Algebra Seminar: A Pieri rule for gl(infinity)

Abstract:If we take two simple finite-dimensional gl(n,C)-modules, say L and F, Weyl's theorem implies that their tensor product is semisimple. If F is a symmetric or exterior power of the natural representation (or its dual), the simples appearing in this decomposition are determined by a combinatorial rule known as the Pieri rule, and the decomposition is multiplicity free, that is, each simple appears exactly once.

In this talk I will present an extension of this result for the infinite-dimensional Lie algebra gl(infinity). Finite-dimensional representations of this algebra are trivial, so the setting is the

following: L is a simple integrable module with a highest weight (this is not automatic!), and F is a symmetric or exterior power of the natural representation or of its dual. I will present several structural results about these tensor products: they are rarely are semisimple and typically indecomposable; they have a canonical filtration, which in most cases coincides with the socle and radical filtration; and, although they are not Jordan-Holder modules in general, they are "multiplicity free" in a suitable sense.

The seminar will be held via Zoom link (meeting ID: 645 5572 6999, please contact the organizer for passcode)

This is a seminar in our algebra seminar series.