6,993 research outputs found
Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in
\cite{Chi08}, we use quiver theory to represent the generalized
Littlewood-Richardson coefficients for the branching rule for the diagonal
embedding of \gl(n) as the dimension of a weight space of semi-invariants.
Using this, we prove their saturation and investigate when they are nonzero. We
also show that for certain partitions the associated stretched polynomials
satisfy the same conjectures as single Littlewood-Richardson coefficients. We
then provide a polytopal description of this multiplicity and show that its
positivity may be computed in strongly polynomial time. Finally, we remark that
similar results hold for certain other generalized Littlewood-Richardson
coefficients.Comment: 28 pages, comments welcom
On the complete perturbative solution of one-matrix models
We summarize the recent results about complete solvability of Hermitian and
rectangular complex matrix models. Partition functions have very simple
character expansions with coefficients made from dimensions of representation
of the linear group , and arbitrary correlators in the Gaussian phase
are given by finite sums over Young diagrams of a given size, which involve
also the well known characters of symmetric group. The previously known
integrability and Virasoro constraints are simple corollaries, but no vice
versa: complete solvability is a peculiar property of the matrix model
(hypergeometric) -functions, which is actually a combination of these two
complementary requirements.Comment: 8 page
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
Fermionic Matrix Models
We review a class of matrix models whose degrees of freedom are matrices with
anticommuting elements. We discuss the properties of the adjoint fermion one-,
two- and gauge invariant D-dimensional matrix models at large-N and compare
them with their bosonic counterparts which are the more familiar Hermitian
matrix models. We derive and solve the complete sets of loop equations for the
correlators of these models and use these equations to examine critical
behaviour. The topological large-N expansions are also constructed and their
relation to other aspects of modern string theory such as integrable
hierarchies is discussed. We use these connections to discuss the applications
of these matrix models to string theory and induced gauge theories. We argue
that as such the fermionic matrix models may provide a novel generalization of
the discretized random surface representation of quantum gravity in which the
genus sum alternates and the sums over genera for correlators have better
convergence properties than their Hermitian counterparts. We discuss the use of
adjoint fermions instead of adjoint scalars to study induced gauge theories. We
also discuss two classes of dimensionally reduced models, a fermionic vector
model and a supersymmetric matrix model, and discuss their applications to the
branched polymer phase of string theories in target space dimensions D>1 and
also to the meander problem.Comment: 139 pages Latex (99 pages in landscape, two-column option); Section
on Supersymmetric Matrix Models expanded, additional references include
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