations in coordinate system Consider a set of linearly independent functions \{(x\in\mathbb{R},y\in\mathbb{R})\mapsto f_i(x,y)\in\mathbb{R}\} with the property that for any given \theta\in\mathbb{R} and any given \{a_i\in\... 0answers 80 views ### Invariant subspace of a nonlinear map First please see this very simple fact: Fact: \  Any linear map T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 has a proper invariant linear subspace. By an invariant subspace we mean a space M ... 0answers 49 views ### propagation of a invariance along some PDE Consider the following non linear PDE on \mathbb{R}^n$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$with given initial condition u_0(x). Assume that: u_0 is rotation invariant, ... 0answers 69 views ### Characterization of all-orthogonal tensors In the paper [1], it is proven in Theorem 2 that any n-tensor \mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n} can be decomposed as$$ \mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n $$... 0answers 58 views ### Invariant theory of the indefinite orthogonal groups I believe the following statements are true: Let V be a finite-dimensional real vector space with a positive-definite inner product g. Let g_{\otimes n} denote the natural extension of g to ... 1answer 315 views ### Moduli of smooth curves in |\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)|  and their invariants It is well known that any smooth curve C\in |\mathcal{O}_{\mathbf{P}^1\times\mathbf{P}^1}(2,2)|  has geometric genus equal to 1, so its isomorphism class is determined by its j-invariant. ... 1answer 103 views ### Consequences of invariant-subspace problem to Li–Yorke chaos [closed] The invariant-subspace problem is probably an open problem for reflexive spaces which asks: Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ... 1answer 98 views ### Equivalence classes of a circle of n bits upon flipping 3 consecutive 0s to 1s or vice versa Consider a circle of n-bits and define the equivalence relation as follow: Two configurations A and B of the n-bits circle are equivalent if they can be transformed into each other by performing a ... 1answer 265 views ### Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable? I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book Greco, Silvio, ... 1answer 310 views ### Invariants of symmetric forms with respect to the symplectic group Take a 6-dimensional vector space V (for simplicity, over \mathbb{C}) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space S^2V^* of ... 1answer 194 views ### Highest weight vector as a global section of an affine scheme Let G be a connected, reductive quasi-split group over a field k, acting on an afffine k-variety X. Let B = TU be a Borel subgroup of G with maximal torus T and unipotent radical U. ... 2answers 283 views ### Rosenlicht's theorem and fundamental domain for unipotent group acting on \mathbb A_k^n I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very ... 1answer 321 views ### Lifting G-invariants from characteristic p\gg 0 to characteristic 0 for a reductive algebraic group G Let S\subset \mathbb{C} be a finitely generated ring, let R be a finitely generated commutative ring over S. Let G be a linear algebraic group over S, such that G_{\mathbb{C}} is reductive.... 1answer 279 views ### Invariants for SO_n \backslash \mathfrak{gl}_n / SO_n Is there a nice theorem about the algebra of invariants \mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}, where the action is by left and right multiplication? I'm hoping for something along the lines ... 0answers 116 views ### On Shephard-Todd theorem There is an excellent Torsten Ekedahl's answer to Roman Fedorov's question here: Chevalley–Shephard–Todd theorem. Does anyone know any articles or books where this approach is outlined? I didn't ... 0answers 67 views ### Rational torus invariants Let T=(\mathbb{C}^{\times})^n be the n-dimensional torus acting on the polynomial algebra \mathbb{C}[x_1,x_2, \ldots,x_n] diagonally, i.e.$$ diag(t^{a_1},t^{a_2},\ldots,t^{a_n})x_i=t^{a_i}x_i, ...
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$...