Title:A mystery group action and the mystery statistic
| Speaker: | Stephan Pfannerer-Mittas |
| Affiliation: | University of Waterloo |
| Location: | MC 5501 |
Abstract: In 2010, B. Rhoades proved that promotion on rectangular standard Young tableaux together with the associated fake-degree polynomial shifted by an appropriate power, provides an instance of the cyclic sieving phenomenon.
Motivated in part by this result, we show that we can expect a cyclic sieving phenomenon for m-tuples of standard Young tableaux of the same shape and the m-th power of the associated fake-degree polynomial, for fixed m, under mild and easily checked conditions. However, we are unable to exhibit an appropriate group action explicitly.
Put differently, we determine in which cases the mth tensor power of a character of the symmetric group carries a permutation representation of the cyclic group.
To do so, we use a method proposed by N. Amini and P. Alexandersson, which amounts to establishing a bound on the number of border-strip tableaux.
Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group. In particular, we prove the existence of a statistic on permutations, which is equidistributed with the RSK-shape and invariant under rotation.
This is based on joint work with Per Alexandersson, Martin Rubey and Joakim Uhlin.