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The column sums of the character table of a finite group are always integers, which may also be negative. However, by Fields' conjecture, the combined sum of the entries in all columns except the first (which records the degrees of the irreducible characters) is always non-negative. On the other hand, we know that the first column sum 'D' is always larger than any individual column sum, but whether 'D' dominates the combined sum is a matter of investigation.
In this talk, we explore the validity of this property for finite irreducible Coxeter groups (such as the symmetric group) while also highlighting notable exceptions and proposing further conjectures in this direction.
Time permitting, we will also present explicit generating functions for the character table sums, expressed as infinite products of continued fractions, for the infinite one-parameter families of irreducible Coxeter groups. This is a recent work with Arvind Ayyer and Hiranya Kishore Dey. |