Commutative algebra seminar

Speaker: Vaibhav Pandey
Affiliation: Purdue University
Venue: Ramanujan Hall
Date: Thursday, 27 July 2023
Title: On the natural null cones of the classical invariant rings.

Abstract: The generic determinantal ring can be seen as a subring of a polynomial ring in a natural manner. When the field is infinite, this subring is, in fact, the invariant ring of the natural action of the general linear group on the ambient polynomial ring. Over a field of characteristic zero, the general linear group is linearly reductive, so the invariant ring splits from the polynomial ring. This immediately implies a wealth of nice properties for the generic determinantal rings including the Cohen-Macaulay property and rational singularities.

When the field has positive characteristic, the general linear group is typically not linearly reductive. It is natural to ask if the above embedding continues to split over an infinite field of positive characteristic. We show that strikingly, this embedding does NOT split in ANY positive characteristic if the general linear group is NOT linearly reductive.

Broadly speaking, this non-splitting is due to the Cohen-Macaulay property of the irreducible components of the null cone of this representation. We analyze the irreducible components of the null cones of the general linear group and also the null cone of the symplectic group in greater depth. We show that they are all F-regular in positive characteristic and calculate their divisor class groups independent of characteristic. This talk is based on two separate projects: One with Mel Hochster, Jack Jeffries, and Anurag Singh and another with Uli Walther and Yevgeniya Tarasova.