# Tagged Questions

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.

121 views

### Smoothness of a (given) global section of a vector bundle over G(2,6)

Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong$ det$(S^*)$, $S$ being the ...
245 views

### Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$. Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...
68 views

### Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group. Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
53 views

### Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
489 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 ...
198 views

### How to compute the tangent space of a quotient by a finite group

Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...
186 views

### Checking smoothness of the components of a highly symmetric scheme via quotient?

Setting Let $I\subseteq\mathbb C[x_0,\ldots,x_n]=:S$ be a homogeneous ideal and $X\subseteq\mathbb P^n$ the scheme defined by $I$. Consider the action of the symmetric group $\mathfrak S_{n+1}$ on ...
166 views

### $\text{PGL}_2(\mathbb{Q})$-equivalence versus $\text{PGL}_2(\mathbb{Z})$-equivalence

Let $V_{\mathbb{R}}$ be the space of binary quartic forms with real coefficients (so in particular $V_{\mathbb{R}}$ is a 5-dimensional vector space over $\mathbb{R}$), and define the twisted action of ...
165 views

### How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE. Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...
Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$. Is it always possible to construct $k$ functions $f_1, ... 0answers 59 views ### Diagonal invariants of$SO(n)$Consider a Lie algebra$\mathfrak g$(I am mostly interested in the case$\mathfrak g=so(n)$), its universal enveloping algebra$U$and its center$C$. There is an adjoint action of$\mathfrak g$on ... 1answer 259 views ### Is the Veronese variety “enough” to describe all the$SL(V)$-orbits in$\mathbb{P}(\textrm{Sym}^dV)$? I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts. Let$V$be a complex vector space of dimension$n$, ... 0answers 165 views ### The density of quartic polynomials whose Galois group is a subgroup of$D_4$Let$D_4$denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form$f$with integer coefficients, define$\text{Gal}(f)$to be the Galois group of the Galois ... 2answers 646 views ### How to make the Capelli's identity less mysterious? The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ... 2answers 398 views ### Cyclically symmetric functions Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)? E.g., do the functions$x+y+z$,$xy+yz+zx$, and$x^2y+y^2z+z^2x$generate the ... 1answer 415 views ### Most discriminants are almost squarefree Write, for$f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$,$H(f) := \max(|a_i|^{\frac{1}{i}})$. Does anyone know of a reference that would allow me to show that the proportion of$f$with ... 1answer 152 views ### Existence of$SO(n)$-isotropic inner products which are not$O(n)$-isotropic$\newcommand{\al}{\alpha}$Let$M_n$be the space of$n \times n$real matrices. Question: For which$n$, is there an inner product on$M_n$which satisfies: $$(*) \, \, \langle Q^TXQ,Q^TYQ ... 0answers 65 views ### Hausdorff limits of fibers of affine maps Let \mathbb{K}=\mathbb{R} or \mathbb{C}, and let$$ F=(P_1,\ldots, P_m):\mathbb{K}^n\to \mathbb{K}^m $$be a polynomial map. I would like to know under what conditions the preimages F^{-1}(y) of ... 1answer 246 views ### Invariant ring of S_5 The irreducible representations of the Symmetric group S_5 are classified by the partitions of 5. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ... 2answers 245 views ### Fundamental invariants for root subsystems Let \Phi be an irreducible root system of rank \ell. The fundamental invariants of \Phi is a set of \ell integers d_1, \cdots, d_\ell canonically attached to \Phi. Now suppose \Psi is ... 5answers 1k views ### 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 ... 1answer 130 views ### Tensor bundles as G structures [closed] For a smooth, real surface \Sigma, its bundle of symmetric, bi-linear forms S^2T\Sigma reduced to a PGL(2,\mathbb{R}) structure. A similar reduction(with different structure group) can be done ... 1answer 249 views ### Ring of invariants for the regular representation The symmetric group S_n acts on \mathbb C^n by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ... 1answer 144 views ### Common zero of invariants of finite groups Let G be a finite group n = |G|. Let \sigma : G \rightarrow GL(n,\mathbb{C}) be the regular representation. Hence every element of G can be seen as a permutation matrix. Let ... 0answers 75 views ### Irreducible real curves on {\mathbb C}P^1 invariant under the finite group action Let G be a finite subgroup of a Möbius group with a standart action on a real algebraic variety {\mathbb C}{\mathbf P^1}. How one can describe G-invariant irreducible real algebraic curves? ... 2answers 189 views ### Can a general binary sextic form be put into the following form? Let F(x,y) = a_6 x^6 + a_5 x^5 y + \cdots + a_0 y^6 be a binary sextic form with real coefficients and non-zero discriminant. Can one always find an element U = \begin{pmatrix} u_1 & u_2 \\ u_3 ... 3answers 369 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 of all n\times n-matrices, ... 1answer 185 views ### Description of the algebra of G-invariant polynomials by generators and relations Fix n > 1 and let \zeta \in \mathbb{C} be a primitive n-th root of unity. Let G \subset \text{SL}_2(\mathbb{C}) be a cyclic subgroup of order n generated by the diagonal matrix g = ... 