Timeline for Faithful characters of finite groups
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 19, 2016 at 13:15 | review | Suggested edits | |||
| Jun 19, 2016 at 13:22 | |||||
| Sep 14, 2015 at 7:05 | comment | added | Duchamp Gérard H. E. | @SebastianBurciu The space considered is exactly the space of central functions $G\rightarrow \mathbb{C}$. It is a product $\mathbb{C}^n$. The remaining question is "can it happen that this algebra be generated by a single irreducible character ?". This I do not know. | |
| May 17, 2010 at 19:08 | vote | accept | Sebastian Burciu | ||
| May 31 at 12:33 | |||||
| Jan 9, 2010 at 10:15 | comment | added | Sebastian Burciu | Next natural question. What is the polynomial space generated by two characters, $\chi$ and $\mu$? Is possible a similar description? | |
| Jan 9, 2010 at 10:10 | comment | added | Sebastian Burciu | Yes, the space of all polynomials in $\chi$ is exactly the space you mentioned. Basically given distinct complex numbers $a_1, a_2, ... a_r$ there are polynomials $P_i$ with $P_i(a_j)=\delta_{i,j}$. Then $P_i(\chi)$ give a basis of central idempotents for the space you mentioned. One chooses the $a_i$'s as all distinct group values of $\chi$. And indeed the answer of my first question follows from here. | |
| Jan 8, 2010 at 17:01 | comment | added | Richard Stanley | As for your first question, the space of all polynomials in a class function $\chi$ is the space of all functions $f$ that satisfy $f(u)=f(v)$ whenever $\chi(u)=\chi(v)$, a simple consequence of the evaluation of the Vandermonde determinant. Since a faithful character has a different value at the identity element than at any other element of the group, the proof follows. | |
| Jan 8, 2010 at 16:18 | comment | added | Richard Stanley | You are right. I meant to say that the character is irreducible. | |
| Jan 7, 2010 at 17:34 | comment | added | Sebastian Burciu | I suppose you keep the assumption that $\chi$ is an irreducible character, or at least a "real character"(i.e the character of a representation). Virtual characters with this property of course exist. | |
| Jan 7, 2010 at 1:28 | comment | added | Richard Stanley | As a followup to my answer, it's not hard to show that if $\chi$ is a character of a finite group $G$, then any character is a complex polynomial in $\chi$ if and only if $\chi$ takes distinct values on distinct conjugacy classes. I suspect that this property is quite rare. Can anyone make this suspicion more precise? For instance, I feel that a "typical" character takes on small integer values such as $0,1,-1$ quite often. | |
| Jan 4, 2010 at 17:17 | comment | added | Sebastian Burciu | What about question 1? Did anyone see this result before? | |
| Jan 2, 2010 at 18:25 | comment | added | Sebastian Burciu | Thank you! $\chi$ is constant on the conjugacy classes of (12) and (1234). | |
| Jan 2, 2010 at 18:05 | vote | accept | Sebastian Burciu | ||
| May 17, 2010 at 19:07 | |||||
| Jan 2, 2010 at 17:45 | history | answered | Richard Stanley | CC BY-SA 2.5 |