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On the sum of the entries in a character table

Algebra & Discrete Mathematics

Speaker: Digjoy Paul, Indian Institute of Science, Bangalore
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Location: 2112 MSB
Start time: Tue, Nov 28 2023, 4:10PM

 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 conjecture here – this sum is at most twice the sum of degrees of the irreducible characters. We establish the validity of this conjecture for symmetric,
hyperoctahedral and demihyperoctahedral groups using the standard relation between
column sums in the character table and the number of square roots of conjugacy class
representatives. Using these techniques, we are able to show that the asymptotics of
the character table sums is the same as the number of involutions in these groups.
Finally, we derive generating functions for the character tables sum for these groups
as well as generalized symmetric groups as infinite products of continued fractions.

This is a joint work with Arvind Ayyer and Hiranya Kishore Dey.