Enumeration of quarter-turn symmetric alternating-sign matrices of odd order

It was shown by Kuperberg that the partition function of the square-ice model related to the quarter-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with the quarter-turn symmetric alternating-sign matrices of odd order, and show that the partition function of this model can be also written in a similar way. This allows to prove, in particular, the conjectures by Robbins related to the enumeration of the quarter-turn symmetric alternating-sign matrices.Comment: 11 pages, 13 figures; minor correction

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.Comment: 33 pages. Has color figure

Four dimensional topological quantum field theory, Hopf categories, and the canonical bases

We propose a new mwthod of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf Category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.Comment: 38 page

Impairments in probabilistic prediction and Bayesian learning can explain reduced neural semantic priming in schizophrenia

It has been proposed that abnormalities in probabilistic prediction and dynamic belief updating explain the multiple features of schizophrenia. Here, we used electroencephalography (EEG) to ask whether these abnormalities can account for the well-established reduction in semantic priming observed in schizophrenia under nonautomatic conditions. We isolated predictive contributions to the neural semantic priming effect by manipulating the prime’s predictive validity and minimizing retroactive semantic matching mechanisms. We additionally examined the link between prediction and learning using a Bayesian model that probed dynamic belief updating as participants adapted to the increase in predictive validity. We found that patients were less likely than healthy controls to use the prime to predictively facilitate semantic processing on the target, resulting in a reduced N400 effect. Moreover, the trial-by-trial output of our Bayesian computational model explained between-group differences in trial-by-trial N400 amplitudes as participants transitioned from conditions of lower to higher predictive validity. These findings suggest that, compared with healthy controls, people with schizophrenia are less able to mobilize predictive mechanisms to facilitate processing at the earliest stages of accessing the meanings of incoming words. This deficit may be linked to a failure to adapt to changes in the broader environment. This reciprocal relationship between impairments in probabilistic prediction and Bayesian learning/adaptation may drive a vicious cycle that maintains cognitive disturbances in schizophrenia

Factorized domain wall partition functions in trigonometric vertex models

We obtain factorized domain wall partition functions for two sets of trigonometric vertex models: 1. The N-state Deguchi-Akutsu models, for N = {2, 3, 4} (and conjecture the result for all N >= 5), and 2. The sl(r+1|s+1) Perk-Schultz models, for {r, s = \N}, where (given the symmetries of these models) the result is independent of {r, s}.Comment: 12 page

On the correlation functions of the domain wall six vertex model

We propose an (essentially combinatorial) approach to the correlation functions of the domain wall six vertex model. We reproduce the boundary 1-point function determinant expression of Bogoliubov, Pronko and Zvonarev, then use that as a building block to obtain analogous expressions for boundary 2-point functions. The latter can be used, at least in principle, to express more general boundary (and bulk) correlation functions as sums over (products of) determinants.Comment: LaTeX2e, requires eepic, 25 pages, including 29 figure

Topological Qubit Design and Leakage

We examine how best to design qubits for use in topological quantum computation. These qubits are topological Hilbert spaces associated with small groups of anyons. Op- erations are performed on these by exchanging the anyons. One might argue that, in order to have as many simple single qubit operations as possible, the number of anyons per group should be maximized. However, we show that there is a maximal number of particles per qubit, namely 4, and more generally a maximal number of particles for qudits of dimension d. We also look at the possibility of having topological qubits for which one can perform two-qubit gates without leakage into non-computational states. It turns out that the requirement that all two-qubit gates are leakage free is very restrictive and this property can only be realized for two-qubit systems related to Ising-like anyon models, which do not allow for universal quantum computation by braiding. Our results follow directly from the representation theory of braid groups which means they are valid for all anyon models. We also make some remarks on generalizations to other exchange groups.Comment: 13 pages, 3 figure

Spatiotemporal signatures of lexical–semantic prediction

Although there is broad agreement that top-down expectations can facilitate lexical-semantic processing, the mechanisms driving these effects are still unclear. In particular, while previous electroencephalography (EEG) research has demonstrated a reduction in the N400 response to words in a supportive context, it is often challenging to dissociate facilitation due to bottom-up spreading activation from facilitation due to top-down expectations. The goal of the current study was to specifically determine the cortical areas associated with facilitation due to top-down prediction, using magnetoencephalography (MEG) recordings supplemented by EEG and functional magnetic resonance imaging (fMRI) in a semantic priming paradigm. In order to modulate expectation processes while holding context constant, we manipulated the proportion of related pairs across 2 blocks (10 and 50% related). Event-related potential results demonstrated a larger N400 reduction when a related word was predicted, and MEG source localization of activity in this time-window (350-450 ms) localized the differential responses to left anterior temporal cortex. fMRI data from the same participants support the MEG localization, showing contextual facilitation in left anterior superior temporal gyrus for the high expectation block only. Together, these results provide strong evidence that facilitatory effects of lexical-semantic prediction on the electrophysiological response 350-450 ms postonset reflect modulation of activity in left anterior temporal cortex

Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem

Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem.Comment: 21 page

Combinatorial point for higher spin loop models

Integrable loop models associated with higher representations (spin k/2) of U_q(sl(2)) are investigated at the point q=-e^{i\pi/(k+2)}. The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries.Comment: latest version adds some reference