25 research outputs found
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 of a Coxeter system has a
"standard basis" indexed by the elements of 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 is a canonical basis. Lusztig and Vogan have defined a
representation of a modified Iwahori-Hecke algebra on the free
-module generated by the set of twisted involutions in
, 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 -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 -orbits and Bruhat--Chevalley order on involutions
Let be the group of invertible upper-triangular complex
matrices, the space of upper-triangular complex matrices with
zeroes on the diagonal and its dual space. The group acts
on by , , ,
.
To each involution in , the symmetric group on letters, one
can assign the -orbit . 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 -orbits on .
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
Fusing contextual word embeddings for concreteness estimation
Natural Language Processing (NLP) has a long history, and recent research has focused in particular on encoding meaning in a computable way. Word embeddings have been used for this specific purpose, allowing language tasks to be treated as mathematical problems. Real valued vectors have been generated or employed as word representations for several NLP tasks. In this work, different types of pre-trained word embeddings are fused together to estimate word concreteness. In the evaluation of this task, we have taken into account how much contextual information can affect final results, and also how to properly fuse different word embeddings in order to maximize their performance. The best architecture in our study surpasses the winning solution in the Evalita 2020 competition for the word concreteness task