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Published April 2013 | public
Journal Article

The Young Bouquet and its Boundary

Abstract

The classification results for the extreme characters of two basic "big" groups, the infinite symmetric group S(∞) and the infinite-dimensional unitary group U(∞), are remarkably similar. It does not seem to be possible to explain this phenomenon using a suitable extension of the Schur–Weyl duality to infinite dimension. We suggest an explanation of a different nature that does not have analogs in the classical representation theory. We start from the combinatorial/probabilistic approach to characters of "big" groups initiated by Vershik and Kerov. In this approach, the space of extreme characters is viewed as a boundary of a certain infinite graph. In the cases of S(∞) and U(∞), those are the Young graph and the Gelfand–Tsetlin graph, respectively. We introduce a new related object that we call the Young bouquet. It is a poset with continuous grading whose boundary we define and compute. We show that this boundary is a cone over the boundary of the Young graph, and at the same time it is also a degeneration of the boundary of the Gelfand–Tsetlin graph. The Young bouquet has an application to constructing infinite-dimensional Markov processes with determinantal correlation functions.

Additional Information

© 2013 Independent University of Moscow. Received October 19, 2011; in revised form September 18, 2012. To the memory of I. M. Gelfand. The first named author supported in part by NSF-grant DMS-1056390. The second named author supported in part by a grant from Simons Foundation (Simons–IUM Fellowship), the RFBR-CNRS grant 10-01-93114, and the project SFB 701 of Bielefeld University.

Additional details

Created:
August 19, 2023
Modified:
October 23, 2023