Representations of reductive normal algebraic monoids

The rational representation theory of a reductive normal algebraic monoid (with one-dimensional center) forms a highest weight category, in the sense of Cline, Parshall, and Scott. This is a fundamental fact about the representation theory of reductive normal algebraic monoids. We survey how this result was obtained, and treat some natural examples coming from classical groups.Comment: 10 pages. To appear in a volume of the Fields Communications Series: "Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics," edited by Mahir Can, Zhenheng Li, Benjamin Steinberg, and Qiang Wan

Perceptions of randomness in binary sequences: Normative, heuristic, or both?

When people consider a series of random binary events, such as tossing an unbiased coin and recording the sequence of heads (H) and tails (T), they tend to erroneously rate sequences with less internal structure or order (such as HTTHT) as more probable than sequences containing more structure or order (such as HHHHH). This is traditionally explained as a local representativeness effect: Participants assume that the properties of long sequences of random outcomes—such as an equal proportion of heads and tails, and little internal structure—should also apply to short sequences. However, recent theoretical work has noted that the probability of a particular sequence of say, heads and tails of length n, occurring within a larger (>n) sequence of coin flips actually differs by sequence, so P(HHHHH) < P(HTTHT). In this alternative account, people apply rational norms based on limited experience. We test these accounts. Participants in Experiment 1 rated the likelihood of occurrence for all possible strings of 4, 5, and 6 observations in a sequence of coin flips. Judgments were better explained by representativeness in alternation rate, relative proportion of heads and tails, and sequence complexity, than by objective probabilities. Experiments 2 and 3 gave similar results using incentivized binary choice procedures. Overall the evidence suggests that participants are not sensitive to variation in objective probabilities of a sub-sequence occurring; they appear to use heuristics based on several distinct forms of representativeness

Steinberg modules and Donkin pairs

We prove that in positive characteristic a module with good filtration for a group of type E6 restricts to a module with good filtration for a subgroup of type F4. (Recall that a filtration of a module for a semisimple algebraic group is called good if its layers are dual Weyl modules.) Our result confirms a conjecture of Brundan for one more case. The method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, in characteristic not 2, with a computer-aided variation on the old method of Donkin.Comment: 16 pages; proof of Brundan's conjecture adde

Carter-Payne homomorphisms and Jantzen filtrations

We prove a q-analogue of the Carter-Payne theorem in the case where the differences between the parts of the partitions are sufficiently large. We identify a layer of the Jantzen filtration which contains the image of these Carter-Payne homomorphisms and we show how these homomorphisms compose.Comment: 30 page

Homomorphisms and Higher Extensions for Schur algebras and symmetric groups

This paper surveys, and in some cases generalises, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined

Super duality and irreducible characters of ortho-symplectic Lie superalgebras

We formulate and establish a super duality which connects parabolic categories $O$ between the ortho-symplectic Lie superalgebras and classical Lie algebras of $BCD$ types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category $O$, which includes all finite-dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.Comment: 30 pages, Section 5 rewritten and shortene

Relative judgement is relatively difficult: evidence against the role of relative judgement in absolute identification

A variety of processes have been put forward to explain absolute identification performance. One difference between current models of absolute identification is the extent to which the task involves accessing stored representations in long-term memory (e.g. exemplars in memory, Kent & Lamberts, Journal of Experimental Psychology: Learning Memory and Cognition, 31, 289–305, 2005) or relative judgement (comparison of the current stimulus to the stimulus on the previous trial, Stewart, Brown & Chater, Psychological Review, 112, 881–911, 2005). In two experiments we explored this by tapping into these processes. In Experiment 1 participants completed an absolute identification task using eight line lengths whereby a single stimulus was presented on each trial for identification. They also completed a matching task aimed at mirroring exemplar comparison in which eight line lengths were presented in a circular array and the task was to report which of these matched a target presented centrally. Experiment 2 was a relative judgement task and was similar to Experiment 1 except that the task was to report the difference (jump-size) between the current stimulus and that on the previous trial. The absolute identification and matching data showed clear similarities (faster and more accurate responding for stimuli near the edges of the range and similar stimulus-response confusions). In contrast, relative judgment performance was poor suggesting relative judgement is not straightforward. Moreover, performance as a function of jump-size differed considerably between the relative judgement and absolute identification tasks. Similarly, in the relative judgement task, predicting correct stimulus identification based on successful relative judgement yielded the reverse pattern of performance observed in the absolute identification task. Overall, the data suggest that relative judgement does not underlie absolute identification and that the task is more likely reliant on an exemplar comparison process

Diffusion versus linear ballistic accumulation: different models but the same conclusions about psychological processes?

Quantitative models for response time and accuracy are increasingly used as tools to draw conclusions about psychological processes. Here we investigate the extent to which these substantive conclusions depend on whether researchers use the Ratcliff diffusion model or the Linear Ballistic Accumulator model. Simulations show that the models agree on the effects of changes in the rate of information accumulation and changes in non-decision time, but that they disagree on the effects of changes in response caution. In fits to empirical data, however, the models tend to agree closely on the effects of an experimental manipulation of response caution. We discuss the implications of these conflicting results, concluding that real manipulations of caution map closely, but not perfectly to response caution in either model. Importantly, we conclude that inferences about psychological processes made from real data are unlikely to depend on the model that is used

Joint effects of alcohol use, smoking and body mass index as an explanation for the alcohol harm paradox : causal mediation analysis of eight cohort studies

Background and aims Lower socio-economic status (SES) is associated with higher alcohol-related harm despite lower levels of alcohol use. Differential vulnerability due to joint effects of behavioural risk factors is one potential explanation for this 'alcohol harm paradox'. We analysed to what extent socio-economic inequalities in alcohol-mortality are mediated by alcohol, smoking and body mass index (BMI), and their joint effects with each other and with SES. DesignCohort study of eight health examination surveys (1978-2007) linked to mortality data. Setting Finland.ParticipantsA total of 53 632 Finnish residents aged 25+ years.MeasurementsThe primary outcome was alcohol-attributable mortality. We used income as an indicator of SES. We assessed the joint effects between income and mediators (alcohol use, smoking and BMI) and between the mediators, adjusting for socio-demographic indicators. We used causal mediation analysis to calculate the total, direct, indirect and mediated interactive effects using Aalen's additive hazards models. Findings During 1 085 839 person-years of follow-up, we identified 865 alcohol-attributable deaths. We found joint effects for income and alcohol use and income and smoking, resulting in 46.8 and 11.4 extra deaths due to the interaction per 10 000 person-years. No interactions were observed for income and BMI or between alcohol and other mediators. The lowest compared with the highest income quintile was associated with 5.5 additional alcohol deaths per 10 000 person-years (95% confidence interval = 3.7, 7.3) after adjusting for confounders. The proportion mediated by alcohol use was negative (-69.3%), consistent with the alcohol harm paradox. The proportion mediated by smoking and BMI and their additive interactions with income explained 18.1% of the total effect of income on alcohol-attributable mortality. Conclusions People of lower socio-economic status appear to be more vulnerable to the effects of alcohol use and smoking on alcohol-attributable mortality. Behavioural risk factors and their joint effects with income may explain part of the alcohol harm paradox.Peer reviewe