The semester is starting soon, but today I got sucked into the whirlpool of pure math. You read one thing and do some calculations and then you find another thing and then you follow a link to another and there you are: it’s 4 pm and you’ve found all kinds of things about multidegrees but you didn’t answer the original question.

So what did I learn? The following is clearly aimed mostly at myself, as I’m not exactly explaining anything for an audience here.

- Multidegrees: they’re polynomials that sort of generalize the concept of “degree” in algebraic geometry, which in turn generalized the idea of “degree” that you might remember from algebra (x^2 + 3x is a degree two polynomial, because it intersects with a generic line in two places, more or less).
- The current cool reference is Miller and Sturmfels’ Combinatorial Commutative Algebra, in chapter 8. Their approach is very algebraic, to me, and I got stuck on some of the simple examples right away and had to go refresh my memory on how they work with multigraded rings. I learned some things about Smith normal form that I don’t think I ever knew.
- However, I really love Chriss and Ginzburg’s Representation Theory and Complex Geometry, and they discuss the same exact polynomials but they call them equivariant Hilbert polynomials. They make the point that it’s a generalization of the Hilbert polynomial. Cool. But I have to admit I still found parts of their exposition somewhat difficult as well.
- In 2011 a guy named Gergely Berczi gave some Impanga lectures that covered these polynomials, too, and in his summary I found the most natural presentation of the polynomials. In the correct situation, looking at a T-invariant subset C of a G-representation V, the multidegree (polynomial) mdeg(C,V) is just the product of the equivariant weights in the normal direction. That, to me, is a very nice intuitive description, and explains why the multidegrees are double Schubert polynomials in some very special situations!
- Some people call multidegrees “equivariant Poincare duals” and others call them “Joseph polynomials,” it looks like.

Fine. I guess I will turn to my administrative stuff now for a short time and then it is time for dinner and the rest of life!

*Right: some of these links to Amazon are affiliate links so I might make some tiny amount of money if you buy one of these lovely math books via such a link. I don’t recommend any math books in this post I don’t have myself 😉*