Have a ball with Young tableaux

Allen Knutson
Department of Mathematics, UC Berkeley

Abstract: Young introduced his tableaux in 1900, in order to study polynomials which are symmetric under exchanging variables. His formula for Schur functions (a particularly nice basis for the symmetric polynomials), which uses them, is not obviously symmetric; we'll see why it's nonobviously symmetric.

Very recently, Ezra Miller, Alex Yong, and I observed that Young tableaux are better than a naked combinatorial set; they index a set of n-dimensional tetrahedra that naturally glue together. While most any reasonable space, and many unreasonable spaces, arise by such gluings, this one turns out to have the topology of an n-dimensional ball! I'll explain why this is true, surprising, and quite convenient for some things.