A theorem on Pfaffians

Elliott H. Lieb

doi:10.1016/S0021-9800(68)80078-X

A theorem on Pfaffians

Elliott H. Lieb

Mathematics

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Abstract

We prove the Pfaffian analog of the well-known diagonal expansion theorem for determinants. If P_k is the sum of the Pfaffians of all the k-square principal triangular subarrays of a given N-square triangular array A then P(λ)=Σ₁^NP_kλ^k=Pfaffian [A(λ)] with a_ij(λ)=a_ij-(-1)^j-iλ². Our proof is an application of Wick's theorem.

Original language	English (US)
Pages (from-to)	313-319
Number of pages	7
Journal	Journal of Combinatorial Theory
Volume	5
Issue number	3
DOIs	https://doi.org/10.1016/S0021-9800(68)80078-X
State	Published - Nov 1968

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title = "A theorem on Pfaffians",

abstract = "We prove the Pfaffian analog of the well-known diagonal expansion theorem for determinants. If Pk is the sum of the Pfaffians of all the k-square principal triangular subarrays of a given N-square triangular array A then P(λ)=Σ1NPkλk=Pfaffian [A(λ)] with aij(λ)=aij-(-1)j-iλ2. Our proof is an application of Wick's theorem.",

author = "Lieb, \{Elliott H.\}",

note = "Funding Information: * Work supported by National Science Foundation Grant GP-6851. \textasciitilde{} Present address: Mathematics Department, M.I.T., Cambridge, Massachusetts, 02139.",

year = "1968",

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doi = "10.1016/S0021-9800(68)80078-X",

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journal = "Journal of Combinatorial Theory",

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T1 - A theorem on Pfaffians

AU - Lieb, Elliott H.

N1 - Funding Information: * Work supported by National Science Foundation Grant GP-6851. ~ Present address: Mathematics Department, M.I.T., Cambridge, Massachusetts, 02139.

PY - 1968/11

Y1 - 1968/11

N2 - We prove the Pfaffian analog of the well-known diagonal expansion theorem for determinants. If Pk is the sum of the Pfaffians of all the k-square principal triangular subarrays of a given N-square triangular array A then P(λ)=Σ1NPkλk=Pfaffian [A(λ)] with aij(λ)=aij-(-1)j-iλ2. Our proof is an application of Wick's theorem.

AB - We prove the Pfaffian analog of the well-known diagonal expansion theorem for determinants. If Pk is the sum of the Pfaffians of all the k-square principal triangular subarrays of a given N-square triangular array A then P(λ)=Σ1NPkλk=Pfaffian [A(λ)] with aij(λ)=aij-(-1)j-iλ2. Our proof is an application of Wick's theorem.

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U2 - 10.1016/S0021-9800(68)80078-X

DO - 10.1016/S0021-9800(68)80078-X

M3 - Article

AN - SCOPUS:0000182194

SN - 0021-9800

VL - 5

SP - 313

EP - 319

JO - Journal of Combinatorial Theory

JF - Journal of Combinatorial Theory

IS - 3

ER -

A theorem on Pfaffians

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