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More minor summation formulae

Shane Chern, Theresia Eisenkölbl, Ilse Fischer, Moritz Gangl, Mona Gatzweiler, Álvaro Gutiérrez, Christian Krattenthaler, Nishu Kumari, Markus Reibnegger, Marcus Schönfelder, Atsuro Yoshida

Abstract

We prove determinantal-Pfaffian formulae that simultaneously generalise the Pfaffian minor summation formula of Ishikawa and Wakayama and Byun's recent minor summation formula. These formulae are based on factorisation formulae for the determinant of the sum of a skew-symmetric matrix and a rank-1 matrix. Applications include a Cauchy-type identity for skew Schur functions.

More minor summation formulae

Abstract

We prove determinantal-Pfaffian formulae that simultaneously generalise the Pfaffian minor summation formula of Ishikawa and Wakayama and Byun's recent minor summation formula. These formulae are based on factorisation formulae for the determinant of the sum of a skew-symmetric matrix and a rank-1 matrix. Applications include a Cauchy-type identity for skew Schur functions.
Paper Structure (9 sections, 18 theorems, 69 equations)

This paper contains 9 sections, 18 theorems, 69 equations.

Table of Contents

  1. Introduction and statement of main results
  2. The determinant of a skew-symmetric plus a rank-$1$ matrix
  3. Proof of Theorem \ref{['thm:main2']}, and an auxiliary result
  4. Proof of Theorem \ref{['thm:main1']}
  5. Applications
  6. The "symmetric" cases of the factorisation identities from Section \ref{['sec:Y+M']}
  7. The Pfaffian minor summation formula of Ishikawa and Wakayama
  8. Asymmetric extensions of the minor summation formulae of Byun and of Okada
  9. A Cauchy-type identity for skew Schur functions

Key Result

Theorem 1

Let $m$ and $n$ be positive integers, where $m$ is even. Furthermore, let $A$ be an $m\times n$ matrix. Then we have where, by definition, $[n]:=\{1,2,\dots,n\}$, where $A^I$ denotes the submatrix of $A$ consisting of the columns indexed by $I$, and $U_n$ is the upper triangular matrix with all entries above the diagonal equal to $1$ and all other entries equal to $0$.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 25 more