Timeline for Derivations for symmetric functions
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 11, 2020 at 19:23 | comment | added | lambda | @ZacharyHamaker You can always just conjugate everything by the algebra automorphism that sends the $p$'s to some other generating set. | |
| Sep 11, 2020 at 18:29 | comment | added | darij grinberg | @lambda: You're right! It's a direct product, not a direct sum. | |
| Sep 11, 2020 at 17:48 | comment | added | Zach H | @darijgrinberg I suppose you can just differentiate with respect to the $h_i$'s (or the $e_i$'s for that matter). One thing that's interesting to me is that all of the $\partial / \partial p_k$'s come from $\partial /\partial p_1$ and $\nabla$. Maybe the other families can be expressed similarly? | |
| Sep 11, 2020 at 17:32 | comment | added | Richard Stanley | If you differentiate the Jacobi-Trudi matrix with respect to $h_i$, then one expresses $\frac{\partial}{\partial h_i}s_\lambda$ (or more generally $s_{\lambda/\mu}$) as a "canonical" linear combination with $\pm 1$-coefficients of skew Schur functions $s_{\nu/1^k}$. E.g., $\frac{\partial}{\partial h_2}s_{2211}=s_{211}+s_{311/1}-s_{22}$. However, this does not seem so interesting to me. | |
| Sep 11, 2020 at 16:40 | comment | added | Sam Hopkins | Maybe to restrict attention, focus on derivations that interact with $\omega$ in a simple way: e.g., I think we have $\Delta \circ \omega = -\omega \circ \Delta$, right? | |
| Sep 11, 2020 at 16:34 | comment | added | lambda | I think "free $\Lambda$-module" is not quite right, because you can also have infinite sums. | |
| Sep 11, 2020 at 14:58 | comment | added | darij grinberg | As a commutative $\mathbb{k}$-algebra, $\Lambda$ is just a polynomial ring in (for example) $h_1, h_2, h_3, \ldots$. Thus, its derivations form a free $\Lambda$-module with basis $\partial / \partial h_1, \partial / \partial h_2, \partial / \partial h_3, \ldots$. That's a lot of derivations :) The derivations that decrease degree form a free $\mathbb{k}$-module with basis $\partial / \partial h_1, \partial / \partial h_2, \partial / \partial h_3, \ldots$; that's still a lot. The question is which of them are combinatorially interesting. | |
| Sep 11, 2020 at 13:40 | history | asked | Zach H | CC BY-SA 4.0 |