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Degree bound conjecture for transfer quotients of modular invariant rings

About 11 years old · traced to

Let GG be a finite group, let \b5\b5 be a field of characteristic p>0p>0, let VV be a \b5G\b5 G-module, and let PP be a Sylow-pp-subgroup of GG. For each subgroup H≤GH\leq G, let IHGI^G_H be the image of the relative transfer map

Tr⁡HG:\b5[V]H⟶\b5[V]G,\operatorname{Tr}^G_H:\b5[V]^H\longrightarrow \b5[V]^G,

and set

I<PG=∑Q<PIQG.I^G_{<P}=\sum_{Q<P}I^G_Q.

Here \b5[V]G/I<PG\b5[V]^G/I^G_{<P} is the corresponding transfer quotient. Degree bound conjecture. The algebra \b5[V]G/I<PG\b5[V]^G/I^G_{<P} is generated by invariants of degree at most ∣G∣|G|. This conjecture concerns the extent to which transfer quotients retain the bounded-generation behaviour of non-modular invariant rings; the source reports it as a conjecture of Wehlau, while Cohen–Macaulayness of the quotient is known from work of Totaro.

References

Primary source

Jonathan Elmer, “Symmetric powers and modular invariants of elementary abelian p-groups”, arXiv:1503.07797 (2017).

invariant-theorymodular-representation-theorytransfer
Posted

Progress summary

Refreshed
Claimed progress

The general degree bound remains unproved, but it is known in several restricted representation-theoretic cases.

The conjecture, attributed to Wehlau and collaborators, asserts that the transfer quotient k[V]G/I<PGk[V]^G/I^G_{<P} is generated in degrees at most ∣G∣|G|. No posing date is given in the retrieved sources.

Known results

  • Totaro proved that k[V]G/I<PGk[V]^G/I^G_{<P} is always Cohen–Macaulay; this does not establish the degree bound.
  • For an elementary abelian group EE and the specified faithful indecomposable 22-dimensional module construction V=Sm(W)V=S^m(W) with m<∣E∣m<|E|, the quotient is generated in degrees at most ∣E∣|E|.
  • The earlier cyclic-group result covers indecomposable representations of a cyclic group of order pp.

Current status (as of August 2026): The conjectured bound is proved only in reported special cases, while the assertion for arbitrary finite groups and modules remains open.

Sources

Solutions 0

No solutions have been posted yet.