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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

6 votes
1 answer
311 views

Prove that $\overline{a}_{11}$ is a prime element in $R$

Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB- …
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  • 839
5 votes
0 answers
350 views

What representation theoretic properties does the semi-invariant ring tell us?

I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)? I have been studying about semi-in …
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  • 839
4 votes
2 answers
861 views

Research topics in representation theory of algebras [closed]

I was wondering what are some of the hot topics in quiver representation or representation theory of algebras that can lead to good mathematics and is important to many mathematicians and top mathemat …
3 votes
2 answers
392 views

Cohen-Macaulay Representations

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 …
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  • 839
3 votes
0 answers
109 views

Example of an irreducible component with an open set of infinitely many codimension 2 (codim...

Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, wh …
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  • 839
2 votes
1 answer
245 views

Possible "algebraic" direction in hyperplane arrangements

I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the " …
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  • 839
2 votes
1 answer
178 views

Orbits in the open set given by Rosenlicht's result

Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we …
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  • 839
2 votes
1 answer
159 views

Rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting infini …
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  • 839
2 votes
0 answers
85 views

Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an …
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  • 839
2 votes
0 answers
262 views

Understanding a proof of a result of Schofield

I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's underlined in t …
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  • 839
1 vote
0 answers
126 views

Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$

I'm trying to compute some examples and I'm unable to come up with a following example: What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a su …
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  • 839
1 vote
0 answers
142 views

Non-empty stable locus of an irreducible component

I have a vague question: Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T. quotient $X{/\!/}G$. Is there any result (maybe in some special case) which tells u …
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  • 839
1 vote
0 answers
52 views

Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module a …
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  • 839
1 vote
0 answers
65 views

Structure of tame concealed algebra of Euclidean type

I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ki …
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  • 839
1 vote
1 answer
214 views

A result of Schofield in the case of quivers with relations

Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\bet …
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