Timeline for The zero entries in the character table of a finite group
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2021 at 10:49 | answer | added | Nicky Hekster | timeline score: 1 | |
| Jul 5, 2020 at 20:39 | vote | accept | Sebastien Palcoux | ||
| Jun 22, 2020 at 17:19 | history | became hot network question | |||
| Jun 22, 2020 at 16:25 | answer | added | John Shareshian | timeline score: 5 | |
| Jun 22, 2020 at 15:48 | comment | added | Sebastien Palcoux | @JohnShareshian: Yes, you got a counter-example for Question 1, perhaps the smallest one. | |
| Jun 22, 2020 at 15:31 | comment | added | John Shareshian | Maybe I misunderstand something, but it seems to me that if $\chi$ is the irreducible character of $S_6$ corresponding to the partition $(3,3)$ and $g \in S_6$ is a $6$-cycle, then $\chi(g)=0$ as $(3,3)$ does not have hook shape, $\chi(1)=5$ by the hook-length formula or counting standard Young tableaux or a well-know fact about Catalan numbers, and $C_{S_n}(g)=\langle g \rangle$ has order six. | |
| Jun 22, 2020 at 14:26 | answer | added | Alex B. | timeline score: 7 | |
| Jun 22, 2020 at 13:57 | comment | added | John Murray | See (3.7) in Isaacs, Character Theory of Finite Groups, Academic Press or G. James, M. Liebeck, Representations and Characters of Groups, Academic Press. | |
| Jun 22, 2020 at 13:48 | comment | added | Sebastien Palcoux | @JohnMurray: your comment is very useful and deserves to be posted in answer. What is the reference for (1)? | |
| Jun 22, 2020 at 13:27 | comment | added | Sebastien Palcoux | @TheoJohnson-Freyd: This is the statement that Question 1 asks for, but for now it is just an observation on (very) small groups, not a general conjecture (yet). | |
| Jun 22, 2020 at 12:55 | comment | added | John Murray | Dear Sebastien, here are two facts which partially explain your observations: (1) The central character $\frac{|G|\chi(g)}{|C_G(g)|\chi(1)}$ of $\chi$ at the conjugacy class of $g$ is an algebraic integer. When gcd$(\chi(1),|C_G(g)|)\ne1$, this makes it more likely that $\chi(g)=0$. (2) Let $p$ be a prime and suppose that $|G|/\chi(1)$ is coprime to $p$. Then Richard Brauer proved that $\chi(g)=0$ if $p$ divides the order of $g$. Note that in this situation $p$ divides gcd$(\chi(1),|C_G(g)|)$. | |
| Jun 22, 2020 at 12:51 | comment | added | Theo Johnson-Freyd | Just to parse the logic, you make the following conjecture: If $G$ is a finite group, $\chi$ a character of $G$, and $g \in G$ such that $|C_G(g)|$ and $\deg(\chi)$ are coprime, then $\chi(g) \neq 0$. Did I get it right? | |
| Jun 22, 2020 at 9:13 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |