Natalia V. Tsilevich

We define a graded graph, called the Schur–Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical duality between representations of the symmetric and general linear groups. As one of the first applications of this graph, we give a new proof of the completeness of the list of discrete indecomposable characters of the infinite symmetric group.

For every irreducible complex representation~$\pi_\lambda$ of the symmetric group~$\S_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement~$\A_{\lambda}$ in the space of~$\pi_\lambda$. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of~$\S_n$), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.

Following the general idea of Schur--Weyl scheme and using two suitable symmetric groups (instead of one), we try to make more explicit the classical problem of decomposing tensor representations of finite and infinite symmetric groups into irreducible components.