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Let $h_r$ and $e_r$ be the complete and elementary symmetric functions of degree $r \in \mathbb{N}_0$ and let $s_\lambda$ be the Schur function labelled by the partition $\lambda$. Let $e_r^\bot$ denote the adjoint of multiplication by $e_r$. By Pieri's rule, $e_r^\bot s_\lambda = \sum_\mu s_\mu$ where the sum is over all partitions $\mu$ obtained from $\lambda$ by removing $r$ boxes, no two in the same row. Similarly $h_r s_\lambda = \sum_\mu s_\mu$ where the sum is over all partitions $\mu$ obtained from $\lambda$ by adding $r$ boxes, no two in the same column.

For $a \in \mathbb{N}_0$, the Bernstein vertex operation $B_a$ may be defined by $B_a(s_\lambda) = \sum_{r \ge 0} (-1)^r h_{r+a} e_r^\bot s_\lambda$. It is well known that if $a \ge \lambda_1$ then $B_a(s_\lambda) = s_{\lambda \sqcup (a)}$, where $\lambda \sqcup (a) = (a,\lambda_1,\ldots, \lambda_{\ell(\lambda)})$. For instance this follows the equation at the bottom of page 96 in I.5.29 in Macdonald's textbook.

More generally, let $c \in \mathbb{N}_0$ be least such that $a + c \ge \lambda_{c+1}$. We take $\lambda_{\ell(\lambda) + 1} = 0$, so $c$ is well-defined and $c \le \ell(\lambda)$. I can show that if $c = 0$ or $a + c < \lambda_c$ then

$$ B_a(s_\lambda) = (-1)^c s_\mu \quad\text{where $\mu = (\lambda_1-1,\ldots,\lambda_c-1,a+c, \lambda_{c+1}, \ldots, \lambda_{\ell(\lambda)})$} $$

while if $a+c \ge \lambda_c$ then $B_a(s_\lambda) = 0$. Since, by definition of $c$, $a+(c-1)\not\ge \lambda_c$, an equivalent condition for the latter is $a+c = \lambda_c$.

Has this generalization appeared in the literature on symmetric functions?

It was new to me even that $B_a(s_\lambda)$ has at most one summand. An important corollary of the special case for $a \ge \lambda_1$ is that $B_{\lambda_1}\ldots B_{\lambda_{\ell(\lambda)}} 1 = s_\lambda$. This can be interpreted via raising operators as a version of the Jacobi–Trudi identity; again this is in Macdonald. As a follow-up, can the generalization above be interpreted in a similar way?

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The one place where this appears almost explicitly is Exercise 2.9.1 of Darij Grinberg, Victor Reiner, Hopf algebras in combinatorics, 17 November 2020 (archived version in case the numbering changes one day; I still have plans to extend the notes...). Your operator $B_{a}$ is called $\mathbf{B}_{a}$ in that exercise, and is defined for all $a\in\mathbb{Z}$, not only for $a\in\mathbb{N}_{0}$.

The claim that $B_{a}\left( s_{\lambda}\right) =s_{\lambda\sqcup\left( a\right) }$ if $a\geq\lambda_{1}$ is part (b) of this exercise. Part (d) generalizes it in a rather straightforward way: It says that $B_{a}\left( s_{\lambda}\right) $ (for all $a\in\mathbb{Z}$) is the "non-partition Schur function" $\overline{s}_{\left( m,\lambda_{1},\lambda_{2},\lambda_{3} ,\ldots\right) }$, where we define "non-partition Schur functions" by the usual Jacobi--Trudi formula applied to non-partitions: If $\left( \alpha _{1},\alpha_{2},\ldots,\alpha_{k}\right) $ is a sequence of integers, then we set \begin{align*} \overline{s}_{\left( \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right) } :=\det\left( h_{\alpha_{i}-i+j}\right) _{i,j\in\left[ k\right] }. \end{align*} Now, part (c) explains that any "non-partition Schur function" $\overline {s}_{\left( \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right) }$ is either $0$ or $\pm s_{\nu}$ for an actual partition $\nu$. The construction of this partition $\nu$ appears in the solution to this exercise (Section 13.81, but that numbering is prone to changes). From the viewpoint of the Jacobi--Trudi formula, it is clear how this happens: If the numbers $\alpha_{i}-i$ for $i\in\left[ k\right] $ are not distinct, then $\overline{s}_{\left( \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right) }=0$, because the matrix $\left( h_{\alpha_{i}-i+j}\right) _{i,j\in\left[ k\right] }$ has two equal rows. If they are distinct, then you permute them's so that you have $\alpha_{1}-1>\alpha_{2}-2>\cdots>\alpha_{k}-k$ (on the level of the original $k$-tuple $\left( \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right) $, this is what you call the "dot action" of $S_{k}$ in §4.3 of your symmetric functions notes), and then you add $1,2,\ldots,k$ to the values again, and take $\nu$ to be the resulting partition. The $\pm$ sign is the sign of the permutation you have applied. Note that this way of re-expressing $\overline{s}_{\left( \alpha_{1},\alpha_{2},\ldots,\alpha _{k}\right) }$ as $0$ or $\pm s_{\nu}$ is known as the "slinky rule" among the physicists (see, e.g., pp. 5--6 in B. G. Wybourne, Notes on Symmetric functions and the Symmetric Group), but I find their language hard to understand.

