Enumerative Coding for Grassmannian Space

The Grassmannian space \Gr is the set of all $k-$dimensional subspaces of the vector space~\smash{\F_q^n}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of $k$-dimensional subspaces of \F_q^n. One enumerative coding method is based on a Ferrers diagram representation and on an order for \Gr based on this representation. The complexity of this enumerative coding is $O(k^{5/2} (n-k)^{5/2})$ digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is $O(nk(n-k)\log n\log\log n)$ digits operations. A combination of the two methods reduces the complexity on average by a constant factor.Comment: to appear in IEEE Transactions on Information Theor

On the number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function $f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.Comment: 17 pp, 4 figs, accepted for Physical Review

On Optimally Partitioning Variable-Byte Codes

The ubiquitous Variable-Byte encoding is one of the fastest compressed representation for integer sequences. However, its compression ratio is usually not competitive with other more sophisticated encoders, especially when the integers to be compressed are small that is the typical case for inverted indexes. This paper shows that the compression ratio of Variable-Byte can be improved by 2x by adopting a partitioned representation of the inverted lists. This makes Variable-Byte surprisingly competitive in space with the best bit-aligned encoders, hence disproving the folklore belief that Variable-Byte is space-inefficient for inverted index compression. Despite the significant space savings, we show that our optimization almost comes for free, given that: we introduce an optimal partitioning algorithm that does not affect indexing time because of its linear-time complexity; we show that the query processing speed of Variable-Byte is preserved, with an extensive experimental analysis and comparison with several other state-of-the-art encoders.Comment: Published in IEEE Transactions on Knowledge and Data Engineering (TKDE), 15 April 201

On a symbolic representation of non-central Wishart random matrices with applications

By using a symbolic method, known in the literature as the classical umbral calculus, the trace of a non-central Wishart random matrix is represented as the convolution of the trace of its central component and of a formal variable involving traces of its non-centrality matrix. Thanks to this representation, the moments of this random matrix are proved to be a Sheffer polynomial sequence, allowing us to recover several properties. The multivariate symbolic method generalizes the employment of Sheffer representation and a closed form formula for computing joint moments and cumulants (also normalized) is given. By using this closed form formula and a combinatorial device, known in the literature as necklace, an efficient algorithm for their computations is set up. Applications are given to the computation of permanents as well as to the characterization of inherited estimators of cumulants, which turn useful in dealing with minors of non-central Wishart random matrices. An asymptotic approximation of generalized moments involving free probability is proposed.Comment: Journal of Multivariate Analysis (2014

Combinatorial proof for a stability property of plethysm coefficients

Plethysm coefficients are important structural constants in the representation the- ory of the symmetric groups and general linear groups. Remarkably, some sequences of plethysm coefficients stabilize (they are ultimately constants). In this paper we give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose plethysm coefficients as a alternating sum of terms counting integer points in poly- topes, and exhibit bijections between these sets of integer points.Ministerio de Ciencia e Innovación MTM2010–19336Junta de Andalucía FQM–333Junta de Andalucía P12–FQM–269

Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the q-colouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ~ 3^m, where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q^{-79}. On the triangular lattice, we verify Baxter's analytical expression for the bulk free energy (to order q^{-40}), and we are able to conjecture exact product formulae for the surface and corner free energies.Comment: 17 pages. Version 2: added 4 further term to the serie

The Diameter of a Rouquier Block

For my Honors Research Project, I will be researching special properties of Rouquier blocks that represent the partitions of integers. This problem is motivated by ongoing work in representation theory of the symmetric group. For each integer n and each prime p, there is an object called a Rouquier block; this block can be visualized as a collection of points in a plane, each corresponding to a partition. In this group of points, we say a pair of points is “connected” if certain conditions on the partitions are met. We compare each partition with each other partition, add edges when we can, and we end up with a collection of points that are connected by some number of edges (note that two points are not connected by a line if the conditions are not met). In my project, I will be finding a formula that will restrict the diameter of this graph. I want to minimize the distance between the two points that are the furthest away from each other. A formula to give the most efficient path is either impossible to find or is too complicated to be useful; rather, I will set a ceiling on this distance, so that, given any two blocks, I can give the largest “most efficient” path length possible