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Abdelmalek Abdesselam
Abdelmalek Abdesselam
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This is the second fundamental theorem of classical invariant theory for $GL$ acting on vectors and covectors. This is Theorem 8.1 from Ch. 13 of the book Lie Groups: An Approach Through Invariants and Representations by C. Procesi (I still don't have it in front of me but I trust Darij on this).

Alternatively, it is Theorem 5.2.15 (SFT for $GL(n)$) in the book Representations and Invariants of the Classical Groups by Goodman and Wallach. I have that one on front of me, the old 1998 edition.

The problem is about finding equations for the variety of $n\times m$ matrices $Z$ which factor as $XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$. The condition is that the rank is at most $k$. By elementary linear algebra the $(k+1)\times(k+1)$ minor determinants are set-theoretic equations, however Darij's question is about showing these are also ideal-theoretic equations and that needs a bit more work.

This is the second fundamental theorem of classical invariant theory for $GL$ acting on vectors and covectors. This is Theorem 8.1 from Ch. 13 of the book Lie Groups: An Approach Through Invariants and Representations by C. Procesi (I still don't have it in front of me but I trust Darij on this).

Alternatively, it is Theorem 5.2.15 (SFT for $GL(n)$) in the book Representations and Invariants of the Classical Groups by Goodman and Wallach. I have that one on front of me, the old 1998 edition.

The problem is about finding equations for the variety of $n\times m$ matrices $Z$ which factor as $XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$. The condition is that the rank is at most $k$. By elementary linear algebra the $(k+1)\times(k+1)$ are set-theoretic equations, however Darij's question is about showing these are also ideal-theoretic equations and that needs a bit more work.

This is the second fundamental theorem of classical invariant theory for $GL$ acting on vectors and covectors. This is Theorem 8.1 from Ch. 13 of the book Lie Groups: An Approach Through Invariants and Representations by C. Procesi (I still don't have it in front of me but I trust Darij on this).

Alternatively, it is Theorem 5.2.15 (SFT for $GL(n)$) in the book Representations and Invariants of the Classical Groups by Goodman and Wallach. I have that one on front of me, the old 1998 edition.

The problem is about finding equations for the variety of $n\times m$ matrices $Z$ which factor as $XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$. The condition is that the rank is at most $k$. By elementary linear algebra the $(k+1)\times(k+1)$ minor determinants are set-theoretic equations, however Darij's question is about showing these are also ideal-theoretic equations and that needs a bit more work.

Source Link
Abdelmalek Abdesselam
Abdelmalek Abdesselam
  • 23k
  • 1
  • 67
  • 127

This is the second fundamental theorem of classical invariant theory for $GL$ acting on vectors and covectors. This is Theorem 8.1 from Ch. 13 of the book Lie Groups: An Approach Through Invariants and Representations by C. Procesi (I still don't have it in front of me but I trust Darij on this).

Alternatively, it is Theorem 5.2.15 (SFT for $GL(n)$) in the book Representations and Invariants of the Classical Groups by Goodman and Wallach. I have that one on front of me, the old 1998 edition.

The problem is about finding equations for the variety of $n\times m$ matrices $Z$ which factor as $XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$. The condition is that the rank is at most $k$. By elementary linear algebra the $(k+1)\times(k+1)$ are set-theoretic equations, however Darij's question is about showing these are also ideal-theoretic equations and that needs a bit more work.