Lab7-RSK_Littlewood_Richardson.sagews

RSK and the Littlewood-Richardson rule

Math 737 - Lab 7

Exercise 1:

• Google search the phrase 'sage math rsk' to find the Sage documentation for the RSK and RSK_inverse functions.

• Then create one or more examples (different from the examples in the documentation) to demonstrate how to use these functions. One of your examples should check you work on a Team Activity from the class notes.

• Demonstrate in your examples what happens to the matrix when you swap the tableaux P and Q.

Exercise 2: Finish the 'for' loop below to check that for every involution of $S_3$, its corresponding insertion and recording tableaux are equal. Do the same for $n=4,5,6$.

for perm in Permutations(3):
    if perm
    

Exercise 3: Calculate the sum of $(f^{\lambda})^2$ for all $\lambda$ with $3$ boxes. Do the same for $n=4,5,6$. Conjecture a formula in terms of $n$. How does RSK prove this formula? (The answer is in Fulton Section 4.3.)

In the rest of the lab, we will calculate the Littlewood-Richardson coefficents $c_{\lambda\mu}^{\nu}$ in two different ways.

Start by evaluating the block below to define the operator s, which is the built-in Schur polynomial function in Sage.

s = SymmetricFunctions(QQ).s()
s([2,1])

Exercise 4: Multiply the Schur polynomials for $\lambda=[3,2,1]$ and $\mu=[2,2]$. Note that the code automatically expands the product as a sum of Schur polynomials. By looking at the coefficients, find the Littlewood-Richardson coefficient $c^{\nu}_{\lambda\mu}$ corresponding to $\nu=[4,3,2,1]$.

Exercise 5: Evaluate the code below to see how to create skew SSYT of a given shape and content, create the row word, then check whether it is Yamanouchi.

Then use these functions to help you find $c^{\nu}_{\lambda\mu}$ Littlewood-Richardson tableaux of shape $\nu / \lambda$ and content $\mu$, with $\lambda$, $\mu$, and $\nu$ as above. Then find $c^{\nu}_{\lambda\mu}$ for different partitions $\lambda$, $\mu$, and $\nu$.

S = SemistandardSkewTableaux([[3,2,1],[2,1]],[1,1,1])
S

Semistandard skew tableaux of shape [3, 2, 1] / [2, 1] and weight [1, 1, 1]

for s in S:
    s.pp()

s.pp()
s.to_word_by_row()
s.to_word_by_row().is_yamanouchi()

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