Surprising Relations Between Sums-Of-Squares of Characters of the Symmetric Group Over Two-Rowed Shapes and Over Hook Shapes

Amitai Regev ; Doron Zeilberger

Séminaire lotharingien de combinatoire, Tome 75 (2016-2019)

Résumé

In a recent article, we noted (and proved) that the sum of the squares of the characters of the symmetric group, \chi^\lambda(\mu), over all shapes \lambda with two rows and n cells and \mu = 31^n-3, equals, surprisingly, to 1/2 of that sum-of-squares taken over all hook shapes with n+2 cells and with \mu = 321^n-3. In the present note, we show that this is only the tip of a huge iceberg! We will prove that, if $\mu$ consists of odd parts and (a possibly empty) string of consecutive powers of 2, namely 2,4,...,2^t-1 for t >= 1, then the sum of \chi^\lambda(\mu)² over all two-rowed shapes \lambda with n cells equals exactly 1/2 times the analogous sum of \chi^\lambda(\mu')² over all shapes \lambda of hook shape with n+2 cells, where \mu' is the partition obtained from $\mu$ by retaining all odd parts but replacing the string 2,4,...,2^t-1 by 2^t.