1answer 118 views ### Generate harmonic polynomials for a finite group Let G be a finite group acting on a complex vector space V. Let \mathcal{D} denote the differential operators with constant coefficients and \mathcal{D}^{G} be the G-invariant operators. A ... 1answer 71 views ### Generalization of a Result about degree bounds of invariant rings A theorem of Knop states that if G is semisimple and connected acting on a vector space V over a field K of characteristic 0, then the degree of the Hilbert series of K[V]^G is less than or ... 1answer 188 views ### Invariant Laurent polynomials under cyclic group action Start with the cyclic group G:=\mathbb{Z}/p of prime order p and and an integer lattice P:=\mathbb{Z}^p. Let G act on P by cyclic permutation of coordinates. There is an induced action on ... 1answer 129 views ### Equivariant polynomial maps Let V be a complex vector spaces and assume that a compact group G acts linearly on V. Then look at the G-equivariant polynomial maps from V to V. Denote this by Mor_G(V,V). In the case ... 2answers 118 views ### Invariant polynomials under the action of H\le\operatorname{GL}_n(\mathbb{F}_p) Let n be a positive integer, and p a prime. Any subgroup H\le \operatorname{GL}_n(\mathbb{F}_p) acts on the polynomial ring \mathbb{F}_p[x_1,\ldots,x_n] via A\cdot x_i=\sum_j a_{ji}x_j for ... 0answers 53 views ### Involutions of binary sextic forms Let F(x,y) = a_6 x^6 + a_5 x^5 y + \cdots + a_1 xy^5 + a_0 y^6 be a binary sextic form with complex coefficients. Let V_\mathbb{C} be the space over \mathbb{C} of binary sextic forms. It is ... 0answers 66 views ### Is there a natural covariant of sextic polynomials with the following coefficients? Let$$\displaystyle f(x) = a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$$be an irreducible sextic polynomial with integer coefficients. Write$\theta_1, \cdots, \theta_6$for the ... 0answers 168 views ### What is the status of this fifty-year-old conjecture of Kostant? On page 3.27 of his 1963 thesis on the cohomology of homogeneous spaces as approached through the Eilenberg–Moore spectral sequence, Paul Baum states the following conjecture, which he attributes to ... 1answer 106 views ### Orbits in the adjoint representation of$SU(2,1)$How can one describe the orbits of the Lie group$G=\mathrm{SU}(2,1)$in its Lie algebra$\mathfrak{g}=\mathfrak{su}(2,1)$with respect to the adjoint representation? 0answers 93 views ### What are the E7(7) invariants in the adjoint representation? Take a real vector space$R$transforming in the adjoint representation of the${\rm E}_7(7)$Lie group as$R \rightarrow G R G^{-1}$. One can define invariants using traces of products of$R$as ... 0answers 136 views ### Does the functor of taking invariants commute with tensor products? [closed] Suppose that$G$is a group acting on a commutative ring$R$, inducing an action on each$R$-module. For any$R$-module$M$, let$M^G$denote the collection of elements of$M$invariant under the ... 1answer 138 views ### Invariant polynomials with respect to group actions on matrices Let$\mathfrak{gl}_n(\mathbb{R})$be the Lie algebra of matrices with real entries and$GL_n(\mathbb{R})$its associated Lie group. Recall that a linear subgroup$G \subseteq GL_n(\mathbb{R})$acts by ... 2answers 1k 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 ... 1answer 155 views ### Algebraically independent matrix invariants Let$V$be the space of pairs of$n \times n$matrices over$\mathbb{C}$and let$G$be the space of$n \times n$permutation matrices which acts on$(A,B) \in V$by simultaneous conjugation. It is ... 2answers 641 views ### Is there a topological Chevalley-Shephard-Todd Theorem? Is the following true: For a representation of a finite group$G$on$\mathbb{C}^n$, the quotient$\mathbb{C}^n/G$is a topological manifold if and only if$G$is generated by pseudo-reflections. ( ... 2answers 236 views ### Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices Let$G=GL(n,\mathbb{C})$and let$U\subset G$be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let ... 0answers 31 views ### Terminology for research on distributions of inner products Consider a set of vectors$M$from an inner product space$V$. The ordered set of inner products of all pairs of elements in$M$uniquely characterizes$M$up to isomorphism. Suppose now that$V$is ... 1answer 136 views ### Classification of 3-forms in dimension 7 I'm looking for a classification of$3$-forms over a real vector space of dimension$7$as for the$3$-forms in dimension$6$. References on the latter case are R. Bryant On the geometry of almost ... 1answer 124 views ### A vector version of the Segre embedding: what is the kernel of the ring map? TL;DR version. Given a commutative ring$\mathbf{k}$and$n+m$"generic" vectors$\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$in$\mathbf{k}^k$... 2answers 327 views ### Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables Pick$p,q,r$complex numbers (I am most interested in the case when they are positive integers). Define the function$P_i = px^i + qy^i + rz^i$where$x,y,z$are coordinates. I have a few related ... 1answer 248 views ### Does the ring of invariants inherit normality? Let$A$be a normal ring (in the sense that its localizations at prime ideals are normal domains), and suppose that a finite group$G$acts on$A$by ring automorphisms. Form the subring$A^G \subset ...
Given permutation module with three generators and corresponding Galois action of symmetric group $\mathfrak S_3$ I am interested in computing corresponding dual torus $T$ (which should be of ...