The Alexander polynomial as quantum invariant of links

In these notes we collect some results about finite dimensional representations of $U_q(\mathfrak{gl}(1|1))$ and related invariants of framed tangles which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite dimensional $U_q(\mathfrak{gl}(1|1))$-representation and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a nonzero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_q(\mathfrak{gl}(1|1))$.Comment: This is a corrected and revised version, to be published on Arkiv f\"or Matematik. This is part of the author's PhD Thesi

A diagram algebra for Soergel modules corresponding to smooth Schubert varieties

Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category $\mathcal{O}(\mathfrak{gl}_n)$ which show up in categorification of $\mathfrak{gl}(1|1)$-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras.Comment: A previous version of this paper was originally included as Part 3 in arXiv:1305.6162, which now has been replaced. This paper contains many corrections and improvements. This is part of the author's PhD thesi

Complementary Actions

Human beings come into the world wired for social interaction. At the fourteenth week of gestation, twin fetuses already display interactive movements specifically directed towards their co- twin. Readiness for social interaction is also clearly expressed by the newborn who imitate facial gestures, suggesting that there is a common representation mediating action observation and execution. While actions that are observed and those that are planned seem to be functionally equivalent, it is unclear if the visual representation of an observed action inevitably leads to its motor representation. This is particularly true with regard to complementary actions (from the Latin complementum ; i.e. that fills up), a specific class of movements which differ, while interacting, with observed ones. In geometry, angles are defined as complementary if they form a right angle. In art and design, complementary colors are color pairs that, when combined in the right proportions, produce white or black. As a working definition, complementary actions refer here to any form of social interaction wherein two (or more) individuals complete each other\u2019s actions in a balanced way. Successful complementary interactions are founded on the abilities:\ua0 (1)\ua0 to simulate another person\u2019s movements; (2)\ua0 to predict another person\u2019s future action/ s; (3)\ua0to produce an appropriate congruent/ incongruent response that completes the other person\u2019s action/ s; and (4)\ua0to integrate the predicted effects of one\u2019s own and another person\u2019s actions. It is the neurophysiological mechanism that underlies this process which forms the main theme of this chapte

Empirical and Simulated Adjustments of Composite Likelihood Ratio Statistics

Composite likelihood inference has gained much popularity thanks to its computational manageability and its theoretical properties. Unfortunately, performing composite likelihood ratio tests is inconvenient because of their awkward asymptotic distribution. There are many proposals for adjusting composite likelihood ratio tests in order to recover an asymptotic chi square distribution, but they all depend on the sensitivity and variability matrices. The same is true for Wald-type and score-type counterparts. In realistic applications sensitivity and variability matrices usually need to be estimated, but there are no comparisons of the performance of composite likelihood based statistics in such an instance. A comparison of the accuracy of inference based on the statistics considering two methods typically employed for estimation of sensitivity and variability matrices, namely an empirical method that exploits independent observations, and Monte Carlo simulation, is performed. The results in two examples involving the pairwise likelihood show that a very large number of independent observations should be available in order to obtain accurate coverages using empirical estimation, while limited simulation from the full model provides accurate results regardless of the availability of independent observations.Comment: 15 page

Complementary actions

Complementary colors are color pairs which, when combined in the right proportions, produce white or black. Complementary actions refer here to forms of social interaction wherein individuals adapt their joint actions according to a common aim. Notably, complementary actions are incongruent actions. But being incongruent is not sufficient to be complementary (i.e., to complete the action of another person). Successful complementary interactions are founded on the abilities: (i) to simulate another person's movements, (ii) to predict another person's future action/s, (iii) to produce an appropriate incongruent response which differ, while interacting, with observed ones, and (iv) to complete the social interaction by integrating the predicted effects of one's own action with those of another person. This definition clearly alludes to the functional importance of complementary actions in the perception-action cycle and prompts us to scrutinize what is taking place behind the scenes. Preliminary data on this topic have been provided by recent cutting-edge studies utilizing different research methods. This mini-review aims to provide an up-to-date overview of the processes and the specific activations underlying complementary actions