Mathematics

Young Tableaux

Tableaux (fr. "table") are mathematical objects formed by stacked, numbered boxes, like the ones depicted on the left. They are only ever numbered to n, the number of boxes in the tableau.

Row Group

All permutations in Sn that do not alter the rows of the tableau.

Column Group

All permutations in Sn that do not alter the columns of the tableau.

Univariate Symmetrizer

We define y(T) as a helpful intermediary between the true Young Symmetrizer (see the paper below) and the row and column groups.

Notice that y(T) is an expression with coefficients that are in the set { -k, 0, k }. Within the scope of the our project, we were interested in the efficient computation of these values after a transformation was applied.

For more information in how y(T) is used to construct the true (bivariate) Young Symmetrizer, look at the full paper at the bottom of the page.

The Artifact

For the digital artifact, we are only concerned about attempting to determine the components gamma and rho of some permutation sigma, where gamma and rho are elements of the column and row groups. In particular, we are interested in the coefficient of that sigma in y(T).

A fast algorithm to determine if a tableau generates a non-zero coefficient for a particular permutation is still undiscovered! The hope is that the game allows players to gain insight in how fast algorithm to play the game might work.

The objective of the game is to try and identify the proper gamma and then rho that sends a generated tableau to the naturally ordered target.

This is a large simplification of the actual problem, but captures the essence of its difficulty.

Start

Target

Controls

You are allowed to make only column swaps, then rows.

Try to get the current tableau to look like the target in as few swaps as possible!

Try the artifact on Github now!

The Paper

Young Symmetrizers Capstone Paper.pdf

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