"Twisted" Invariant Theory

Question

Let $ V $ be a finite dimensional representation $ \pi $ of a group $ G $. For any $ k $, there is a natural representation of $ G $ on $$ V^{\otimes k} $$ There is also a natural representation of $ G $ on $$ Sym^k(V) $$ If $ V $ is $ n $ dimensional then, after a choice of basis, $ Sym^k(V) $ can be thought of as the space of homogeneous degree $ k $ polynomials in $ n $ variables. Some polynomials are left invariant by this natural action of $ G $. Such a polynomial is called an invariant of $ G $ (really an invariant of the representation $ \pi $). The span of such a polynomial is a trivial irreducible representation of $ G $. Now what if we have a polynomial whose span is invariant under the $ G $ action but the polynomial itself is not invariant with respect to the $ G $ action. In other words, the span of the polynomial is a nontrivial irreducible one. dimensional representation of $ G $.

Is there a name for a polynomial which is a "twisted" (transforms as a nontrivial character) invariant in this sense?

The (regular) invariants form a (finitely generated for many cases of interest) graded subring of $$ Sym(V)=\oplus_k Sym^k(V) $$

These "twisted" invariants would be closed under sums but almost never closed under products. It seems like a major drawback that there is no ring structure for such a theory of "twisted" invariants.

But if we take all "twisted" invariants for all 1d irreps of $ G $ together then this should be a ring since the product of two characters is another character.

Question: Is there a theory of these "twisted" invariants, say for $ G $ a finite group and $ V\cong \mathbb{C}^n $? If so where would I go to learn about such a thing? Also does anyone study the ring of "twisted" invariants I describe above?

Answer 1

This is exactly the ring of invariants of the commutator subgroup $[G, G]$, with an additional grading given by the character group $\text{Hom}(G, \mathbb{C}^{\times})$ of $G$, so all the standard invariant theory stuff continues to apply. A well-known example is $G = S_n$ acting on $\mathbb{C}^n$ by permutation matrices, whose commutator subgroup is $A_n$, and where the extra "twisted" invariant polynomial is the Vandermonde determinant.