Timeline for Bivariate polynomials with special properties
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27, 2011 at 19:06 | comment | added | ARupinski | So I thought about it and $A$ is indeed closely related to the representations of $SU(3)$ for (almost) trivial reasons. Despite this, I think I might be able to use the general idea of such matrices to solve a few open questions I've been pondering for awhile, so thanks for the inspiration. | |
| Mar 25, 2011 at 1:47 | comment | added | ARupinski | @Johann: Based on your results on $A^n$, its almost certain that $A$ is somehow closely related to the representation theory of $SU(3)$ but offhand I'm not sure how as I have never really looked at such matrices (although based on your results it seems that I should sometime). As far as using the recurrence and initial values to prove remarkability, there is probably some way (independent of the relationship to $SU(3)$), but offhand I don't know how. Definitely something I will keep thinking about when I have free time. | |
| Mar 24, 2011 at 15:27 | comment | added | Johann Cigler | @ARupinski. As I already wrote I am not familiar with group representations. So I did not quite understand your argument that the above sequence is indeed "remarkable". Is there perhaps also a direct way using only the recurrence and initial values to prove this? The matrix $A$ is a useful construction associated with such a recurrence. But I am astonished that all entries of $A^n$ are related to representations of $SU(3)$. What about the matrix $A$ itself or the sequence $Tr(A^n)$? Are they also in some form related to $SU(3)$ or are the above results only accidental coincidencies? | |
| Mar 23, 2011 at 21:50 | comment | added | ARupinski | I thought about it a bit more, and your $r_n$'s are also related to the $P_{a,b}$'s I defined by $r_k(u,v) = -P_{1,k-1}(u,v)$. In light of my third edit, this implies the $r_{2k}$ are all divisible by $(1-uv)$ as you already noted; but since $1-uv$ does not satisfy remarkability, the $r_{2k}$ cannot satisfy it either; however in light of my third edit, I am pretty sure you are right that the cofactors $r_{2k}(u,v)/(1-uv)$ do satisfy remarkability. | |
| Mar 22, 2011 at 13:39 | history | edited | Johann Cigler | CC BY-SA 2.5 |
added 788 characters in body
|
| Mar 22, 2011 at 8:10 | history | answered | Johann Cigler | CC BY-SA 2.5 |