# "Twisted" Invariant Theory

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?

• Oh of course! Feel free to add that as an answer. Aug 5 at 18:13

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.
• This is essentially the only example I understand so that would be difficult :P The Wikipedia article says that the $S_n$ example is a special case of a more general phenomenon related to Weyl groups and the Weyl character formula but I can't claim to understand this material in enough detail to explain it. Aug 5 at 18:28