# The (possibly negative) integers arising from sum of elements in any column of the character table

Let $$G$$ be a finite group and $$A$$ be the character table for the irreducible complex representations. The sum of elements in any row of the character table is a positive integer as its equals to the multiplicity of irreducible representation corresponding to that row inside $$V$$, where $$V$$ is the representation on the group algebra induced by the conjugacy action of $$G$$.

But how to understand the sum of elements in any column of the character table in terms of representation theory? It's an integer by Galois theory but might not be positive (e.g $$G=M_{11}$$). In the case of Weyl groups, is there a simple formula? At least, can we determine its sign from $$G$$?

• There is also the possibility of a Stembridge $q=-1$ phenomenon (aka cyclic sieving): maybe the absolute value of a column sum counts the fixed points of some set with involution. We can see this for the sum of all characters evaluated at the identity, since in some cases (when all representations are real) this counts the number of elements of the group that are involutions. Also, this picture wouldn't necessarily help with an interpretation for the sign. Dec 27, 2019 at 18:45

If $$G$$ is a finite group whose complex irreducible representations are all realizable over $$\mathbb{R}$$, (eg $$G = S_{n}$$), then for any $$x \in G$$, the sum of the entries of the column corresponding to (the conjugacy class of ) $$x$$ is precisely the number of square roots of $$x$$ in $$G$$, so is, in particular, a non-negative integer (and is strictly positive if $$x$$ has odd order).