Kempf.spring.lecture.07

Abstract: Littlewood-Richardson coefficients are fundamental constants in several fields of mathematics (and in nature). In combinatorics, they appear in the ring of symmetric functions; in representation theory, they appear in the representations of groups such as $GL(n)$ and $S_n$. In geometry they turn up in the topology of the Grassmannian, which parametrizes sub-vector spaces of an $n$-dimensional vector spaces. (This is the ``geometry behind linear algebra''.) I will describe how to interpret Littlewood-Richardson numbers in this way, and show you the key idea behind being able to understand them with pictures (the ``geometric Littlewood-Richardson rule''). I'll conclude with a list of applications in several fields, but the main goal of this talk will be to communicate the flavor of the ideas involved. In particular, no background will be assumed.