7,443 research outputs found
Computing Multiplicities of Lie Group Representations
For fixed compact connected Lie groups H \subseteq G, we provide a polynomial
time algorithm to compute the multiplicity of a given irreducible
representation of H in the restriction of an irreducible representation of G.
Our algorithm is based on a finite difference formula which makes the
multiplicities amenable to Barvinok's algorithm for counting integral points in
polytopes.
The Kronecker coefficients of the symmetric group, which can be seen to be a
special case of such multiplicities, play an important role in the geometric
complexity theory approach to the P vs. NP problem. Whereas their computation
is known to be #P-hard for Young diagrams with an arbitrary number of rows, our
algorithm computes them in polynomial time if the number of rows is bounded. We
complement our work by showing that information on the asymptotic growth rates
of multiplicities in the coordinate rings of orbit closures does not directly
lead to new complexity-theoretic obstructions beyond what can be obtained from
the moment polytopes of the orbit closures. Non-asymptotic information on the
multiplicities, such as provided by our algorithm, may therefore be essential
in order to find obstructions in geometric complexity theory.Comment: 10 page
Eigenvalue Distributions of Reduced Density Matrices
Given a random quantum state of multiple distinguishable or indistinguishable
particles, we provide an effective method, rooted in symplectic geometry, to
compute the joint probability distribution of the eigenvalues of its one-body
reduced density matrices. As a corollary, by taking the distribution's support,
which is a convex moment polytope, we recover a complete solution to the
one-body quantum marginal problem. We obtain the probability distribution by
reducing to the corresponding distribution of diagonal entries (i.e., to the
quantitative version of a classical marginal problem), which is then determined
algorithmically. This reduction applies more generally to symplectic geometry,
relating invariant measures for the coadjoint action of a compact Lie group to
their projections onto a Cartan subalgebra, and can also be quantized to
provide an efficient algorithm for computing bounded height Kronecker and
plethysm coefficients.Comment: 51 pages, 7 figure
Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2
The characters of irreducible finite dimensional representations of compact
simple Lie group G are invariant with respect to the action of the Weyl group
W(G) of G. The defining property of the new character-like functions ("hybrid
characters") is the fact that W(G) acts differently on the character term
corresponding to the long roots than on those corresponding to the short roots.
Therefore the hybrid characters are defined for the simple Lie groups with two
different lengths of their roots. Dominant weight multiplicities for the hybrid
characters are determined. The formulas for "hybrid dimensions" are also found
for all cases as the zero degree term in power expansion of the "hybrid
characters".Comment: 15 page
Generating functions and multiplicity formulas: the case of rank two simple Lie algebras
A procedure is described that makes use of the generating function of
characters to obtain a new generating function giving the multiplicities of
each weight in all the representations of a simple Lie algebra. The way to
extract from explicit multiplicity formulas for particular weights is
explained and the results corresponding to rank two simple Lie algebras shown
Dimensions of Imaginary Root Spaces of Hyperbolic Kac--Moody Algebras
We discuss the known results and methods for determining root multiplicities
for hyperbolic Kac--Moody algebras
On the generating function of weight multiplicities for the representations of the Lie algebra
We use the generating function of the characters of to obtain a
generating function for the multiplicities of the weights entering in the
irreducible representations of that simple Lie algebra. From this generating
function we derive some recurrence relations among the multiplicities and a
simple graphical recipe to compute them.Comment: arXiv admin note: text overlap with arXiv:1304.720
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