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nLab > Latest Changes: Exterior Algebra -- general definition

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- CommentRowNumber1.
- CommentAuthorsgarush
- CommentTimeApr 10th 2018
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Author: sgarush
Format: MarkdownItexIn the article [[exterior+algebra]], in sect. 2, subsection In General, it says that the nth exterior power of an object $V$ is the cokernel of the antisymmetrization operator $$ P_{A} = \frac{1}{n!} \sum_{\sigma \in S_{n}} \mathrm{sgn}(\sigma) \sigma. $$ Shouldn't the exterior algebra be the image of this operator? If so, I am happy to edit the article accordingly.

In the article exterior+algebra, in sect. 2, subsection In General, it says that the nth exterior power of an object $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>$ is the cokernel of the antisymmetrization operator
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi> <mi>A</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><munder><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow></munder><mi mathvariant="normal">sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mi>σ</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> P_{A} = \frac{1}{n!} \sum_{\sigma \in S_{n}} \mathrm{sgn}(\sigma) \sigma. </annotation></semantics></math>$
Shouldn’t the exterior algebra be the image of this operator? If so, I am happy to edit the article accordingly.
- CommentRowNumber2.
- CommentAuthorsgarush
- CommentTimeApr 10th 2018
- PermaLink
Author: sgarush
Format: MarkdownItexIt looks like in [[schur+functor]] what is used is the cokernel of $1 - P_A$ -- is this correct?

It looks like in schur+functor what is used is the cokernel of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msub><mi>P</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">1 - P_A</annotation></semantics></math>$ – is this correct?
- CommentRowNumber3.
- CommentAuthorOscar_Cunningham
- CommentTimeApr 10th 2018
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Author: Oscar_Cunningham
Format: MarkdownItexYes you're correct. At least in the context of vector spaces over a field of characteristic $0$, $\Lambda^{n}V$ is the image of $P_A$, the coimage of $P_A$, the kernel of $1-P_A$ and the cokernel of $1-P_A$. So the statement at [[Schur functor]] is correct.

Yes you’re correct. At least in the context of vector spaces over a field of characteristic $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n}V</annotation></semantics></math>$ is the image of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">P_A</annotation></semantics></math>$ , the coimage of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">P_A</annotation></semantics></math>$ , the kernel of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msub><mi>P</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">1-P_A</annotation></semantics></math>$ and the cokernel of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msub><mi>P</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">1-P_A</annotation></semantics></math>$ . So the statement at Schur functor is correct.
- CommentRowNumber4.
- CommentAuthorsgarush
- CommentTimeApr 10th 2018
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Author: sgarush
Format: MarkdownItexGreat, thanks. I guess I'll edit the article to bring it in line with [[Schur functor]].

Great, thanks. I guess I’ll edit the article to bring it in line with Schur functor.

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nLab > Latest Changes: Exterior Algebra -- general definition