Talk:First and second fundamental theorems of invariant theory

Wikipedia:Miscellany for deletion/Draft:First and second fundamental theorems of invariant theory

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Any editor nominating this for Deletion, I attempted to have this page deleted under MFD and the author of the page moved it to mainspace during the MFD. Per accepted convention I withdrew the MFD, however I would like to firmly register that I oppose any return to Draftspace without the stick of CSD:G13 being in place including a no REFUND allowed rule. Either the page survives in mainspace, it is returned to Draft namespace and it stays of the G13 rail, or it gets deleted without any allowance for WP:REFUND. Hasteur (talk) 01:18, 9 November 2019 (UTC)

Now that is it in mainspace, I believe it falls under the normal procedures for a topic that a) is obviously notable and has plenty of sources to draw on, and b) isn't a complete trainwreck of an article. This is no beauty but it has enough vital organs to survive for some time while the WikiMagic does its thing (hopefully). Marked reviewed as such --Elmidae (talk · contribs) 02:11, 9 November 2019 (UTC)

No other editor suggested such a restriction on this page, and it cannot therefore be considered to have been imposed by consensus. I do not see any reason why this should be returned to draft status, but if it should be, it will be in precisely the same position as any other draft. The availability of refund is a general policy and cannot be foreclosed in advance by a single editor, although an admin asked for a future REFUND on this page might consider this history. DES ^(talk)_{DESiegel Contribs} 03:45, 9 November 2019 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

The first fundamental theorem isn't the same as the one in "Algorithms in Invariant Theory" by Sturmfels

Additionally, both theorem statements in the article seem ambiguous.

In matrix language, the 1st Fundamental Theorem should say: If the group $SL_{n}(\mathbb {C} )$ acts on $\mathbb {C} ^{n\times m}$ by $g.M=gM$, then the corresponding invariant ring (denoted $\mathbb {C} [\mathbb {C} ^{n\times m}]^{SL_{n}(\mathbb {C} )}$) is generated by the $n\times n$ minors of $M\in \mathbb {C} ^{n\times m}$.

By "a minor" in this particular context, we mean the determinant of a matrix obtained from $M$ by picking $n$ columns.

I can't state the 2nd Fundamental Theorem as clearly at the moment: The 2nd fundamental theorem states that the corresponding syzygies, which are the relations between the above generators, are given by the Grassmann-Plucker syzygies. These are a special case of the Van der Waeden syzygies. See bracket algebra. --Svennik (talk) 18:12, 26 March 2024 (UTC)

I followed the citation, and it looks like the article's 1st fundamental theorem is in fact the 1st fundamental theorem for GL(V). Presumably, if we were to replace this with other reductive groups then we would get other "1st fundamental theorems". Svennik (talk) 19:41, 27 March 2024 (UTC)