# Commutative Algebra Seminar: On the Natural Nullcone of the Symplectic Group, by Jonah Tarasova

April 10, 2024

3:00 PM - 3:50 PM

Jonah Tarasova (University of Michigan): On the Natural Nullcone of the Symplectic Group

Consider a group acting on a polynomial ring $S$ over a field $K$ by degree-preserving $K$-algebra automorphisms. The invariant ring $R$ is a graded subring of $S$; let $\mathfrak{m}_R$ denote the homogeneous maximal ideal of $R$. Several key properties of the invariant ring and its embedding in $S$ can be deduced by studying the nullcone $S/\mathfrak{m}_R S$ of the group action. This includes, for example, the finite generation of the invariant ring and the purity of the embedding. In this talk, we study the nullcone arising from the natural action of the symplectic group.

For the natural representation of the symplectic group (via copies of the standard representation), the invariant ring is the ring defined by the principal Pfaffians of a fixed even size of a generic alternating matrix. We show that the nullcone of this embedding is a strongly $F$-regular ring in positive characteristic, and hence in characteristic zero, a ring of strongly $F$-regular type. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein.

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## Date posted

Apr 15, 2024

## Date updated

Apr 15, 2024