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Questions tagged [invariant-theory]

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. (Def: http://en.m.wikipedia.org/wiki/Invariant_theory)

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Reference: Wikipedia.

Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group.

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Is an SL-invariant rational function necessarily a quotient of two SL-invariant polynomials?

Let $\mathbf x = (x_{i,j})_{1\leq i \leq n, 1\leq j \leq N}$ denote a collection of indeterminates. The algebraic group $\mathrm{SL}_n(\mathbb C)$ acts on $\mathbb C[\mathbf x]$ by "matrix multiplication", and invariant theory guarantees that the…
Linus S
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The anti-commutative Molien series

Suppose $V$ is a finite dimensional complex vector space and $f:V\to V$ is an automorphism. There is a natural extension $\Lambda^\bullet(f):\Lambda^\bullet(V)\to\Lambda^\bullet(V)$ to the exterior algebra $\Lambda^\bullet(V)$. If $f$ is of finite…
Mariano Suárez-Álvarez
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Invariants of $K\left[\bigoplus_{k=1}^n V^{\bigodot k} \right]$

What are the generators of invariant ring $K\left[\bigoplus_{k=1}^n V^{\bigodot k} \right]^G$, where $G$ is subgroup of $GL(V)$ with natural representation on $\bigoplus_{k=1}^n V^{\bigodot k} $i.e. for $g\in G$ is $g\cdot (v_1 \oplus (v_2 \odot…
tom
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How does Molien series describe polynomial invariants?

As I understood from wiki page, Given a finite group acting on a vector space, Molien series gives a generating function, although I am not sure what this means. And how is this related to the polyomial invariants of the group? Could anyone help…
user166467
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Condition for a polynomial invariant ring to be a UFD

Let $W$ be a finite-dimensional vector space over $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $. Let $\rho $ $:G\rightarrow $ GL$(W)$ be a representation of any finite group $G$. How can we show that if all characters of $G$…
Kabalan Gaspard
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Does anyone know of Invariant Theory enough to comment on this question?

I am trying to find a minimal set of invariants for the binary homogenous form $$\displaystyle ax^7 + bx^{6}y + cx^{5}y^{2} + dx^{4}y^{3} + ex^{3}y^{4} + fx^{2}y^{5} + gxy^{6} + hy^{7}$$ What is the basis for all of the invariants for this form? Is…
Low Scores
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What is a concrete example of a perpetuant (in classical invariant theory)?

I am trying to determine whether an object of my recent research is actually a "perpetuant" in the sense of Sylvester and classical invariant theory. There are a few papers on the topic, including Kraft and Procesi's very recent one, but the…
WQE
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What is the significance of the quotient of two Hilbert series?

Just a disclaimer, I am not a mathematician so I am sorry for being less than formal. Let us say we have a Hilbert series $\mathcal{H}(K[V]^G)$. Now I know that by eliminating some parameters in a given theory I get the freedom of a subgroup $H <…
M.Π.B
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$\mathbb{C}(V)$ is a finite module over $\mathbb{C}(V)^G$?

Is it true that $\mathbb{C}(V)$ is a finite module over $\mathbb{C}(V)^G$ for any finite subgroup $G \subset GL(V)$ and, moreover, $\dim_{\mathbb{C}(V)^G} \mathbb{C}(V) = |G|$? It possibly follows from the well-known theorems of Hilbert and Noether…
Timur Bakiev
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Ring of invariant of finite groups and its subgroups

I am reading the book "Algorithms in invariant theory" of Bernd Sturmfels. As Prop 1.1.3/p5, we can see that for every $f\in C{[x]^{{A_n}}}$, $f$ can be written uniquesly in the form $$f = g + h.D$$ where $g,h \in [C{[x]^{{S_n}}}$ and $D =…
Nguyen Dang Son
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When is $S^G \subset \text{Frac}(R)$.

Let $S/R$ be an extension of rings where $R$ is a domain, and let $G$ be a finite group acting on $S$, fixing $R$. When do we have $S^G \subset \text{Frac}(R)$? For instance, if $S$ is a domain, $G$ acts on $\text{Frac}(S)$, and $\text{Frac}(S) /…
Ronald J. Zallman
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Does $k[B]^{B'}=k[\det(b),\det(b)^{-1}]$?

Let $k=\overline{k}$. Suppose $GL_2(k)$ acts on $GL_2(k)$ by left multiplication. Then $k[GL_2]^{SL_2} = k[\det(g),\det(g)^{-1}]$. Now for $B=\left\{ \left( \begin{array}{cc} b_{11} & b_{12} \\ 0 & b_{22} \\ \end{array}\right) : b_{ii}\not=0,…
user57127
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Invariant-theory

An evil wizard has imprisoned 64 math geeks. The wizard announces, "Tomorrow I will have you stand in a line, and put a hat on each of your heads. The hat will be colored either white or black. You will be able to see the hats of everyone in front…
Guwanc Beserow
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$GL_2(\mathbb{C})$-invariant ring for $M_2(\mathbb{C})\times M_2(\mathbb{C})$

For $\mathbb{C}^*$-action on $\mathbb{C}^2$ by $t\circ(x,y)=(t^{-1}x,ty)$, the ring of invariant polynomials is $\mathbb{C}[xy]$. For $\mathbb{C}^*$-action on $\mathbb{C}^4$ by $t\circ(x,y,z,w)=(t^{-1}x,t^{-1}y,tz,tw)$, then the ring of invariants…
math-visitor
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Invariant Problem on colored chips

In a course we were once given the following question There is a finite stack of chips on a table, each chip having one of three different colors $a,b$ and $c$ . At any time, you may choose two chips of different color and replace them by a chip of…
Léreau
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