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The diagonal derivative of a skew Schur polynomial

Darij Grinberg, Nazar Korniichuk, Kostiantyn Molokanov, Severyn Khomych

Abstract

We prove a formula for the image of a skew Schur polynomial $s_{λ/μ}\left( x_{1}, x_{2}, \ldots, x_{N}\right) $ under the differential operator $\nabla:= \dfrac{\partial}{\partial x_{1}} +\dfrac{\partial}{\partial x_{2}}+\cdots+\dfrac{\partial}{\partial x_{N}}$. This generalizes a formula of Weigandt for $\nabla\left( s_λ\right) $.

The diagonal derivative of a skew Schur polynomial

Abstract

We prove a formula for the image of a skew Schur polynomial under the differential operator . This generalizes a formula of Weigandt for .
Paper Structure (7 sections, 11 theorems, 48 equations)

This paper contains 7 sections, 11 theorems, 48 equations.

Table of Contents

  1. Notations and definitions
  2. The results
  3. A lemma on $\nabla\left( h_{n}\right)$
  4. Lemmas on determinants
  5. The last lemma
  6. Proofs of the main results
  7. Final remarks

Key Result

Theorem 2.1

Theorems & Definitions (24)

  • Theorem 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Corollary 2.5
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem.Nabla-h']}.
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['lem.det.s']}.
  • Lemma 4.2
  • ...and 14 more