Exercises #6

1. There is a better RSK correspondence:
{matrices of natural numbers with row sums mu₁ and column sums mu₂}
correspond to
{pairs (A,B) of two semistandard Young tableaux of the same shape, A content mu₁ and B content mu₂}
(semistandard means weakly increasing in each row, strictly in each column).
a. Show how the RSK we defined in class is a special case.
b. Show how to derive this from the hive associator, by tensoring with Altⁱ Cⁿ at each step (for various i) rather than just Cⁿ as we did.

2. Find all the puzzles of size 3.

3. Assume that a puzzle has only one 1 on one side. Give a description of all such puzzles. Show that for each size n, there are {n+1 choose 2} of them.

4. Consider a positive linear combination of rhombus inequalities where no two rhombi overlap in a triangle. If the dependence on the interior all cancels, show that the coefficients on the rhombi are all equal (so may as well be scaled to 1).