Invariants in divided power algebras

TL;DR

This paper investigates the structure of invariants in divided power algebras associated with general linear groups over fields of positive characteristic, providing bases, conjectures, and exploring the restriction property.

Contribution

It offers a basis for G-invariants in divided power algebras up to degree n, introduces the restriction property, and conjectures its validity for s=1, advancing understanding of invariants in positive characteristic.

Findings

Basis for G-invariants in D_s up to degree n

Restriction property does not hold for s>1

Dimensions of filtration subspaces of the center of the hyper algebra of Dist(G_s)

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$, let G=GL_n be the general linear group over $k$, let g=gl_n be its Lie algebra and let $D_s$ be subalgebra of the divided power algebra of g^* spanned by the divided power monomials with exponents $<p^s$. We give a basis for the $G$-invariants in $D_s$ up to degree $n$ and show that these are also the g-invariants. We define a certain natural \emph{restriction property} and show that it doesn't hold when $s>1$. If $s=1$, then $D_1$ is isomorphic to the truncated coordinate ring of g of dimension p^{dim(g)} and we conjecture that the restriction property holds and show that this leads to a conjectural spanning set for the invariants (in all degrees). We give similar results for the divided power algebras of several matrices and of vectors and covectors, and show that in the second case the restriction property doesn't hold. We also give the dimensions of the filtration subspaces of degree $\le n$ of the centre of the hyper algebra of Dist(G_s).