Two examples of mixed lattice point enumerators
Richard Stanley
MIT
Febrary 12,
4:15pm
refreshments at 3:45pm
2338
ABSTRACT

Let ${\cal P}_1,\dots,{\cal P}_k$ be convex polytopes in
$\mathbb{R}^n$ with integer vertices, and let $t_1,\dots,t_k$ be
nonnegative integers. Define the ``Minkowski sum''
$$ t_1{\cal P}_1 +\cdots+t_k{\cal P}_k=\{t_1v_1+\cdots+t_kv_k\,:\,
v_i\in{\cal P}_i\}. $$
The \emph{mixed lattice point enumerator} of $P_1,\dots,P_k$
is the function of $t_1,\dots,t_k$ defined by
$$ F(t_1,\dots,t_k) = \#\left( (t_1{\cal P}_1 +\cdots+t_k{\cal P}_k
)\cap\mathbb{Z}^n\right). $$
It was first shown by McMullen that $F(t_1,\dots,t_k)\in
\mathbb{Q}[t_1,\dots,t_k]$. The function $F(t_1,\dots,t_k)$ contains a
lot of interesting information about the ${\cal P}_i$'s, such as their
volumes, mixed volumes, and Ehrhart polynomials.
We will discuss two recently discovered examples of mixed lattice
point enumerators. The first (in joint work with A. Postnikov) is
closely connected with Kostant's partition function for the root
system $A_n$. The second (with J. Pitman) is closely connected with
parking functions.

Speaker's Contact Info: rstan(atsign)math.mit.edu
Return to seminar home page
Page loaded on February 01, 1999 at 09:44 AM.

Copyright © 199899, Sara C. Billey.
All rights reserved.

