Symmetric Polynomials of the Weights of a Lie Group Representation

Learn about symmetric polynomials of weights in Lie group representations in this 55-minute lecture from the Combinatorial Methods in Enumerative Algebra program at ICTS Bengaluru. Explore how algebraic counting problems generate integer sequences and their encoding through generating functions, with particular focus on zeta and L-functions. Delve into the connections between Dirichlet's zeta function for number field ideals, Witten's zeta function for Lie group representations, and Hasse-Weil zeta functions for algebraic varieties. Examine how zeta functions of groups and rings serve as tools in asymptotic theory, featuring rational local factors in their Euler product decompositions that reveal underlying structural patterns. Part of a broader program bringing together experts in zeta functions and combinatorial areas to address key challenges in enumerative algebra while training new researchers in this dynamic field.