99 research outputs found
A bideterminant basis for a reductive monoid
We use the rational tableaux introduced by Stembridge to give a bideterminant
basis for a normal reductive monoid and for its variety of noninvertible
elements. We also obtain a bideterminant basis for the full coordinate ring of
the general linear group and for all its truncations with respect to saturated
sets. Finally, we deduce an alternative proof of the double centraliser theorem
for the rational Schur algebra and the walled Brauer algebra over an arbitrary
infinite base field which was first obtained by Dipper, Doty and Stoll
ANALYTIC AND TOPOLOGICAL COMBINATORICS OF PARTITION POSETS AND PERMUTATIONS
In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible
Improved one-way rates for BB84 and 6-state protocols
We study the advantages to be gained in quantum key distribution (QKD)
protocols by combining the techniques of local randomization, or noisy
preprocessing, and structured (nonrandom) block codes. Extending the results of
[Smith, Renes, and Smolin, quant-ph/0607018] pertaining to BB84, we improve the
best-known lower bound on the error rate for the 6-state protocol from 14.11%
for local randomization alone to at least 14.59%. Additionally, we also study
the effects of iterating the combined preprocessing scheme and find further
improvements to the BB84 protocol already at small block lengths.Comment: 17 pages, to appear in Quantum Information & Computation. Replaced by
accepted versio
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
Delta and Theta Operator Expansions
We give an elementary symmetric function expansion for and when
in terms of what we call -parking functions and lattice
-parking functions. Here, and are certain
eigenoperators of the modified Macdonald basis and . Our main
results in turn give an elementary basis expansion at for symmetric
functions of the form whenever is expanded
in terms of monomials, is expanded in terms of the elementary basis, and
is expanded in terms of the modified elementary basis . Even the most special cases of this general Delta
and Theta operator expression are significant; we highlight a few of these
special cases. We end by giving an -positivity conjecture for when is
not specialized, proposing that our objects can also give the elementary basis
expansion in the unspecialized symmetric function.Comment: 38 pages, 12 figure
Valence bond approach and Verma bases
The unitary group approach (UGA) to the many-fermion problem is based on the Gel’fand–Tsetlin (G–T) representation theory of the unitary or general linear groups. It exploits the group chain U(n)⊃U(n−1)⊃⋯⊃U(2)⊃U(1) and the associated G–T triangular tableau labeling basis vectors of the relevant irreducible representations (irreps). The general G–T formalism can be drastically simplified in the many-electron case enabling an efficient exploitation in either configuration interaction (CI) or coupled cluster approaches to the molecular electronic structure. However, while the reliance on the G–T chain provides an excellent general formalism from the mathematical point of view, it has no specific physical significance and dictates a fixed Yamanouchi–Kotani coupling scheme, which in turn leads to a rather arbitrary linear combination of distinct components of the same multiplet with a given orbital occupancy. While this is of a minor importance in molecular orbital based CI approaches, it is very inconvenient when relying on the valence bond (VB) scheme, since the G–T states do not correspond to canonical Rumer structures. While this shortcoming can be avoided by relying on the Clifford algebra UGA formalism, which enables an exploitation of a more or less arbitrary coupling scheme, it is worthwhile to point out the suitability of the so-called Verma basis sets for the VB-type approaches
Cellularity and the Jones basic construction
We establish a framework for cellularity of algebras related to the Jones
basic construction. Our framework allows a uniform proof of cellularity of
Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras,
partition algebras, and others. Our cellular bases are labeled by paths on
certain branching diagrams rather than by tangles. Moreover, for the class of
algebras that we study, we show that the cellular structures are compatible
with restriction and induction of modules. Applied to cyclotomic BMW algebras,
our method allows a new a shorter proof of the finite spanning result and
isomorphism with cyclotomic Kauffman tangle algebras.Comment: Revised introduction and improved treatment of cyclotomic BMW
algebras. To appear in Advances in Applied Mathematics
Giant Graviton Oscillators
We study the action of the dilatation operator on restricted Schur
polynomials labeled by Young diagrams with p long columns or p long rows. A new
version of Schur-Weyl duality provides a powerful approach to the computation
and manipulation of the symmetric group operators appearing in the restricted
Schur polynomials. Using this new technology, we are able to evaluate the
action of the one loop dilatation operator. The result has a direct and natural
connection to the Gauss Law constraint for branes with a compact world volume.
We find considerable evidence that the dilatation operator reduces to a
decoupled set of harmonic oscillators. This strongly suggests that
integrability in N=4 super Yang-Mills theory is not just a feature of the
planar limit, but extends to other large N but non-planar limits.Comment: 72 page
- …