# 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.

287 questions
Filter by
Sorted by
Tagged with
36 views

### Characterization of all-orthogonal tensors

In the paper , 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 $$\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n$$ ...
40 views

267 views

### higher Casimirs for $\mathfrak{sl}$

The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
197 views

### On a paper of Formanek about $PGL_4$

In his paper "The center of the ring of 4 × 4 generic matrices" (Journal of Algebra, 1980) Formanek shows that, if $V$ is the representation of $PGL_4$ given by simultaneous conjugation of pairs of ...
Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity: $$J_{ij,k\ell} := \operatorname{... 1answer 121 views ### Rotation invariant of surface Let (x, y, f(x,y)) be a surface in \mathbb{R^3}. It is written in a book without proof that all rotation invariant (rotating around z-axis) of f are combinations of the following four ... 0answers 26 views ### Does the G-norm coincide with the ordinary norm for “quasi-G-Galois” extensions Let S be a commutative ring, let G be a finite group acting on S via automorphisms (not necessarily faithfully), and let R be a subring of S consisting of elements fixed G. The extension ... 1answer 123 views ### Kernel of restriction for ring of functions on reductive groups Let H \subset G be an inclusion of reductive groups over an algebraically closed field k of char 0. For simplicity, let's assume that G is split and H contains a maximal torus for G. Then ... 0answers 268 views ### A question related to Conways 99 graph problem I have observed that the number of triangles \frac{vk}{6} of a strongly regular graph with parameters (v,k,1,2) is given by the coefficient 2(k-1) in the molien series of the "4-D extraspecial ... 0answers 123 views ### Length of fibers of (\mathbb{A}^n)^d\to\mathrm{Sym}^d(\mathbb{A}^n) Let k be a field, consider the canonical morphism f\colon (\mathbb{A}_k^n)^d\to\mathrm{Sym}^d\mathbb{A}_k^n. Is there an explicit bound on the length of fibers of f in terms of n,d,\mathrm{... 1answer 112 views ### SU(2)\times SU(2) invariant SU(3)-structure on \{t\} \times M^6 I am reading Jason Lotay and Goncalo Oliveira's paper -SU(2)^2 invariant G_2-instantons, and have few questions from the same. If we consider the space M = S^3 \times S^3. Then the cone metric ... 1answer 112 views ### Software for computing equivariants If \Gamma is a finite group with action on two vector spaces \mathbb R^n and \mathbb R^m denoted by \gamma_n and \gamma_m respectively, the fundamental equivariants are the polynomials f: \... 0answers 135 views ### Semisimple Lie groups admitting a free algebra of invariants Assume we work over an algebraically closed field of characteristic zero. I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ... 1answer 85 views ### Sufficient conditions for secondary invariants Let G be a finite group, k be a field whose characteristic divides |G|, and \rho:G\hookrightarrow\operatorname{GL}_n(k) be a faithful representation of G. Let V be a k-space of dimension ... 3answers 401 views ### Explicit formulas for invariants of binary quintic forms I am looking for explicit formulas for the four basic invariants I_4, I_8, I_{12}, I_{18} of a generic binary quintic form, either given in the shape$$\displaystyle F(x,y) = ax^5 + 5bx^4y + 10cx^...
Let $X$ be a complex algebraic variety and $G$ a complex reductive group acting on $X$. Let $L$ be a linearization of this action, i.e. a line bundle with a linear action of $G$ covering that of $X$. ...