| Abstract:
A central topic in representation theory and combinatorics is the
decomposition of tensor products of irreducible representations of
GLn(C). In particular, what inequalities must a triple of
high weights (l,m,
n) satisfy in order for the triple tensor
product to have an invariant vector? A list of inequalities derived
from Schubert calculus has recently been found, which were shown by
Klyachko and Helmke-Rosenthal to be necessary, and by Klyachko
to be asymptotically sufficient.
In this talk we introduce a combinatorial model, the honeycomb, to study
invariant vectors in triple tensor products (reinterpreting work of
Berenstein and Zelevinsky). With this we show that the inequalities
are sufficient even for small weights (not just asymptotically) -- this
latter one was known as the "saturation conjecture".
In particular this implies Horn's conjecture from 1962 on the spectrum
of the sum of two Hermitian matrices. |
alexsuciu@neu.edu