2,881 research outputs found
Schur dynamics of the Schur processes
We construct discrete time Markov chains that preserve the class of Schur
processes on partitions and signatures.
One application is a simple exact sampling algorithm for
q^{volume}-distributed skew plane partitions with an arbitrary back wall.
Another application is a construction of Markov chains on infinite
Gelfand-Tsetlin schemes that represent deterministic flows on the space of
extreme characters of the infinite-dimensional unitary group.Comment: 22 page
Representations of classical Lie groups and quantized free convolution
We study the decompositions into irreducible components of tensor products
and restrictions of irreducible representations of classical Lie groups as the
rank of the group goes to infinity. We prove the Law of Large Numbers for the
random counting measures describing the decomposition. This leads to two
operations on measures which are deformations of the notions of the free
convolution and the free projection. We further prove that if one replaces
counting measures with others coming from the work of Perelomov and Popov on
the higher order Casimir operators for classical groups, then the operations on
the measures turn into the free convolution and projection themselves.
We also explain the relation between our results and limit shape theorems for
uniformly random lozenge tilings with and without axial symmetry.Comment: 43 pages, 4 figures. v3: relation to the Markov-Krein correspondence
is updated and correcte
Commutator maps, measure preservation, and T-systems
Let G be a finite simple group. We show that the commutator map is almost equidistributed as the order of G goes to infinity. This
somewhat surprising result has many applications. It shows that for a subset X
of G we have , namely is almost measure
preserving. From this we deduce that almost all elements can be
expressed as commutators where x,y generate G. This enables us to
solve some open problems regarding T-systems and the Product Replacement
Algorithm (PRA) graph. We show that the number of T-systems in G with two
generators tends to infinity as the order of G goes to infinity. This settles a
conjecture of Guralnick and Pak. A similar result follows for the number of
connected components of the PRA graph of G with two generators. Some of our
results apply for more general finite groups, and more general word maps. Our
methods are based on representation theory, combining classical character
theory with recent results on character degrees and values in finite simple
groups. In particular the so called Witten zeta function plays a key role in
the proofs.Comment: 28 pages. This article was submitted to the Transactions of the
American Mathematical Society on 21 February 2007 and accepted on 24 June
200
Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials
There are several common ways to encode a tree as a matrix, such as the
adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of
the natural random walk), and the matrix of pairwise distances between leaves.
Such representations involve a specific labeling of the vertices or at least
the leaves, and so it is natural to attempt to identify trees by some feature
of the associated matrices that is invariant under relabeling. An obvious
candidate is the spectrum of eigenvalues (or, equivalently, the characteristic
polynomial). We show for any of these choices of matrix that the fraction of
binary trees with a unique spectrum goes to zero as the number of leaves goes
to infinity. We investigate the rate of convergence of the above fraction to
zero using numerical methods. For the adjacency and Laplacian matrices, we show
that that the {\em a priori} more informative immanantal polynomials have no
greater power to distinguish between trees
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