Title: Where are the bijections? Plane Partitions and Alternating Sign Matrices
Abstract: `For about 35 years, combinatorialists have not been able to find bijections between three classes of objects that are all counted by the product formula $\\prod\\limits_{i=0}^{n-1} \\frac{(3i+1)!}{(n+i)!}$.These objects are $n \\times n$ alternating sign matrices, totally symmetric self-complementary plane partitions in a $2n \\times 2n \\times 2n$ box, and cyclically symmetric rhombus tilings of a hexagon of side lengths $n+2,n,n+2,n,n+2,n$ with a central hole of size $2$. Recently, we have added a fourth class of objects to this list, namely alternating sign triangles, and, even more recently, we have extended this class to alternating sign trapezoids, and have shown that they are equinumerous with \; cyclically symmetric rhombus tilings of a hexagon with a central hole of size $k$. At about the same time, we have constructed the first (complicated) bijection relating alternating sign matrices to cyclically symmetric rhombus tilings with a central hole of size two.
In my talk, I shall tell the fascinating story of this search for explicit bijections. I will also report on joint work with Arvind Ayyer, Roger Behrend and Matjaz Konvalinka.k.

Where: Seminar Room @ Villa and Zoom:
https://sfsu.zoom.us/j/82165524617?pwd=Vkwzd1RhaEpqcFVncTRaWmNpUjBCQT09

Meeting ID: 821 6552 4617
Passcode: 180147

Title: Polytope parametrization of bases of representations
Abstract: The representation theory of finite-dimensional complex Lie algebras is well-understood. There are dimension formulas, character Formulas and many parametrizations of their bases, e.g. using Young tableaux or GT pattern. The goal of this talk is to overview some known results and to present new parametrizations by convex polytopes. At the end of the talk we will discuss how these results can be extended to the representation theory of Lie superalgebras or quantum groups which is less understood


Where: Seminar Room @ Villa and Zoom:
https://sfsu.zoom.us/j/82165524617?pwd=Vkwzd1RhaEpqcFVncTRaWmNpUjBCQT09

Meeting ID: 821 6552 4617
Passcode: 180147