Projection constants measure, in a quantitative way, how well a finite-dimensional Banach space can sit as a complemented subspace of larger spaces: they ask for projections onto the space with minimal possible norm. They are classical invariants in Banach space theory, but their relevance reaches well beyond it, touching questions in approximation theory, convex geometry, harmonic analysis, operator theory, and the geometry of high-dimensional structures.

In this talk I will discuss a project devoted to the concrete determination of projection constants for classical spaces of polynomials on finite-dimensional Banach spaces, with particular emphasis on their asymptotic behaviour in high dimensions. The guiding aim is to obtain concrete formulas and to draw from them deeper structural and asymptotic conclusions. This requires a solid theoretical framework: it is precisely the interplay between explicit computation and abstract structure that leads to effective results.

The main point of the talk is to explain how representation theory of compact groups can be used to exploit these symmetries: it identifies the right invariant subspaces, produces canonical minimal projections, and reduces projection constants to computable integral formulas involving reproducing kernels or characters.

After outlining the general framework, I will illustrate its strength in a nontrivial example: the determination of projection constants for spaces of homogeneous polynomials on $\mathcal L(\ell_2^n)$. This example shows how abstract harmonic-analytic methods lead not only to exact formulas, but also to sharp high-dimensional asymptotics.

This talk is mainly based on joint work with D. Galicer, M. Mansilla, M. Mastyło, and S. Muro.