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I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties. I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ne …

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below) I want to understand …

I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the follow …

I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below) It says that "every invariant rational function can be ex …

Consider the grassmannian $\mathrm{Gr}(2,5)$. We know there is an embedding of $\mathrm{Gr}(2,5)$ into $\mathbb{P}^9$ by using the 10 Plucker coordinates, and they satisfy 5 Plucker relations. And, so …

Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y) …

Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a …

Let $\mathbb{K}$ be an algebraically closed field of characteristics zero. Let $X$ be an irreducible affine variety, with a rational action of a linearly reductive algebraic group $G$. Also, assume th …

I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research. If yes, then what are some of the active …

I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^ …

Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U …

Let $G$ be a linearly reductive algebraic group acting regularly on an affine space over $\mathbb{A}^n$ an algebraically closed field $\mathbb{K}$. Let $Y$ be a $G$-invariant (closed) affine subvariet …

Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here) …

Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ s …

Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also …