Suppose $G$ is a finite group, and $H$ a subgroup. For an irreducible character $\chi$ of $G$, there is a central idempotent in the group algebra $\mathbb{C}[G]$: $$ e_\chi=\frac{\chi(1)}{|G|}\sum_{g\in G}\chi(g^{-1})g. $$

I write $e_H:=e_{1_H}=\frac{1}{|H|}\sum_{h\in H}h$ for the idempotent in $\mathbb{C}[H]$ corresponding to the trivial character on $H$. If $[G/H]$ denotes a complete set of left coset representatives of $H$ in $G$, then by construction the element $$ \sum_{g\in [G/H]}ge_Hg^{-1} $$ is central in $\mathbb{C}[G]$. Is there any way to somewhat explicitly extract which characters have their corresponding central idempotents in this linear combination, and/or their multiplicity? Just looking at the Mackey formula and since $ge_Hg^{-1}=e_{gHg^{-1}}$, my guess is it may be characters that are constituents of $\operatorname{Ind}^G_H(1_H)$, or something similar, but I am unsure.

  • 2
    $\begingroup$ Yes, your central element is $\operatorname{Ind}_H^G\left(1_H\right)$ when regarded as a character of $G$. Indeed, look at the formula (4.1.3) in Darij Grinberg, Victor Reiner, Hopf Algebras in Combinatorics, arXiv:1409.8356v5 and set $U = 1_H$ (so that $\chi_U$ is the function that is constantly $1$). The right hand side of the formula differs from your sum in that it sums over all $g \in G$ as opposed to only over left coset representatives; but to balance this out, it is being divided by $\left|H\right|$. $\endgroup$ – darij grinberg Apr 12 at 5:35
  • $\begingroup$ @darijgrinberg Thank you, Darij, I'll take a look. This book looks like a valuable resource to boot. $\endgroup$ – Denise Gi Apr 12 at 6:53

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