Hugh Thomas, UQAM
Title:聽Parking functions via piecewise-linear transformations of R^m.
Abstract:聽Parking functions are certain simple combinatorial objects which play an important role in the study of symmetric functions. I will give a very brief indication of their importance, but my main topic will be a new characterization of parking functions. A (rational) parking function can be encoded as a word of length n on the alphabet {0, ..., m-1}, but not all words correspond to parking functions. To any word, we associate a piecewise linear transformation of R^m. We show that this transformation has a fixed point if and only if the word corresponds to a parking function. This is useful because it allows us to prove that a certain map (usually called "zeta") from parking functions to parking functions, defined when m and n are relatively prime, is in fact a bijection, verifying a conjecture of Gorsky, Mazin, and Vazirani. Perhaps more relevantly to the audience, the geometry of these piecewise-linear transformations seems as if it may also contain further interesting information. This talk is based on joint work with Jon McCammond and Nathan Williams.