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I am computing some combinatorial parameter associated with the complex simple Lie algebras of type $A_n$ using sage and the output will be a rectangular integer matrix.

All I need to prove to solve my research problem is to prove that this matrix has distinct rows.

I thought I shall prove the matrix is of full rank. But clearly, it is an over-expectation and turns out to be false.

I have some understanding of the matrix entries and I am looking for an algebraic method to employ on this matrix to detect the distinctness of rows.

I hope checking by hand is not the only way to do it as the matrix size grows very fast with $n$. If techniques like Jordan form, row echelon form or something like them can be used, then it will be good.

Kindly help me with this. Thank you.

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  • $\begingroup$ I know this is not what you're asking for, but note also that the "checking by hand" complexity can be reduced from naive O(n^2) to O(n log n) by sorting and to O(n) using a hash table. $\endgroup$
    – Yoav Kallus
    Commented Aug 8, 2019 at 13:16
  • $\begingroup$ Are you asking for techniques to prove that the rows are distinct or are you asking for programming techniques that will help you check that the rows are distinct? In the former case, I think you need to be more specific and give us the definition of the matrix. $\endgroup$
    – Vít Tuček
    Commented Aug 8, 2019 at 13:20
  • $\begingroup$ I think you are asking for a theoretical answer, but I'll give a programming note: the SAGE function uniq() is meant for this task doc.sagemath.org/html/en/reference/misc/sage/misc/misc.html . $\endgroup$
    – David E Speyer
    Commented Aug 8, 2019 at 14:01

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The obvious "algebraic" condition is that the left null space of your matrix contains no vector of the form $e_i - e_j$. Does that help?

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