# Four stages of invariant theory

#
by

Roger Howe

These talks will survey progress on classical invariant theory since Weyl’s book,
with an emphasis on more recent developments.

They will begin with a brief review of Weyl’s results, including the notion of classical action,
and Weyl’s First Fundamental Theorem for such actions. This will be followed by the introduction
of the Weyl algebra, and the notion of dual pair of Lie subalgebras of the symplectic Lie algebra.
These ideas permit description of the full isotopic decomposition for classical actions,
not just the invariants, and also give rise to decomposition of polynomial rings into invariants
and harmonics relative to a given classical action. These phenomena suggest the concept of stable
range: when relevant parameters are restricted appropriately, various phenomena of interest become
more tractable. For example, in the stable range, the full polynomial ring can be expressed as the tensor
product of the invariants and the harmonics. This is referred to as separation of variables. Seeing classical
actions as part of a dual pair structure enables consideration of the relationship between different actions,
and leads to the study of branching rules and to reciprocity laws. These are formulated in terms of certain
algebras, the branching algebras. Finer investigation of these issues is aided by ideas from commutative
algebra, especially term orders, SAGBI theory and Hibi rings. These allow detailed description
of the rings of invariants and of harmonics, and provide a natural generalization of Hodge’s standard
monomial theory. They also enable detailed descriptions of branching algebras, including
a representation-theoretic approach to the Littlewood-Richardson Rule.

Finally, the extent to which separation of variable fails outside the stable range will be considered.
This is the subject of current research.

Lecture
I (Due to a technical glitch, the video is frozen after eighth minute. Slides and sound are OK.)

Lecture
IIa
Lecture
IIb
Lecture
in Trest
Lecture
IIIa
Lecture
IIIb
Lecture
IIIc
Lecture
IVa
Lecture
IVb