6
$\begingroup$

A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra of symmetric functions with coefficents in $\mathbb{Q}$ and $\Lambda_n$ be the subspace of homogeneous degree $n$ symmetric functions. Recall, a derivation is a a linear map $D: \Lambda \to \Lambda$ satisfying the product rule: $$ D(f \cdot g) = (Df) \cdot g + f \cdot (Dg). $$

I'm aware of several examples of derivations for $\Lambda$:

  • Differentiating with respect to a power-sum $\partial/\partial p_k$ where $p_k = x_1^k + x_2^k + \dots$.

  • In this paper, Nenashev introduces(?) the operator $\nabla$, which acts on the Schur basis by $$ \nabla s_\lambda = \sum_{(i,j) \ \text{an outer corner of $\lambda$}} (j-i) s_{\lambda \setminus (i,j)} $$ (here we view $\lambda$ as a subset of $\mathbb{N} \times \mathbb{N}$), and shows it is a derivation.

Additionally, Nenashev proves $\partial/\partial p_k = (\partial/\partial p_{k-1}\nabla - \nabla \partial/\partial p_k)/(k-1)$.

Question: Are there other derivations of $\Lambda$? If yes, have the derivations of $\Lambda$ been classified?

The derivations $\partial/\partial p_1$ and $\nabla$ are both of degree 1, in that they map from $\Lambda_n \to \Lambda_{n-1}$. It'd be especially interesting to identify other degree 1 derivations.

Note $\partial/\partial p_k$ has a nice combinatorial formulation when acting on the Schur basis closely related to the Murnaghan-Nakayama rule. If there are other derivations, do they also have nice combinatorial formulations when acting on the Schur basis? Can derivations be characterized by their actions on the Schur basis?

Improve this question
$\endgroup$
7
  • 5
    $\begingroup$ 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. $\endgroup$
    – darij grinberg
    Commented Sep 11, 2020 at 14:58
  • $\begingroup$ I think "free $\Lambda$-module" is not quite right, because you can also have infinite sums. $\endgroup$
    – lambda
    Commented Sep 11, 2020 at 16:34
  • $\begingroup$ 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? $\endgroup$
    – Sam Hopkins
    Commented Sep 11, 2020 at 16:40
  • 1
    $\begingroup$ 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. $\endgroup$
    – Richard Stanley
    Commented Sep 11, 2020 at 17:32
  • 1
    $\begingroup$ @ZacharyHamaker You can always just conjugate everything by the algebra automorphism that sends the $p$'s to some other generating set. $\endgroup$
    – lambda
    Commented Sep 11, 2020 at 19:23

0

Reset to default

You must log in to answer this question.