Lab4-Tableaux.sagews

Tableau and row-insertion (bumping)

Math 737 - Lab 4

Note about functions. You do not need to redefine a function to use it on a different object. Define the function once and then you can use it on any object of the right type. An example is given below.

def square_a_number(num):
    return num^2

square_a_number(5)

square_a_number(7)

25 49

SYT and SSYT are classes of objects in Sage. Evaluate the cells below to see how to do some things with SYT. Then in Exercise 2, you'll do similar things with SSYT.

S = StandardTableaux([3,3])
for tab in S:
    show(tab)
    print("----")

----

----

----

----

----

StandardTableaux([6,5,4,3,2,1]).cardinality()

Exercise 1: How many SYT are there of shape $[n,n]$, for $n$ ranging from $1$ to $8$? What numbers are these?

Hint: Use a 'for' loop. An example of a 'for' loop is given below.

for i in [1..10]:
    i^2

Exercise 2: Use the command SemistandardTableaux to create (not print!) the set of SSYT there are of shape [3,3] (with entries bounded above by 6). How many tableaux are in this set?

Then use the optional parameter max_entry to create the set of SSYT of shape [3,3] with entries bounded above by 3. Print this smaller set of SSYT.

(If you need help with this, evaluate the cell below to see the documentation.)

SemistandardTableaux?

Evaluate the cells below to see how to create a SSYT and row-insert numbers into it.

T = SemistandardTableau([[1,2,4,11],[3,5,9,12],[6,8],[7],[10]])
show(T)

show(T.schensted_insert(4))

U = SemistandardTableau([[1,2],[3]])
show(T.bump_multiply(U))

Exercise 3: Use the above commands to verify the examples of SSYT mutiplication you did in class on Wednesday in two ways: first, use schensted_insert to insert each number in the second tableau. Then, show that you get the same thing when you use bump_multiply.

Exercise 4: Is SSYT multiplication commutative? Prove or give a counterexample.

Exercise 5:

- Create a random standard Young tableau with 12 boxes. Compute its promotion by hand and by computer (find the correct function using dot+tab). Verify that you got the same thing both ways. Note that there is a promotion command and a promotion_inverse command, so you may need to try both.

- Verify the orbits of promotion of tableaux of shape (3,3) that you computed by hand.

syt = StandardTableaux(12).random_element()
show(syt)

If you have time: Play with the built-in tableaux functions to find some fun things to do with tableaux. Pick your favorite, demonstrate using it, and write a sentence to say what it does. You will share this with the class at the start of the next lab.

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