Four symmetry classes of plane partitions under one roof

In previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2,C). The result is a unified proof of the four enumerations

Non-involutory Hopf algebras and 3-manifold invariants

We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H) is closely related to well-known quantum link invariants such as the HOMFLY polynomial, but it is not a topological quantum field theory.Comment: 36 page

Circumscribing constant-width bodies with polytopes

Makeev conjectured that every constant-width body is inscribed in the dual difference body of a regular simplex. We prove that homologically, there are an odd number of such circumscribing bodies in dimension 3, and therefore geometrically there is at least one. We show that the homological answer is zero in higher dimensions, a result which is inconclusive for the geometric question. We also give a partial generalization involving affine circumscription of strictly convex bodies.Comment: 6 pages. This version has minor changes suggested by the referee. Note that Makeev, and independently Hausel, Makai, and Szucs, also obtained the main resul

A tracial quantum central limit theorem

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity [math.CO/9906120,math.PR/9909104].Comment: 7 page

Random words, quantum statistics, central limits, random matrices

Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved [math.CO/9906120] that the expected shape \lambda of the semi-standard tableau produced by a random word in k letters is asymptotically the spectrum of a random traceless k by k GUE matrix. In this article we give two arguments for this fact. In the first argument, we realize the random matrix itself as a quantum random variable on the space of random words, if this space is viewed as a quantum state space. In the second argument, we show that the distribution of \lambda is asymptotically given by the usual local limit theorem, but the resulting Gaussian is disguised by an extra polynomial weight and by reflecting walls. Both arguments more generally apply to an arbitrary finite-dimensional representation V of an arbitrary simple Lie algebra g. In the original question, V is the defining representation of g = su(k).Comment: 11 pages. Minor changes suggested by the refere