The Schur–Weyl graph and Thoma’s theorem
Asymptotic representation theory, initiated by Vershik and Kerov in
the early 1970s, is now a very active and important research area at
the intersection of functional analysis, algebra, probability, ergodic
theory, etc. In the case of symmetric groups, it is most closely
related to combinatorics. My talk will show a typical interplay of
ideas from all these different fields leading to interesting notions
and results.
Thoma's theorem, one of the cornerstones of asymptotic representation
theory, gives the list of indecomposable characters of the infinite
symmetric group, or, which is essentially the same, the list of
central measures on the Young graph. There are several proofs of this
important theorem, but they are rather indirect and technically
involved. We suggest a new approach to proving Thoma's theorem, based
on using the famous RSK algorithm and introducing a new graded graph,
called the Schur--Weyl graph, which arises naturally from a "dynamic"
view on RSK, when it is applied to growing sequences of symbols and
one considers the dynamics of the arising tableaux.
Based on joint work with Anatoly Vershik.