The ubiquity of Howe duality

I’m been thinking recently about multiple tensor products of SL_n representations and I keep running into the strange phenomenon of Howe duality. Consider in particular the problem of finding the dimension of the space of SL_n invariants inside a tensor product \Lambda^{k_1} V \otimes \cdots \otimes \Lambda^{k_m} V where V = \mathbb{C}^n is the standard representation of SL_n.

Such a tensor product has no invariants unless k_1 + 2k_2 + \ldots + mk_m =  pn for some integer p and in this case, the dimension is the number of semistandard young tableaux of shape an n x p rectangle filled with k_1 1s, k_2 2s, ..., k_m ms . This can be proven using the theory of Littelmann paths, but the reader will probably as recognize that this is also the dimension of a certain weight space of a certain representation of GL_m. This strange coincidence can be explained using Howe duality.

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Spheres in Higher Dimensions

I just reread this problem on the Harvey Mudd Math Fun Facts website. It’s about your basic intuition in higher dimensions. I remember when I first heard it I was so shocked.

Suppose you have a square in the plane. Cut it into four quadrants, then inscribe a circle in each. Finally add a fifth circle which is just tangent to the other circles.

Square with inscribed circles.

As you can see, it is much smaller then the other circles and the square.

You can repeat this in higher dimensions. For example, you can cut the cube into octants and inscribe eight spheres in these, then add a last final sphere, tangent to the eight inscribed spheres.

So how does the volume of this final sphere change with respect to the volume of the cube when we change dimensions? How does its size compare? It’s always smaller right? WRONG!

As you increase dimension, the final sphere gets bigger and bigger. In dimension 9 it is tangent to the original n-cube and in dimensions bigger than 9 it pushes outside the original cube. Eventually as the dimension increases, the volume of the final sphere exceeds that of the n-cube!

The details are pretty easy and can be found here:
Su, Francis E., et al. “High-Dimensional Spheres in Cubes.” Mudd Math Fun Facts.

There are all sorts of other cool math fun facts on this webpage. You should check it out.

Fun ring theory problem

Right now I’m teaching a class on unique factorization and its failure. One of my hobbies is finding fun problems like “Find a domain where every finitely generated ideal is principle, but not every ideal is finitely generated.” Today one of my students found an answer to the following fun question (which was cute enough that I thought I’d pass it on to all of you).

A weaker statement than unique factorization is just that the number of irreducible factors is independent of factorization. For example, in \mathbb{Z}[\sqrt{-5}], even though 6 factors nonuniquely as 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5}) both of these factorizations have 2 irreducible factors. How badly can this fail? It turns out the answer is “really badly.” Find a domain R where every nonzero element factors as a unit times a product irreducibles, but which has a fixed element that can be written as a product of more than N irreducibles for any arbitrarily large N.