I've joined the 21st century and reformatted everything in PDF.No man but a blockhead ever wrote, except for money.
--Samuel Johnson
I've joined the 21st century and reformatted everything in PDF.
Perron-Frobenius Theory is a widely applicable collection of facts about the eigenvalues and eigenvectors of real nonnegative matrices. In these notes we provide complete proofs of the main results; the one non-trivial thing we take for granted is the existence of Jordan Canonical Form over the complex field.
It is "well-known", or at least folklore, that every crystallographic root system (finite or infinite) may by constructed by folding up a suitably chosen simply-laced root system by a diagram automorphism. For example, E6 can be folded into F4, D4 into G2, etc.
It is surprisingly often the case that theorems in representation theory/root systems/Weyl groups are easier to prove in the simply-laced case. Folding can be a valuable tool in these situations, since it provides the possibility of deducing the general case from the simply-laced case by arguing that one has a construction that behaves compatibly with automorphisms.
However, while I was familiar with the ad hoc cases that occur in finite root systems, I have never seen details in the literature about how and why this works in general, so I have written my own explanation here for posterity.
These are lecture notes for a two-hour survey talk that I gave in July 2005 at a Workshop on Generalized Kostka-Foulkes Polynomials at the American Institute of Mathematics. These notes can be used as a crash course on Hall-Littlewood functions for general root systems; as I explained in the introduction, it is aimed at someone who has read Chapter III of Macdonald's book, and wants to know what happens when (s)he leaves the comfort of the type A world.
These notes are also archived as part of the permanent homepage of the Kostka-Foulkes Workshop hosted at AIM.
These rough notes report on some calculations and observations I made in connection with a study of symmetric functions associated to stable Schubert polynomials and the combinatorics of reduced words. I know that hand-written copies of it did circulate---for example, it is cited by Sara Billey in her paper on Kostant polynomials and the cohomology ring for G/B, but I hadn't planned on publishing it.
Recently, Fernando Delgado generously volunteered to convert the handwritten notes to TeX, so it is now available to the public. Thanks, Fernando!
This unfinished article is a report of some observations and conjectures I made several years ago concerning weighted enumerations of cyclically symmetric and totally symmetric plane partitions. The section concerned with the totally symmetric case is still unwritten and needs to be extracted from my hand-written notes. Greg Kuperberg has recently (October 1998) reported progress on some of these conjectures.
Update: Ciucu, Eisenkolbl, Krattenthaler and Zare have a 2001 paper Enumeration of lozenge tilings of hexagons with a central triangular hole that includes proofs of the conjectures listed under "Case 9" (also proved by Kuperberg) and "Case 10" in this manuscript.