Talk:Alternating multilinear map

(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)

I rewrote the definition. From it follows that a linear map $f:V\to W$ is alternating iff it meets the following condition:

If $x$ is linearly dependent then $f(x)=0$, which is equivalent to

If $x=0$ then $f(x)=0$

Since this condition is always met one can deduce from the given definition of alternation that linear maps of the form $f:V\to W$ are alternating. Bomazi (talk) 16:26, 29 April 2019 (UTC)

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)

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)

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)

The two definitions are equivalent (as are others, e.g. using linear dependency), so it is not surprising if different versions are chosen by different authors. I prefer a definition that does not depend on the unnecessary structure of argument ordering (i.e., we can treat the cartesian product as an indexed family, for which adjacency is undefined). At least we should state the property in terms of every pair, otherwise we incorrectly create the impression that they are not equivalent. —Quondum 12:47, 17 November 2016 (UTC)

Serge Lang immediately states the equivalence (Proposition 4.1: If x_i=x_j for i≠j then f(x₁,...,x_n)=0). Dummit and Foote do the same (Proposition 22(3): If v_i=v_j for any pair of distinct i,j∈{1,2,...,n} then φ(v₁,v₂,...,v_n)=0). It is thus a reliably sourced property of alternating multilinear maps. Since the most reliable sources use adjacency in their definition, but they then derive the any pair property, I think we can follow suit (despite my preference): defined in terms of adjacency, but perhaps using the derived property in the lead to describe it. —Quondum 02:53, 18 November 2016 (UTC)

Yes, that was my point. I did not say that one is correct and the other is not. I just wanted to know which statement is the definition, and which statement is derived from the definition. We seem to agree that the definition is "adjacent elements" and the derived property is "any two elements". The current text makes it crystal clear, so the problem is closed as far as I am concerned. J.P. Martin-Flatin (talk) 14:22, 30 November 2016 (UTC)

Cool. I've tweaked the lead in line with my suggestion. The definition remain as is. —Quondum 14:44, 30 November 2016 (UTC)

No, I think we should keep the definition and the property as agreed above. J.P. Martin-Flatin (talk) 15:35, 30 November 2016 (UTC)

I find your communication unclear. We agreed on the definition and the property. You have said absolutely nothing about which should be in the lead, despite my raising the point. I said that I feel the lead should contain the property, not the definition. You cannot interpret this as my agreement that the definition belongs in the lead. I have only been able to infer that you prefer the lead to contain the definition by your revert of my edit, not anything you have said on the talk page up to this point. Do as you wish – I'm not interested in arguing the point. —Quondum 17:40, 30 November 2016 (UTC)

Rename proposal, to Alternating multilinear map

The article at the moment is about an alternating multilinear map, and all the notable sources used do not seem to use the term alternating map on its own. I propose a name change, unless someone can find a notable source for a (more general) term alternating map. —Quondum 03:43, 18 November 2016 (UTC)

You are right. alternating multilinear map is the proper term. I have made the move. Bomazi (talk) 02:06, 29 November 2016 (UTC)

Error in the lead

An alternating multilinear map is always antisymmetric, but the converse is only true for fields of characteristic different from 2. Bomazi (talk) 20:51, 18 November 2016 (UTC)

Yep, I had this flagged for as couple of months now. I've now deleted the statement. —Quondum 21:41, 18 November 2016 (UTC)

Bad edit summary for revision 06:37, 29 November 2016‎

The correct edit summary is:

Removed the 'expert-subject' tag. It unnecessarily scares the reader. The definition of an alternating multilinear map is *not* disputed, only the choice between two equivalent formulations is. The article can thus be trusted on this topic.

Bomazi (talk) 05:46, 29 November 2016 (UTC)

Residual tweaks

Thanks Bomazi for the definite improvements and polish. I have some additional comments for consideration:

• The definition of alternatization is not universal; definitions vary by a factor. I have seen the division by n! being included to make it into a projection, though this obviously does not work when the characteristic of the base ring has a prime factor p ≤ n.
• I gather that the concept of alternatization applies to more than multilinear maps: multilinearity is not required. It might make sense to merge the content relating to this into Symmetrization, which could be renamed Symmetrization and antisymmetrization. I suspect that there is no difference between alternatization and antisymmetrization.
• As per a previous comment of mine, I suggest keeping the existing definition (on grounds of reliable sourcability), but to remove the word "adjacent" from the lead. I argue that a description of a useful property belongs in the lead in preference to the formal definition.

—Quondum 23:59, 29 November 2016 (UTC)

Serious problems

This article has some serious problems. It states (correctly) that alternating multilinear maps can be defined over any commutative ring. Then they are defined by three "equivalent" conditions, but these are only equivalent if the ring is a field. For instance, consider the determinant on $(\mathbb {Z} _{6}^{2})^{\wedge 2}$, that is, $f((a,b),(c,d))=ad-bc$. It satisfies conditions 1 and 2 but not 3, since $0(1,0)+3(0,2)=(0,0)$ but $f((1,0),(0,2))=2\neq 0$. Later, it states (correctly) that every antisymmetric form is alternating if $n!$ is invertible. However, this holds under the much weaker condition that 2 is invertible.

98.128.166.193 (talk) 05:17, 19 August 2021 (UTC)