Anyway, combining parts (d) and (c) gives your formula for $B_{a}\left( s_{\lambda}\right) $, modulo having to check that the resulting $\pm s_{\nu }$ is your $\left( -1\right) ^{c}s_{\mu}$. But this is not hard: Inserting $a$ at the front of the partition $\lambda=\left( \lambda_{1},\lambda _{2},\ldots,\lambda_{\ell\left( \lambda\right) }\right) $ does not completely mess up the order of the $\lambda_{i}-i$ but merely puts a new entry $a-1$ at the front that does not necessarily belong there, and subtracts $1$ from the remaining entries because their positions have now shifted by $1$ to the right. It is easy to re-sort the resulting tuple by moving the $a-1$ to its proper place. This is the $c+1$-st place, where $c$ is as you defined it. Each of the entries $\lambda_{1},\lambda_{2},\ldots,\lambda_{c}$ thus moves by $1$ position to the left, whence it ends up decremented by $1$ in the resulting partition $\mu$. Thus your formula for $\mu$.

Now who has actually discovered these facts? I don't really know, but I give a few references in Remark 2.9.2 in op. cit.. I have learnt it from Theorem 2.3 in Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano, Mike Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, arXiv:1208.5191v3, which states an equivalent claim without proof (note that Proposition 2.2 is once again part (c) of my exercise) and uses it to lift the Schur functions to $\operatorname*{NSym}$. Earlier, the operator $B_{a}$ appeared under the name of "Schur row adder" (presumably not the snake) in Adriano M. Garsia, Permutation q-enumeration with the Schur row adder. PU. M. A. (Pure Mathematics and Applications) 21 (2010), No. 2, 233--248, but only the case $a\geq\lambda_{1}$ is explicitly stated there. I have seen these operators used a lot by Mike Zabrocki and Mark Shimozono, and I would not be surprised if it appears somewhere in their works generalized to the Hall--Littlewood case, or earlier in some unpublished notes of Garsia, or perhaps long ago in the works of MacMahon or Littlewood. My historical research usually ends at the point where I no longer expect readable proofs, and that cutoff is pretty recent in algebraic combinatorics (occasional exceptions such as Rutherford's book notwithstanding). If I find nothing, I put it in my notes with Vic; this is the story behind most of the exercises in said notes :)

EDIT: I think the formula $B_a\left(s_\lambda\right) = \left(-1\right)^c s_\mu$ in the general case appears as Corollary 5 in Mike Zabrocki, Vertex Operators for Standard Bases of the Symmetric Functions, Journal of Algebraic Combinatorics 13 (2001), pp. 83--101. Please check that it really is the same.

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    $\begingroup$ Thank you, this is a great answer. I'll hold off accepting just in case someone posts a prior reference in the next day, but I suspect your answer will be authoritative! $\endgroup$ Commented Mar 5 at 15:35
  • $\begingroup$ For the slinky rule, I have a few examples here: symmetricfunctions.com/gessel.htm#gesselSlinkyRule $\endgroup$ Commented Mar 6 at 7:24

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