Sami Assaf sometimes uses a Rubik’s cube to demonstrate symmetry, among the most crucial ideas in mathematics.
Twist the face of a Rubik’s cube and the cube keeps its symmetry — all sides are the mirror image of one another.
Try any arbitrary sequence of turns and the result does not alter the shape of the cube or disturb its symmetry.
“I’m looking for symmetries of objects and spaces,” Assaf said, holding up a Rubik’s cube. “So if you take this cube and flip it around, it still looks the same. I can rotate it, flip it and it’s still the same cube. So what I look at is how many ways you can change it yet it stays the same.”
But when you consider the colors of the nine squares on each of the cube’s six sides, everything changes.
“Once you take the colors into account it looks different, right?” she said. “Now each face is a different color, so when I flip the cube, the top changes from red to orange, so it doesn’t look the same. Most of my projects have this property at their root: Add more features, like colored faces, and there is less symmetry.”
Assaf applied symmetric functions toward a fundamental problem in representation theory that experts have tried to solve for hundreds of years. The problem is to understand the irreducible decomposition of the tensor product of representations of the symmetric group.
Her approach took creativity with a dash of moxie.
“In order to make any progress you can’t just do what everyone else has done,” Assaf said. “You have to come up with an original approach. Previously, people thought you couldn’t apply symmetric functions to the problem for many good reasons. But we found a way.”