Kazhdan-Lusztig polynomials, tight quotients and Dyck superpartitions

We give an explicit combinatorial product formula for the parabolic Kazhdan–Lusztig polynomials of the tight quotients of the symmetric group. This formula shows that these polynomials are always either zero or a monic power of q and implies the main result in [F. Brenti, Kazhdan–Lusztig and R-polynomials, Youngʼs lattice, and Dyck partitions, Pacific J. Math. 207 (2002) 257–286] on the parabolic Kazhdan–Lusztig polynomials of the maximal quotients. The formula depends on a new class of superpartitions

Comparing and characterizing some constructions of canonical bases from Coxeter systems

The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a "standard basis" indexed by the elements of $W$ and a "bar involution" given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan-Lusztig basis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan have defined a representation of a modified Iwahori-Hecke algebra on the free $\mathbb{Z}[v,v^{-1}]$-module generated by the set of twisted involutions in $W$, and shown that this module has a unique pre-canonical structure satisfying a certain compatibility condition, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan-Lusztig basis. One can modify the parameters defining Lusztig and Vogan's module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this fashion, and explain the relationships between their resulting canonical bases. While some of these canonical bases are related in a trivial fashion to Lusztig and Vogan's construction, others appear to have no simple relation to what has been previously studied. Along the way, we also clarify the differences between Webster's notion of a canonical basis and the related concepts of an IC basis and a $P$-kernel.Comment: 32 pages; v2: additional discussion of relationship between canonical bases, IC bases, and P-kernels; v3: minor revisions; v4: a few corrections and updated references, final versio

Combinatorics of BB-orbits and Bruhat--Chevalley order on involutions

Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $\sigma$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $\Omega_{\sigma}\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of rook placements and is dual to A. Melnikov's results on $B$-orbits on $\mathfrak{u}$. Using results of F. Incitti, we also prove that this partial order coincides with the restriction of the Bruhat--Chevalley order to the set of involutions.Comment: 27 page

Effective-Range Expansion of the Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential

The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.Comment: 21 pages, 6 figures and 6 tables, submitted to Prog. Theor. Phy

Lattice paths, lexicographic correspondence and Kazhdan-Lusztig polynomials

In this paper we give a new closed formula for the Kazhdan-Lusztig polynomials of finite Coxeter groups and affine Weyl groups. This formula is computationally more efficient than any existing one, and it conjecturally holds for all Coxeter groups. (c) 2006 Elsevier Inc. All rights reserved

Beyond word embeddings: A survey

The goal of this paper is to provide an overview of the methods that allow text representations with a focus on embeddings for text of different lengths, specifically on works that go beyond word embeddings. Analyzing pieces of text can be more challenging in comparison to the analysis of single words, because several additional factors come into play. For this reason, representations of longer pieces of text can be obtained with different strategies, leveraging additional information with respect to what is done for single words. A text is defined by its components and how these are combined together, and this should be taken into account when integrating information to obtain a single document embedding. In addition, multimodal approaches are described to show how it is possible to fuse information of different nature (aural, visual and knowledge) in order to obtain enriched representations. The aim of this survey is to help navigate through the existing methods proposed in the literature and understand which strategies are most suitable to specific needs

Long-term benefit of adeno-associated virus/antisense-mediated exon skipping in dystrophic mice

Many mutations and deletions in the dystrophin gene, responsible for Duchenne muscular dystrophy (DMD), can be corrected at the posttranscriptional level by skipping specific exons. Here we show that long-term benefit can be obtained in the dystrophic mouse model through the use of adeno-associated viral vectors expressing antisense sequences: persistent exon skipping, dystrophin rescue, and functional benefit were observed 74 weeks after a single systemic administration. The therapeutic benefit was sufficient to preserve the muscle integrity of mice up to old age. These results indicate a possible long-term gene therapy treatment of the DMD pathology