The Capacity of Quiver Representations and Brascamp–Lieb Constants

@article{Chindris2019TheCO,
  title={The Capacity of Quiver Representations and Brascamp–Lieb Constants},
  author={Calin Chindris and Harm Derksen},
  journal={arXiv: Representation Theory},
  year={2019}
}
Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp-Lieb operator $T_{V,\sigma}$ associated to $(V,\sigma)$ and study its capacity, denoted by $\mathbf{D}_Q(V, \sigma)$. When $Q$ is the $m$-subspace quiver, the capacity of quiver data is intimately related to the Brascamp-Lieb constants that occur in the $m$-multilinear… 
4 Citations
Edmonds' problem and the membership problem for orbit semigroups of quiver representations
TLDR
A quiver invariant theoretic approach to Edmonds' problem, which asks to decide if the span of a given $l$-tuple of complex matrices contains a non-singular matrix, and shows that if $A/\Ann_A(\V)$ is a tame algebra then any weight in the weight semigroup of $\V$ is $V$-saturated.
A quiver invariant theoretic approach to Radial Isotropy and Paulsen's Problem for matrix frames
TLDR
This paper first proves a far reaching generalization of Barthe’s Radial Isotropy Theorem to matrix frames and provides a quiver invariant theoretic approach to the Paulsen problem for matrix frames.
R T ] 5 M ar 2 02 0 SIMULTANEOUS ROBUST SUBSPACE RECOVERY AND SEMI-STABILITY OF QUIVER REPRESENTATIONS
ABSTRACT. We consider the problem of simultaneously finding lower-dimensional subspace structures in a given m-tuple (X , . . . ,Xm) of possibly corrupted, high-dimensional data sets all of the same

References

SHOWING 1-10 OF 31 REFERENCES
Edmonds' problem and the membership problem for orbit semigroups of quiver representations
TLDR
A quiver invariant theoretic approach to Edmonds' problem, which asks to decide if the span of a given $l$-tuple of complex matrices contains a non-singular matrix, and shows that if $A/\Ann_A(\V)$ is a tame algebra then any weight in the weight semigroup of $\V$ is $V$-saturated.
Sylvester–Gallai for Arrangements of Subspaces
TLDR
If every pair V_a, V_b of subspaces is contained in a dependent triple, then the entire arrangement must becontained in a subspace whose dimension depends only on k (and not on n), which generalizes the Sylvester-Gallai theorem, which proves the k=1 case.
Rational points of quiver moduli spaces
For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\overline{k}$-valued points of moduli spaces of quiver representations over $k$; the fixed locus is the set of
Operator Scaling: Theory and Applications
TLDR
A complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004 ), who proved it was polynomial time for certain classes of inputs, that is extended to actually approximate capacity to any accuracy in polynometric time.
Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via operator scaling
TLDR
The application of operator scaling algorithm to BL-inequalities further connects analysis and optimization with the diverse mathematical areas used so far to motivate and solve the operator scaling problem, which include commutative invariant theory, non-commutative algebra, computational complexity and quantum information theory.
Classical complexity and quantum entanglement
The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals
Abstract.We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the
MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian
Operator scaling with specified marginals
TLDR
The completely positive maps, a generalization of the nonnegative matrices, are a well-studied class of maps from n× n matrices to m× m matrices and a central ingredient in this analysis is a reduction from operator scaling with specified marginals to operator scaling in the doubly stochastic setting.
...
...