Optimality and comparison of monomial Lie-additive diameters

Determine whether every irreducible representation of a complex Lie algebra has monomial Lie-additive diameter equal to the dimension lower bound, and whether the monomial Lie-additive diameter is always at most the corresponding group-additive diameter.

Background

For a Lie algebra representation, the paper defines the monomial Lie-additive diameter using translates by products of operators arising from Lie algebra elements. The dimension-counting lower bound remains valid because monomial operators cannot increase the dimension of the translated subspace.

The paper proves optimality for the family $sl_3(\mathbb C)\curvearrowright \Sym^k\mathbb C^3$ with respect to symmetric powers of planes, and notes that these examples are also consistent with the conjectured comparison between Lie-additive and group-additive diameters. The general questions remain unresolved.

References

It is unclear how the group-additive and the Lie-additive diameters are related. In Questions 6.6 and 6.7 we wondered whether irreducible representations of Lie algebras always have optimal monomial diameters, and whether the monomial diameter is always at most the group diameter.

— Additive diameters and covering complexity of irreducible representations  (2609.03882 - Jezernik et al., 3 Sep 2026) in Section 5, subsection “Lie algebras”

Is \diamp[sl_n(C),mon]\bigl(\Symk Cn, \Symk C{n-1}\bigr) = \Bigl\lceil \frac{n+k-1}{n-1} \Bigr\rceil for all n \geq 3 and all k \geq 1?

— Additive diameters and covering complexity of irreducible representations  (2609.03882 - Jezernik et al., 3 Sep 2026) in Question 4, Section 9, subsection “Monomial diameters in higher rank”