Conference Agenda

For a finite-dimensional simple complex Lie algebra $\mathfrak{g}$ and its finite-dimensional irreducible representation $V(\lambda)$, finding an explicit vector space basis of $V(\lambda)$ is one of the most important questions in representation theory. In the literature, there have been many proposals, some of which are restricted to certain types or other properties. Fang-Fourier-Littelmann generalized some of these geometrically inspired approaches to essential bases, a family of monomial bases with particular convenience.

We present a divide-and-conquer approach to compute such essential bases by solving the problem for smaller weights $\lambda$ and gluing the results together. In cases where this does not suffice, additional, more expensive measures have to be taken. This algorithm heavily relies on machinery from both polyhedral geometry and algebraic Lie theory, and is therefore a perfect candidate to be implemented in OSCAR. We will give some insights into the implementation and describe how we adapted it to also solve the basis finding problem for Demazure modules.

In his paper ``Calculating discriminants by higher direct images'' from '94 J. Weyman introduces a method for computing so-called $A$-resultants. However, his techniques seem to not have been widely used for practical computations beyond some special cases like projective space and products thereof. Picking up on Weyman's ideas, we render his partially non-constructive approach into a full algorithm, which realizes a functor for direct images in the derived category of coherent sheaves on toric varieties. A fairly competitive prototype of our algorithm is implemented in the computer algebra system OSCAR.

The expected signature of a family of paths need not be a signature of a path itself. Motivated by this, we consider the notion of a Lie group barycenter introduced by Buser and Karcher to propose a barycenter on path signatures. We show that every element of the free nilpotent Lie group is a barycenter of a group sample, where all but one sample element can be fixed arbitrarily. In the case of piecewise linear paths, we study the problem of recovering an underlying path corresponding to the barycenter of signatures. We determine the minimal number of segments required to learn from signature matrices, providing explicit transformations to the associated congruence normal forms. Our methods are accompanied by an efficient implementation in the computer algebra system OSCAR.