Effective Invariant Theory of Permutation Groups using Representation Theory

Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at http://www.springer.com

A Nonparametric Bayesian Approach to Copula Estimation

We propose a novel Dirichlet-based P\'olya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Further, the continuity of the imposed D-P tree prior leads to a more favorable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S\&P 500 index and the IBM stock prices during the 2007-08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviors

Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

For a space X acted by a finite group \G, the product space $X^n$ affords a natural action of the wreath product \Gn. In this paper we study the K-groups K_{\tG_n}(X^n) of \Gn-equivariant Clifford supermodules on $X^n$. We show that \tFG =\bigoplus_{n\ge 0}K_{\tG_n}(X^n) \otimes \C is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on \tFG, when \G is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups.Comment: 33 pages, latex, references updated, to appear in Commun. Math. Phy