Séminaire Lotharingien de Combinatoire, 91B.99 (2024), 12 pp.
Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul
On The Sum of The Entries in a Character Table
Abstract.
In 1961, Solomon proved that the sum of all the entries in the character table of a finite group does not exceed the cardinality of the group.
We state a different and incomparable property here - this sum is at most twice the sum of dimensions of the irreducible characters. We establish the validity of this property for all finite irreducible Coxeter groups.
The main tool we use is that the sum of a column in the character table of such a group is given by the number of square roots of the corresponding conjugacy class representative. We then show that the asymptotics of character table sums is the same as the number of involutions in symmetric, hyperoctahedral and demihyperoctahedral groups.
Finally, we derive generating functions for the character table sums for these latter groups as well as generalized symmetric groups as infinite products of continued fractions.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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