Remarks on classical invariant theory

Remarks on Classical Invariant Theory

A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.

Transcending Classical Invariant Theory

Let SP2n(R) = SP2n = Sp be the real symplectic group of rank n. Let Sp denote the two-fold cover of Sp. There is a unitary representation, constructed by Shale [Sh] and Weil [WA], of Sp that is of considerable interest in the theory of automorphic forms. We denote this representation by wand call it the oscillator representation. The purpose of this paper is to establish about w a fact that sho...

Some remarks on generalizations of classical prime submodules

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

Graph Theory and Classical Invariant Theory

Classical invariant theory has died and been resurrected many times. Its golden age in the last century was marked with the flowering of unsurpassed computational ability, and the explicit determination of the invariants of most of the elementary polynomials. The computational approach is commonly acknowledged to have been dealt a death-blow by Hilbert’s celebrated Basis Theorem, an existential...

Standard Monomial Theory for Harmonics in Classical Invariant Theory