Additive diameters and covering complexity of irreducible representations

Abstract

Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.