3,316 research outputs found
Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers
This is the first of three papers that develop structures which are counted
by a "parabolic" generalization of Catalan numbers. Fix a subset R of
{1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are
determined by R. These are the "inverses" of (parabolic) multipermutations
whose multiplicities are determined by R. The standard forms of the ordered
partitions are refered to as "R-permutations". The notion of 312-avoidance is
extended from permutations to R-permutations. Let lambda be a partition of N
such that the set of column lengths in its shape is R or R union {n}. Fix an
R-permutation pi. The type A Demazure character (key polynomial) in x_1, ..,
x_n that is indexed by lambda and pi can be described as the sum of the weight
monomials for some of the semistandard Young tableau of shape lambda that are
used to describe the Schur function indexed by lambda. Descriptions of these
"Demazure" tableaux developed by the authors in earlier papers are used to
prove that the set of these tableaux is convex in Z^N if and only if pi is
R-312-avoiding if and only if the tableau set is the entire principal ideal
generated by the key of pi. These papers were inspired by results of Reiner and
Shimozono and by Postnikov and Stanley concerning coincidences between Demazure
characters and flagged Schur functions. This convexity result is used in the
next paper to deepen those results from the level of polynomials to the level
of tableau sets. The R-parabolic Catalan number is defined to be the number of
R-312-avoiding permutations. These special R-permutations are reformulated as
"R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless
R-tuples"; the latter n-tuples arise in multiple contexts in these papers.Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the
publication data for the DMTCS journal (dates and volume/issue/number). This
is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v
Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)
Fix an integer partition lambda that has no more than n parts. Let beta be a
weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur
function indexed by lambda and beta is a polynomial generating function in x_1,
.., x_n for certain semistandard tableaux of shape lambda. Let pi be an
n-permutation. The type A Demazure character (key polynomial, Demazure
polynomial) indexed by lambda and pi is another such polynomial generating
function. Reiner and Shimozono and then Postnikov and Stanley studied
coincidences between these two families of polynomials. Here their results are
sharpened by the specification of unique representatives for the equivalence
classes of indexes for both families of polynomials, extended by the
consideration of more general beta, and deepened by proving that the polynomial
coincidences also hold at the level of the underlying tableau sets. Let R be
the set of lengths of columns in the shape of lambda that are less than n.
Ordered set partitions of {1,..,n} with block sizes determined by R, called
R-permutations, are used to describe the minimal length representatives for the
parabolic quotient of the nth symmetric group specified by the set
{1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to
these set partitions. The R-parabolic Catalan number is defined to be the
number of these. Every flagged Schur function arises as a Demazure polynomial.
Those Demazure polynomials are precisely indexed by the R-312-avoiding
R-permutations. Hence the number of flagged Schur functions that are distinct
as polynomials is shown to be the R-parabolic Catalan number. The projecting
and lifting processes that relate the notions of 312-avoidance and of
R-312-avoidance are described with maps developed for other purposes.Comment: 27 pages, 2 figures. Identical to v.2, except for the insertion of
the publication data for the DMTCS journal (dates and volume/issue/number).
This is two-thirds of our preprint "Parabolic Catalan numbers count flagged
Schur functions; Convexity of tableau sets for Demazure characters",
arXiv:1612.06323v
Household Stock Market Beliefs and Learning
This paper characterizes heterogeneity of the beliefs of American households about future stock market returns, provides an explanation for that heterogeneity and establishes its relationship to stock holding behavior. We find substantial belief heterogeneity that is puzzling since households can observe the same publicly available information about the stock market. We propose a simple learning model where agents can invest in the acquisition of financial knowledge. Differential incentives to learn about the returns process can explain heterogeneity in beliefs. We check this explanation by using data on beliefs elicited as subjective probabilities and a rich set of other variables from the Health and Retirement Study. Both descriptive statistics and estimated relevant heterogeneity of the structural parameters provide support for our explanation. People with higher lifetime earnings, higher education, higher cognitive abilities, defined contribution as opposed to defined benefit pension plans, for example, possess beliefs that are considerably closer to what historical time series would imply. Our results also suggest that a substantial part of the reduced form relationship between stock holding and household characteristics is due to differences in beliefs. Our methodological contribution is estimating relevant heterogeneity of structural belief parameters from noisy survey answers to probability questions.
Cognition and Wealth: The Importance of Probabilistic Thinking
This paper utilizes a large set of subjective probability questions from the Health and Retirement Survey to construct an index measuring the precision of probabilistic beliefs (PPB) and relates this index to household choices about the riskiness of their portfolios and the rate of growth of their net worth. A theory of uncertainty aversion based on repeated sampling is proposed that resolves the Ellsberg Paradox within a conventional expected utility model. In this theory, uncertainty aversion is implied by risk aversion. This theory is then used to propose a link between an individual’s degree of uncertainty and his propensity to give “focal” answers of “0”, “50_50” or “100” or “exact” answers to survey questions and the validity of this interpretation is tested empirically. Finally, an index of the precision of probabilistic thinking is constructed by calculating the fraction of probability questions to which each HRS respondent gives a non-focal answer. This index is shown to have a statistically and economically significant positive effect on the fraction of risky assets in household portfolios and on the rate of growth of these assets longitudinally. These results suggest that there is systematic variation in the competence of individuals to manage investment accounts that should be considered in designing policies to create individual retirement accounts in the Social Security system.
Who Becomes a Stockholder? Expectations, SUbjective Uncertainty, and Asset Allocation
We develop a model of portfolio selection with subjective uncertainty and learning in order to explain why some people hold stocks while others don’t. We model heterogeneity in information directly, which is an alternative to the existing explanations that emphasized heterogeneity in transaction costs of investment. We plan to calibrate the model to survey data (when available) on people’s perception about the distribution of stock market returns. Our approach also leads to a model of learning with new implications such as zero optimal risky assets, or ex post correlation of uncorrelated labor income and optimal portfolio composition. It also points to two factors in probabilistic thinking that should have a major impact on stock ownership. These are the level and the precision of expectations. We construct proxy measures for the two parameters from the 1992-2000 waves of the Health and Retirement Study (HRS). We use a large battery of the subjective probability questions administered in each wave of HRS to construct an overall “index of optimism” (the correlated factor between all subjective probabilities) and “index of precision” (the fraction of nonfocal probability answers, following Lillard and Willis, 2001). We also construct measures for how people forecast the weather, their cognitive capacity, wealth, and basic demographics. Our results indicate that stock ownership and the probability of becoming a stockholder are strongly positively correlated with the indices of the level and precision of expectations. Interpretation of the former is quite challenging and further research is needed to understand its full content.
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