Talk:Alternating multilinear map

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(uni-)linear case

The linear case (a linear map V → W), is an alternating map by any sensible definition, as may be seen by the statement that every p-vector is alternating. The generalized Kronecker delta is a useful mechanism for producing a fully alternating tensor of any order, for example, but process this leaves scalars and order-1 tensors unchanged. I can imagine a reader seeking to answer the question "Is a vector alternating?" or "Is a linear map alternating?" Does anyone have language from a reference that allows us to naturally answer this question in the affirmative? —Quondum 22:43, 1 September 2016 (UTC)[reply]

Two adjacent elements or any two elements?

There seems to be no consensus in the community regarding the definition of an alternating multilinear map.

For some authors, it is zero if any two adjacent elements are equal. References:

Serge Lang, "Algebra", revised 3rd ed., GTM 211, Springer, 2002, page 511, §4, lines 13-15.
N. Bourbaki, "Eléments de mathématique", Algèbre Chapitres 1 à 3, Springer, 2007 reprint, page A III.80, §4, lines 1-5.
David S. Dummit and Richard M. Foote, "Abstract Algebra", 3rd ed., Wiley, 2004, page 436, lines 1-3.

For others, it is zero if any two elements are equal, be they adjacent or not. References:

Thomas W. Hungerford, "Algebra", GTM 73, Springer, 1974, page 349, Definition 3.1., last line.
Anthony W. Knapp, "Basic Algebra", Birkhäuser, 2006, page 67, lines 15-16.
Article Multilinear form on English Wikipedia.
Article Application multilinéaire on French Wikipedia.
Article Multilineare Abbildung on German Wikipedia.

Until recently, this article gave the second definition. Yesterday, a contributor replaced it with the first definition. Should we "choose our camp" in this article, at the risk of infuriating the other camp, or should we give both definitions and say that there is no consensus? Please vote. J.P. Martin-Flatin (talk) 09:36, 22 September 2016 (UTC)[reply]

Added tags to this article requesting the help of an expert and documenting the cause of the dispute. J.P. Martin-Flatin (talk) 09:58, 27 September 2016 (UTC)[reply]

I'm partial to having simpler definitions, so I think we should say a form is alternating if it is zero when any two adjacent inputs are equal. Further, I know that those three sources you mentioned (Lang, Bourbaki, and Dummit & Foote) are all very widely used and highly respected, whereas Hungerford's book is more obscure (and I've never heard of Knapp's book). Indeed, Bourbaki was really the text that established much of the foundation of abstract algebra as we know it today, so if we want to be completely precise/pedantic we should always be following Bourbaki. Zdorovo (talk) 09:27, 17 November 2016 (UTC)[reply]