Linked Questions
11 questions linked to/from What are the current breakthroughs of Geometric Complexity Theory?
39
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Who invented diagrammatic algebra?
There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations ...
- 22.1k
49
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4
answers
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How to constructively/combinatorially prove Schur-Weyl duality?
How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring
$\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
- 33.8k
10
votes
4
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Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,541
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3
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Product of the entries of a matrix
Given a $n \times n$ matrix $A = (a_{ij})$, I was wondering if there was any theory or research interest relevant to the term
$$ \prod_{i,j} a_{ij}$$
the product of all the entries of the matrix.
- 401
11
votes
3
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Nonnegativity of an integral over the unitary group
For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
- 1,593
20
votes
1
answer
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Symmetric polynomial from graphs
Let $g$ be a directed, connected multigraph, on $n$ vertices, without loops.
Define
$$P_g(x_1,\dots,x_n) := Sym\left[ \prod_{(i,j) \in g} (x_i-x_j) \right]$$
where $(i,j)$ is the directed edge ...
- 15.8k
19
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2
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Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 148k
18
votes
1
answer
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Why should algebraic geometers and representation theorists care about geometric complexity theory?
Geometric complexity theory has demonstrated that complexity theorists should care about algebraic geometry and representation theory, but, why should algebraic geometers and representation theorists ...
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8
votes
1
answer
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Steps in Geometric Complexity Theory
GCT purports to provide a program to show that $NP \not \subset P/poly$.
At the high level what are the steps involved in the program and what stage is each step in?
What difficulties currently are ...
- 13.9k
9
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Generalization of Pascal's theorem to higher dimensions
Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
- 4,484
9
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1
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Current state of the art in geometric complexity theory
I came across this interesting question from almost 7 years ago:
What are the current breakthroughs of Geometric Complexity Theory?
My question is quite simple: Have there been any breakthroughs in ...
